NONDESTRUCTIVE TESTING HANDBOOK -
Electromagnetic Testing
Manual de Ensaio Não Destrutivo - Ensaio Eletromagnético
- Parte 1. Modêlo do Fenômeno do Ensaio Eletromagnético
- Introdução
- Equações Diferenciais Básicas para Campos Eletromagnéticos
- Modêlos Analítico e Numérico
- Parte 2. Modêlo do Meio Condutor Homogêneo
- Fundamentos
- Modêlos Analíticos
- Modêlos de Dodd e Deeds
- Extensões dos Modêlos de Dodd e Deeds
- Modêlos Tridimensionais
- Perturbação e Expansão da Função de Engen
- Conclusões
- Parte 3. Modêlos Analíticos e Integral para Simular Trincas
- Introdução
- Elementos da Teoria de Trincas
- Plano de Impedâncias
- Dipolo da Corrente
- Mono polo da Corrente Estática
- Dipolo de Campo Estático
- Pequena Inclusão Esférica
- Dipolo Dinâmico da Corrente
- Resposta da Sonda
- Pequenas Descontinuidades
- Trincas Longas
- Técnicas Avançadas
- Campo Elétrico na Abertura da Trinca
- Trinca Impenetrável
- Distribuição do Dipolo da Corrente na Superfície
- Formulação Integral
- Resultados dos Elementos de Contorno
- Teoria da Trinca de Pouca Penetração
- Formulações Alternativas
- Regime de Pouca Penetração
- Parte 4. Modêlo Computacional do Campo de Correntes Parasitas
- Bases Matemáticas do Modelo
- Tipos de Modêlo
- Visão Geral da Modelagem Analítica e Numérica
- Modêlo Analítico
- Técnica de Solução Integral
- Modêlo Numérico
- Técnica das Diferenças Finitas
- Representação das Diferenças Finitas
- Formulação das Diferenças Finitas para Problemas de Campo Bidimensional e Axissimétricos
- Contornos e Condições de Contorno
- Malhas Não Uniformes e Não Retangulares
- Solução do Sistema de Equações
- Solução Interativa
- Solução por Matriz de Inversão
- Técnica de Elementos Finitos
- Formulação de Elementos Finitos para Geometrias Bidimensionais e Axissimétricas
- Energia Funcional para Problemas de Correntes Parasitas
- Discretização de Elementos Finitos
- Formulação de Elementos Finitos
- Elementos Isoparamêtricos Quadrilaterais
- Minimização Funcional
- Condições de Contorno
- Cálculos com Vetor Magnético Potencial
- Modelagem da Física do Ensaio de Correntes Parasitas
- Modelagem para Projeto de Sondas
- Projeto por Elementos Finitos de Sondas de Correntes Parasitas Absoluta e Diferencial
- Modelagem para Simulação
- Conclusões
1 MODÊLO DO FENÔMENO DO ENSAIO ELETROMAGNÉTICO
1.1 INTRODUÇÃO
Modelos
matemáticos são usados para simular o fenômeno das correntes
parasitas e suas aplicações em ensaios não destrutivos. Os modelos
tipicamente simulam um ensaio de correntes parasitas e predizem o sinal
da sonda associado a uma descontinuidade específica (uma região onde a
condutividade ou permeabilidade muda abruptamente) sob diferentes
condições experimentais. Os resultados desses estudos paramétricos são
úteis no projeto de sondas, na visualização da interação do campo com
as descontinuidades, na otimização da configuração do ensaio e na
geração de assinaturas de descontinuidade que podem ser usadas para
desenvolver algoritmos de interpretação de sinais. Os modelos de
simulação são relativamente baratos em comparação com os dados
adquiridos experimentalmente a partir de descontinuidades artificiais.
Todos
os fenômenos eletromagnéticos, incluindo aqueles relacionados ao
vazamento de fluxo magnético e aos ensaios de correntes parasitas, são
governados por equações diferenciais. (R01)
1.2 EQUAÇÕES DIFERENCIAIS BÁSICAS PARA CAMPOS ELETROMAGNÉTICOS (R02)
As
equações diferenciais que governam campos eletromagnéticos gerais,
variáveis no tempo, em baixas frequências, em regiões que incluem
materiais magnéticos e condutores e densidades de corrente aplicadas,
são derivadas das equações de Maxwell: (R01)
onde B é a densidade de fluxo magnético (tesla), D é a densidade de fluxo elétrico (coulomb por metro quadrado), E é a intensidade do campo elétrico (volt por metro), H é a intensidade do campo magnético (ampère por metro), J é a densidade de corrente (ampère por metro quadrado), t é o tempo (segundo) e ρ é a densidade de carga (coulomb por metro cúbico).
A
Equação 2 depende da aproximação quase-estática, que negligencia a
corrente de deslocamento. A técnica de micro-ondas necessita da
corrente de deslocamento, mas sua omissão é justificável na técnica de
correntes parasitas, pois as frequências mais altas encontradas são da
ordem de alguns megahertz. Nessas frequências, a corrente de condução
em metais é tipicamente muitas ordens de magnitude maior que a corrente
de deslocamento. A carga pode se acumular em limites de descontinuidade
e na superfície de condutores, causando um salto na componente normal
do campo elétrico. No entanto, a Eq. 2 implica que \/.J = 0, o que
significa, por exemplo, que a corrente normal a uma superfície que
adquire carga é desprezível. Embora a corrente de carga possa ser
desprezada, o efeito da carga no campo elétrico não pode ser ignorado.
Se o limite não for abrupto, a carga incidente se distribui por um
volume.
Observe
que, ao igualar todas as derivadas temporais a zero, essas equações
podem ser usadas para descrever fenômenos de fuga de fluxo magnético. O
mesmo modelo numérico usado para ensaios de correntes parasitas pode
ser aplicado a ensaios de fuga de fluxo magnético, igualando-se a
frequência da corrente da fonte a zero.
Além das equações de Maxwell, as seguintes relações descrevem meios lineares e isotrópicos:
A permissividade ou constante dielétrica ε (farad por metro), a permeabilidade magnética μ (henry por metro) e a condutividade elétrica σ
(siemens por metro) são tratadas aqui como constantes escalares. Em
meios anisotrópicos, cada uma se torna um tensor 3 x 3. O comportamento
não linear de qualquer uma das três propriedades pode existir em uma
determinada situação. Embora a não linearidade na condutividade e na
permissividade seja raramente encontrada em problemas de correntes
parasitas, a não linearidade de materiais magnéticos é comum e se
expressa como a dependência da permeabilidade em relação ao campo. Para
aplicações práticas de correntes parasitas, os níveis de excitação
geralmente são baixos o suficiente para justificar a suposição de
linearidade para materiais magnéticos.
Usando essa suposição e substituindo a Eq. 5, a Eq. 2 se torna:
Isso,
no entanto, não é suficiente para especificar completamente os campos
dentro da região da solução, pois a densidade de corrente J contém duas
fontes diferentes. A primeira e mais óbvia é a densidade de corrente
aplicada Js. Uma segunda componente é a densidade de corrente parasita induzida Je. Assim, a Eq. 8 torna-se:
Neste ponto, é útil introduzir o potencial vetor magnético A, que é definido como segue:
Substituindo isso na Eq. 8 e na Eq. 1, obtemos as Eqs. 11 e 12 para uma região livre de fontes:
O campo elétrico na Eq. 12 é:
A
Eq. 13 mostra que o campo elétrico pode ser dividido em um termo de
potencial vetor magnético e uma contribuição escrita como o gradiente
de um potencial escalar. O gradiente do potencial é incluído para
expressar o campo elétrico como uma forma geral que satisfaz a Eq. 12.
O potencial escalar é eliminado quando a Eq. 13 é substituída na Eq. 1,
porque o rotacional do gradiente é identicamente zero.
Portanto, o campo eletromagnético está definido para qualquer problema físico específico, mas A e Φ
ainda não estão definidos. Por exemplo, um gradiente de potencial
diferente poderia ser adicionado ao termo do potencial vetor em vez do \/Φ original, e A
poderia ser ajustado para fornecer o campo elétrico correto. A
expressão resultante satisfaria a Eq. 12 e produziria o mesmo fluxo
magnético da Eq. 1. Portanto, há flexibilidade na escolha de A e Φ.
Para garantir que os potenciais sejam definidos de forma única, a
partição do campo deve ser fixada de alguma forma. Isso geralmente é
feito completando a definição de A.
Um
campo vetorial pode ser definido, além de uma constante arbitrária,
especificando seu rotacional e sua divergência. No caso do potencial
vetor magnético, o rotacional é dado pela Eq. 10. É necessário apenas
decidir sobre a divergência para que ela esteja totalmente
especificada. A especificação da divergência é chamada de condição de
calibre.
Substituindo a Eq. 13 na Eq. 12 fornece:
Expandindo o lado esquerdo com a identidade vetorial \/ x \/ x = \/\/-\/, obtemos:
A divergência de A
é comumente definida como zero (condição de calibre de Coulomb), mas
isso, em geral, não separaria os potenciais escalar e vetorial. Em vez
disso, a condição de calibre é escolhida:
o que elimina os dois últimos termos da equação 15, resultando em:
A equação 17 se assemelha à equação de difusão para fluxo de calor e possui soluções semelhantes no domínio do tempo.
A
maioria dos ensaios de correntes parasitas, no entanto, é realizada com
corrente alternada, cuja dependência temporal é simplesmente uma
oscilação harmônica no tempo. A oscilação harmônica é caracterizada por
uma amplitude e uma fase, que podem ser convenientemente representadas
na forma fasorial: A(r,t) = R{A(r) ejwt}, onde A(r)
é um vetor complexo que representa a amplitude e a fase das componentes
do potencial vetor magnético e onde j = \/(-1), R denota a operação de
extrair a parte real e ω
é a frequência angular (radianos por segundo). Observe que o mesmo
símbolo é usado aqui para representar tanto a quantidade real
dependente do tempo A(r,t) quanto a quantidade complexa A(r),
mas elas são distinguidas por seus argumentos. Em outros lugares, os
argumentos não serão fornecidos e a distinção entre as duas deve ser
reconhecida pelo contexto. A derivada temporal fornece:
Portanto, para a teoria harmônica temporal, jω é substituído por δ.(δt)-1 na Eq. 17 e o potencial vetor pode ser visto como um fasor complexo. Dessa forma, a Eq. A equação 17 torna-se a equação 19:
1.3 MODÊLOS ANALÍTICO E NUMÉRICO
Existem
diferentes tipos de modelos. Alguns são analíticos e outros numéricos.
Os modelos analíticos são computacionalmente mais eficientes do que os
modelos numéricos. No entanto, os modelos numéricos são muito mais
flexíveis e podem ser usados para modelar geometrias complexas de
descontinuidades, não linearidade do material e outras complexidades
associadas a cenários de ensaiosreais.
A
seguir, são descritos modelos analíticos que caracterizam o
comportamento de correntes parasitas em meios condutores homogêneos
livres de descontinuidades, particularmente o modelo proposto por Dodd
e Deeds (R10) e
suas extensões. Soluções analíticas e integrais, técnicas numéricas que
abrangem descontinuidades em materiais, também são descritas a seguir,
assim como técnicas numéricas baseadas em análise de diferenças finitas
e elementos finitos.
2. MODÊLO DO MEIO CONDUTOR HOMOGÊNEO
2.1 FUNDAMENTOS
Os ensaios quantitativos de correntes parasitas baseados em modelos
evoluíram de forma constante com as melhorias na capacidade
computacional. O foco na modelagem precisa levou a uma compreensão
completa dos ensaios de correntes parasitas e à automação total dos ensaios de campo. (R02) (R07) A
modelagem é realizada resolvendo as equações de Maxwell, e as soluções
podem ser expressas analiticamente ou numericamente. As soluções
analíticas fornecem expressões de forma fechada para os parâmetros de
interesse nos ensaios de correntes parasitas e são o tema da presente
discussão.
Os
modelos de ensaios de correntes parasitas podem ser usados para o
projeto da bobina, seleção da frequência de ensaio e interpretação dos
dados de ensaio. Grandezas importantes a serem calculadas são a
distribuição de correntes parasitas induzidas no espécime submetido ao
ensaio, bem como a mudança de impedância resultante da bobina. O
cálculo
e a visualização do padrão de fluxo de correntes parasitas podem ser
usados para avaliar a profundidade real de penetração no material e a
interação com descontinuidades específicas. Dessa forma, a configuração
da bobina pode ser otimizada para garantir a máxima interação com
determinados tipos de descontinuidade, levando em consideração
adequadamente a frequência e os parâmetros do material. O cálculo e a
visualização dos planos de impedância podem ser usados para
comparação com sinais de ensaios reais. Essa comparação proporciona uma
melhor compreensão das variações de impedância decorrentes de
descontinuidades conhecidas de tamanho e orientação específicos, bem
como de características materiais e espaciais particulares do objeto de
ensaio.
Problemas
relacionados à indução de correntes parasitas são formulados por meio
de equações diferenciais, que determinam o campo magnético e grandezas
correlatas em um determinado ponto em função da densidade de corrente
de uma fonte existente. O fluxo de correntes parasitas é calculado
utilizando-se a equação diferencial de difusão, que é convenientemente
expressa em termos do potencial vetor magnético. Existem duas maneiras
de resolver essa equação diferencial: técnicas analíticas e numéricas.
Analiticamente,
a equação é resolvida pela técnica de separação de variáveis dentro
de uma região da geometria. A influência de fontes externas à região é
considerada pela imposição de condições de contorno apropriadas.
Soluções analíticas podem lidar com problemas bidimensionais, problemas
axisimétricos e, em certos casos, tridimensionais, desde que as
equações correspondentes sejam lineares e a geometria das fronteiras e
fontes seja relativamente simples. Como a classe de geometrias que
podem ser tratadas geralmente se restringe a problemas com fronteiras
canônicas (regiões planas, cilíndricas e esféricas), essas técnicas
permitem apenas uma aproximação para problemas com fronteiras não
canônicas ou descontinuidades. As soluções obtidas por técnicas
analíticas são gerais e exatas, proporcionando uma compreensão mais
profunda do problema. Elas são normalmente obtidas na forma de uma
relação matemática, que pode então ser usada para análise, estudos
paramétricos e calibração de sistemas de ensaio. Um aspecto importante
dos modelos analíticos é que as expressões de forma fechada são
facilmente codificadas, seja com linguagens de programação de alto
nível ou com pacotes matemáticos comerciais, exigindo, portanto, um
esforço mínimo do desenvolvedor. Quando as soluções são codificadas,
elas são muito mais rápidas do que as técnicas numéricas, que exigem
tempos de computação significativamente maiores.
Soluções
analíticas também são usadas para validação de soluções obtidas por
técnicas numéricas mais complexas. Estas últimas produzem resultados
numéricos em vez de expressões de forma fechada, e sua precisão pode
ser confirmada independentemente por modelos analíticos, que fornecem
uma alternativa de baixo custo à verificação experimental de resultados
numéricos.
Modelos
para problemas com fronteiras canônicas são descritos abaixo, começando
com os modelos bem estabelecidos desenvolvidos por Dodd e Deeds.
Extensões desses modelos, bem como modelos tridimensionais e modelos
semianalíticos para problemas envolvendo fronteiras canônicas, são
apresentados. Soluções aproximadas com aplicação à modelagem de
descontinuidades são apresentadas em outro local, abaixo.
2.2 MODÊLOS ANALÍTICOS
No
caso de uma geometria axisimétrica bidimensional com simetria
rotacional em torno do eixo Z, a Eq. 17 em uma região livre de fontes
torna-se:
A
equação acima é resolvida adotando-se a técnica de separação de
variáveis. Embora muitas aplicações possam ser modeladas com uma
geometria axisimétrica, muitas aplicações são descritas por uma
geometria tridimensional que apresenta dificuldades específicas. Essas
dificuldades surgem ao usar coordenadas curvilíneas para a descrição do
problema, porque os componentes de A (Eq. 17) estão acoplados nas
equações diferenciais escalares resultantes. Nesse caso, a técnica de
separação de variáveis não pode ser aplicada. O inconveniente é
evitado usando o potencial vetorial de segunda ordem W, que foi
introduzido por Smythe (R08). Para o caso de um solenoidal A, com
divergência zero como na Eq. 17, W é definido como:
onde μ é um vetor unitário fixo e Wa e Wb são duas funções escalares ortogonais que satisfazem a equação escalar:
Como
a equação acima é separável em vários sistemas de coordenadas,
formulações baseadas em W podem ser usadas efetivamente para a
separação da equação diferencial vetorial da Eq. 17.
Modelos
analíticos adequados para ensaios de correntes parasitas foram
desenvolvidos ao longo dos anos por pesquisadores em ensaios não
destrutivos e em geofísica, bem como por projetistas de ímãs, motores e
aceleradores. Inicialmente, o problema básico estudado era o de uma
fonte de corrente filamentar próxima a um objeto condutor de ensaio. Uma
revisão e uma lista de soluções são apresentadas por Tegopoulos e
Kriezis (R09) para
uma variedade de configurações em relação à forma das fontes e à
geometria dos meios condutores. Os problemas bidimensionais são
estudados usando o potencial vetor magnético A, enquanto os problemas
tridimensionais são tratados usando o potencial vetor de segunda ordem
W.
2.3 MODÊLOS DE DODD E DEEFS
Na teoria do ensaios por correntes parasitas, o trabalho de Dodd e Deeds (R10) forneceu
a base para um dos modelos mais populares. Com base em uma série de
trabalhos anteriores, eles apresentaram soluções para distribuições de
correntes parasitas, na forma de integrais de Fourier-Bessel, para
diversas configurações de bobinas axissimétricas frequentemente
encontradas em aplicações de ensaios por correntes parasitas. Essas
soluções foram aplicadas ao cálculo de correntes parasitas produzidas
por bobinas cilíndricas em condutores planos e cilíndricos, na análise
de mudanças de impedância da bobina causadas pela presença de tais
condutores e na previsão de mudanças de impedância causadas por
descontinuidades no seu interior. (R11) (R12) Uma
característica essencial da análise de Dodd e Deeds é que, em
frequências típicas de correntes parasitas, uma bobina de múltiplas
espiras enrolada com fio isolado circular pode ser aproximada por uma
lâmina de corrente, obtendo-se o campo eletromagnético por superposição.
A
equação diferencial resolvida foi a Eq. 20 e a impedância da bobina foi
calculada a partir da seguinte expressão para simetria axial:
onde Acs é
a área da seção transversal (metros quadrados) e N é o número de
espiras na bobina. O princípio da superposição é aplicado integrando o
potencial vetor magnético sobre a área da seção transversal da bobina.
Closed form expressions for the
electromagnetic field and the coil
impedance were obtained for a variety of
common test object geometries (Fig. 1):
for a cylindrical coil of rectangular cross
section above a layered plane, encircling a
layered rod or inside a cylindrically
layered bore hole. The spherical
configuration of Fig. 1c was also
considered but the particular case of a
rectangular cross section coil was analyzed
by Nikitin.(R13)(R14) Once the calculations are
performed using a single coil, the analysis
can be extended to multiple coil
configurations simply by superimposing
the solutions.(R11)(R15) Dodd’s models were
also extended to an arbitrary number of
layers, by using the matrix technique
proposed by Cheng, Dodd and Deeds. (R16)(R18)
Expressões
analíticas para o campo eletromagnético e a impedância da bobina foram
obtidas para uma variedade de geometrias comuns de objetos de teste
(Fig. 1): para uma bobina cilíndrica de seção transversal retangular
acima de um plano estratificado, circundando uma haste estratificada ou
dentro de um furo cilíndrico estratificado. A configuração esférica da
Fig. 1c também foi considerada, mas o caso particular de uma bobina de
seção transversal retangular foi analisado por Nikitin. (R13) (R14) Uma
vez realizados os cálculos usando uma única bobina, a análise pode ser
estendida para configurações de múltiplas bobinas simplesmente pela
superposição das soluções. (R11) (R15) Os
modelos de Dodd também foram estendidos para um número arbitrário de
camadas, utilizando a técnica matricial proposta por Cheng, Dodd e
Deeds. (R16) (R18)
 
Ficure 1. Test object geometries for models
of Dodd and Deeds:
(a) layered half space;
(b) layered bore hole;
(c) layered sphere.
Figura 1. Geometrias de objetos de teste para modelos de Dodd e Deeds:
(a) semi-espaço estratificado;
(b) furo estratificado;
(c) esfera estratificada.
For the case of a coil over a
homogeneous conducting half space
(Fig. 2a), the analytical expression for the
coil impedance is given:
Para
o caso de uma bobina sobre um semi-espaço condutor homogêneo (Fig. 2a),
a expressão analítica para a impedância da bobina é dada por:
onde:
e:
where a is the integration variable, J1(x) is
the bessel function of the first kind and
first order, l is the width of the coil
(meter), lo is the liftoff (meter), r1 is the
inner radius of the coil (meter), r2 is the
outer radius of the coil (meter), μ is
relative magnetic permeability
(dimensionless), μo is magnetic
permeability (henry per meter) of free
space and σ is conductivity (siemens per
meter).
onde a é a variável de integração, J 1 (x) é a função de Bessel de primeira espécie e primeira ordem, l é a largura da bobina (metro), lo é a distância (metro), r 1 é o raio interno da bobina (metro), r 2 é o raio externo da bobina (metro), μ é a permeabilidade magnética relativa (adimensional), μ o é a permeabilidade magnética (henry por metro) do vácuo e σ é a condutividade (siemens por metro).
 
Legenda
r1 = coil inner radius = 2 mm (0.08 in.)
r2 = coil outer radius = 4 mm (0.16 in.)
l = coil width = 1 mm (0.04 in.)
μr = relative magnetic permeability of half space (ratio) = 1
σ = 35,4 MS-m-1 (61 percent International Annealed Copper Standard)
Ficure 2. Coil above metal plate:
(a) geometric
configuration;
(b) normalized impedance plane display.
Legenda:
r 1 = raio interno da bobina = 2 mm (0,08 pol.)
r 2 = raio externo da bobina = 4 mm (0,16 pol.)
l = largura da bobina = 1 mm (0,04 pol.)
μ r = permeabilidade magnética relativa do semi-espaço (razão) = 1
σ = 35,4 MS-m -1 (61% do Padrão Internacional de Cobre Recozido)
Figura 2. Bobina acima da placa de metal:
(a) configuração geométrica;
(b) exibição do plano de impedância normalizado.
The eddy current density is calculated
from the magnetic vector potential:
A densidade de corrente parasita é calculada a partir do potencial vetor magnético:
In the case of a normal coil above a
half-space conductor (Fig. 2a), the
induced current density is as follows:
No caso de uma bobina normal acima de um condutor de semi-espaço (Fig. 2a), a densidade de corrente induzida é dada por:
where J is the root mean square of the coil
current.
onde J é a raiz quadrada média da corrente na bobina.
Equations 24 and 28 involve the
numerical computation of an infinite
integral. Numerical integration techniques
available in most numerical analysis
software packages can be used to calculate
the integrals.
As
equações 24 e 28 envolvem o cálculo numérico de uma integral infinita.
Técnicas de integração numérica disponíveis na maioria dos softwares de
análise numérica podem ser usadas para calcular as integrais.
Figure 2b is a computer generated
impedance display for a surface coil. The
impedance is depicted normalized, using
the inductive reactance of the coil in air
as the normalizing factor. (This quantity
can also be computed from Eq. 24 by
setting conductivity to zero, a1 = a). Such
impedance displays demonstrate the
optimum frequency for a specific test.
A
Figura 2b é uma representação de impedância gerada por computador para
uma bobina de superfície. A impedância é representada normalizada,
usando a reatância indutiva da bobina no ar como fator de normalização.
(Essa grandeza também pode ser calculada a partir da Eq. 24, definindo
a condutividade como zero, a₁ = a). Essas representações de impedância demonstram a frequência ideal para um teste específico.
This frequency is usually the one that
produces the best phase difference
between the loci of two parameters. The
conducting half-space material is
aluminum and the solid curve represents
the locus produced by varying the
excitation frequency. Because the
conductivity and frequency always appear
as a product in Eq. 22, the same curve
would have been produced for a constant
excitation frequency and a varying
conductivity. The dashed lines are the
liftoff curves and represent the impedance
variation with coil liftoff. The dotted
curves show the impedance variation with
frequency for different magnetic
permeabilities of the half-space material.
Essa
frequência geralmente é aquela que produz a melhor diferença de fase
entre os lugares geométricos de dois parâmetros. O material condutor do
semi-espaço é o alumínio e a curva sólida representa o lugar geométrico
produzido pela variação da frequência de excitação. Como a
condutividade e a frequência sempre aparecem como um produto na Eq. 22,
a mesma curva teria sido produzida para uma frequência de excitação
constante e uma condutividade variável. As linhas tracejadas são as
curvas de afastamento e representam a variação da impedância com o
afastamento da bobina. As curvas pontilhadas mostram a variação da
impedância com a frequência para diferentes permeabilidades magnéticas
do material do semi-espaço.
Figure 3 is an example of a computer
generated display of eddy current
contours induced bya surface coil at
various frequencies. As expected, the
higher frequencies result in a smaller
penetration of the eddy currents in the
conducting object. Using Eq. 28 for a
variety of coils reveals that peak eddy
current densities associated with larger
coils fall off more slowly with depth than
those produced by smaller coils. A similar
investigation conducted by Mottl (R19)
showed that the standard depth of
penetration and linear-with-depth phase
delay, obtained as solutions for the plane
wave case, very rarely approximate the
eddy current distribution in conducting
samples beneath a real coil. The standard
depth of penetration remains a material
parameter rather than a real measure of
penetration.
A
Figura 3 é um exemplo de uma representação gerada por computador dos
contornos das correntes parasitas induzidas por uma bobina de
superfície em várias frequências. Como esperado, as frequências mais
altas resultam em uma menor penetração das correntes parasitas no
objeto condutor. Usando a Equação 28 para uma variedade de bobinas,
observa-se que as densidades de pico das correntes parasitas associadas
a bobinas maiores diminuem mais lentamente com a profundidade do que
aquelas produzidas por bobinas menores. Uma investigação semelhante
conduzida por Mottl (R19) mostrou
que a profundidade de penetração padrão e o atraso de fase linear com a
profundidade, obtidos como soluções para o caso de onda plana,
raramente se aproximam da distribuição de correntes parasitas em
amostras condutoras sob uma bobina real. A profundidade de penetração
padrão permanece um parâmetro do material, e não uma medida real de
penetração.
Ficure 3. Contours of eddy currents induced by surface coil
at various frequencies:
(a) 1 kHz;
(b) 10 kHz;
(c) 100 kHz.
Figura 3. Contornos das correntes parasitas induzidas por bobina de superfície em várias frequências:
(a) 1 kHz;
(b) 10 kHz;
(c) 100 kHz.
The Dodd and Deeds models have been
proven very useful because they were
successful in predicting experimental data
from eddy current measurements. Since
the 1970s, they have been widely used by
the nondestructive testing community in
the design of eddy current tests. More
specifically, they have been used to
optimize general types of eddy current
tests such as thickness and conductivity
measurements, to optimize specific tests
for specific problems and to help design
general induction instrumentation for
process control.
Os
modelos de Dodd e Deeds provaram ser muito úteis, pois foram
bem-sucedidos na previsão de dados experimentais de medições de
correntes parasitas. Desde a década de 1970, eles têm sido amplamente
utilizados pela comunidade de ensaios não destrutivos no projeto de
testes por correntes parasitas. Mais especificamente, têm sido usados
para otimizar tipos gerais de testes por correntes parasitas, como
medições de espessura e condutividade, para otimizar testes específicos
para problemas específicos e para auxiliar no projeto de instrumentação
de indução geral para controle de processos.
2.4 EXTENSÕES DOS MODÊLOS DE DODD E DEEDS
The Dodd and Deeds models assume a
harmonic time variation for the solution
of the diffusion equation. Similar
modeling techniques can be used in the
case of transient coil excitations, such as
step time functions or rectangular pulses.
These current excitations are used in the
pulsed eddy current technique, which is
applied to either metal loss or crack
detection at greater depths.
Os
modelos de Dodd e Deeds assumem uma variação harmônica no tempo para a
solução da equação de difusão. Técnicas de modelagem semelhantes podem
ser usadas no caso de excitações transientes de bobinas, como funções
de tempo em degrau ou pulsos retangulares. Essas excitações de corrente
são usadas na técnica de correntes parasitas pulsadas, que é aplicada à
detecção de perdas de metal ou trincas em maiores profundidades.
Além
da superposição de bobinas, diferentes frequências também podem ser
sobrepostas para obter a resposta de um sistema de correntes parasitas
transientes. Uma técnica simples para avaliar campos transientes é
obter, por meio de uma transformada de Fourier, o espectro de
frequência do pulso de corrente de excitação e calcular a resposta de
tensão em cada frequência, adquirindo assim o espectro de
tensão-frequência. A resposta de tensão transiente é então obtida por
uma transformada inversa de Fourier. Uma vantagem distinta dessa
técnica é que ela pode ser aproximada numericamente usando a
transformada rápida de Fourier. Bowler (R20) usa
essa abordagem para uma excitação pulsada com a forma de uma função
degrau com a bobina localizada acima de um sistema estratificado.
consistindo em duas placas. A configuração imita geometrias encontradas
na detecção e identificação de metal em juntas sobrepostas de aeronaves.
Outra
técnica para avaliar campos transientes é calcular a transformada de
Laplace das equações de campo, resolver as equações transformadas e
recuperar o comportamento no domínio do tempo por meio de uma
transformada inversa de Laplace. Essa abordagem é seguida por Waidelich (R21) , Ludwig (R22) , Sapunov (R23) e Bowler (R24) para
obter a resposta de tensão de uma bobina situada acima de um plano
condutor estratificado. No caso de um semi-espaço condutor homogêneo ou
para sistemas de placas finas simples (R25) ,
a transformada inversa de Laplace pode ser obtida analiticamente, mas
no caso de um semi-espaço estratificado isso não é possível e técnicas
numéricas são necessárias para obter a resposta em função do tempo.
Nessa situação, uma rotina numérica robusta deve ser usada para
calcular a transformada inversa de Laplace. Em outras situações, é
preferível trabalhar com a solução no domínio da frequência, como já
descrito, usando a transformada de Fourier.
In addition to coil superposition,
different frequencies can also be
superimposed to obtain the response of a
transient eddy current system. A simple
technique of evaluating transient fields is
to obtain, through a fourier transform,
the frequency spectrum of the excitation
current pulse and to calculate the voltage
response at each frequency, thus acquiring
the voltage frequency spectrum. The
transient voltage response is then
obtained by an inverse fourier transform.
A distinct advantage of this technique is
that it can be approximated numerically
using the fast fourier transform. Bowler (R20)
uses this approach for a pulsed excitation
having the form of a step function with
the coil located above a layered system.
consisting of two slabs. The configuration
mimics geometries encountered in the
detection and identification of the metal in lap joints of aircraft.
Another technique of evaluating
transient fields is to compute the laplace
transform of the field equations, solve the
transformed equations and recover the
time domain behavior through an inverse
laplace transform. This approach is
followed by Waidelich(R21), Ludwig(R22),
Sapunov(R23) and Bowler(R24) to obtain the
voltage response of a coil situated above a
layered conducting plane. In the case of a
homogeneous conducting half space or
for simple thin plate systems(R25), the
inverse laplace transform can be obtained
analytically but in the case of a layered
half space this is not possible and
numerical techniques are needed to
obtain the response as a function of time.
In the above situation, a robust numerical
routine should be used for computing the
inverse laplace transform. In other
situations, it is preferable to work with the
frequency domain solution, as already
described, using the fourier transform.
As
Figuras 4 a 6 mostram as respostas de tensão obtidas para o caso
descrito por Bowler.² A resposta de tensão é calculada avaliando-se
numericamente a transformada inversa de Laplace. Observa-se que certas
características do pulso, como a amplitude, o tempo de chegada do
máximo e o ponto de cruzamento, são sensíveis a diferentes
características geométricas, possibilitando assim a estimativa da perda
no metal.
Figures 4 to 6 show voltage responses
derived for the case described by Bowler.2°
The voltage response is computed by
numerically evaluating the inverse laplace
transform. It is observed that certain
features of the pulse, such as the
amplitude of the pulse, the time of arrival
of the maximum and the cross point, are
sensitive to different geometry
characteristics, thus making possible the
estimation of metal loss.
Legenda:
Ficure 4. Top plate metal loss in system of two plates:
(a) setup;
(b) transient electric potential.
Depicted signal is
coil voltage subtracted from response of same coil due to
conducting half space.
Percentage of parameter variation is
in terms of thickness of one slab.
Figura 4. Perda metálica na placa superior em um sistema de duas placas:
(a) configuração;
(b) potencial elétrico transiente.
O sinal apresentado é a tensão da bobina subtraída da resposta da mesma bobina devido ao semi-espaço condutor.
A porcentagem de variação do parâmetro é em função da espessura de uma placa.
Legenda:
Ficure 5. Plate separation in system of two plates:
(a) setup;
(b) transient electric potential.
Depicted signal is coil voltage
subtracted from response of same coil due to conducting
half space.
Percentage of parameter variation is in terms of
thickness of one slab.
Figura 5. Separação entre placas em um sistema de duas placas:
(a) configuração;
(b) potencial elétrico transiente.
O sinal apresentado é a tensão da bobina subtraída da resposta da mesma bobina devido ao semi-espaço condutor.
A porcentagem de variação do parâmetro é em função da espessura de uma placa.
Legenda:
Ficure 6. Bottom plate metal loss above system of two plates:
(a) setup;
(b) transient electric potential.
Depicted signal is coil voltage
subtracted from response of same coil due to conducting
half space.
Percentage of parameter variation is in terms of
thickness of one slab.
Figura 6. Perda metálica na placa inferior acima do sistema de duas placas:
(a) configuração;
(b) potencial elétrico transiente.
O sinal apresentado é a tensão da bobina subtraída da resposta da mesma bobina devido ao semi-espaço condutor.
A porcentagem de variação do parâmetro é em função da espessura de uma placa.
Other extensions of Dodd’s modeling
technique concern the conductivity and
permeability profiles of the test objects.
Applications include case hardening, heat
treatment, ion bombardment or chemical
processes, which produce smoothly
varying near surface conductivity and
permeability profiles. In these cases,
where for example the conductivity σ(z)
in Eq. 20 is a continuous function of
depth, the electromagnetic field and the
impedance of the coil can be evaluated in
two ways.
Outras
extensões da técnica de modelagem de Dodd dizem respeito aos perfis de
condutividade e permeabilidade dos objetos de teste. As aplicações
incluem têmpera superficial, tratamento térmico, bombardeio iônico ou
processos químicos, que produzem perfis de condutividade e
permeabilidade próximos à superfície com variação suave. Nesses casos,
onde, por exemplo, a condutividade σ (z)
na Eq. 20 é uma função contínua da profundidade, o campo
eletromagnético e a impedância da bobina podem ser avaliados de duas
maneiras.
The first is to solve Eq. 20 analytically
for special forms of conductivity
variations. Such solutions that result in
closed form expressions involving higher
transcendental functions have been
derived by many researchers for specific
functions not only of conductivity but
also of magnetic permeability profiles.(R26)(R29)
This approach is much faster than the
more general piecewise approach
described next.
A
primeira é resolver a Eq. 20 analiticamente para formas especiais de
variações de condutividade. Tais soluções, que resultam em expressões
de forma fechada envolvendo funções transcendentais de ordem superior,
foram derivadas por muitos pesquisadores para funções específicas não
apenas de condutividade, mas também de perfis de permeabilidade
magnética. (R26) (R29) Essa abordagem é muito mais rápida do que a abordagem por partes mais geral descrita a seguir.
Como discutido acima, Cheng (R17) estendeu
os modelos de Dodd e Deed para regiões estratificadas com um número
arbitrário de camadas. Se os perfis contínuos de condutividade e
permeabilidade forem substituídos por perfis constantes por partes,
então é possível aproximar numericamente a impedância da bobina
implementando a técnica acima. Quanto maior o número de camadas, melhor
a aproximação. Usando esta técnica, Uzal (R26)estudou
o problema de um condutor revestido cuja condutividade do revestimento
variava continuamente com a profundidade e a permeabilidade. Embora
esta técnica seja mais lenta do que a baseada na solução analítica para
cada perfil específico, ela é mais geral e particularmente útil quando
se deseja resolver o problema inverso, ou seja, avaliar o perfil a
partir de medições de frequência variável. A abordagem por partes
também foi estendida a objetos de teste cilíndricos e esféricos por
Uzal e Theodoulidis, respectivamente. (R30) (R31)
As discussed above, Cheng (R17) extended
Dodd and Deed’s models to layered
regions with an arbitrary number of
layers. If continuous conductivity and
permeability profiles are replaced with
piecewise constant profiles, then it is
possible to approximate numerically the
coil impedance by implementing the
above technique. The greater the number
of layers, the better the approximation.
Using this technique, Uzal (R26) studied the
problem of a coated conductor whose
coating conductivity varied continuously
with depth and permeability. Although
this technique is slower than the one
based on the analytical solution for each
specific profile, it is more general and
particularly useful when it is desired to
solve the inverse problem, that is, to
evaluate the profile from variable
frequency measurements. The piecewise
approach was also extended to cylindrical
and spherical test objects by Uzal and
Theodoulidis, respectively. (R30)(R31)
2.5 MODÊLOS TRIDIMENSIONAIS
Os
modelos descritos até agora são bidimensionais e axissimétricos. Sua
simplicidade reside no fato de o potencial vetor magnético ter apenas
uma componente e a técnica de separação de variáveis ser aplicável.
Uma quantidade significativa de trabalhos aborda modelos de bobinas com
formatos diferentes da bobina cilíndrica clássica ou posições que
destroem a axissimetria. Um problema de grande interesse é a avaliação
do campo eletromagnético tridimensional para uma bobina com formato e
orientação arbitrários sobre um semi-espaço condutor.
The models described so far are
two-dimensional and axisymmetric. Their
simplicity lies in the fact that the
magnetic vector potential has only one
component and the technique of
separation of variables is applicable.
A significant amount of work concerns
models of coils that have shapes other
than the classical cylindrical coil or
positions that destroy the axisymmetry. A
problem of great interest is the evaluation
of the three-dimensional electromagnetic
field for a coil with an arbitrary shape and
orientation above a conducting half space.
Weaver (R32) apresentou
uma teoria geral da indução eletromagnética em um semi-espaço condutor
por uma fonte magnética externa usando os vetores de Hertz elétrico e
magnético, enquanto Hannakam (R33) forneceu
soluções para uma bobina filamentar usando a formulação similar do
potencial vetor de segunda ordem. Com base nesta última formulação,
Kriezis (R34) avaliou
a densidade de corrente parasita induzida em um semi-espaço condutor
por uma bobina filamentar cujo eixo é paralelo à superfície.
Weaver (R32) presented a general theory of
electromagnetic induction in a
conducting half space by an external
magnetic source using the electric and
magnetic hertz vectors whereas
Hannakam (R33) provided solutions for a
filamentary coil using the similar second
order vector potential formulation. Based
on the latter formulation, Kriezis (R34)
evaluated the eddy current density
induced in a conducting half space by a
filamentary coil whose axis is parallel to
the surface.
Outros pesquisadores, como Beissner e Bowler (R35), têm
preferido as funções diádicas de Green na resolução do problema. Bowler
conseguiu apresentar expressões analíticas para a densidade de
correntes parasitas de uma bobina cilíndrica orientada verticalmente
sobre um semi-espaço condutor, estendendo assim os resultados de
Kriezis para uma bobina de sonda de correntes parasitas de seção
transversal finita. Beissner (R37) e Tsaknakis (R38) apresentaram
fórmulas para a distribuição de correntes parasitas provenientes de
fontes cilindricamente simétricas inclinadas em um ângulo arbitrário em
relação à normal da superfície. A solução geral para uma fonte não
simétrica assume a forma de uma integral de Fourier bidimensional.
Other researchers like Beissner?S and
Bowler (R35) have favored Green’s dyadic
functions in solving the problem. Bowler
was able to present analytical expressions
for the eddy current density of a vertically
oriented cylindrical coil over a conducting
half space, thus extending the results of
Kriezis to an eddy current probe coil of
finite cross section. Beissner (R37) and
Tsaknakis (R38) presented formulas for the
eddy current distribution from.
cylindrically symmetric sources inclined
at an arbitrary angle with respect to the
surface normal. The general solution for a
nonsymmetric source is in the form of a
two-dimensional fourier integral.
Os
cálculos numéricos para o caso não simétrico são, portanto, mais
exigentes do que aqueles necessários para avaliar os campos a partir
das fórmulas de Dodd e Deeds, onde as integrais são unidimensionais. Um
modelo semianalítico também foi apresentado por Juillard (R39) para
o mesmo problema, onde a bobina é dividida em vários elementos
denominados fontes de corrente pontuais. O problema é resolvido para
cada fonte de corrente pontual e a superposição é aplicada para
calcular o campo eletromagnético de toda a bobina. Outra técnica para
calcular o campo magnético, baseada na transformada de Fourier, foi
apresentada por Panas (R40) e Sadeghi (R41), que resolveram o problema de uma bobina elíptica e de uma bobina retangular em posição inclinada, respectivamente.
Numerical computations for the
nonsymmetric case are therefore more
demanding than those needed to evaluate
fields from Dodd and Deeds formulas,
where the integrals are one-dimensional.
A semianalytical model was also presented
by Juillard (R39) for the same problem where
the coil is divided in a number of
elements called point current sources. The
problem is solved for each point current
source and superposition is applied to
compute the electromagnetic field from
the whole coil. Another technique for
computing the magnetic field, based on
the fourier transform, was presented by
Panas (R40) and Sadeghi, (R41) who solved the
problem of an elliptical coil and a
rectangular coil in an inclined position,
respectively.
Uma
conclusão importante de todos esses estudos é que as correntes
parasitas induzidas no condutor fluem paralelamente à superfície do
condutor, independentemente da forma da bobina indutora. As Figuras 7 e
8 mostram as correntes parasitas induzidas na superfície de um
semi-espaço metálico condutor por uma bobina retangular quando a bobina
está paralela e perpendicular ao metal.
An important conclusion of all these
studies is that the eddy currents induced
in the conductor flow parallel to the
surface of the conductor, irrespective of
the shape of the inducing coil. Figures 7
and 8 show the eddy currents induced on
the surface of a conducting metal half
space from a rectangular coil when the
coil is parallel and perpendicular to the
metal.
 
Ficure 7. Eddy current testing with rectangular coil parallel
to test object:
(a) setup;
(b) eddy current pattern.
Figura 7. Teste de correntes parasitas com bobina retangular paralela ao objeto de teste:
(a) configuração;
(b) padrão de correntes parasitas.
 
Ficure 8. Eddy current testing with rectangular coil
perpendicular to test object:
(a) setup;
(b) eddy current
pattern.
Figura 8. Teste de correntes parasitas com bobina retangular perpendicular ao objeto de teste:
(a) configuração;
(b) padrão de correntes parasitas.
O problema de uma bobina de formato arbitrário adjacente a um sistema condutor cilíndrico foi estudado por Hannakam (R42) com o potencial vetorial de segunda ordem e por Grimberg (R43) (R44) com funções de Green diádicas. Hannakam (R45) , Theodoulidis (R46) e Mrozynski (R47) estenderam
a formulação do potencial vetorial de segunda ordem no sistema de
coordenadas esféricas para resolver o problema de uma bobina de formato
arbitrário adjacente a uma esfera condutora. Uma conclusão importante
foi que as correntes parasitas fluem em superfícies esféricas
concêntricas à superfície do condutor.
The problem of an arbitrarily shaped
coil beside a cylindrical conducting
system was studied by Hannakam (R42) with
the second order vector potential and by
Grimberg (R43)(R44) with dyadic Green’s
functions. Hannakam, (R45) Theodoulidis (R46)
and Mrozynski (R47) extended the second
order vector potential formulation in the
spherical coordinate system to solve for
an arbitrarily shaped coil beside a
conducting sphere. An important
conclusion was that the eddy currents
flow in spherical surfaces concentric with
the conductor’s surface.
Todas
as soluções analíticas acima mencionadas referem-se ao campo
eletromagnético, com ênfase na densidade de correntes parasitas
induzidas. A variação da impedância da bobina, por outro lado, é
calculada em duas etapas: (1) primeiro, o problema tridimensional da
avaliação do campo eletromagnético é resolvido analiticamente e (2) em
seguida, aplica-se a expressão geral da variação da impedância de uma
bobina. Uma expressão para a variação da impedância foi derivada por
Auld (R48) .
Demonstrou-se, por meio do teorema da reciprocidade de Lorenz, que a
variação da impedância de uma sonda de correntes parasitas na presença
de uma descontinuidade é expressa em termos de uma integral avaliada
sobre qualquer superfície fechada S que contenha a descontinuidade.
All of the above analytical solutions
concern the electromagnetic field with
emphasis on the induced eddy current
density. The impedance change of the
coil, on the other hand, is calculated in
two steps: (1) first the three-dimensional
problem of evaluating the electromagnetic
field is solved analytically and (2) then
the general expression of the impedance
change of a coil is applied. An impedance
change expression was derived by Auld.(R48)
It was shown, through the lorenz
reciprocity theorem, that the change in
the impedance of an eddy current probe
in the presence of a discontinuity is
expressed in terms of an integral
evaluated over any closed surface S
containing the discontinuity.
where n is the unit vector normal to the surface and where E and H are the electric
and magnetic field intensities; the primed
symbols denote the fields in the presence
of the discontinuity and the unprimed
symbols denote the fields in the absence
of the discontinuity. The ΔZ formula is
well suited to derivation of general
expressions and can also be used
effectively to compute the impedance
change of a coil in canonical problems.5
This development is significant because
the coil geometry does not appear
explicitly (no integrals appear over the
volume of the coil) and allows the choice
of planar, cylindrical and spherical
boundaries in keeping with the symmetry
of the problem.
onde n é o vetor unitário normal à superfície e onde E e H são
as intensidades dos campos elétrico e magnético; os símbolos com
apóstrofo denotam os campos na presença da descontinuidade e os
símbolos sem apóstrofo denotam os campos na ausência da
descontinuidade. A fórmula ΔZ é
adequada para a derivação de expressões gerais e também pode ser usada
efetivamente para calcular a variação de impedância de uma bobina em
problemas canônicos.⁵ Este desenvolvimento é significativo porque a
geometria da bobina não aparece explicitamente (nenhuma integral
aparece sobre o volume da bobina) e permite a escolha de condições de
contorno planas, cilíndricas e esféricas, em consonância com a simetria
do problema.
No
caso particular de uma bobina com forma e orientação arbitrárias, sobre
um semi-espaço condutor, a superfície de integração coincide com a
superfície do semi-espaço, fechada por uma superfície no infinito, que
não contribui. Seguindo essa abordagem e resolvendo analiticamente o
campo eletromagnético tridimensional, Burke* apresentou a seguinte
expressão geral para a impedância de qualquer bobina sobre um
semi-espaço condutor:
In the particular case of a coil with
arbitrary shape and orientation, above a
conducting half space, the surface of
integration coincides with the surface of
the half space, closed by a surface at
infinity, which makes no contribution.
Following this approach and solving
analytically for the three-dimensional
electromagnetic field, Burke*?>° presented
the following general expression for the
impedance of any coil over a conducting
half space:
where uw
andv are integration variables,
onde μ e v são variáveis de integração,
e:
O termo B^5z(μ,v) denota
a transformada dupla de Fourier da componente normal do campo magnético
da fonte na superfície do plano metálico. Para formas de bobina
simples, possui uma expressão analítica em termos de μ e
v. Para formas mais complexas, deve ser calculado numericamente usando
a lei de Biot-Savart. A mesma abordagem foi seguida por Theodoulidiss (R51) (R52) para
avaliar a impedância de uma bobina retangular sobre um semi-espaço
condutor e foi posteriormente estendida a coordenadas cilíndricas para
avaliar a impedância de uma bobina em posição deslocada em relação a um
tubo, simulando assim o sinal de oscilação presente durante os testes
de tubo.
The term B^5z(μ,v) denotes the double
fourier transform of the normal
component of the source magnetic field
on the surface of the metal plane. For
simple coil shapes, it has an analytical
expression in terms of μ and v. For more
complex shapes, it has to be calculated
numerically using the Biot-Savart law. The
same approach was followed by
Theodoulidiss (R51)(R52) for evaluating the
impedance of a rectangular coil over a
conducting half space and was further
extended to cylindrical coordinates for
evaluating the impedance of a bobbin coil
in an offset position to a tube, thus
simulating the wobble signal present
during tube tests.
2.6 PERTURBAÇÃO E EXPANSÃO DA FUNÇÃO DE ENGEN
The class of problems that can be solved
analytically can be extended with the aid
of perturbation techniques, which are
often used to provide solutions to
physical problems that would otherwise
be difficult or time consuming to treat.
Perturbation techniques are inherently
approximate and their main applicability
is in the modeling of discontinuities. Such
techniques can be used by assuming that
the conductivities of the discontinuity
and the surrounding medium do not
differ very much or by considering
limiting cases such as a high frequency
limit. (R53)
A
classe de problemas que podem ser resolvidos analiticamente pode ser
estendida com o auxílio de técnicas de perturbação, frequentemente
utilizadas para fornecer soluções a problemas físicos que, de outra
forma, seriam difíceis ou demorados de tratar. As técnicas de
perturbação são inerentemente aproximadas e sua principal
aplicabilidade reside na modelagem de descontinuidades. Tais técnicas
podem ser utilizadas assumindo-se que as condutividades da
descontinuidade e do meio circundante não diferem muito ou
considerando-se casos limite, como um limite de alta frequência. (R53)
Nevertheless, perturbation techniques
have also been applied to models of
canonical problems. A technique called
the layer approximation, based on the
analytic transfer matrix solution for the
electric field in a layered metal, was used
by Satveli (R54) to calculate the impedance
change in a number of canonical
problems. Burkes (R55) also has presented a
perturbation technique, which enables
the impedance computation in the high
frequency limit when the conducting
region is canonical. The technique was
applied to the cases of a two-dimensional
conducting wedge anda slot in a
conducting half space.
Não
obstante, as técnicas de perturbação também têm sido aplicadas a
modelos de problemas canônicos. Uma técnica denominada aproximação de
camadas, baseada na solução analítica da matriz de transferência para o
campo elétrico em um metal estratificado, foi utilizada por Satveli (R54) para calcular a variação de impedância em diversos problemas canônicos. Burkes (R55) também
apresentou uma técnica de perturbação que permite o cálculo da
impedância no limite de alta frequência quando a região condutora é
canônica. A técnica foi aplicada aos casos de uma cunha condutora
bidimensional e uma fenda em um semi-espaço condutor.
Expansões
das funções de Eigen também podem ser usadas para ampliar ainda mais a classe
de problemas que podem ser resolvidos analiticamente. (R56) (R58) O
problema é resolvido novamente usando separação de variáveis; como a
região de interesse é finita, condições de contorno adicionais limitam
o domínio da solução. Como resultado, a solução envolve séries em vez
de integrais. Os coeficientes da série são calculados resolvendo-se um
sistema matricial, formado pela imposição das condições de interface e
de contorno do problema.
O cálculo numérico dos coeficientes classifica a técnica como semianalítica. A técnica foi efetivamente usada por Theodoulidis (R59) para
derivar uma expressão para a impedância de uma bobina de sonda com
núcleo de ferrite sobre um semi-espaço condutor em camadas.
2.7 CONCLUSÕES
As
soluções analíticas em ensaios por correntes parasitas, embora
restritas a certas geometrias em comparação com as soluções numéricas
mais gerais, possuem uma forma explícita e fechada. Os modelos não são
computacionalmente intensivos e oferecem soluções precisas. Eles têm
escopo limitado, mas não valor limitado.
Sempre
que plausível, as soluções analíticas são preferíveis às numéricas
porque são mais fáceis de aplicar, menos dispendiosas
computacionalmente, mais precisas e, finalmente, permitem estudos
paramétricos fáceis da geometria do ensaio.
3. MODÊLOS ANALÍTICOS E INTEGRAL PARA SIMULAR TRINCAS
3.1 INTRODUÇÃO
Eddy current nondestructive testing uses
inductive probes to excite currents in
electrical conductors. The simple fact that
the coil carrying an alternating current
can sense a discontinuity in a metal is
intuitively easy to understand but
evaluating the signal for a given
configuration of coil and discontinuity is
not always easy. The present discussion
describes calculations of probe signals
from cracks, starting with a review of the
basic theoretical concepts and moving on
to a number of related techniques for
evaluating probe response.
Early investigators applied concepts
from other fields of electromagnetism to
problems in eddy current testing. The
researcher in relatively unexplored areas
of electromagnetic theory inevitably
brings concepts from the parent discipline
and adapts them for the new field of
investigation. As advances in the new area
begin to mature, the new discipline
adopts distinct themes and approaches
that are successful and rewarding. At the
end of the twentieth century, eddy
current nondestructive testing was at a
point of early maturity. Basic problems
had been solved satisfactorily yet many
problems remained open and relatively
underdeveloped.
This discussion of crack theory briefly
reviews a few significant early
developments relevant to the treatment of
crack problems in eddy current testing,
including the analysis of the spherical
inclusion and the penny shaped crack.
Recent advanced developments in the
evaluation of crack signals are then briefly
outlined. Two approaches are described:
(1) integral techniques that represent the
effect of a discontinuity in terms of dipole
distribution and (2) approaches valid at
high frequencies that use small
approximations of standard depth of
penetration.
3.2 ELEMENTOS DA TEORIA DE TRINCAS
The pioneering achievements of Friedrich
Forster and his colleagues in eddy current
nondestructive testing resulted from
extensive theoretical and experimental
investigations,© laying the foundations
on which others have built over the
intervening half century. Early uses of
eddy current testing investigated by
Forster are metal sorting, hardness
measurement and the evaluation of heat
treatments through the effects of electrical
resistivity variations. In developing
instruments for these measurements,
Forster recorded the impedance change of
a solenoid when it was near an electrically
conducting material. In the initial
investigations, the solenoid impedance
changes due to the cylindrical rods acting
as cores were measured using an
inductance bridge. It soon became
apparent that the measurements yielded
results dependent on the dimensions of
the rod. Consequently, much effort would
be devoted to the problem of separating
the effects of variations in the sample
dimensions and the variation in
resistivity. Forster’s ultimate success was
made possible by his willingness and
ability to analyze the problem
theoretically. (R61)
Forster used analytical expressions for
the impedance of an infinite solenoid in
the presence of a conducting rod to
account for the effects of variations in rod
diameter and material properties. Later
Dodd and Deeds derived closed form
integral expressions for the field and
impedance of an axial coil of finite length
encircling an infinitely long rod.!° In
addition, they derived integral expressions
for the impedance and field of a
normal
coil above a layered half-space conductor,
a normal coil being one whose axis is
normal to the surface of the conductor.
Although ferrite cored probes may be
preferred for discontinuity detection
because of their enhanced sensitivity, air
cored coils have been widely used in
calculations because of the ease of
evaluating the field using the formulas of
Dodd and Deeds. Usually numerical
techniques are needed to calculate the
fields of probes with ferrite cores.!,2
However, Theodoulidis has shown that
solutions satisfying Maxwell’s equations
for axially symmetric ferrite cored probes
can be found.5?3 Other significant and
interesting results using the analytical
solutions of Maxwell’s equations are
described elsewhere in this chapter.
More thana decade after Férster’s work
became widely known, an embryonic
discontinuity theory was given in the
dissertation of Michael Burrows.** Central
to the thesis is the idea that a small
discontinuity in a conductor, such as a
tiny spherical cavity, produces a perturbed
field that is the same as that of a suitably
chosen dipole. Because the discontinuity
is small compared with the standard
depth of penetration and small on the
scale of other spatial variations of the
unperturbed field, the field can be
approximated as locally uniform and the
polarization of a spheroidal discontinuity
can be found by using standard textbook
theory.®> Having determined the dipole
intensity, Burrows found the induced
electromotive force in a pickup coil due to
the discontinuity by using an expression
derived from reciprocity principles.
(R66)
Because key elements of this approach
arise in more advanced treatments of
discontinuities, the dipole analysis will be
summarized later.
The small discontinuity analysis is
itself of limited practical application but
the principle of representing the effect of
a discontinuity by an equivalent
electromagnetic source distribution can be
applied to arbitrary discontinuities using
either multipole expansions or a dipole
distribution. Multipole techniques for
representing the field have not been
pursued®’ extensively in nondestructive
testing although they may be fruitful.
However, numerous approaches have
been developed based on the
representation of a discontinuity in terms
of a current dipole distribution. Early
developments in which a volume dipole
density was expanded in terms of volume
elements were made by the geophysics
community,°*-7° followed by an adaption
of the technique by McKirdy’! and by
Bowler, Jenkins, Sabbagh and Sabbagh”*73
to the solution of problems in eddy
current testing. An account of the volume
element technique is given in this
handbook and elsewhere.
Although the equivalent source
representation is a common feature of a
number of crack response calculations, a
seminal article by Kahn, Spal and
Feldman”! can be seen as a significant
initial step for developments that have
taken a different path. In the essentially
two-dimensional problem, the field is
uniform along the length of a crack of
constant depth and negligible opening. If
the standard depth of penetration is small
compared with the crack depth, the
current flow follows stream lines parallel
to the crack faces except at the corners
where the crack meets the surface of the
conductor and in the region of the crack
edge. Kahn gives local solutions for the
corner, face and edge field, each of which
contribute to the impedance change. An
interesting feature of the edge field is that
it has the same mathematical form as that
given by Sommerfeld’ for the diffraction
of a plane wave bya half-plane barrier.
The diffraction of an electromagnetic
wave at a thin conducting barrier and the
flow of eddy currents around the edge of
a crack are physically distinct phenomena
but both are governed by Maxwell’s
equations and are subject to comparable
boundary conditions. For a time
harmonic field, the physical difference
between the two cases is manifest in the
wave number, a number that is real in a
lossless medium but complex in a
conductor. Hence the solutions are
essentially the same, differing only in the
nature of the wave number.
Before describing in more detail the
implications of the equivalent source
approaches, typical examples of the
outcomes of such calculations in the form
of probe signals due to cracks are
reviewed.
3.2.1 PLANO DE IMPEDÂNCIAS
The impedance of an eddy current probe
varies with frequency and with its
proximity to the conductor as measured
bya liftoff parameter, defined here as the
distance from the surface of the
conductor to the base of the coil. Energy
dissipated by induced current is related to
an increase in the resistive part of the
driving point impedance whereas the
reactive component of impedance is
reduced by the induced current as a
consequence of Lenz’s law. Following
Forster, the probe impedance
Zn = Ry + jXn, Normalized with respect to
the magnitude of the free space coil
reactance Xo, varies with frequency as
shown on the impedance plane diagram
(Fig. 9),”6 where the normalized reactance,
X, =X-Xo" is plotted against the
normalized resistance Ry= (R —
Ro)-(Xo)"!,
Ro being the free space coil resistance. In
the low frequency limit, X, = 1 and
Rn = 0, as represented by a point at the
top of the main curve. In the high
frequency limit, the curve intersects the
reactance axis at a value of X,, about 0.68
in this case, which depends on the coil
geometry. For flat pancake coils with a
small liftoff, the limiting value of the
normalized reactance has a lower value
than for a longer solenoidal coil with
larger liftoff. Thus, the high frequency
intersection point is a measure of the
coupling between the probe and the work
piece, having a low normalized reactance
for greater coupling.
The data for the main curve in Fig. 9
were calculated from a Dodd and Deeds
formula!® and hence represent results of
an idealization that neglects interwinding
capacitance and the effects of a finite
penetration depth in the windings. The
parameters of a coil are taken from a
benchmark experiment on simulated
cracks in aluminum.’° Superimposed on
the diagram are two signals calculated
using the lowest and highest frequencies
of the experiments, 250 Hz and 50 kHz
respectively, for the same simulated
planar crack.
Ficure 9. Calculated normalized impedance
variation with frequency of normal coil. Two
discontinuity signals from semielliptical
simulated crack are also shown.
Discontinuity responses were calculated for
excitation frequencies of 250 Hz, upper
trace, and 50 kHz, lower trace, for same
simulated crack. Details of coil parameters
and simulated crack are given by Harrison
and Burke.76
In Fig. 10, the calculated discontinuity
signals are displayed with the background
coil impedance removed. The response is
for a normal coil whose axis is in the
plane of the crack. Taking the crack plane
to be the x = 0 plane, then the impedance
variation shown occurs as the coil is
moved in the horizontal Y direction from
one end of the crack to the other. The
numerical techniques used for calculating
these impedance variations are described
elsewhere in this chapter. First some
general comments are in order,
concerning the nature of numerical
schemes.
Ficure 10. Normalized impedance due to semielliptical
simulated crack shown as impedance plane locus. Trace is
obtained from impedance variation as coil position is varied
along crack. Note that impedance of discontinuity has been
normalized by dividing by free space coil reactance at
designated frequency. Details of coil parameters and
simulated crack are given by Harrison and Burke.76
The discontinuity impedance is
calculated from the electromagnetic field
in the presence of the discontinuity.
Simple cases that can be dealt with
analytically are discussed first. For more
complicated geometries, numerical
techniques are needed. Numerical
techniques for solving electromagnetic
field problems are traditionally
categorized as differential or integral
techniques. Finite element and finite
difference techniques are the most
common in the differential category
whereas the integral techniques can be
classified as boundary element and
volume element techniques.
Most numerical schemes introduce a
set of localized functions defined with
respect to a grid or mesh. Often these
functions are low order polynomials,
which interpolate between nodal points
or the edges of cells. Typically, they do
not satisfy Maxwell’s equations (or the
integral equivalent) nor does a linear
superposition of them forma solution.
Nevertheless, it is postulated that a
superposition of such functions gives a
reasonably accurate numerical
approximation of a solution.
The numerical results rarely come with
a guarantee of accuracy. Because of the
way in whicha solution is constructed,
the results are dependent on a mesh or
grid. In the absence of error estimates,
and these are rarely given, it is important
that code is validated because, even if it is
bug free, the onus is on the author to
demonstrate that the results are reliable.
Elementary techniques, on the other
hand, provide a means of predicting
limited results. A
number of simple
formulas for evaluating discontinuity
signals are given below, preceded by a
summary of the basic expressions for a
current dipole field. The dipole theory is
presented in a way that anticipates the
more advanced numerical techniques for
homogeneous conducting media, in
which integral formulations are used.
3.3 DIPOLO DA CORRENTE
3.3.1 MONO POLO DA CORRENTE ESTATÍSTICA
The current dipole is formed from two
monopoles of opposite polarity adjacent
to one another. A current monopole is a
point source of current with intensity I. In
an unbounded homogeneous region, the
current spreads uniformly in all directions
from the source. Hence, the current
density obeys an inverse square law and is
directed radially from the source. Suppose
a current monopole located at a position
represented by the vector r’ gives rise to a
current density J at some other point
whose coordinate is r. Then:
where R = Ir-r’l and R is a radial unit
vector. Expressing the electric field as
E =-V®, then the current monopole
potential is:
where 6p is the electrical conductivity of
the medium. The potential satisfies the
laplace equation, V7® = 0, except at the
singular point where the point source is
located. The (4nR)-! dependence ofa static
potential due to a point source is
identified as a scalar Green’s function for
a laplacian problem in three dimensions.
3.3.2 DIPOLO DE CAMPO ESTÁTICO
Let two current monopoles of opposite
polarity approach one another while
keeping constant the product of their
source intensity and their separation.
With initial separation dr, the dipole
potential is:
In general, the limit of f(r’ + 5r) — f(r’)
as the separation dr’ tends to zero can be
written as 6r’-V f(r’). Hence the limit
above can be related to the gradient of R!
with respect to the primed source
coordinates. The gradient may be written
in terms of the unprimed field
coordinates with a reversal of sign. Also
expressing the dipole moment as the
(finite) limit of p = dr’ gives the static
current dipole potential:
where p is the dipole moment (ampere
meter).
By taking the negative gradient to find
the electric field and multiplying by the
conductivity, the current density can be
written:
Although the scalar product here can
be seen as producinga scalar function on
which the first gradient acts, the above
expression can also be interpreted as a
dyadic operator, VV(4nR)"!, acting on the
vector p. The final result is the same but
the second viewpoint prompts the idea
that the dyad may be detached from the
vector on which it acts and given a
separate mathematical life. Studying the
properties of dyadic Green’s functions’”
leads to distinct ways of finding solutions
of Maxwell's equations as outlined below.
Before returning to the role of the
dyadic Green’s functions, a simple
illustration of the fundamental utility of
the current dipole is given. The dipole
field is used to express the solution of a
problem in which a uniform current in an
otherwise homogeneous conductor of
electrical conductivity 69 encounters a
spherical inclusion of uniform
conductivity σ.
3.3.3 PEQUENA INCLUSÃO ESFÉRICA
The spherical inclusion problem, usually
found in textbooks as a problem in
electrostatics involving a dielectric sphere,
has a solution that satisfies the laplace
equation inside and outside the sphere.
Interface conditions on its surface ensure
that the normal current and tangential
electric field are continuous. Given a
uniform field Ep in the Z direction, which
is also the polar direction of a spherical
coordinate system (z = R cos 8) and
defining the parameter s as the
conductivity ratio s = o-o9', the internal
potential (volt) is:65
whereas outside the sphere the potential
is:
where Θ is the polar angle (radian). The
external potential can also be written:
where the dipole intensity and direction
are given by:
Perhaps of greater interest here is the
fact that the external electric field can be
written:
where Eo = £o2. This goes beyond the basic
textbook account by expressing the field
of the dipole in terms of a dyadic Green’s
function, 69 !VV(42R)"1.
Equation 42 can apply to a dipole of
arbitrary orientation. Figure 11 shows the
current associated with the perturbed field
that when added to the unperturbed
current 6oEo2 gives the total current
density.
Ficure 11. Perturbed current at small spherical inclusion in
metal.
An additional point of interest is that
the dipole intensity can be related to a
uniform current dipole density P
distributed in the spherical region. By
puttingp = 4.3"! x na8P, it is found that:
where E is the electric field in the sphere
given by taking the negative gradient of
Eq. 38.
3.3.4 DIPOLO DINÂMICO DA CORRENTE
In eddy current testing, the fields are
dynamic rather than static. Therefore, the
dynamic current dipole has a more
significant elemental discontinuity field
than does the field of static current
dipole. The dynamic current dipole for a
time harmonic field is described by
essentially the same equations as those
used for the textbook treatment of the
hertzian dipole.’® The difference arises
from the fact that in eddy current
applications the host medium is a
conductor, not air. In a good conductor
such as a metal, the charge current is
much larger than the displacement
current. Consequently, the latter can be
neglected. This means that Ampére’s law
(Eq. 2), V x
H =J, is adequate and
Maxwell’s addition of the displacement
current j@D to the right hand side of this
relationship is not needed. Here, the field
is expressed in terms of complex phasors,
which means, for example, that the
magnetic field varies in time as the real
part of He/, m being the angular
frequency (radian per second) of the
excitation. The neglect of displacement
current means that solutions are sought in
the quasistatic limit. As a short cut from
the description of waves in air to fields in
a conductor, the displacement current
j@eE, which appears in standard hertzian
dipole theory,”* can usually be replaced
with the charge current ooE.
It is convenient to express the dynamic
field in terms of a magnetic vector
potential A, related to the magnetic flux
density:
and having a gage condition:
replacing the usual lorenz condition. For a
current dipole in an unbounded
conductor of conductivity op and
permeability of vacuum, the magnetic
vector potential is:
where k = V(-j@oL09), taking the root with
a positive real part and I-p = p. The
parameterk is related to the standard
depth of penetration 6 (meter):
where:
The identity dyad I in Eq. 46 has been
inserted to express the magnetic vector
potential as a dot product of a dyadic
operator acting on a vector source, this
being the appropriate general form for the
relationship between a vector source and
a vector field A. The magnetic field due to
the current dipole is found from:
The electric field is found from Ampére’s
law in the form:
Combining Eqs. 46, 49 and 50 gives:
Equation 51 has been derived from the
identity V x V x = VV--V? and from the
fact that the vector potential satisfies
Helmholtz’s equation.”? A discussion of
the dyadic form between the braces has
been given by Tai.’78°
Clearly, the dynamic dipole field
reduces to the static case, Eq. 42 in the
limit, as angular frequency @ goes to zero.
It also reduces to the static case in the
near field, where the first term of the
dyadic operator is negligible. This is a
reminder of the fact that a local field on a
scale small compared with the standard
depth of penetration 6 can often be
analyzed using electrostatic or
magnetostatic theory.
Equation 51 may be generalized to give
the perturbed field due to a volumetric
discontinuity by representing the effect of
such a discontinuity as a general dipole
distribution P(r’). Then the perturbed
electric field is found by replacing the
point dipole p in Eq. 51 by P(r’) and
integrating with respect to the source
coordinate r’ over the region of the dipole
density. This field representation is used
in volume integral formulations and is a
preliminary step toward a volume element
calculation of the dipole density.”2
Similarly, the effects of a thin crack can be
represented by a surface dipole layer and
form the basis of a boundary element
formulation.®4 In either case, the dipole
density is determined by an integral
equation. Having founda solution, the
probe signal due to the discontinuity can
be calculated from the probe response
formulas below.
3.3.5 RESPOSTA DA SONDA
An eddy current probe senses
discontinuities through changes of
impedance. There are a
number of
techniques for calculating the
discontinuity response depending on the
details of the approach used. For example,
Kahn and others used the integration of
the poynting vector over a surface.”4 Auld
uses a reciprocity relationship attributed
to Lorenz*® whereas others use a
reciprocal relationship associated with
Rumsey.°°8! Rumsey’s relationship is used
next.
The coil current density can be
represented by a function J. With E®
defined as the perturbed field due to the
discontinuity, the probe impedance
change due to the discontinuity is:
where the integration is over the coil
region denoted by Q,.
The coil current can be used as a phase
reference and taken to be real. Although
the coil current is confined to the coil
windings, these are usually on such a
small scale that the current density can be
approximated as a smooth function,
usually a constant, over the coil cross
section. In a calculation in which the
effects of the discontinuity are
represented by a dipole volume
distribution P, Rumsey’s reciprocal
relations may be invoked to write the
impedance change in terms of the
unperturbed electric field at the
discontinuity E:72
where the integration is now over the
region Qp where the discontinuity
conductivity differs from that of the host.
Equation 53 is advantageous because P is
usually calculated directly by an integral
equation technique whereas the
evaluation of E) for Eq. 52 requires an
additional step once the dipole density
has been found. In general:
which defines P(f) for an arbitrary
discontinuity whose conductivity o(r)
differs from that of the host conductivity
Go. For the special case of the small
spherical region with constant
conductivity, a similar relationship
(Eq. 43) is used.
3.3.6 PEQUENAS DISCONTINUIDADES
For a small spherical discontinuity, such
as a gas bubble or spherical inclusion in a
conductor, the impedance change sensed
by a probe is given by:
An explicit expression for the response
can be found using a suitable unperturbed
field, for example the normal coil field in
a half-space conductor.!° A simpler case is
one where the field at the surface of the
conductor is uniform. This approximation
may in practice be reasonable if the probe
dimensions are larger than the standard
depth of penetration. With Ho as the
tangential magnetic field in the
(horizontal) Y direction and the Z
direction normal to the surface of the
conductor, the unperturbed
electromagnetic field in a conductor
below the plane z =0 is given by:
e:
By substituting the expression for the
dipole density of a small spherical
inclusion given in Eq. 41 into the relation
Eq. 55 with E® given by Eq. 57, it is
found that, for a small spherical cavity
(o = 0), centered at r = ro and of radius a
(meter), the impedance is:
The impedance change is proportional
to a? simply because the dipole intensity
varies in proportion to the volume of the
sphere. Note that the ratio Ho-I"' is real for
a magnetic field uniform at the surface.
However, it may be useful to estimate the
small sphere response for a
nonuniform
field, for which Eq. 55 applies if the
unperturbed field is known. Note that the
maximum value of the ratio Hy-I-! can be
regarded as a figure of merit for the probe
because the signal intensity depends on
its square. Note also that the factor 2jkz
in the exponential of Eq. 58 indicates that
the signal is attenuated over a path of
length 2z, representing the round trip
distance from the surface to the
discontinuity and back.
Another small discontinuity result that
can be found by elementary means is the
response due to a semicircular surface
crack of negligible opening whose radius
is smaller than the standard depth of
penetration. The assumption of a
relatively large standard depth of
penetration means that the local field can
be treated as static in the sense that it
may be described by a potential satisfying
the laplace equation. The surface of the
conductor acts as a plane of reflection,
allowing a conversion of the semicircular
crack problem to a circular crack problem
by appealing to the technique of images.
The problem can then be solved as if the
crack were a thin disk in a uniform stream
of incompressible fluid. With a uniform
applied electromagnetic field given by
Eqs. 56 and 57, the impedance change
due to such a crack is:82:83
Of practical importance is the question
of what limits the detection of small
cracks. Equation 59 yields insight and
significant basic information in this
regard. First, note that the response
depends on the third power of the crack
radius. Second, the impedance change
increases in proportion to the frequency
because k? = — jmpbtoo:
Thirdly note that for a strictly uniform
field, the change in impedance is purely
inductive (imaginary), HoT"! being real.
Even if the assumptions that went into
the derivation of the simple relation given
by Eq. 59 are not precisely satisfied, the
equation can provide an approximate
answer. If the accuracy is inadequate,
improvements may be made by extending
the results to higher order terms by using
perturbation theory or by taking into
account nonuniformities in the field by
an extension of the basic analytical
technique.*?
3.3.7 TRINCAS LONGAS
A long crack of constant depth d (meter)
may be treated as a two-dimensional
problem provided that the unperturbed
field does not vary along its length. Such
a configuration does not relate directly to
most practical problems but its solution
has had an impact on the understanding
of crack fields. The problem can be solved
analytically in the low and high
frequency regimes that correspond to
small and large standard depths of
penetration compared with the crack
depth.
According to the thin penetration
approach of Kahn, assuming the crack is
in the plane x = 0, the field on the crack
faces has the form:
e:
where n is the characteristic impedance of
the medium:
This field can be used to evaluate the
complex time average poynting vector P
(not to be confused with dipole density P)
at the crack faces from:
where the asterisk (*) denotes the complex
conjugate. The uniform face field means
that:
where the characteristic impedance of the
medium is 1 = jk-(oo)-!. The upper and
lower signs on the right side refer to the
positive and negative sides of the crack,
respectively. Integrating the poynting
vector over the crack surface and equating
the result to the energy transferred at the
drive point of the probe gives an
impedance:
per unit length of the crack. To Eq. 65
must be added the corner and edge effects
that together with Zs give rise to a
combined impedance:
The three contributions to the
impedance per unit length include the
field at the edge (represented by the 1 in
parentheses) and the corner field (the
8-1! term). A complete analysis of the
above expression is given elsewhere.*4 The
impedance in this problem therefore
contains a dominant face term that varies
as the square root of frequency, is
proportional to the crack depth and has a
phase angle of m-4-! with respect to the
drive current. The additional terms due to
the edge and corner are resistive.
Complementary to the kahn thin
penetration result is a formula valid in the
low frequency regime that can be found
froma solution valid in the static, direct
current limit. In this regime, ikd is a small
parameter; this fact can be exploited to
find a field solution in the form of an
ordered series using Rayleigh-Ritz
perturbation theory. Likewise, the
impedance can be expressed as an ordered
series:83
For a uniform excitation field, the
leading term at low frequency is purely
inductive and increases linearly with
frequency and as the square of the crack
depth.
The long crack theory is readily
extended in range from the high
frequency limit to lower frequencies by
accounting for the interaction between
the edge and corner fields through the
Weiner-Hopf technique®s and by applying
the perturbation technique to extend the
range of validity of the low frequency
approximation.®’ The impedance results
of these extensions, shown in Fig. 12,
have been compared with numerical
results of a boundary element code.*? In
these figures, the impedance is
normalized by writing:
Legenda:
- - - = teoria de alta frequência
___ = teoria de baixa frequência
o = elemento limite
Ficure 12. Analytical and numerical results of change in
normalized impedance Z, due to long surface breaking
crack: (a) for inductive, or imaginary, component;
(b) for resistive, or real, component.
Hence, the kahn impedance (Eq. 66) is
written in terms of the normalized
impedance:
The main benefit of the study of the two
dimension problem is that it provides a
simple test bed for new techniques,
including an adaption of the geometrical
theory of diffraction,*® to problems in
eddy current crack interaction.
3.4 TÉCNICAS AVANÇADAS
Two types of advanced techniques for
evaluating probe signals due to cracks are
considered next. First, equivalent source
techniques are discussed, of which the
Burrows small discontinuity theory® is an
elementary precursor. Second, the thin
penetration approaches, prototyped by
Kahn and others” and applicable to both
ferromagnetic and nonferromagnetic
materials, are described.
The equivalent source techniques cover
all frequencies and are closely linked with
field formulations based on integral
equations. They can be used to evaluate
fields at cracks in ferromagnetic material
but here the description will be limited to
materials with the permeability of a
vacuum.
Finding a numerical solution from
integral equations can be more
demanding in the thin penetration regime
because a large number of volumetric cells
or boundary elements may be needed to
give an accurate result. Usually, a grid
containing several cells per standard
depth of penetration is required, so the
number of unknowns and the
computational cost are usually high in the
thin penetration regime. Because this cost
is avoided in approaches that explicitly
take advantage of small penetration depth
approximations, the techniques described
here are complementary. To understand
dipole and thin skin techniques, it is
helpful to consider the behavior of the
electric field near the crack mouth and
the properties of the field at the crack
face.
3.4.1 CAMPO ELÈTRICO NA ABERTURA DA TRINCA
The crack opening is typically much
smaller than the standard depth of
penetration. Therefore, a local field theory
for this region can be based on Maxwell’s
equations in the static limit. Because the
electric field varies relatively slowly along
the crack mouth away from the ends, a
two-dimensional solution in a plane
perpendicular to the mouth direction
adequately captures the significant
features. This approach implies that the
solution of the laplace equation in two
dimensions is suitable for the task.
The geometry of the problem (Fig. 13)
lends itself to the Schwarz-Christoffel
theory,8788 which yields a conformal
transformation to map the domain of the
crack and the adjoining half plane above
it into a half plane. An elementary
solution for the half plane will lead to a
fixed potential difference across the crack.
Then, an inverse transform can be applied
to produce a representation of the electric
field at the crack mouth. In this case, a
suitable analytic inverse transform is
apparently lacking and the mapping must
be done numerically by using, possibly,
the newton-raphson iterative technique or
the brent algorithm.89
Forster?° and others?! have used
conformal mapping to determine the
magnetic flux leakage at the crack mouth.
In fact, the mapping is used widely to
find the magnetic field at the gap between
two pole pieces such as the field at the
gap between the poles of a magnetic
recording head.% In eddy current
problems, the electric field is needed
rather than the magnetic field but the
solution is essentially the same (Fig. 13).
Ficure 13. Electric field at crack opening.
At the corners, the electric field is
singular, varying in magnitude in air close
to the corner as (feorner)!/3, Where (Feorner)
is the radial distance from the apex of the
corner. This behavior is characteristic of
the field in the vicinity of a right angled
wedge.°? Between the crack faces, the field
tends to become more uniform deeper
into the crack. The magnitude of the field
between the faces depends on how deep
and wide the crack is. If the crack is made
narrower while the potential across the
crack remains the same, then the
magnitude of the electric field increases.
In the limit of closure without contact,
the electric field forms a singular layer,
infinitely strong, of infinitesimal
thickness. It is this limiting case that will
be explored here because the singular
layer has a simple mathematical
representation.
3.4.2 TRINCA IMPENETRÁVEL
In calculations of the field perturbation
due to a crack, it is usual and convenient
to apply a boundary condition that states
that the normal component of the current
density in the conductor at the crack face
is zero. Although the surface of the crack
supports a distribution of electrical charge
and the charge must get there somehow,
in the quasistatic approximation the
charging current is neglected. In a
conductor, the displacement current jweoE
is neglected because it is very much
smaller than the charge current o9£. Even
at high eddy current test frequencies,
~10 MHz, where the magnitude of
displacement current is greater than at
lower frequencies, the ratio g&@-09! is on
the order of 10~° for a low conductivity
metal, 0.58 MS-m-! (1 percent of the
International Annealed Copper Standard).
However, the accuracy of a boundary
condition that neglects the charging
current at the crack face is dependent on
crack width. Therefore, it is necessary to
seek a justification for the quasistatic
approximation in this context.
The normal component of the true
current, to use Maxwell’s term for the sum.
of the displacement and charge current, is
continuous across an interface. Therefore,
the displacement current between the
faces and directed across the crack is equal
to the charging current at the conducting
side of the crack face. Hence, the
boundary condition is justified if the
displacement current jwgoE, across the
crack is negligible compared with the
tangential charge current ooE; at the crack
face. In the following argument, these
currents are estimated and compared.
Applying Stokes’ theorem to Faraday’s
induction law in differential form gives an
integral form of the induction law in
which the line integral of the electric field
arounda closed path is equated to the
rate of change of magnetic flux through
the surface S bounded by the path. If it
happens that the rate of change of
magnetic flux through S can be neglected.
Then the line integral is approximately
zero:
whereC is the path bounding S and ds is
an incremental displacement along the
path.
For this case, the path links points
ABCD (Fig. 14) in the limit as the points
approach the crack surface. By
considering an exponential field at the
crack face, it can be shown that the
magnetic flux throughS is less than the
path integral of E over a crack face by a
factor on the order w-8!, wherew is crack
width (meter) and6 is standard depth of
penetration (meter). Hence, if w is small
compared with the standard depth of
penetration, as it usually is, then Eq. 70 is
a reasonable approximation. This
equation indicates that the following are
of roughly comparable order of
magnitude: 2Eod = E,w, where E, is the
normal component (volt per meter) of the
electric field in the crack and Ep is the
tangential field (volt per meter) at the
outer surface. That being the case, the
ratio of the displacement current across
the crack jwegE,, to the tangential face
current O9£p is small if:
Ficure 14. Integration path C, crossing crack.
This condition for the validity of the
quasistatic approximation at the crack is
usually satisfied. For example, if
d-w-! = 104, then meod-(ogw)! = 10-5 at
10 MHz in a conductor with a low
conductivity, 0.5 MS-m-! (1 percent
International Annealed Copper Standard).
Assuming the quasistatic approximation
for a
nonconducting crack, the zero
normal current at the crack face is
written:
where the + sign denotes points on one or
the other crack face approached from the
interior of the conductor.
3.4.3 DISTRIBUIÇÃO DO DIPOLO DA CORRENTE NA SUPERFÍCIE
A basic problem to be considered initially
is the probe response to an ideal crack,
defined as having negligible opening
compared with the standard depth of
penetration but satisfying the requirement
in Eq. 72 for the validity of the quasistatic
approximation. The crack is therefore
impenetrable to the flow of electric
current. The ideal crack is defined, for
example, with respect to an open surface
So bounded by the crack edge and by the
intersection of the crack with the surface
of the conductor (Fig. 15).
Ficure 15. Side view of coil and crack, showing crack in Y,Z
plane of coordinate system. Surface So is part of the Y,Z
plane occupied by the crack.
Eddy currents flow around the buried
crack edge such that the current density is
different at points adjacent to one
another on opposite faces. The fact that
the crack opening is neglected means that
the ideal crack gives rise to a discontinuity
in the tangential current density at So
and, consequently, a discontinuity in the
tangential electric field. The solution of
the ideal crack problem can be found by
evaluating the discontinuity in the field
directly or by expressing the jump in the
field in terms of an equivalent dipole
source distribution, either electric °4 or
magnetic.°> The relationship between the
field and the equivalent current dipole
source is described next.
For an open crack, the volume dipole
density P is defined by Eq. 54 and, like
the electric field in the crack, is larger for
cracks of narrower opening. However, the
integral of P along a path C,, across the
crack is expected to tend to a finite value
in the limit as the crack opening becomes
infinitesimal. With w as the width (meter)
of the crack opening, the limit is written:
where p is the surface dipole density
having the vector representation
p = fp.
For a crack whose interior has zero
conductivity, it can be seen from the
definition (Eq. 54) that P = —oE.
Therefore:
This relation can be used in formulas
for the line integral (Eq. 70) along a path
Co around a segment of the surface of an
ideal crack (Fig. 16) to give:
Ficure 16. Integration path Co crosses crack at points A and
B and is formed in limit as At and B+ approach surface So.
where the subscript t denotes components
tangential to Sp and where $s is a
displacement vector between points A and
B on the surface So (Fig. 16):
Because 4g is arbitrary, it can be seen that:
A similar relationship between the
jump in the electric field at a surface and
the gradient of the surface dipole density
exists for the electrostatic charge dipole
layer.® Here it relates the discontinuity in
the dynamic tangential electric field at an
ideal crack surface Sp to the surface
distribution of dynamic current dipoles
whose orientation is normal to So.
Two properties of the dipole density
are worthy of note at this point. Firstly, it
tends to zero at the buried crack edge.
Secondly, the derivative dP-(dz)"! is zero at
the crack mouth, z being the coordinate
whose axis is normal to the surface of the
conductor (Fig. 15). These properties are
written as:
e:
where re is the coordinate of an edge
point and r,, is the coordinate of a point
at the crack mouth. For example, Eq. 80
gives the dipole density for a long straight
crack of depth d in a uniform unperturbed
field Eo:83
Note that p(z) vanishes at z =-d and
that the derivative with respect to z
vanishes at z = 0 in keeping with the
general properties in Eqs. 78 and 79. In
addition, it is important to be aware that
the electric field has a half-power
singularity at the edge of an ideal crack
varying locally as: 96
wherep is the perpendicular distance
(meter) of a point from the edge and9 is
an angle (radian) measured from the
surface Sp in a plane perpendicular to the
edge. This means that, in general, the
dipole density varies as:
near the edge.
The solution of the eddy current ideal
crack problem has been reduced to one of
finding the surface dipole density p. Thus
a scalar replaces a two-component vector,
the jump in the tangential electric field.
Consequently fewer unknowns are needed
for a
numerical solution. To calculate p, it
is necessary to know the continuity
conditions that apply to the magnetic
field at the crack surface Sy because these
conditions will be needed in the
derivation of an equation from which the
dipole density can be calculated.
Although the details of these derivations
will not be given here, it is useful to
understand the continuity conditions that
apply to the magnetic field at the ideal
crack surface.
The jump in the tangential electric
field at the ideal crack is inseparable from
the singular property of the electric field
between the crack faces, as expressed here
in terms of a current dipole layer.
However, no such singular behavior
occurs in the magnetic field. The truth of
this can be demonstrated by following an
argument like the one for the electric field
but applying Stokes’ theorem to Ampére’s
law rather than to the induction law,
thereby forming the line integral of H
around the path Co. Following this
parallel argument, it can be deduced that
the line integral
of H vanishes as the
closed path A_A,B,B_ (Fig. 16) collapses
onto the crack but no singular behavior of
the magnetic field in the crack could lead
to a discontinuity in the tangential
magnetic field. It is concluded that:
at So. In addition, it may be recalled that
the normal magnetic flux B (tesla) is
continuous at an interface.®> At a crack,
which is in fact a double interface, the
same relationship holds:
To confirm the consistency of the
continuity conditions at So, note that
Faraday’s induction law implies that the
normal magnetic flux density at So is:
By using this relationship to express
the difference B,, — B,_ in terms of the
jump in the tangential electric field and
substituting for the jump in the transverse
electric field using Eq. 77, the transverse
curl acts on the transverse gradient of the
dipole density to give zero. Thus, the
continuity of the normal flux density is
ensured by the fact that the jump in the
tangential electric field is expressed as the
tangential gradient of a scalar function.
Having now defined the continuity
conditions at the surface So, one is
equipped for the task of finding a
governing equation for the dipole
density p.
3.4.4 FORMULAÇÃO INTEGRAL
The most common approach to the
solution of electromagnetic field problems
at low frequencies, such as the modeling
of electrical machines, electromagnets and
eddy current discontinuity detection, is to
use a differential formulation as the basis
of a finite element solution. However, in
the area of antennas and electromagnetic
wave propagation, integral techniques are
used more commonly than the finite
element scheme. In the approaches
described here, the aim is to compute
solutions for simple but realistic
geometries using relatively few unknowns
and adapt the forward problem solver for
the task of iterative inversion. Integral
equation techniques are better suited to
this strategy, particularly if the region of
the required solution can be confined to
the discontinuity. The implication is that
the number of unknowns is small and the
forward solver is fast.
In antenna theory, the hertzian dipole
is used as a fundamental solution from
which the field of a wire antenna is found
by integration over the wire structure, a
step that is justified by the principle of
superposition. The elementary current
dipole field (Eq. 51) like the hertzian
dipole, plays the role of a fundamental
solution in a conductor. It allows the field
of an extensive discontinuity in a
conductor to be expressed as an integral
over a discontinuity region. The
fundamental solution is written here as:
where G(rir’) is a dyadic Green’s function
transforming the current dipole source p
into the electric field. For a dipole
embedded in an unbounded domain, the
dyadic Green’s function is given in the
braces of Eq. 51. A more representative
configuration in eddy current testing is
one in which a probe in air interacts with
a discontinuity in a conducting plate. If
the standard depth of penetration is
smaller than the plate thickness, the
conductor can be considered as occupying
a half space (Fig. 15). The dyadic Green's
function for a half space, like the
fundamental solution, is known in
explicit analytical form.°*°4 Hence the
discontinuity field can be written as an
integral over the discontinuity in the
knowledge that the integral kernel will
ensure that the correct continuity
conditions will be satisfied automatically
at the interface of air and conductor.
For a crack in a half-space conductor
(z = 0), the electric field is written as the
sum of the unperturbed probe field E©
and the discontinuity field:
Here the field due to the crack is
expressed in terms of its equivalent
sourcep as superposition of dipole fields
written as an integral over the crack
surface So. It should be noted that, rather
than simply invoking the principle of
superposition, the formal techniques of
deriving integral equations for the field
are based on Green’s second theorem.’”
Equation 87 is multiplied by the
conductivity o) and the condition (Eq. 72)
is applied so that the normal component
of the current density at a point at the
crack surface is zero:
onde:
e:
It is to be understood that the field point
whose coordinate is r approaches a point
r* on the crack and that this limiting
process takes place after the integration
has been performed. Equation 88
determines the current dipole density on
the surface So.
Rather than seeking a solution of the
integral equation itself, an approximation
is constructed by expanding the unknown
p(r) as a linear superposition of a set of N
basis functions and the expansion
coefficients determined by using the
moment technique.” By this
approximation procedure, a matrix
equation replaces the integral equation as
the means of finding the field. The
solution of the matrix equation can then
be found by standard numerical
techniques.*° The classic text on the
moment technique in electromagnetism is
by Harrington’ and a more recent
volume on the subject, which includes
the treatment of dyadic Green’s functions,
is by Wang.98
Having calculated a discrete estimate of
the dipole density p(ro), ro € So, the coil
impedance change due to the
discontinuity is determined from a variant
of Eq. 53:
where the integration is over the surface
So. In applying the moment technique to
the ideal crack problem,% the discrete
approximation of the dipole density
converts the impedance integral to a
summation.
3.4.5 RESULTADOS DOS ELEMENTOS DE CONTORNO
Results have been calculated using a
version of the moment technique in
which the dipole density is approximated
as a piecewise constant with respect to a
regular grid of rectangular boundary
elements. For a piecewise constant
solution, it is necessary to find the value
of the constant coefficient for each of, say,
N cells. This value is obtained by
expressing the dipole density as a linear
superposition of N rectangular pulse
functions, substituting the expansion into
Eq. 88 and demanding that the resulting
equation is satisfied at the center of each
and every rectangular cell, a step known
as point matching or collocation. The
procedure leads to an N x N matrix
equation for the coefficients of the
piecewise constant approximation.
In general, the moment technique
proceeds by expanding the unknown
function in terms of suitable set basis
functions defined with respect to a grid or
a set of nodal points subdividing the
domain of the solution. Therefore, the
dipole density can be approximated by
using a set of basis functions that lead to
a smoother representation of the solution
than does the piecewise constant
approximation. This approximation.
certainly leads to improved results.°?
However, despite the relatively crude
approximation of the piecewise constant
solution, the results (Fig. 17) agree
reasonably well with experiment on a
semielliptical artificial crack. 76
Legenda:
__ - = plotagem teórica fara 16 x 8 células
- - - = plotagem teórica fara 32 x 16 células
___ = plotagem teórica fara 40 x 20 células
o = observações
Ficure 17. Variation with probe position for coil whose axis is
in plane of semielliptical simulated crack in aluminum:
(a) resistance change; (b) reactance change.76
Incidentally, note that the theoretical
predictions computed with a grid of
16 x 8 elements are also used to generate
the 250 Hz impedance plane plot in
Fig. 10. The computed results in Fig. 17
are plotted for three different rectangular
cell sizes showing the dependence of the
results on the number of unknowns. A
reasonably accurate result can be achieved
with only 128 unknowns and the finer
grid results are consistent with each other.
Figure 18 shows similar low frequency
(250 Hz) results for a simulated crack
whose shape is shown in Fig. 15. At
intermediate frequencies, the crack
opening must be taken into account!
and at high frequencies, the number of
boundary elements must be increased.
However, in the thin penetration regime,
boundary elements can be avoided
altogether as discussed in the following
section.
Legenda:
___ = plotagem teórica fara 32 x 16 células
o = observações
3.4.6 TEORIA DA TRINCA DE POUCA PENETRAÇÃO
A number of approaches have been used
to determine the electromagnetic field at
a crack for the thin penetration regime. In
this regime, in which the standard depth
of penetration is very much smaller than
the length and depth of the crack, eddy
currents are confined to a region close to
the conductor and to the crack surface. It
is found that their distribution over the
crack is governed by the solution of the
laplace equation in the domain of the
crack face. The reduction to a
two-dimensional laplace problem is
theoretically attractive because a number
of standard techniques can be adopted to
solve such problems. From the practical
point of view, it is often desirable to carry
out eddy current testing and experiments
in the thin penetration regime because
the sensitivity to discontinuities is greater
at high frequencies. In testing
ferromagnetic materials for cracks, the
standard depth of penetration is usually
much smaller than the overall
discontinuity dimensions. Hence, the
high frequency limit has important
practical significance.
The main theoretical question to be
faced in seeking a solution of the
two-dimensional laplace problem is,
“What are the boundary conditions?”
Beginning in the early 1980s, a research
group at University College London in the
United Kingdom produceda series of
articles on the alternating current
potential drop technique for measuring
cracks. A
number of these articles were
based on the unfolding model.'0!,102 This
model was successfully applied to the
problem of finding the depth of cracks in
ferromagnetic steel in the thin
penetration regime. The problem domain
can be divided into two equal parts, each
consisting of a half plane at the surface of
the conductor anda crack face at right
angles to it. The line adjoining the half
plane and the crack face is called the fold
line. By unfolding the crack face into the
surface plane of the conductor, a modified
problem domain is formed. A scalar
potential representing the electromagnetic
field in the plane was deemed to be
continuous and have continuous normal
gradient at the fold line. At the crack
edge, a boundary condition on the
potential was deduced from the fact that
the electric field tangential to the tip is
zero. These constraints are sufficient to
form a well posed, two-dimensional
laplace problem that was solved to give
results in agreement with experiment.
Estimates of crack depth in steel
components using alternating current
potential drop were improved as a result
of this work.
The unfolding model is not valid for
nonferrous material but an alternative
thin penetration theory was developed for
eddy current testing in such materials by
Auld and others, who considered cracks in
aluminum alloys.48!93 Auld’s boundary
condition assumes that the external
magnetic field tangential to the conductor
surface is not perturbed by the crack. The
assumption may have been inspired by
Kahn’s two-dimensional long crack
problem’? because it is exact when the
magnetic field is uniform along the length
of a crack of uniform depth but, for a
nonuniform probe field at a finite crack, it
is approximate. The approximation is
reasonable provided the coil diameter is
large compared with the crack size but
this limitation leaves room for
improvement in the predictions.
It became evident in the late 1980s
that the differences between the London
group’s model and Auld’s approach ought
to be reconcilable in a unified theory that
would be valid for arbitrary permeability.
In seeking the unified approach, the
perturbation in the magnetic field at the
crack mouth was taken into account by
Lewis, Michael, Lugg and Collins, 104105
who derived a boundary condition using
a flux conservation argument applied to a
region around the opening. The resulting
theory is applicable to materials of
arbitrary relative permeability and
corroborates the unfolding model in the
high permeability limit.
3.4.7 FORMULAÇÕES ALTERNATIVAS
A more formal approach to obtaining the
unified theory is to start with a technique
valid at an arbitrary frequency and
specialize it systematically for the thin
penetration regime. A suitable
formulation for this strategy is one where
the electromagnetic field in the conductor
is expressed in terms of transverse electric
and transverse magnetic hertz
potentials,!°° y and y’ respectively. Then,
the electric and magnetic fields take the
forms:
e:
where z <(0 and where the preferred
direction 2 is normal to the crack plane.
In a half-space problem formulated
using hertz potentials, it is usual to
choose the preferred direction as the
normal to the interface between the air
and the conductor. In this way the two
potentials are decoupled at the interface.
Although the present choice of preferred
direction leads to coupled interface
conditions, the chosen modes are
decoupled at the crack surface. In fact, the
transverse electric mode does not interact
directly with an ideal crack at all. Instead,
it is perturbed indirectly through its
coupling with the transverse magnetic
mode at the surface of the conductor.
Because direct transverse electric
interaction with the crack is absent, the
transverse electric potential and its
gradients are continuous at the ideal crack
plane. In contrast, the transverse
magnetic potential is subject to a direct
interaction of the crack with the field and
therefore exhibits a discontinuity at the
crack.
‘To examine the discontinuity of the
transverse magnetic hertz potential, it is
necessary to reconsider the properties of
the electromagnetic field at the crack.
First, the fact that the normal
component of the electric field at the
surface of the crack is zero means that:
Second, in the absence of direct
transverse electric interaction with the
crack, the continuity of the tangential
magnetic field (Eq. 83) implies that:
as can be deduced from Eqs. 93 and 83.
Thus the transverse magnetic potential
itself is continuous at the crack surface So.
Third, noting that the jump in the
electric field is due solely to the transverse
magnetic mode, it can be seen from the
form of the transverse magnetic
contribution in Eq. 92 combined with
Eq. 77 that:
It is concluded that the transverse
magnetic potential has a discontinuity in
its normal gradient at the crack surface So.
It can be shown that the transverse
electric hertz potential y, expressed as the
sum of unperturbed and perturbed
components, is given by:
for arbitrary frequency and standard
depth of penetration. The Green’s
function G(r,r’) accounts for the cross
coupling between transverse electric and
transverse magnetic modes. 107
Several approaches for finding a
solution to the ideal crack problem follow
immediately from Eq. 97, both at an
arbitrary frequency and for the thin
penetration regime. For example, without
restricting the frequency, one can use a
symmetry argument to write the jump in
the derivative of the potential at the crack
as 2(0y)-(0x)"!. Differentiating Eq. 97 with
respect to x and assigning the field
coordinate to a point at the crack face
denoted by r* will give an equation for
the normal derivative of y. From the
solution, p can be found from Eq. 96 and
the probe impedance due to the
discontinuity found from Eq. 91. The
following approach has appeared in the
literature.
Setting the field coordinate in Eq. 97 to
r* and using Eq. 96 gives:
Essentially the same equation as Eq. 98 is
found using a magnetic vector potential
formulation.!°8 Findinga solution relies
on the fact that the unknown potential
w(r*) satisfies the laplace equation on So
(Eq. 94) and must be determined
simultaneously with p(f). These two
unknown functions can indeed be found
from the same equation simultaneously
by imposing further constraints. The
additional constraints are not the
boundary conditions on the laplace
problem for y at the crack face, because
these are not defined. Instead, the
boundary conditions at an arbitrary
frequency (Eqs. 78 and 79) are imposed
on p.
In finding a solution using the
moment technique using N equations for
N unknowns, a reduction in the
unknowns needed to approximate p can
be made because the prior knowledge
derived from Eqs. 78 and 79 restricts its
behavior at the perimeter of the crack.
This technique releases some degrees of
freedom that can be used to represent
w(r*) as a solution of the laplace equation
on the crack face. By management of the
unknown coefficients in this way, a
solution can be found that agrees with
experiment. 108
Legenda:
___ = plotagem teórica fara 32 x 16 células
o = observações
Ficure 19. Inductance and resistance variation with probe position for coil whose axis is in
plane of semielliptical artificial crack in aluminum: (a) inductance plot; (b) resistance plot.
Theory (solid line) is compared with experimental results (points) acquired at 50 kHz. See
Harrison and Burke for details of coil parameters and simulated crack. 76
3.4.8 REGIME DE POUCA PENETRAÇÃO
As Auld has shown, a suitable boundary
condition for formulating a well defined
laplace problem on Sp in the thin
penetration regime can be derived from
the magnetic field at the crack mouth.
The transverse magnetic component of
the magnetic field in the Y direction can
be written:
where wy is the perturbed potential (volt)
due to the crack. As it stands, Eq. 99
cannot be used immediately as a
boundary condition because the
perturbed field at the mouth is not known
in advance. Auld got around this problem
by neglecting the perturbation of the
magnetic field at the crack mouth, a
reasonable approximation because it can
be small for nonferromagnetic materials.
Taking the field perturbation into account
increases the complexity of the
problem!°” but improves the accuracy of
the results for nonferrous alloys and gives
results valid for ferromagnetic
materials.
109
Results of impedance predictions!!°
and measurements for a semielliptical
artificial crack are shown in Fig. 19. The
experimental data are taken froma series
of measurements made at
16 frequencies.”° For a comparison with
thin penetration theory, results at the
highest frequency (SO kHz) are shown.
Calculations were performed with
conformal mapping.!!° At this frequency,
the depth of the simulated crack,
8.61 mm (0.339 in.), is more than
18 times the standard depth of
penetration, 0.47 mm (0.019 in.). Note
that the theory underpredicts the resistive
component by about 10 percent.
However, this component, is small
compared with the inductive reactance,
which has a
maximum value over 600 Q.
The overall accuracy of the predictions is
good.
4. MODÊLO COMPUTACIONAL DO CAMPO DE CORRENTES PARASITAS
4.1 BASES MATEMÁTICAS DO MODELO
4.1.1 TIPOS DE MODELO
4.1.2 VISÃO GERAL DA MODELAGEM ANALÍTICA E NUMÉRICA
4.2 MODELO ANALÍTICO
4.2.1 TÉCNICA DE SOLUÇÃO INTEGRAL
4.3 MODÊLO NUMÉRICO
4.3.1 TÉCNICA DAS DIFERENÇAS FINITAS
4.3.2 REPRESENTAÇÃO DAS DIFERENÇAS FINITAS
4.3.3 REPRESENTAÇÃO DAS DIFERENÇAS FINITAS PARA PROBLEMAS DE CAMPO BIDIMENSIONAL E ASSIMÉTRICOS
4.3.4 CONTORNOS E CONDIÇÕES DE CONTORNO
4.3.5 MALHAS NÃO UNIFORMES E NÃO RETANGULARES
4.3.6 SOLUÇÃO DO SISTEMA DE EQUAÇÕES
4.3.7 SOLUÇÃO INTERATIVA
4.3.8 SOLUÇÂO POR MATRIZ DE INVERSÃO
4.4 TÉCNICA DE ELEMENTOS FINITOS
4.4.1 FORMULAÇÃO DE ELEMETOS FINITOS PARA GEOMETRIS BIDIMENSIONAIS E AXISSIMÉTRICAS
4.4.2 ENERGIA FUNCIONAL PARA PROBLEMAS DE CORRENTES PARASITAS
4.4.3 DISCRETIZAÇÃO DE ELEMENTOS FINITOS
4.4.4 FORMULAÇÃO DE ELEMENTOS FINITOS
4.4.5 ELEMENTOS ISOPARAMÉTRICOS QUADRILATEROS
4.4.6 MINIMIZAÇÃO FUNCIONAL
4.4.7 CONDIÇÕES DE CONTORNO
4.4.8 CÁLCULOS COM VETOR MAGNÉTICO POTENCIAL
4.5 MODELAGEM DA FÍSICA DO ENSAIO DE CORRENTES PARASITAS
4.5.1 MODELAGEM PARA PROJETO DE SONDAS
4.5.2 PROJETO POR ELEMENTOS FINITOS DE SONDAS DE CORRENTES PARASITAS ABSOLUTA E DIFERENCIAL
4.5.3 MODELAGEM PARA SIMULAÇÃO
4.5.4 CONCLUSÕES
Autores:
- Lalita S. Upda, Michigan State University, East Lansing, Michigan
- Nathan Ida, University of Akron, Akron, Ohio (Parts 1 and 4)
- John R. Bowler, Iowa State University, Ames, Iowa (Part 3)
- Theodoros Theodoulidis, Aristotle University of Thessaloniki, Thessaloniki, Greece (Part 2)
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