## Abstract

The scattering effect of harmonic flexural waves at a through crack in an elastic plate carrying electrical current is investigated. In this context, the Kirchhoffean bending plate theory is extended as to include magnetoelastic interactions. An incident wave giving rise to bending moments symmetric about the longitudinal *x*-axis of the crack is applied. Fourier transform technique reduces the problem to dual integral equations, which are then cast to a system of two singular integral equations. Efficient numerical computation is implemented to get the bending moment intensity factor for arbitrary frequency of the incident wave and of arbitrary electrical current intensity. The asymptotic behaviour of the bending moment intensity factor is analysed and parametric studies are conducted.

## 1. Introduction

A new concept of multi-functional materials/structures aiming at providing broader capabilities to the next generation of aerospace vehicles/spacecraft was proposed in the last few years. The underlying idea of this concept is to exploit multi-(physical/scale) properties of materials or structures in such a way that besides its major designated functionality, the same structural component should accomplish at least one more function. An example of such design is a smart load carrying structure that can conduct non-destructive crack diagnosis or health monitoring by itself. To implement this concept in various contexts, such as, to name only a few, aerospace vehicles, nuclear reactor constructions, a better understanding of static and dynamic behaviours of thin/thick-walled elastic structures subjected to simultaneous action of mechanical, thermal, electrical, magnetic and other fields becomes necessary. This can lead to truly integrated structures, being able to perform multiple structural, as well as electromagnetic and electromechanical functions. In recent years, since there are many devices that operate in an electrothermal–magnetic field environment, interest in magnetoelastic fracture mechanics has grown rapidly. When an elastic plate with a through crack is under the influence of such fields, singular moments concentrated around the crack are induced.

In spite of the extensive available work devoted to the stability and dynamics of elastic structures carrying electrical currents (see e.g. Chattopadhyay & Moon 1975; Ambartsumian *et al*. 1977; Maugin 1988; Eringen & Maugin 1990; Ambartsumian & Belubekyan 1992; Dolbin & Morozov 1996), to the best of the authors' knowledge, there are no investigations related to the problems featuring a through crack in plates/shells. In the present paper, the scattering effect of flexural waves at a through crack in an elastic plate carrying electrical current is investigated. Similar problems for electroconductive and ferromagnetic plates in a uniform magnetic fields have been addressed by Shindo *et al*. (1997, 2000).

## 2. Statement of the problem and basic equations

An isotropic plate of uniform thickness 2*h*, Poisson's ratio *ν*, Young's modulus *E* (see appendix B for a full list of notations used in this paper) is considered. The coordinate axes *x*, *y* are associated with the middle plane of the plate and the *z*-axis is perpendicular to this plane. It is assumed that the plate carries an electrical current **J**_{0}=(*J*_{0}(*t*), 0, 0) (see figure 1), which is uniformly distributed throughout the cross-section of the plate.

The magnetic field **H**_{0} induced by the applied current **J**_{0} is determined within the plate domain, by the following equations (quasistatic approximation, see e.g. Maugin 1988; Eringen & Maugin 1990; Ambartsumian & Belubekyan 1992):(2.1a–c)and in the external domain (vacuum) by the equations(2.2a–c)In equations (2.1) and (2.2), *μ*_{0} is the magnetic permeability in the vacuum and *μ*_{r} is the relative magnetic permeability of the material within the plate. The associated boundary conditions are(2.3a,b)In (2.3*a*), **N**_{0} is the outer unit vector normal to the undeformed plate surfaces. Solutions of **H**_{0} and fulfilling (2.1)–(2.3) are(2.4a)and(2.4b)where in equation (2.4*b*), Sign(*z*) is the Signum distribution function.

Within the context of the Kirchhoff assumptions that are adopted herein, the displacement components in the *x*, *y* and *z* directions, denoted as *u*_{x}, *u*_{y} and *u*_{z}, respectively, are given as(2.5a–c)where *w*≡*w*(*x*, *y*, *t*) denotes the deflection of the middle plane of the plate. The bending and twisting moments per unit length, *M*_{x}, *M*_{y} and *M*_{xy} are expressed in terms of *w* via the following expressions:(2.6a)(2.6b)(2.6c)where is the plate flexural stiffness. By discarding the tangential inertia terms and integrating the equilibrium equations of the three dimensional elasticity theory through the plate thickness, the transversal shearing forces *Q*_{x} and *Q*_{y} can be expressed in terms of the transverse displacement as(2.7a)(2.7b)where denotes the two dimensional Laplace operator. By neglecting the rotatory inertia effect, the governing equation of the plate including electrodynamic effects reduces to(2.8)In these equations, *R*_{m} (*m*=1, 2, 3) are the three components in the *x*_{i} directions of the effective Lorentz (pondermotive) force vector * R* per unit volume, which is expressed as(2.9)where

*is the electrical current density vector and*

**J***is the magnetic induction vector. The terms*

**B***σ*

_{mn}(±

*h*) in equation (2.8) (

*σ*

_{zz}≡

*σ*

_{33},

*σ*

_{xz}≡

*σ*

_{13},

*σ*

_{yz}≡

*σ*

_{23}) fulfil the following jump conditions(2.10)In equation (2.10),

*F*

_{m}is the surface traction of mechanical origin, while and are Maxwell's stress tensor components in the plate domain and in the vacuum, respectively. These are defined as follows:(2.11a)(2.11b)where

*δ*

_{mn}is the Kronecker delta,

*m*,

*n*=1, 2, 3.

Next, one expresses the electromagnetic field quantities within the domain occupied by the deformed plate in the form(2.12a–c)(2.12d,e)Herein, **J**_{0}, **H**_{0}, **B**_{0}, and are the primary field quantities (corresponding to the undeformed plate); while , , , and are their disturbed counterparts due to the deformation of the plate. Within the assumption of the small disturbance concept implying that , equation (2.9) yields(2.13)Under the assumption of perfectly electroconductivity and omitting the terms of (*z*^{2}), we get(2.14)where * u*≡(

*u*

_{x},

*u*

_{y},

*u*

_{z}) is the displacement vector (see equations (2.5

*a*,

*b*,

*c*)). From the Maxwell's equations, it follows that(2.15)Considering the case of the absence of surface tractions of mechanical origin, i.e.

*F*

_{m}=0 (

*m*=1, 2, 3), neglecting the jumps of Maxwell's stresses which is valid for non-ferromagnetic or some soft-ferromagnetic materials (see e.g. Ambartsumian

*et al*. 1977; Maugin 1988; Eringen & Maugin 1990; Ambartsumian & Belubekyan 1992), discarding the terms of (

*h*

^{3}) or higher in the right hand side of equation (2.8), in conjunction with equations (2.13)–(2.15), (2.8) reduces to the following form:(2.16)where . Worthy of noting is that the governing equation (2.16) is mathematically similar to that of a plate on a fictitious, linear Winkler's foundation with negative foundation modulus (Librescu & Lin 1997). Since our interest in the present article is focused on the dynamic effect of the incident waves, in the following analysis, we will only consider the case that the values of

*J*

_{0}(

*t*) are in the subcritical buckling range, and for further simplification, it is assumed that

*J*

_{0}(

*t*)=

*J*

_{0}=const., thereby

Next, we consider a plate carrying an electrical current and featuring a through crack of length 2*a* as shown in figure 1. It is supposed that the plate is excited by flexural waves that propagate within the plate and are reflected/refracted (i.e. scattered) around the crack. For simplicity, the sources that emit the incident wave (e.g. applied bending moments) are assumed to be symmetrically located about the crack plane *y*=0, and the incident wave arrive to the crack from both sides ±∞ simultaneously. The expression for the incident wave can then be written as (Sih & Chen 1977)(2.17)where , *w*_{0} is a constant, generally complex-valued quantity, *ω* is the frequency, , *γ* is the angle of the incident wave measured from the positive *x*-axis in the range −*π*≤*γ*<*π*, while the parameter *λ* is defined as(2.18)in which .

For brevity, in the following formulation of the time-harmonic problem, the time factor exp[−i*ωt*] as well as the operator Re will be removed and we will work in the complex domain, but the final solution of displacements and stresses corresponds to the real part of the complex-valued counterpart. Furthermore, the spatial part of the corresponding physical parameters (.) will be marked by an overbar, i.e. .

The scattered wave may be added to the incident wave to yield the total wave field:(2.19)where the superscript ‘sc’ stands for the scattered component.

In a form similar to equation (2.19), the bending moments and shear forces can also be represented as(2.20a–c)(2.20d,e)For the traction-free crack, the bending moment and the effective Kirchhoff's shear force vanishes along the crack surface.

The flexural problem is primarily concerned with determination of the scattered field (i.e. ), which will be derived in the following.

We start with the boundary conditions associated with the traction-free crack that are symmetric about the crack plane *y*=0:(2.21a)(2.21b)(2.21c)The boundary conditions associated with the scattered waves are(2.22a)(2.22b)(2.22c)In equation (2.22*a*)(2.23)where *M*_{0}≡*Dw*_{0}*λ*^{2}.

In passing this section, it is remarked that the Kirchhoff plate theory used here gives in general inaccurate results near the plate boundaries. In the problem being addressed here, the crack surfaces are the plate boundaries and the solution very close to the crack boundaries is inconsistent with the corresponding two dimensional elastic solution (see e.g. Joseph & Erdogan 1989; Wu & Erdogan 1989; Joseph & Erdogan 1991; Librescu & Shalev 1992). As shown by these papers, the stress intensity factors of cracked plates can be predicted with acceptable accuracy by the transverse shear deformation theories (e.g. Reissner theory). Knowles & Wang (1960) showed earlier that within the boundary layer of order *h*/(2*a*), the predictions by the Kirchhoff theory are considerably different to those by the Reissner theory. Librescu & Shalev (1992) further shows that in the dynamic case, the Kirchhoff theory yields an underestimation of the stress intensity factor than the counterpart predicted by the transverse shear deformation theories. In this context, it will be interesting to compare the predictions of the magnetoelastic cracked plates via the classical and transverse shear deformation theories. Addressing this issue will be planned by the present authors in a forthcoming paper.

## 3. Solution methodology

Applying to equation (2.16) the Fourier transform with respect to the variable *x*(3.1)yields an ordinary differential equation in terms of the variable *y*, whose solution for the scattered wave field can be represented as(3.2)where(3.3a–c)In equation (3.1), due to the symmetry of the problem about the plane *y*=0, one can restrict the analysis to the domain *y*≥0 only. It has to be mentioned that *γ*_{m}(*α*) (*m*=1, 3) in equation(3.1) are multi-valued functions. For *γ*_{3}(*α*), it is assumed that when *α*=0 the branch *γ*_{3}(0)=*λ*^{2} is taken. Toward determining the proper branches of *γ*_{1} and *γ*_{2}, the following radiation condition is invoked:(3.4)where *r*≡(*x*^{2}+*y*^{2})^{1/2}, and *λ* is defined by equation (2.18). Fulfilment of this condition requires Re(*γ*_{m}(*α*))≥0 (*m*=1, 2).

Using the boundary conditions, (2.22*b*,*c*), for the unknown functions *A*_{m}(*α*) (*m*=1, 2), we get(3.5a)(3.5b)The unknown function *A*(*α*) in equations (3.5*a*,*b*) fulfils the following dual integral equations:(3.6a)and(3.6b)where(3.7)As it will be shown in the following, the unknown function *A*(*α*) can be uniquely determined by the above dual integral equations. Worth noting further is that equation (3.6*a*) is derived from the displacement constraint (2.22*c*), while (3.6*b*) is derived from the moment boundary condition (2.22*a*).

The dual integral equations (3.6*a*,*b*) have been tackled via different approaches (see e.g. Sneddon 1966; Sih & Chen 1977; Shindo 1981; Alexandrov & Kovalenko 1986; Shindo & Horiguchi 1991; Andronov & Belinskii 1995; Shindo *et al*. 2000) Both the displacement constraint (2.22*c*) and the moment boundary condition (2.22*a*) are simultaneously used in these methods. It is straightforward to show that use of these two different types of boundary conditions leads to an integral equation with a logarithmic kernel for the crack opening displacement (COD), whose behaviour follows the rule of (*r*^{1/2}) (*r* is the distance from the crack tip). In the following, instead of using the boundary condition (2.22*c*), we will use(3.8)It then follows that(3.9)Define the auxiliary variable *ψ*(*x*) such that(3.10)By the inverse Fourier transform theorem, we get an expression for *A*(*α*):(3.11)From equation (3.6*b*), it then follows that(3.12)It has to be mentioned that the validity of the interchange of order of integration in equation (3.12) can be rigorously proved in the framework of distribution theory (see e.g. Gel'fand & Shilov 1964; Brychkov & Prudnikov 1989), and such an interchange of order of integration was used, e.g. by Shindo (1981) and Alexandrov & Kovalenko (1986).

As |*α*|→∞, the asymptotic behaviour of the integrand of the inner integral in equation (3.12) is(3.13)For the purpose of simplicity in the following derivation, we will restrict to the case that *ϵ*→0, i.e. when the wavelength ≫*h*. In such a case, by extracting the above asymptotic value in equation (3.13), and using the following identity:(3.14)Equation (3.12) can be regularized and converted to the following Cauchy-type singular integral equation:(3.15)where denotes that the evaluation of the integral associated with the singular kernel is in the sense of the Cauchy principal value (see e.g. Gakhov 1966, pp. 8–9), and the regular kernel *K*_{2}[*λ*(*s*−*x*)] is defined as(3.16)Worth further noting is that the singular integral equations of the type of (3.15) have been extensively studied in literature (see e.g. Gakhov 1966; Muskhelishvili 1992).

As noted previously, in deriving equation (3.15), instead of using the original boundary condition (2.22*c*), we used (3.8). As a result, the behaviour of the solution of *ψ* in equation (3.15) follows the same rule of (*r*^{−1/2}) near the crack tip as the stress components (see e.g. Sih & Chen 1977).

In order to uniquely determine *A*(*α*), or alternatively, *ψ*(*x*), an additional condition is needed. From the boundary condition (3.8), the following relation of *ψ* is obtained:(3.17)Non-dimensionalizing equations (3.15)–(3.17) via the following parameters,(3.18)the scattered wave field and the bending moments , and can be completely determined. In the following, we will restrict our discussion to the quantity . Along the *x*-axis, one obtains:(3.19)in which *ξ*_{x}∈(−∞,∞), , and the kernels *L*_{21} and *L*_{22} are defined in appendix A.

As is known in literature (see e.g. Sih & Chen 1977), the singularity of the bending moment at crack tips follows the rule of (*r*^{−1/2}) (*r* is the distance from the crack tip to a point outside of the crack); therefore, the bending moment intensity factor is defined as(3.20)where the induced bending moment is defined by equation (3.19). From equation (2.23), one obtains(3.21)Worthy of further remark is that the physical meaning of is related to : from (2.20*c*) and (2.6*c*), one obtains(3.22)Equation (3.15) can be solved by various numerical methods (see e.g. Sih & Chen 1977 and the references therefrom). In the present article, *ψ*(*ξ*_{x}) in equation (3.15) is at first expanded in terms of the complete set of Chebyshev functions *T*_{k}(*ξ*):(3.23)in which *T*_{k}(*ξ*) is the first kind Chebyshev function of order *k*. Then with the aid of Gauss–Chebyshev integrations (Erdogan 1978), the fulfilment of equation (3.15) is enforced at the discrete locations , *m*=1, 2,…,*N*_{d}, where *N*_{d} is the number of the discrete points used in the Gauss–Chebyshev integration. Since the numerical procedure is quite standard (see e.g. Qin *et al*. 2003), the implementation details are omitted here. The final expression of the bending moment intensity factor defined in equation (3.21) can be explicitly expressed in terms of the coefficients *B*_{1k} and *B*_{2k} as(3.24)

## 4. Asymptotic behaviour of the bending moment intensity factor when

From equation (3.15), and keeping only the first order of , one obtains the following singular integral equations:(4.1a)in which .

Invoking the boundary condition for *ψ*, equation (3.17), the following solutions that are asymptotically correct up to the 1st order of can be derived:(4.2a)(4.2b)We notice that *T*_{1}(*ξ*_{x})=*ξ*_{x}, ; then matching equations (4.2) with equation (3.23), the coefficients *B*_{1k} and *B*_{2k} in equation (3.24) are obtained as(4.3)As a result, from the solution of *K*_{IY} for arbitrary , i.e. equation (3.24), the bending moment intensity factor in the case can then be expressed as(4.4)The physical meaning of the quantity *M*_{0}(sin^{2} *γ*+*ν* cos^{2} *γ*) is related to the average of over the crack as(4.5)Normalizing the bending moment intensity factor with respect to the parameter , one obtains(4.6)In the case of arbitrary , from equation (3.24), one obtains(4.7)The phase angle *δ*_{Y} (Sih & Chen 1977) in such a case is(4.8)

## 5. Numerical results and discussion

As a first step, the convergence of the numerical approximation of and is investigated. Table 1 lists the test results of *N*_{d} used in the discretization. It is readily seen that the rate of convergence is quite fast and within the range , *N*_{d}=15 is enough to guarantee the numerical convergence.

Figure 2 displays the dependence of the amplitude of the normalized bending moment intensity factor on . It is noted that the amplitude of the normalized bending moment intensity factor is smaller than 1. As shown in the inset of figure 2, we further note that when , the amplitude of the normalized bending moment intensity factor , under the influence of the electrical current, follows the asymptotic rule of . For pure elastic case (i.e. without considering the influence of electrical current), the same asymptotic rule is found in (see e.g. Folias 1970; Alexandrov & Kovalenko 1986).

Figure 3 displays the influence of the angle of incident wave *γ* on the amplitude of the normalized bending moment intensity factor. It is noted that in the case , variation of *γ* has a negligible influence, while when increases, becomes more sensitive to the variation of *γ*.

As mentioned in §3, we restrict to the case where *ϵ*→0; then consistent with the definition of (see (2.18) and (3.18)), we get(5.1)In this equation, *H*_{0}≡*J*_{0}*h* is the magnetic field intensity on the plate surfaces *z*=±*h*, and the reference frequency *ω*_{0} is defined as the fundamental frequency corresponding to the purely elastic plate counterpart (which is obtained in equation (2.16) by discarding the current-related term) and taking the semi-length of the crack *a* as the characteristic length of elastic waves. This leads to the following expression:(5.2)Figure 4 shows the actual values of for an aluminium plate with respect to *J*_{0} and *ω*.

From equation (5.1), it clearly appears that the inclusion of the electrical current will increase the value of . In conjunction with figure 2, it is concluded that the electrical current has the effect of alleviating the normalized bending moment intensity factor. Figures 5 and 6 quantitatively display such alleviating effect. We note that in all tested cases, with the increase of *a*/*h*, the influence of the electrical current on the normalized bending moment intensity factor becomes more prominent. Figure 7 shows some characteristics of the influence of *a*/*h* on the intensity factor. Under the given parameters *μ*_{r}=1, , when *a*/*h≤*2, the frequency ratio *ω*/*ω*_{0} is the parameter which dominates the value of the normalized bending moment intensity factor; however, when *a*/*h*≥10, the influence of *ω*/*ω*_{0} becomes almost immaterial, and *a*/*h* becomes the dominant parameter.

Figures 8 and 9 show the influence of *a*/*h* on the normalized bending moment intensity factor under different values of *μ*_{r}. Comparison of these two figures reveals the significant influence of magnetic permeability on the normalized bending moment intensity factor.

## 6. Conclusions

Issues related with the normalized bending moment intensity factor of a thin plate with a through crack, carrying an electrical current and subjected to incident flexural waves are investigated. Numerical computation to determine the normalized bending moment intensity factor at the tip of the crack and for arbitrary is implemented. The asymptotic analysis in the case is further conducted. The major conclusions are

the normalized bending moment intensity factor decays with the increase of the electrical current or frequency of the incident wave;

the numerical results show that the normalized bending moment intensity factor is smaller than 1 and follows the same asymptotic rule of when as in the case without considering the electrical current;

variation of the angle of the incident wave has negligible influence on the normalized bending moment intensity factor when is small (e.g. ). With the increase of , such an influence becomes stronger.

## Acknowledgments

The partial support from NASA Langley Research Center through Grant NAG-01101 is gratefully acknowledged. The authors wish to extend their indebtedness to Professor B. P. Belinskiy at the University of Tennessee at Chattanooga and to the anonymous reviewers for their constructive and pertinent comments and suggestions.

## Footnotes

As this paper exceeds the maximum length normally permitted, the authors have agreed to contribute to production costs.

- Received June 1, 2004.
- Accepted May 6, 2005.

- © 2005 The Royal Society