ISRN Materials Science

VolumeΒ 2012, Article IDΒ 659352, 19 pages

http://dx.doi.org/10.5402/2012/659352

## Antiplane Shear Crack Normal to and Terminating at the Interface of Two Bonded Piezo-Electro-Magneto-Elastic Materials

Department of Mechanics of Materials, Technical University of Lodz, Al. Politechniki 6, 93-590 Lodz, Poland

Received 15 December 2011; Accepted 13 February 2012

Academic Editor: V.Β Sglavo

Copyright Β© 2012 Bogdan Rogowski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The magnetoelectroelastic analysis of two bonded dissimilar piezo-electro-magneto-elastic ceramics with a crack perpendicular to and terminating at the interface is made. By using the Fourier integral transform (in perpendicular directions in each materials), the mixed boundary conditions and continuity conditions are transformed to a singular integral equation with generalized Cauchy kernel, the solution of which has been well studied, and classical methods are directly applicable here to obtain the closed form solution. The results are presented for a permeable crack under anti-plane shear loading and in-plane electric and magnetic loadings, as prescribed electric displacement and magnetic inductions or electric and magnetic fields. Obtained results indicate that the magnetoelectroelastic field near the crack tip in the homogeneous PEMO-elastic ceramic is dominated by a traditional inverse square-root singularity, while the coupled field near the crack tip at the interface exhibits the singularity of power law , being distant from the interface crack tip and depending on the material constants of a bimaterial. In particular, electric and magnetic fields have no singularity at the crack tip in a homogeneous solid, whereas they are singular around the interface crack tip. Numerical results are given graphically to show the effects of the material properties on the singularity order, field intensity factors and energy release rates. Results presented in this paper should have potential applications to the design of multilayered magnetoelectroelastic structures.

#### 1. Introduction

The newly emerging materials named magnetoelectroelasticity, which exhibit piezoelectric, piezomagnetic, and electromagnetic properties, have found increasing wide engineering applications, particularly in aerospace and automotive industries. Magnetoelectroelastic solids have been widely used as transducer, sensors, and actuators in smart structures. Because of the brittleness of PEMO-elastic materials, a high possibility of material debonding and cracking or sliding of the interface exists. Consequently, this problem has been the subject of research and discussion in the literature on elasticity theory of coupled fields. Li and Kardomateas [1] investigated the mode III interface crack problem for dissimilar piezo-electromagnetoelastic bimaterial media. The extended Strohβs theory and analytic continuation principle of complex analysis have been used to obtain the solution for interfacial cracks between two dissimilar Magnetoelectroelastic half-planes by Li and Kardomateas [2]. The problem for an antiplane interface crack between two dissimilar PEMO-elastic layers was analyzed by Wang and Mai [3]. Gao et al. [4] derived the exact solution for a permeable interface crack between two dissimilar Magnetoelectroelastic solids under general applied loads. Gao et al. [5] derived also the static solution related to antiplane crack problem. The antiplane shear cracks are a class of simple problems. But, for the case of a crack perpendicular to the interface, the problem becomes more complicated. This problem has been subject of research in the classical literature of elasticity theory. Cook and Erdogan [6] and Erdogan and Cook [7] were apparently the first to publish the solution of this problem for two bonded dissimilar isotropic half-planes. For piezoelectric biceramics an arbitrarily oriented plane crack terminating at the interface was extended by Qin and Yu [8]. The antiplane shear crack normal to and terminating at the interface of two piezoelectric ceramics was extended later by Li and Wang [9]. Although the above studies deal strictly with piezoelectric, it is reasonable to assume that the extension of the findings to electromagnetoelastic materials is valid.

To the best of author knowledge, the behaviour of interfacial cracks normal to and terminating at the interface of two bonded piezo-electromagnetoelastic materials has not been addressed yet. Motivated by these considerations, the author investigates the antiplane deformations and in-plane electric and magnetic fields of a PEMO-elastic bi-material with Mode-III interface crack normal to and terminating at the interface.

The crack is assumed to be electrically and magnetically permeable. Under applied electric, magnetic, and mechanical loading, electric, magnetic, and elastic behaviours near both crack tips are obtained. Two kinds of loading conditions are adopted. By using Fourier integral transform, in perpendicular directions in each materials, the associated boundary value problem is transformed to a singular integral equation with generalized Cauchy kernel. Similar types of equations have been studied, and classical methods of their solutions are directly applicable here to obtain the solution in closed form. The results indicate that magnetoelectroelastic field near the crack tip in a homogeneous PEMO-elastic ceramic exhibits an inverse square-root singularity, while singular field near the interface crack tip is dominant by a singularity of power law. The singularity order is dependent on relevant material constants of two ceramics. The effects of magneto-electro-mechanical parameters on the field intensity factors are evaluated by numerical analysis, which could be of particular interest to the analysis and design of smart sensors/actuators constructed from Magnetoelectroelastic composite laminates.

#### 2. Formulation of the Problem

##### 2.1. Basic Equations

For a linearly Magnetoelectroelastic medium under antiplane shear coupled with in-plane electric and magnetic fields, there are only the nontrivial antiplane displacement : strain components and : stress components and , in-plane electrical and magnetic potentials and , which define electric and magnetic field components , and : and electrical displacement components , and magnetic induction components , and with all field quantities being the functions of coordinates and .

The relations (2) and (3) have the following form: where and .

For linearly Magnetoelectroelastic medium, the coupled constitutive relations can be written in the matrix form as follows: where the superscript denotes the transpose of a matrix and where is the shear modulus along the -direction, which is direction of poling and is perpendicular to the isotropic plane and are dielectric permittivity, and magnetic permeability coefficients, respectively, , and are piezoelectric, piezomagnetic and magneto-electric coefficients, respectively.

The mechanical equilibrium equation (called as Euler equation) and the charge and current conservation equations (called as Maxwell equations), in the absence of the body force electric and magnetic charge densities, can be written as Subsequently, the Euler and Maxwell equations take the following form: where is the two-dimensional Laplace operator.

Since , one can decouple the (8) as follows:

If we introduce, for convenience of mathematics in some boundary value problems, two unknown functions where then where

The governing field variables are where

Note that is the piezo-electro-magnetically stiffened elastic constant.

Note also that the inverse of a matrix is defined by parameters and as follows:

These material parameters will appear in our solutions.

##### 2.2. Boundary Conditions

Consider a crack terminating at the interface of two bonded dissimilar PEMO-elastic ceramics polarized in the direction. For convenience, we denote the PEMO-elastic ceramics occupying the right and left half-planes and as piezoceramics I and II, respectively, shown in Figure 1.

Let a crack be perpendicular to the interface and be situated at in the positive -direction in ceramic I. For an antiplane shear crack having no thickness (so-called βmathematical crackβ), the crack surfaces contact each other, in reality; so the crack is electrically and magnetically contacted. Consequently, the electric and magnetic boundary conditions at the crack surfaces can be described according to so-called double permeable conditions, namely,

Note that besides the crack surfaces, the above conditions, in fact, certainly hold at the crack-absent parts of the crack plane. Using the relations (14), it can be shown that the condition (18) may be replaced by conditions as follows:

Let the constant mechanical loads and uniform electric displacement and magnetic induction or electric field and magnetic field be applied at infinity (two cases of electric and magnetic loads), and the following: where or are prescribed constants, a quantity with superscribes I or II that specifies the one in the PEMO-ceramic I or II, respectively. To solve the crack problem in linear elastic solids, the superposition technique is usually used. Thus, we first solve the Magnetoelectroelastic field problem without the cracks in the medium under electric, magnetic, and mechanical loads. This elementary solution is the following: with .

In addition the crack surfaces are traction-free, that is, and owing to the symmetry one can directly write following conditions:

We further consider the situation when the interface under consideration is perfectly bonded, across which the displacement, stress, electric and magnetic potentials, electric displacement, and magnetic induction are continuous

#### 3. Method of Solution

From the symmetry of the problem, it is sufficient to consider the upper half-plane of the bi-ceramic. Consequently, for , it is easily found that an appropriate solution of the problem, which satisfies the boundary conditions (19a) and (20), takes of the following form: for and for , where , and are unknowns to be determined from given boundary conditions and where

Furthermore with the aid of (14), one can give the components of stress, electric displacement, magnetic induction, and electric and magnetic potentials for and for and for and for .

Now, application of the continuity conditions (24), at the interface to (25) to (33), yields β β

The first two equations, that is, (34) and (35), give three constraints for applied remote electro-magneto-mechanical loadings, from which we may determine the loadings of ceramics II, namely, , , and by means of loadings of ceramics I, namely, , , and . In other words, in order to guarantee the continuity of all physical quantities at the perfectly bonded interface, applied electro-magneto-mechanical loadings must obey the relations (34) and (35). The five equations (36) and (37) give the constraints with respect to unknown functions; that is, the disturbed field, due to the presence of a cracks, requires to satisfy those equations.

From the condition (23)_{2} along with (26), one gets

Continuity of at the interface requires so that since

The result (40) in connection with (36) and (37) yieldswhere

In the special cases, we obtain that

for both piezoelectric materials for both piezomagnetic materials

The formulae (45) are equivalent to that derived by Li and Wang [9] who solved the problem of two bonded dissimilar piezoelectric media with an antiplane shear crack perpendicular to and terminated at the interface. Next, we denote that

From the boundary conditions (23), should satisfy the single-value displacement constraint condition, that is,

Utilizing (25)_{1} in (23) leads to
from which together with (47), by using the inverse Fourier transform, can be deduced

Now, we calculate the following:

Using the result we find with the use of (48) that

Substitution of (53) into (42) yields the expressions for , and in terms of .

From fraction-free condition (22) from (28), one can derive

Substituting (50) and (42) with the use of (53) into (54), we have with the help of known integrals the following singular integral equation with generalized Cauchy kernel for : where

For both piezoelectric materials, is obtained as follows:

The value of for both piezomagnetic materials is obtained from formula (58) if we replace by and by . It is noted that, in a usual integral equation with Cauchy kernel, other kernels except Cauchy kernel are continuous over the entire interval involved. In addition to the singularity of the Cauchy kernel terms as in (56), the other term is also unbonded as simultaneously. Particularly for two elastic dielectric, meaning , and diamagnetic, meaning , elastic field and electric field, and elastic field and magnetic field are not coupled as well as when , the electromagnetic field does not occur. In this case, reduces to Then the integral equation is simplified to

This equation is equivalent to that derived by Cook and Erdogan [6] and Erdogan and Cook [7], who were apparently the first to publish the solution of an antiplane shear crack terminating at the interface of two joined purely elastic media.

#### 4. Magnetoelectroelastic Field

##### 4.1. Solution of the Singular Integral Equation

Based on the result derived by Bueckner [10], the desired solution for of (56) subjected to (48) can be obtained as follows: for with where .

Once is determined the crack tearing displacement can be obtained by the following integrations:

##### 4.2. Crack Tearing Displacement

Expanding the expression (63) near the crack tips yields the asymptotic crack tearing displacement as at the right and left crack tip.

Here denotes the infinitesimal terms compared to , being the distance from the crack tip. Only for the behaviours of the crack tearing displacement for both tips are the same.

##### 4.3. Asymptotic Crack-Tip Field

Antiplane shear crack and in-plane electric displacement and magnetic induction may be deduced by evaluating the following integrals: for and for , where is defined by (57) and

For both piezoelectric or piezomagnetic materials, (67) give or

The analytical expressions for physical quantities may be obtained substituting the solution (61) into (65) and (66). We omit full solution and pay our attentions to the asymptotic crack-tip field. This is very interest from the view point of fracture mechanics. From (61), one can write out the singular behaviour of the function near the point and by the following asymptotic expressions: where stands for nonsingular terms.

Now we define the intensity factor at the right crack tip in the homogeneous solid and the left crack tip at the interface of a bimedium as respectively, where stands for one of , and .

###### 4.3.1. Magnetoelectroelastic Field near the Crack Tip in the Homogeneous PEMO-Elastic Ceramics

Using the integral we obtain from (65) that where the stress intensity factor at the right crack tip. Other field intensity factors are related to as follows:

For the crack tip in homogeneous PEMO-elastic medium the elastic, electric, and magnetic fields still exhibit an inverse square-root singularity at the crack tip. Application of electric and magnetic fields does not alter the stress intensity factors. The stress intensity factor depends on the material properties of two PEMO-elastic ceramics involved since it is governed by (75) and by (62). The intensity factors , and are related to and also depend on the material properties, as shown in (76).

###### 4.3.2. Magnetoelectroelastic Field near the Crack Tip at the Interface

Using the known result [11], putting (71) into (66) and using (77), we obtain the asymptotic expressions for the antiplane shear stress and in-plane electric displacement and magnetic induction, as well as elastic strain, electric and magnetic field, ahead on the left crack tip at the interface as follows: where the identity is used as follows: and where for PEMO-elastic bimaterial and for piezoelectric bi-material and for piezomagnetic bi-material.

Note that for piezoelectric bi-material, we have

The material parameters for piezoelectric ceramics coincide, in general, with the ones derived by Li and Wang [9]. But in , defined exactly by (58), the fourth term in numerator of (58) is omitted in (46) of Li and Wang paper. In consequence, the conclusions in Table 2 of Li and Wang paper that vanishes also in the case of ceramics poled in opposite direction are incorrect. The formula (58) shows that only for two bonded piezoelectric ceramics with unchanged poled in the same direction (not opposite) the field singularity at the interface crack tip maintains the inverse square root singularity, since in this case is and . The parameter in this paper has opposite sign to that presented by Li and Wang. This gives that for meaning that piezoelectric ceramic II is more stiffer that piezoelectric ceramic I , in this case , so stands also , and increases with . Also it is seen that decreases with the ratio of to . In the paper Li and Wang [9], the conclusions, associated with , are inverse. The presented conclusions are consistent with physical consideration. The field intensity factors must satisfy the constitutive equations or material parameters must satisfy the equivalent equations

It is easily verified that both constitutive relations (85) are satisfied by the coefficients defined by (68), (81), and (83). In general, for Magnetoelectroelastic ceramic, the field intensity factors must satisfy the constitutive equations as shown in (14). Of course, we have and .

##### 4.4. The Energy Release Rate

For magnetoelectrically permeable crack, the energy release rates are very important to evaluate the behaviours of crack tips. In accordance with the definition of the energy release rate proposed by [12] (the virtual crack closure integral), the energy release rate can finally be derived as where are the energy release rate for homogeneous material (no bi-material) and normalized stress intensity factors at right and left crack tip. One interesting observation from equation (87) is that though the energy release rate, , is independent on the applied electric-magnetic load, it is affected by electric-magnetic properties of two constituents of the bi-material media.

##### 4.5. Electric Displacement and Magnetic Induction inside the Crack

Since the medium inside the crack (usually air or vacuum) allows some penetrations of the some electric and magnetic fields, these fields may not be zero. Suppose that the normal components of the electric displacement and magnetic induction inside the crack are and , respectively. Then from permeable crack boundary conditions (18) and solutions (78), it follows that the quantities and are as follows:

Then, using (21), we obtain that in both cases of loading conditions.

The electric displacement and magnetic induction intensity factors are proportional to and , respectively [13], and the following relations hold which are in agreement with the solutions (78). For piezoelectric bi-materials or piezomagnetic bi-materials we have, for instance,

In particular, for a fully permeable crack considered here, and two identical magneto or electroelastic planes polarized in opposite directions we have (from (94))

Note that the crack tip electric displacement and the electric displacement inside the crack exist only in the piezoelectric plane. Alternatively the crack tip magnetic induction intensity factor and the magnetic induction inside the crack exist only in the piezomagnetic plane. All quantities occur in the PEMO-elastic bimaterial.

#### 5. Results and Discussions

In studying the fracture behaviour of the PEMO-elastic material, the field intensity factors are of significance. In this section, examples are given to illustrate the effects of material properties on the field intensity factor and the order of singularity.

##### 5.1. Effect of Material Constants on the Singularity Order

We now consider the dependence of the singularity order on -constituent independent piezo-electromagnetoelastic constants. Although analytical evaluation of the relative sensitivities is possible, on the basis of the results presented above, it is rather cumbersome. Therefore, the sensitivity is evaluated here in other way.

Firstly, we assume that both materials are piezoelectric and , and , and analyze the situations(a) changes and , that is, not change,(b) changes and ,(c) changes and .

This states that it is analyzed that right half-plane is fixed, and left one contains a fictitious material with only changing or or .

(a) The changes of the ratio of to : we have or

Figure 3 shows the effects of varying elastic stiffness on and with unchanging piezoelectric and piezomagnetic constants or and . Note that and for and if or if and . Note also that for all of .

The singularity order is larger for two of the same ceramics poled in opposite directions together since .

We take six kinds of particular piezoelectric ceramics as representatives, the relevant material constants and parameters , and of which are listed in Table 1 (with materials poling axes aligned in the positive -direction).

(b) The changes of the ratio of to : we have

For , the singularity parameter increases from β1 to maximum and for declines to β1. Then the singularity parameter varies between , respectively. If both poling directions are opposite; that is, one is in the -direction and second is in the ()-direction, then to satisfy the condition must hold or . If the selection of violates the condition , then the electroelastic field near the interface crack tip is dominant by either logarithmic singularity or is bonded. This situation seems unlikely, take place for realistic piezoelectric ceramics, and it is not beyond the scope of abilities of results of this paper.

(c) For varying and other parameter unchanged, it is easily found that and for and varying . But if , then

Figure 5 shows the variation of and with the ratio for .

The parameter assumes negative values and increases from to zero with . The singularity parameter is positive and increases from to with . Note that must be less unity if tends to zero or for . Some materials shown in Table 1 limit the range of ; for example, PZT-4 has , and must be larger 1,35 to ensure that . Of course this situation is addressed to two piezoelectrics poled in opposite directions.

For piezomagnetic materials, the parameter is
and for magnetostrictive material CoFe_{2}O_{4} assumes the value