Exact Solution for an Anti-Plane Interface Crack between Two Dissimilar Magneto-Electro-Elastic Half-Spaces

Abstract

This paper investigated the fracture behaviour of a piezo-electro-magneto-elastic medium subjected to electro-magneto-mechanical loads. The bimaterial medium contains a crack which lies at interface and is parallel to their poling direction. Fourier transform technique is used to reduce the problem to three pairs of dual integral equations. These equations are solved exactly. The semipermeable crack-face magneto-electric boundary conditions are utilized. Field intensity factors of stress, electric displacement, magnetic induction, cracks displacement, electric and magnetic potentials, and the energy release rate are determined. The electric displacement and magnetic induction of crack interior are discussed. Obtained results indicate that the stress field and electric and magnetic fields near the crack tips exhibit square-root singularity.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Bogdan Rogowski 1

Recommended by Ma Jan

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

Received 27 October 2011; Accepted 10 February 2012

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.

1. Introduction

Mechanics of magneto-electro-elastic solid has gained considerable interest in the recent decades with increasable wide application of piezoelectric/piezomagnetic composite materials in engineering, particularly in aerospace and automotive industries. Recently, much attention has been paid to dislocation crack and inclusion problems in magneto-electro-elastic solids, which simultaneously possess piezoelectric, piezomagnetic, and magnetoelectric effects. Therefore, it is of vital importance to investigate the magneto-electro-elastic fields as a result of existence of defects, such as cracks, in these solids.

The mode III interface crack solution for two dissimilar half-planes has been analyzed by Li and Kardomateas [ 1] while for two dissimilar layers by Wang and Mai [ 2] and by Li and Wang [ 3]. Li and Kardomateas [ 1] solved corresponding plane problem by means of Stroh's formalism and complex variable methods. Li and Wang [ 3] investigated the problems involving an antiplane shear crack perpendicular to and terminating at the interface of a bimaterial piezo-electro-elastic ceramics. Wang and Mai [ 2] investigated the mode III-crack problem for a crack at the interface between two dissimilar magneto-electro-elastic layers. Extension of those investigation problems interested in fracture theory, on dielectric and magnetostrictive crack behaviour, is very important, and results and conclusions could have applications in the failure of PEMO-elastic devices and in smart intelligent structures [ 4].

However, to the authors' best knowledge, no researches dealing with the interface-dielectric crack in PEMO-elastic two-phase composite have been reported in literature. When subjected to mechanical, electrical, and magnetic loads in service, these magneto-electro-elastic composites can fail prematurely due to some defects, namely, cracks, holes and others, arising during their manufacturing process. Therefore, it is of great importance to study the magneto-electro-elastic interaction and fracture behaviours of PEMO-elastic materials.

In mechanic where two-phase composites have twelve material constants only exact solutions are useful. This is motivation for this study. For electrical, magnetic, and mechanical loads (two cases of electrical and magnetic excitations) and semipermeable electrical and magnetic boundary conditions at the interface crack, exact analytical solutions are obtained here for full field interesting in fracture mechanics.

2. Basic Equations

For a linearly magneto-electro-elastic medium under antiplane shear coupled with in-plane electric and magnetic fields, there are only the nontrivial antiplane displacement w [figure omitted; refer to PDF] strain components _{γ xz} and _{γ yz} [figure omitted; refer to PDF] stress components _{τ xz} and _{τ yz} , in-plane electrical and magnetic potentials [varphi] and ψ , which define electrical and magnetic field components _{E x} , _{E y} , _{H x} and _{H y} [figure omitted; refer to PDF]

and electrical displacement components _{D x} , _{D y} , and magnetic induction components _{B x} , _{B y} with all field quantities being the functions of coordinates x and y .

The generalized strain-displacement relations ( 2) and ( 3) have the form [figure omitted; refer to PDF] where α =x ,y and _{w , α} = ∂w / ∂ α .

For linearly magneto-electro-elastic medium, the coupled constitutive relations can be written in the matrix form [figure omitted; refer to PDF] where the superscript T denotes the transpose of a matrix, and [figure omitted; refer to PDF] is the material property matrix, where _{c 44} is the shear modulus along the z -direction, which is direction of poling and is perpendicular to the isotropic plane ( x ,y ), _{[straight epsilon] 11} and _{μ 11} are dielectric permittivity and magnetic permeability coefficients, respectively, _{e 15} , _{q 15} , and _{d 11} are piezoelectric, piezomagnetic, and magneto-electric coefficients, respectively.

The mechanical equilibrium equation (called as Euler equation), 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 [figure omitted; refer to PDF] Subsequently, the Euler and Maxwell equations take the form [figure omitted; refer to PDF] where ^{∇ 2} = ^{∂ 2} / ∂ ^{x 2} + ^{∂ 2} / ∂ ^{y 2} is the two-dimensional Laplace operator.

Since | C | ...0;0 , one can decouple( 8) [figure omitted; refer to PDF]

If we introduce, for convenience of mathematics in some boundary value problems, two unknown functions [figure omitted; refer to PDF] where the matrix _{C 0} , a principal submatrix of C , is [figure omitted; refer to PDF] then [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

The relevant field variables are [figure omitted; refer to PDF] [figure omitted; refer to PDF]

The two last equations ( 15) are equivalent to ( 9) since _{c 1 c 3} - ^{c 2 2} =1 / ( _{[straight epsilon] 11 μ 11} - ^{d 11 2} ) ...0;0 . In ( 14) the material parameters are defined as follows: [figure omitted; refer to PDF]

Note that _{c ... 44} is the piezo-electro-magnetically stiffened elastic constant.

Note also that the inverse of a matrix ^C is defined by parameters α , β , _{c ... 44} and _{c 1} , _{c 2} , _{c 3} as follows: [figure omitted; refer to PDF] and is the matrix generalized compliances of PEMO-elastic material. These material parameters will be appear in our solutions.

3. Formulation of the Crack Problem

Let the medium I occupy the upper half-space and medium II be in the lower half-space; the interface crack is assumed to be located in the region from -a to +a along the x -axis. The two-phase composite is subjected to electric, magnetic, and mechanical loads applied at infinity. These are ( _{τ 0} , _{D 0} , _{B 0} ) or ( _{τ 0} , _{E 0} , _{H 0} ). Under applied external loading, the crack, filled usually by vacuum or air, accumulated an electric and magnetic field, denoted by _{d 0} and _{b 0} , would be built up. By the superposition principle, the interface crack problem is equivalent to the one under the applied loading on the upper surface (Figure 1): [figure omitted; refer to PDF] where [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Figure 1: The interface crack under antiplane mechanical and in-plane electric and magnetic load. Inside the crack the unknown electromagnetic fields _{d 0} and _{b 0} appear. (a) Perturbation problem, (b) Elementary solution for bimaterial without the crack ( 19) and ( 20).

[figure omitted; refer to PDF]

Similarly is for lower crack surface, where the material parameters and electro-magnetic loadings are denoted by prime.

To guarantee the continuity of physical quantities at the perfectly bonded interface, applied electro-magnetic loadings E and H must obey the relations from which the loadings of upper material, namely, _{E 0} and _{H 0} may be determined by means of loading of lower material, namely, ^{E 0 [variant prime]} and ^{H 0 [variant prime]} , using ( 19) and ( 20). Of course, _{D 0} = ^{D 0 [variant prime]} and _{B 0} = ^{B 0 [variant prime]} in Case I of loading and _{E 0} = ^{E 0 [variant prime]} and _{H 0} = ^{H 0 [variant prime]} for homogeneous medium only.

At the interface y =0 ± , we have the conditions [figure omitted; refer to PDF] where the notation [ | f | ] = ^{f +} - ^{f -} and ^{f +} denotes the value for ^{0 +} while ^{f -} for ^{0 -} .

The electric displacement _{d 0} and magnetic induction _{b 0} inside the crack are obtained from semipermeable crack-face boundary conditions [ 5]. For two different magnetoelectric media: PEMO-elastic material I and notch space, we have the continuity condition of electric and magnetic potential in both materials at interfaces, similarly for interface between second PEMO-elastic material and crack interior. The semi-permeable crack-face magnetoelectric boundary conditions are expressed as follows: [figure omitted; refer to PDF] where δ ( x ) describes the shape of the notch, and _{[straight epsilon] c} , _{μ c} are the dielectric permittivity and magnetic permeability of crack interior. If we assume the elliptic notch profile such that [figure omitted; refer to PDF] where _{δ 0} is the half-thickness of the notch at x =0 , we obtain that [figure omitted; refer to PDF]

Equations ( 24) form two coupling linear equations with respect to _{d 0} and _{b 0} since [ | [varphi] | ] and [ | ψ | ] depend linearly on these quantities as shown boundary conditions ( 18) and ( 21).

4. The Solution for Two-Phase Medium with the Discontinuity at Interface

Define the Fourier transform pair by the equations [figure omitted; refer to PDF]

Then ( 15) are converted to ordinary differential equations and their solutions [figure omitted; refer to PDF]

These solutions satisfy the regularity conditions at infinity and the conditions of vanishing jumps of electric displacement and magnetic induction at interface and crack surfaces.

From ( 21)₁ we obtain that [figure omitted; refer to PDF]

The material parameters of the lower material are denoted by prime.

The mixed boundary conditions on the crack plane and outside lead to [figure omitted; refer to PDF]

Using the integrals [figure omitted; refer to PDF] we see that the solutions of ( 28) are [figure omitted; refer to PDF]

We calculate that [figure omitted; refer to PDF] where ^{C -1} is defined by the matrix ( 17) and ^{C [variant prime] -1} by the same matrix with material parameters of second material.

From the condition ( 24), we obtain that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] [figure omitted; refer to PDF] and similarly for ^{C 1 [variant prime] -1} (second material).

5. Field Intensity Factors

The singular behaviour of _{τ zy} , _{D y} , and _{B y} at y =0 , | x | [arrow right] ^{a +} are: [figure omitted; refer to PDF]

Defining the stress, electric displacement and magnetic induction intensity factors as follows: [figure omitted; refer to PDF] we obtain that [figure omitted; refer to PDF]

Furthermore, we obtain the displacement, electric and magnetic potentials intensity factors [figure omitted; refer to PDF]

In view of results ( 31) and ( 37), we have [figure omitted; refer to PDF]

The energy release rate is defined as [figure omitted; refer to PDF] and is the following: [figure omitted; refer to PDF] or in explicit form [figure omitted; refer to PDF]

For fully impermeable case, we have _{d 0} =0 and _{b 0} =0 , and the solutions are obtained from ( 37) and ( 39). For fully permeable case, we have _{[straight epsilon] 0} [arrow right] ∞ and _{μ 0} [arrow right] ∞ and [figure omitted; refer to PDF]

The energy release rate is [figure omitted; refer to PDF]

Note that energy release rate ( 44) for fully permeable crack problem is defined by the harmonic mean of the shear moduli of both materials, that is,

[figure omitted; refer to PDF]

The remaining field intensity factors are obtained, in this case, as follows: [figure omitted; refer to PDF]

In particular, for fully permeable crack between two PEMO-elastic materials polarized in opposite directions, we have _{K D} =0 and _{K B} =0 , since _{e 15} = - ^{e 15 [variant prime]} and _{q 15} = - ^{q 15 [variant prime]} in this case.

For electrically impermeable and magnetically permeable crack, the solutions are independent of the applied magnetic field ( _{d 0} =0 and B - _{b 0} is independent on B for _{[straight epsilon] 0} [arrow right]0 and _{μ 0} [arrow right] ∞ as shown in ( 32)).

Alternatively, the solutions for the electrically permeable and magnetically impermeable crack are independent on the applied electric displacement.

In practical applications the following cases appear:

(i) Let _{[straight epsilon] 0} tends to infinity and _{μ 0} is finite.

Then [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

(ii) Let _{μ 0} tends to infinity and _{[straight epsilon] 0} is finite.

Then [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

The functions of permittivity ( _{[straight epsilon] c} ) and permeability ( _{μ c} ) approaches zero as _{[straight epsilon] c} and _{μ c} tend to zero and are unity as _{[straight epsilon] c} and _{μ c} tend to infinity. The solution perfectly matches exact solution in both limiting cases, namely, permeable and/or impermeable electric and/or magnetic boundary conditions.

In above equations the notation ^{K perm ·imp} denotes the intensity factor for electrically permeable and magnetically impermeable crack boundary conditions.

The electric displacement _{d 0} and magnetic induction _{b 0} in the crack region depend on the matrix [figure omitted; refer to PDF] as well as _{K D} , _{K B} , _{K [varphi]} , _{K ψ} , and _{K w} depend on the matrix [figure omitted; refer to PDF] where again "-1" denotes the inverse matrix.

Thus, [figure omitted; refer to PDF] is the matrix of material property of equivalent homogeneous material after homogenization in our problem. The generalized effective electroelastic compliances of bi-material system are obtained as arithmetically mean of compliances of single materials constituents. If the lower medium and the upper medium have the same properties but are poled in opposite directions, then α = - ^{α [variant prime]} and β = - ^{β [variant prime]} (see ( 16)). In consequence from ( 17), we have [figure omitted; refer to PDF]

Then [figure omitted; refer to PDF]

Magneto-electro-elastic materials usually comprise alternating piezoelectric material and piezomagnetic material. If upper material is piezomagnetic and lower material is piezoelectric (or otherwise), we have [figure omitted; refer to PDF]

The bi-material matrix ^{C ... -1} defined by ( 51) has the form [figure omitted; refer to PDF]

In the solutions also appears electric and magnetic field components [figure omitted; refer to PDF] where the matrix ^{C 1 -1} is defined by ( 34). Of course, for lower material, we have ^{C 1 [variant prime] -1} matrix (the material parameters are denoted by prime). This states that, in general, electric and magnetic fields are also singular. The electric and magnetic field intensity factors _{K E} and _{K H} are related to _{K τ} , _{K D} , and _{K B} , as shown ( 57)₁ . In particular, for a fully permeable crack between two materials polarized in opposite directions, we have _{K D} =0 = _{K B} and [figure omitted; refer to PDF]

The particular solutions [figure omitted; refer to PDF] with [figure omitted; refer to PDF] complete the full fields in both materials.

6. Numerical Results

The basic data for the material properties selected here are similar to those in Sih and Song [ 6, 7]. These constants read as: _{c 44} =43 ,7 ×1 ^{0 9} N / ^{m 2} , _{e 15} =8,12 C / ^{m 2} , _{[straight epsilon] 11} =7 ,86 ×1 ^{0 -9} C /Vm , _{d 11} =0,0 , _{q 15} =165,0 N /Am , and _{μ 11} =180 ,5 ×1 ^{0 -6} N / ^{A 2} for first material and _{c 44} =44 ,6 × 1 ^{0 9} N / ^{m 2} , _{e 15} =3,48 C / ^{m 2} , _{[straight epsilon] 11} =3 ,42 ×1 ^{0 -9} C /Vm , _{d 11} =0,0 , _{q 15} =385,0 N /Am , _{μ 11} =414 ,5 ×1 ^{0 -6} N / ^{A 2} for second material.

Using these properties of both materials, the material property matrix ^{C ... -1} is obtained as (the matrix of generalized "compliances"): [figure omitted; refer to PDF]

The matrix of generalized stiffness is obtained as follows: [figure omitted; refer to PDF]

Therefore, the properties of composite, obtained by averaging the properties of single-phase materials using its volume fractions, as in the literature (see [ 8]) gives erroneous result, since give (if ratio is roughly 50 : 50) [figure omitted; refer to PDF]

The nonzero material constants for BaTiO₃ -piezoelectric and CoFe₂ O₄ -magnetostrictive medium are given in Table 1[ 9].

Table 1: The material constants for BaTiO₃ and CoFe₂ O₄ .

properties	BaTiO3 piezoelectric	CoFe2 O4 piezomagnetic
c 44 ( 1 0 9 N / m 2 )	43,00	45,30
e 15 ( C / m 2 )	11,60	0,00
[straight epsilon] 11 ( 1 0 -9 C /Vm )	11,20	0,08
q 15 ( N /Am )	0,00	550,00
μ 11 ( 1 0 -6 N / A 2 )	5,00	590,00
d 11 ( 1 0 -9 C /Am )	0,00	0,00

The bi-material matrix ^{C ... -1} defined by ( 51) is ("compliance" matrix) [figure omitted; refer to PDF]

The matrix C ... is obtained as follows: ("stiffness" matrix) [figure omitted; refer to PDF]

Using the mixture rule [ 6], ^{κ c} = κ _{V f} + ^{κ [variant prime]} ( 1 - _{V f} ) , for _{V f} =0,5 , where κ with superscripts c without prime or prime denotes the corresponding constants _{c 44} , e , [straight epsilon] , q , μ , d of the composite, first and second material, respectively, and _{V f} is the volume fraction of the first material (piezoelectric), we obtain that [figure omitted; refer to PDF]

which completely differs from C ... .

Note that in both examples the sums of corresponding material parameters are constant. In consequence the matrix, _{C ... aver} and ^{C ... *} have the same elements. Of course, the matrices of generalized stiffness are dissimilar in both bi-material composites.

Due to the absence of magnetoelectric coupling coefficient in a single-phase piezoelectric and piezomagnetic material, the magnetoelectric constant _{d 11} , existing only in the piezoelectric/piezomagnetic composite as a significant new feature, cannot be determined by the above mixture rule. Therefore, based on the analysis of micromechanics, this coefficient is obtained as _{d 11} =1 ,36 ×1 ^{0 -12} C /Am for first combination of materials and 7 ,55 ×1 ^{0 -12} C /Am for barium titanate-cobalt iron oxide bi-material. This is magnetoelectric coupling effect in composite of piezoelectric and piezomagnetic phases.

7. Conclusions

The mode III interface crack in a bi-material magneto-electro-elastic medium subjected to mechanical, electrical, and magnetic loads on the surfaces is studied in this paper, and the following points are noted.

(i) Closed form solution has been obtained for a crack between two dissimilar PEMO-elastic materials. Expressions for the crack-tip field intensity factors, the electromagnetic fields inside the crack, are given. The semipermeable crack-face magneto-electric boundary conditions are investigated.

(ii) The energy release rate can be explicitly expressed in terms of the external loadings (by ( 42)). It is affected by electric-magnetic properties of the two constituents of the bi-material media.

(iii): Applications of electric and magnetic fields do not alter the stress intensity factor of mode III. The values of SIF are identical for any kind of crack-face electric and magnetic boundary condition assumptions. In other words, the crack-face electric and magnetic boundary conditions have no effects on SIF.

(iv) For electrically impermeable and magnetically permeable crack, the solutions for field intensity factors are independent of the applied magnetic field. Alternatively, these solutions for the electrically permeable and magnetically impermeable crack are independent on the applied electric displacement.

(v) For fully permeable crack between two PEMO-elastic materials polarized in opposite directions, the electric displacement and magnetic induction intensity factors vanish. In this case electric and magnetic field intensity factors _{K E} and _{K B} are related to _{K τ} (by ( 58)).

(vi) The matrices of "generalized" compliances or stiffness cannot be determined by the mixture rule since it is a significant new feature in interface crack problem considered in this paper.

(vii): From the reviewing of literature dealing with interface crack, problem may be concluded that the characterization of bonded dissimilar materials with interface crack is still an open problem.

References

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