Mathematical analysis of fracture behavior for multiple offset interfacial cracks in FGMEE-FGM layers

1. Introduction

The concept of FGMs was originally introduced by researchers in Japan in the 1980s for aerospace thermal-barrier applications, where extreme temperature gradients of approximately 2000 K on the outer surface and 1000 K on the inner surface across a 10 mm thickness necessitated materials with continuously varying thermo-mechanical properties. FGMs are engineered such that the volume fractions of constituent phases vary smoothly along a prescribed direction, enabling the material composition to transition from 0% to 100% ceramic across the thickness. Functionally graded magneto-electro-elastic (FGMEE) composites are the result of extending the concept of FGM to magneto-electro-elastic materials in order to improve the reliability of composites. One benefit of these materials is that they lack both desirable boundaries and internal seams. Nevertheless, due to their intrinsic brittleness, crack initiation remains unavoidable in FGMEE materials. For this kind of material under magneto-electro-mechanical loading, strong field concentrations arise near crack fronts, thereby accelerating crack growth. Furthermore, it is inevitable to have holes, cracks, inclusions, dislocations, and other flaws to start during the manufacturing process. The investigation of the fracture behavior of FGMEE materials was prompted by these flaws, which can cause composites or structures to break prematurely at any time.

To date, several studies have contributed to the research domain of FGM [1–4]. Singh and Das [5] investigated the fracture response of multiple collinear Griffith cracks in an FGM medium under thermal, mechanical, and thermomechanical environments. Singh and Das [6] investigated the transient behavior of multiple collinear antiplane cracks in an FGM layer bonded between dissimilar elastic layers and subjected to shear-impact loading. For arbitrarily oriented interface-crossing cracks under combined thermal and mechanical fields, a comprehensive analysis was done in the work by Singh and Das [7]. An angled crack under steady-state temperature conditions was further investigated by Singh and Das [8]. The article by Singh [9] presents an analytical method for the random dynamic investigation of two parallel interfacial cracks in a strip of FGM bonded between two different elastic strips.

Furthermore, many researchers have focused on layered FGM and FGMEE composites. A cracked functionally graded piezoelectric–piezomagnetic strip under the impact of in-plane magnetic and electric and anti-plane mechanical loads was analyzed using the distributed dislocation technique proposed in the work by Mousavi and Paavola [10]. In the work by Bagheri and Lak [11], an FGM layer under antiplane loading that is sandwiched between two isotropic layers and deteriorated by several interface cracks is examined. The anti-plane problem of an FGMEE strip positioned between two functionally graded strips is analyzed in the work by Guo et al. [12]. In the work by Hu et al. [13], the problem of a crack in a functionally gradient piezoelectric interlayer between two different homogeneous piezoelectric half-planes that is subjected to both an in-plane electric loading and an anti-plane mechanical loading is studied. The dynamic anti-plane problem for an FGMEE strip is investigated in the work by Feng and Su [14], with an internal crack that is perpendicular to the boundary. The integral transform approach is used in the work by Hu and Chen [15] to analyze a moving crack that moves along the interface of magnetoelectroelastic and functionally graded elastic layers under in-plane magnetic and electric and anti-plane shear stress. In the work by Hu et al. [16], the same crack model was examined for dynamic analysis. Singh and Das [17] investigated the fracture properties of several parallel cracks in an FGMEE plane subjected to both permeable and impermeable boundary conditions. In the works by Singh [18, 19], the dynamic study of several interfacial permeable cracks originating from circular holes in two bonded semi-infinite functionally graded piezoelectric materials under the effect of steady-state SH waves is examined. The Green’s function method is utilized to get the solution of the boundary conditions.

In general, the cracks are modeled as electrically and magnetically impermeable in order to establish an analytically tractable framework for the fracture behavior of the layered FGMEE composite. This assumption represents a limiting case in which the crack surfaces completely obstruct the transmission of electric displacement and magnetic flux across the crack gap. From a physical standpoint, such a situation can be justified where the crack opening is occupied by voids, micro-gaps, oxidation layers, debonded interfaces, or non-conducting damaged regions, all of which possess significantly lower electric and magnetic permeability compared to the surrounding magneto-electro-elastic medium. Under these conditions, the continuity of coupled electromagnetic fields across the crack faces becomes negligible, and the impermeable idealization becomes meaningful. Furthermore, the impermeable crack model generally produces stronger field perturbations and higher crack-tip intensity measures compared to the permeable case [17]. For this reason, impermeable crack assumptions are widely employed in investigations of piezoelectric, piezomagnetic, and magneto-electro-elastic fracture problems as benchmark formulations for evaluating crack-tip singular behavior.

The combined influence of functional grading, crack offset geometry, interfacial interaction effects, layered structure, and loading behavior on the fracture characteristics is not considered to date. This study contributes toward filling this gap by exploring the fracture initiation likelihood of multiple offset interfacial cracks in a multi-layered composite having a core FGMEE strip sandwiched between two different FGM layers of identical thickness. The configuration includes one of the offset cracks located at the center of the interface of FGMEE and one of the FGM strips, whereas the other two interfacial cracks are situated at the interface of FGMEE and another FGM strip, as elaborated in subsection “Schematic description.” The offset cracks are under the influence of in-plane magnetic and electric, as well as anti-plane mechanical loadings. The interfacial cracks are impermeable in nature. The governing equations and other boundary-continuity conditions are discussed in subsection “Governing equations and related boundary conditions.” Section “Mathematical analysis” of the article describes the reduction of the governing field equations to a set of first-kind singular integral equations. The resultant integral equation is solved numerically using an approach including the Chebyshev polynomial because of the complex nature of the kernel, as explained in subsection “Numerical solution.” Finally, the expressions of crack-tip driving force parameters in terms of SIFs are obtained in section “Driving force parameters.” In section “Numerical outcomes and discussion,” some important results and discussion based on the geometrical and material parameters are analyzed. A verification of the findings with existing results is also presented in this article. With the aid of pictorial illustration, the consequence of strips’ width ratio, non-homogeneity parameter ratio, interfacial crack length ratios, and electric–magnetic loadings on the normalized SIFs is examined. Section “Concluding remarks” concludes the article with some concluding comments. At the end of the article, Appendix 1 and references are given.

2. Problem formulation

2.1. Schematic description

The geometrical diagram of the problem is illustrated in Figure 1. The composite under discussion is composed of two different FGM strips of thickness h and an FGMEE strip of thickness h 0 , as displayed in the figure. For convenience, the upper FGM, FGMEE, and lower FGM strips are also denoted as (1), (2), and (3), respectively. A Cartesian coordinate system is defined so that the y-axis is perpendicular to and the x-axis corresponds with the interface of strips (2) and (3). The multistriped system contains three interfacial offset cracks. One of the cracks of length 2a is placed centrally at the interface of strips (2) and (3), while the other two cracks, each of length 2b, are placed at the interface of strips (1) and (2) at a horizontal offset distance e from the y-axis. It is anticipated that the FGMEE strip will behave in a transversely isotropic manner with a polarization direction along the z-axis. Concurrent in-plane magnetic, electric, and anti-plane mechanical loadings are applied to the crack surfaces.

OPEN IN VIEWER

Figure 1.

Geometric configuration of the problem.

The mechanical parametric functions, namely shear modulus for strips (1) and (3), are given by

μ i ( y ) = μ i 0 e β i y , i = 1 , 3 ,

(1)

and for the strip (2) magnetic permeability, shear modulus, dielectric, piezoelectric, magneto-electric, and piezomagnetic constants, respectively, are given by

{ e 15 ( y ) = e 150 e β 2 y , c 44 ( y ) = c 440 e β 2 y , f 15 ( y ) = f 150 e β 2 y , ϵ 11 ( y ) = ϵ 110 e β 2 y , μ 11 ( y ) = μ 110 e β 2 y , g 11 ( y ) = g 110 e β 2 y ,

(2)

where β i ( i = 1 , 2 , 3 ) are the functionally graded parameters and μ i 10 ( i = 1 , 3 ) , c 440 , e 150 , f 150 , ϵ 110 , g 110 , μ 110 represent the material constants at y = 0 .

2.2. Governing equations and related boundary conditions

The constitutive equation for FGM strips is given by

τ zki = μ i ∂ w i ∂k , i = 1 , 3 ; k = x , y ,

(3)

and those for the FGMEE strip are given by

{ τ zk 2 = c 44 ∂ w 2 ∂k + e 15 ∂ϕ ∂k + f 15 ∂ψ ∂k , D k = e 15 ∂ w 2 ∂k − ϵ 11 ∂ϕ ∂k − g 11 ∂ψ ∂k , k = x , y , B k = f 15 ∂ w 2 ∂k − g 11 ∂ϕ ∂k − μ 11 ∂ψ ∂k ,

(4)

where τ zki ( i = 1 , 2 , 3 ) , D k , B k represent the anti-plane shear stress, in-plane electric displacement, and magnetic inductions, respectively. The symbols w i ( i = 1 , 2 , 3 ) , ψ, and ϕ represent the mechanical displacement, magnetic potential, and electric potential, respectively. The equilibrium equation for the FGM strips in the absence of free charge and body force is provided by

∂ τ xzi ∂x + ∂ τ yzi ∂y = 0 , i = 1 , 3 ,

(5)

and those for the FGMEE strip are given by

∂ τ xz 2 ∂x + ∂ τ yz 2 ∂y = 0 , ∂ D x ∂x + ∂ D y ∂y = 0 , ∂ B x ∂x + ∂ B y ∂y = 0 .

(6)

Defining auxiliary functions χ and ζ as

χ = e 150 w 2 − ϵ 110 ϕ − g 110 ψ , ζ = f 150 w 2 − g 110 ϕ − μ 110 ψ ,

(7)

Equation (4) reduces to

τ zk 2 = e β 2 y [ m 10 ∂ w 2 ∂k + m 20 ∂ϕ ∂k + m 30 ∂ψ ∂k ] , D k = e β 2 y ∂χ ∂k , B k = e β 2 y ∂ζ ∂k , k = x , y ,

(8)

where m 10 , m 20 , m 30 are given by equation (36) of Appendix 1. Substituting equations (3) and (8) into equations (5) and (6), respectively, gives

∇ ⁡ 2 w i + β i ∂ w i ∂y = 0 , i = 1 , 3 ,

(9)

{ ∇ ⁡ 2 w 2 + β 2 ∂ w 2 ∂y = 0 , ∇ ⁡ 2 χ + β 2 ∂χ ∂y = 0 , ∇ ⁡ 2 ζ + β 2 ∂ζ ∂y = 0 .

(10)

It is clear from Figure 1 that the considered geometry is symmetric about the y-axis. Consequently, it suffices to take into account the region { 0 ≤ x < ∞ , − ∞ < y < ∞ } . The boundary conditions for the considered problem are given by

τ yz 1 ( x , h 0 + h ) = 0 , τ yz 3 ( x , − h ) = 0 , 0 < x < ∞ .

(11)

The continuity conditions about the interfaces are as follows

τ yz 1 ( x , h 0 ) = τ yz 2 ( x , h 0 ) , w 1 ( x , h 0 ) = w 2 ( x , h 0 ) , D y ( x , h 0 ) = 0 , B y ( x , h 0 ) = 0 , } 0 < x < e − b , e + b < x ,

(12)

τ yz 2 ( x , 0 ) = τ yz 3 ( x , 0 ) , w 2 ( x , 0 ) = w 3 ( x , 0 ) , D y ( x , 0 ) = 0 , B y ( x , 0 ) = 0 , } x > a .

(13)

Conditions on the magneto-electrically impermeable crack surfaces are given by

τ yz 1 ( x , h 0 ) = τ yz 2 ( x , h 0 ) = − τ 1 , D y ( x , h 0 ) = − D 1 , B y ( x , h 0 ) = − B 1 , e − b < x < e + b ,

(14)

τ yz 2 ( x , 0 ) = τ yz 3 ( x , 0 ) = − τ 2 , D y ( x , 0 ) = − D 2 , B y ( x , 0 ) = − B 2 , 0 < x < a ,

(15)

ϕ ( x , h 0 ) = 0 , ψ ( x , h 0 ) = 0 , 0 < x < e − b , e + b > x ,

(16)

ϕ ( x , 0 ) = 0 , ψ ( x , 0 ) = 0 , x > a .

(17)

3. Mathematical analysis

Equations (9) and (10) using the Fourier cosine transformation with regard to x yield

w i ( x , y ) = ∫ 0 ∞ ∑ j = 1 2 a ji ( s ) cos ⁡ ( sx ) e n ji ( s ) y ds , i = 1 , 3 ,

(18)

{ w 2 ( x , y ) = ∫ 0 ∞ ∑ j = 1 2 a j 2 ( s ) cos ⁡ ( sx ) e n j 2 ( s ) y ds , χ ( x , y ) = ∫ 0 ∞ ∑ j = 1 2 b j ( s ) cos ⁡ ( sx ) e n j 2 ( s ) y ds , ζ ( x , y ) = ∫ 0 ∞ ∑ j = 1 2 c j ( s ) cos ⁡ ( sx ) e n j 2 ( s ) y ds ,

(19)

where a ji ( s ) ( i = 1 , 2 , 3 ) , b j ( s ) , c j ( s ) ( j = 1 , 2 ) are the unknowns in s to be determined and n ji ( s ) ( i = 1 , 2 , 3 ; j = 1 , 2 ) is given by equation (34) of Appendix 1. With the aid of equations (3) and (18), the expression of anti-plane shear stress for FGM strips (1) and (3) takes the form

τ yzi ( x , y ) = μ i ( y ) ∫ 0 ∞ ∑ j = 1 2 a ji ( s ) n ji ( s ) cos ⁡ ( sx ) × e n ji ( s ) y ds , i = 1 , 3 ,

(20)

and for the FGMEE strip (2), equations (8) and (19) give the expressions of anti-plane shear stress, in-plane electric displacement, and magnetic inductions as

{ τ yz 2 ( x , y ) = e β 2 y ∫ 0 ∞ ∑ j = 1 2 [ m 10 a ji ( s ) + m 20 b j ( s ) + m 30 c j ( s ) ] n j 2 ( s ) cos ⁡ ( sx ) e n j 2 ( s ) y ds , D y ( x , y ) = e β 2 y ∫ 0 ∞ ∑ j = 1 2 b j ( s ) n j 2 ( s ) cos ⁡ ( sx ) e n j 2 ( s ) y ds , B y ( x , y ) = e β 2 y ∫ 0 ∞ ∑ j = 1 2 c j ( s ) n j 2 ( s ) cos ⁡ ( sx ) e n j 2 ( s ) y ds .

(21)

Defining new functions as

g 1 ( x ) = { ∂ ∂x [ w 1 ( x , h 0 ) − w 2 ( x , h 0 ) ] , e − b ≤ x ≤ e + b , 0 , 0 < x < e − b , e + b < x , g 2 ( x ) = { ∂ ∂x [ ϕ ( x , h 0 ) ] , e − b ≤ x ≤ e + b , 0 , 0 < x < e − b , e + b < x , g 3 ( x ) = { ∂ ∂x [ ψ ( x , h 0 ) ] , e − b ≤ x ≤ e + b , 0 , 0 < x < e − b , e + b < x ,

(22)

f 1 ( x ) = { ∂ ∂x [ w 2 ( x , 0 ) − w 3 ( x , 0 ) ] , 0 ≤ x ≤ a , 0 , x > a , f 2 ( x ) = { ∂ ∂x [ ϕ ( x , 0 ) ] , 0 ≤ x ≤ a , 0 , x > a , f 3 ( x ) = { ∂ ∂x [ ψ ( x , 0 ) ] , 0 ≤ x ≤ a , 0 , x > a , }

(23)

that under the following conditions

∫ e − b e + b g i ( x ) dx = 0 , ∫ 0 a f i ( x ) dx = 0 , i = 1 , 2 , 3 .

(24)

From equations (11) to (13) and (20) to (23), 10 equations are found that have the following matrix form

X = A − 1 . B , A = [ A 1 A 2 A 3 A 4 ] , A 1 = [ A 11 A 12 A 13 A 14 0 0 0 A 21 A 22 A 23 0 0 A 31 A 32 0 0 0 A 41 A 42 0 0 0 A 51 A 52 0 ] ,

(25)

A 2 = [ 0 0 0 0 0 A 24 0 0 0 0 0 B 11 B 12 C 11 C 12 0 B 21 B 22 C 21 C 22 0 B 31 B 32 C 31 C 32 ] , A 3 = [ 0 0 A 61 A 62 0 A 71 A 72 0 0 0 0 0 0 0 A 81 A 91 A 92 A 93 A 94 0 0 0 A 10 1 A 10 2 A 10 3 ] , A 4 = [ 0 B 41 B 42 C 41 C 42 0 0 0 0 0 A 82 0 0 0 0 0 B 51 B 52 C 51 C 52 A 10 4 B 61 B 62 C 61 C 62 ] , X = [ a 11 a 21 a 12 a 22 a 13 a 23 b 1 b 2 c 1 c 2 ] T , B = [ G 1 F 1 G 2 F 2 G 3 F 3 0 0 0 0 ] T ,

(26)

A ij , B ij , C ij , G i ( s ) , F i ( s ) are given by equation (35) of Appendix 1. The expressions of a ji ( s ) , b j ( s ) , c j ( s ) in terms of G i ( s ) and F i ( s ) can be given as follows

a ji ( s ) = ∑ m = 1 3 [ P 2 m − 1 ji ( s ) G m ( s ) + P 2 m ji ( s ) F m ( s ) ] , b j ( s ) = ∑ m = 1 3 [ Q 2 m − 1 j ( s ) G m ( s ) + Q 2 m j ( s ) F m ( s ) ] , c j ( s ) = ∑ m = 1 3 [ R 2 m − 1 j ( s ) G m ( s ) + R 2 m j ( s ) F m ( s ) ] ,

(27)

where P mji represents the j th row for i = 1 , ( j + 2 ) th row for i = 2 , ( j + 4 ) th row for i = 3 , Q mj represent the ( j + 6 ) th row, R mj represents the ( j + 8 ) th row, and m th column of the matrix A − 1 ( j = 1 , 2 ; m = 1 , … , 6 ) . With the aid of crack surface conditions (14) and (15), it is possible to obtain the subsequent set of singular integral equations

∫ e − b e + b ∑ i = 1 3 [ K mi ( x , t ) + ∫ 0 ∞ r mi { sin ⁡ s ( t − x ) + sin ⁡ s ( t + x ) } ds ] g i ( t ) dt + ∫ 0 a ∑ i = 1 3 [ K m i + 3 ( x , t ) + ∫ 0 ∞ r m i + 3 { sin ⁡ s ( t − x ) + sin ⁡ s ( t + x ) } ds ] f i ( t ) dt = S m , m = 1 , . . . , 6 ,

(28)

where K mi ( x , t ) , r mi , S m ( m = 1 , … , 6 ; i = 1 , 2 , 3 ) are given by equations (35) to (41) of Appendix 1. From equations (35) to (41), the kernels K mi contained semi-infinite integrals and are calculated by using an asymptotic analysis as s → ∞ . The dominant asymptotic contribution is separated analytically, which regularized the kernel behavior and ensured stable numerical evaluation of the remaining integrals.

3.1. Numerical solution

Equation (28) is solved numerically using a truncated series of Chebyshev polynomials of the first kind [5] as a result of the complex nature of the kernel that emerged in the set of singular integral equations

g i ( x ) = ∑ n = 0 N i c in T n ( x − e b ) 1 − ( x − e b ) 2 , f i ( x ) = ∑ n = 0 M i d in T n ( 2 x − a a ) 1 − ( 2 x − a a ) 2 , i = 1 , 2 , 3 ,

(29)

where c in , d in represent the unknown constants and N i , M i indicate the number of summation terms. Utilizing [5], the unknown constants can be found that help to find the driving force parameters in terms of SIFs. Utilizing (29), equation (28) reduces to

∑ i = 1 3 [ ∑ n = 0 N i ( c in U mni ( x ) + c in V mni ( x ) ) + ∑ n = 0 M i ( d in W mn i + 3 ( x ) + d in X mn i + 3 ( x ) ) ] = S m ,

(30)

where m = 1 , … , 6 , U mni , V mni , W mn i + 3 , X mn i + 3 are given by equation (42) of Appendix 1. The collocation points are given by [20] x j = cos ( jπ N + 1 ) , N = N i , M i .

4. Driving force parameters

The expression for the stress intensity factor (SIF) κ 3 τ at the crack tips is

{ κ 3 τ ( a ) = lim ⁡ x → a + 2 π ( x − a ) τ yz 2 ( x , 0 ) , κ 3 τ ( e − b ) = lim ⁡ x → ( e − b ) − 2 π ( e − b − x ) τ yz 2 ( x , h 0 ) , κ 3 τ ( e + b ) = lim ⁡ x → ( e + b ) + 2 π ( x − e − b ) τ yz 2 ( x , h 0 ) .

(31)

In terms of c in and d in ( i = 1 , 2 , 3 ) , equation (31) can be expressed as

{ κ 3 τ ( a ) = πa 2 [ c 440 ∑ n = 0 M 1 d 1 n + e 150 ∑ n = 0 M 2 d 2 n + f 150 ∑ n = 0 M 3 d 3 n ] , κ 3 τ ( e − b ) = πb 2 [ c 440 ∑ n = 0 N 1 ( − 1 ) n c 1 n + e 150 ∑ n = 0 N 2 ( − 1 ) n c 2 n + f 150 ∑ n = 0 N 3 ( − 1 ) n c 3 n ] , κ 3 τ ( e + b ) = πb 2 [ c 440 ∑ n = 0 N 1 c 1 n + e 150 ∑ n = 0 N 2 c 2 n + f 150 ∑ n = 0 N 3 c 3 n ] .

(32)

Note that the present analysis is based on the framework of linear elasticity and linearized constitutive relations, under which the classical r − 1 2 crack-tip singularity naturally emerges. This mathematical singularity implies unbounded stresses in the vicinity of the crack tip. This unboundedness is not physically realizable and lies beyond the validity range of the linearized theory. In practical application, nonlinear effects such as plastic deformation, microstructural damage, cohesive-zone behavior, or other crack-tip regularization mechanisms become significant in this region and alleviate the singular response. In recent years, alternative approaches such as strain-limiting elasticity theories, gradient-enhanced continua [21], cohesive-zone formulations [22, 23], and diffuse phase-field fracture models [24, 25] have been developed to regularize the classical fracture mechanics singularity. The aim of the current work is restricted to capturing the dominant asymptotic behavior away from the immediate crack-tip core rather than a complete physical description of the local fracture process zone. Therefore, the linear approximation remains meaningful for determining the SIFs.

5. Numerical outcomes and discussion

Here, the effect of strips’ width ratio, non-homogeneity parameter ratio, interfacial crack length ratios, and applied magnetic-electric loadings on the normalized SIFs is shown. For calculation purpose, c 440 = 44 GPA , e 150 = 5 . 8 C ∕ m 2 , ϵ 110 = 5 . 6 × 1 0 − 9 C 2 ∕ Nm 2 , f 150 = 275 N ∕ Am , μ 110 = − 297 × 1 0 − 6 Ns ∕ C 2 , g 110 = 0 . 005 × 1 0 − 1 Ns ∕ (VC) , τ 0 = 4 . 2 × 1 0 6 N ∕ m 2 , λ d = ( e 150 D 0 ) ∕ ( ϵ 110 τ 0 ) = 0 . 0 , λ b = ( f 150 B 0 ) ∕ ( μ 110 τ 0 ) = 0 . 0 [12], a ∕ ( e + b ) = 0 . 5 , ( e − b ) ∕ ( e + b ) = 0 . 6 , h 0 ∕ h = 1 , β 1 = β 3 = β 0 , μ 10 = μ 30 = μ 0 , β 2 ∕ β 0 = 1 , μ 0 = c 440 + ( ϵ 110 f 150 2 − 2 e 150 f 150 g 110 + μ 110 e 150 2 ) ∕ ( μ 110 ϵ 110 − g 110 2 ) have been taken. The normalized crack-tip SIFs are chosen as K 3 τ ( a ) = κ 3 τ ( a ) ∕ πa 2 , K 3 τ ( e − b ) = κ 3 τ ( e − b ) ∕ πb 2 , K 3 τ ( e + b ) = κ 3 τ ( e + b ) ∕ πb 2 ; N i and M i are chosen such that three decimal digit accurate values of normalized SIFs are acquired. If one of them is varying, the others remain the same.

5.1. Verification

A verification to compare the found results of this study with [15] is done in this section. For thickness of FGM strip (1) tends to zero, e = 0 , b = 0 , μ 10 = 0 , c 440 = μ 30 = 4 . 53 × 1 0 10 N ∕ m 2 , e 150 = 11 . 6 C ∕ m 2 , ϵ 110 = 0 . 8 × 1 0 − 10 C 2 ∕ Nm 2 , f 150 = 550 N ∕ Am , μ 110 = 15 . 9 × 1 0 − 4 Ns 2 ∕ C 2 , g 110 = 0 . 5 × 1 0 − 11 Ns ∕ (VC) , h 0 = h , a ∕ h = 0 . 4 , β 2 = 0 , K 3 τ ( a ) = κ 3 τ ( a ) ∕ ( T 0 πa 2 ) , the findings of this study are compared to the work by Hu and Chen [15] for different values of β 3 . Clearly, Figure 2 shows that the results obtained by the current study are in good match with the findings in the work by Hu and Chen [15].

OPEN IN VIEWER

Figure 2.

Comparison of this study with the work by Hu and Chen [15].

5.2. Effect of strips’ width ratio h 0 ∕ h

Figure 3 illustrates how the width of strips affects the likelihood of crack initiation. It is detected from the figure that with the increasing width of strips, the normalized SIF at all the crack tips started to decrease. In other words, the fracture initiation likelihood decreases.

OPEN IN VIEWER

Figure 3.

Effect of strips’ width ratio on normalized SIFs at the offset interfacial crack tips κ 3 τ ( a ) , κ 3 τ ( e − b ) , and κ 3 τ ( e + b ) .

By increasing the strips’ width, the influence of boundary constraints reduces. This redistributes the coupled stress fields over a larger region and lowers stress concentration near the crack tips.

Therefore, it can be said that by considering thicker FGM strips crack-tip driving forces in terms of SIFs, cracks can be reduced.

5.3. Effect of non-homogeneity parameter ratio β 2 ∕ β 0

The influence of non-homogeneity parameter ratio β 2 ∕ β 0 on normalized SIFs for crack tips a, ( e − b ) , ( e + b ) is illustrated by Figure 4. From the figure, it is clear that as β 2 ∕ β 0 increases from negative to positive values, K 3 τ at crack tip a and ( e + b ) increases, whereas at the crack tip ( e − b ) it decreases. This varying nature at the crack tips is obtained due to the non-homogeneity of the chosen composites.

OPEN IN VIEWER

Figure 4.

Effect of non-homogeneity parameter ratio on normalized SIFs at the offset interfacial crack tips κ 3 τ ( a ) , κ 3 τ ( e − b ) , and κ 3 τ ( e + b ) .

Thus, upon examination of the results, it can be concluded that for negative values of the non-homogeneity parameter ratio, the crack-tip driving forces in terms of SIFs of the central crack are less compared to the positive ones. Whereas for the offset crack of length ( c − b ) , the SIFs of the crack can be controlled when the non-homogeneity parameter ratio has a magnitude close to zero.

5.4. Effect of interfacial crack length ratios a ∕ ( e + b ) & ( e − b ) ∕ ( e + b )

Figures 5 and 6 demonstrate how the interfacial crack lengths affect the fracture initiation likelihood of cracks.

OPEN IN VIEWER

Figure 5.

Effect of central crack length variation on normalized SIFs at the offset interfacial crack tips κ 3 τ ( a ) , κ 3 τ ( e − b ) , and κ 3 τ ( e + b ) .

OPEN IN VIEWER

Figure 6.

Effect of offset parallel crack length variation on normalized SIFs at the offset interfacial crack tips κ 3 τ ( a ) , κ 3 τ ( e − b ) , and κ 3 τ ( e + b ) .

It is clear from Figure 5 that the normalized crack-tip SIFs began to rise if the length of the central crack a ∕ ( e + b ) is increasing while maintaining the offset parallel crack length unchanged, i.e., ( e − b ) ∕ ( e + b ) = 0 . 6 . Stated differently, the likelihood of fracture grows up.

Similarly, Figure 6 shows that if the length of the central crack is fixed, i.e., a ∕ ( e + b ) = 0 . 5 and ( e − b ) ∕ ( e + b ) is increasing, then the normalized crack-tip SIFs decrease, indicating the reduction in the crack fracture initiation likelihood.

In other words, when offset cracks are located close to each other, strong crack-interaction effects arise due to stress and coupled-field disturbances. This interaction amplifies the intensity of stress near the crack tips and thereby accelerates crack initiation. It increases the probability of interfacial failure.

Therefore, it may be inferred that the likelihood of crack initiation increases as the central and offset parallel cracks approach one another.

5.5. Effect of electric λ d and magnetic λ b loadings

The influence of electric and magnetic loadings is depicted in Figures 7 and 8.

OPEN IN VIEWER

Figure 7.

Effect of electric load λ d on normalized SIFs at the offset interfacial crack tips κ 3 τ ( a ) , κ 3 τ ( e − b ) , and κ 3 τ ( e + b ) .

OPEN IN VIEWER

Figure 8.

Effect of magnetic load λ b on normalized SIFs at the offset interfacial crack tips κ 3 τ ( a ) , κ 3 τ ( e − b ) , and κ 3 τ ( e + b ) .

According to Figure 7, maintaining the magnetic load unchanged, if the electric load increases, then K 3 τ starts to decrease initially and then near λ d = 1 . 5 it starts to increase at the crack tip a and ( e + b ) . Opposite behavior is obtained at the crack tip ( e − b ) .

On the other hand, the fracture initiation likelihood of offset parallel cracks decreases for fixed electric load and increasing magnetic load, but increases for the central crack. It shows that normalized SIFs are significantly impacted by both electric and magnetic loads.

The magnitude and nature of the applied electric and magnetic loadings affect the crack-tip SIF by either increasing or decreasing, as per the load nature. Therefore, electromagnetic loading conditions play a significant role in the fracture stability and reliability of the considered layered structures.

6. Concluding remarks

The influence of geometrical and material factors on the offset interfacial type cracks placed in an FGMEE-FGM layered composite under the multiple loading of in-plane magnetic, electric, and anti-plane mechanical kinds is analyzed in this article. The approach employed includes the integral transformation technique and Chebyshev polynomials. Moreover, the findings of the present investigation are also verified. The following are some major findings of this study.