A Modified Strip-Yield-Saturation-Induction Model Solution for Cracked Piezoelectromagnetic Plate

A strip-yield-saturation-induction model is proposed for an impermeable crack embedded in piezoelectromagnetic plate. The developed slide-yield, saturation, and induction zones are arrested by distributing, respectively,mechanical, electrical, andmagnetic loads over their rims. Two cases are considered: when saturation zone exceeds induction zone and vice-versa. It is assumed that developed slide-yield zone is the smallest because of the brittle nature of piezoelectromagnetic material. Fourier integral transform technique is employed to obtain the solution. Closed form analytic expressions are derived for developed zones lengths, crack sliding displacement, crack opening potential drop, crack opening induction drop, and energy release rate. Case study presented for BaTiO 3 –CoFe 2 O 4 shows that crack arrest is possible under small-scale mechanical, electrical, and magnetic yielding.


Introduction
The work on magnetoelectroelastic (MEE) fracture problem was started late back in the last century.The field is a natural extension of piezoelectric media since electricity and magnetism go in hand.Due to coupling effect of magneto-, electro-, and elastic fields, MEE materials become more popular than piezoelectric materials and serve as the excellent sensor, actuator, and transducer.
Wang and Shen [1] obtained energy release rate for a mode-III magnetoelectroelastic media based on the concept of energy-momentum tensor.Based on the extended Stroh formalism combined with complex variable technique, Green's function is obtained for an infinite two-dimensional anisotropic MEE media containing an elliptic cavity which degenerates into a slit crack, by Jinxi et al. [2].Sih and Song [3] proposed a model which showed that crack growth in a magnetoelectroelastic material could be suppressed by increasing the magnitude of piezomagnetic constants in relation to these for piezoelectricity.They [4] further derived energy density function for cracked MEE medium and studied the additional magnetic strictive effect which could influence crack initiation as applied field direction is altered.Wang and Mai [5] addressed the problem of a crack in a MEE medium possessing coupled piezoelectric, piezomagnetic, and magnetoelastic effects.Wang and Mai [6] further extended above problem to calculate a conservative integral based on governing equations for MEE media.Gao et al. [7] investigated the fracture mechanics for an elliptic cavity in a MEE solid under remotely applied uniform in-plane electromagnetic and/or antiplane mechanical loadings.Reducing cavity into a crack they considered two extreme cases for impermeable crack and permeable crack cases.Hu and Li [8] obtained singular stress, electric and magnetic fields in MEE strip containing a Griffith crack under longitudinal shear for a crack situated symmetrically and oriented in a direction parallel to the edges of the strip.Tian and Rajapakse [9] obtained the solution for single, multiple, and slowly growing impermeable cracks in a MEE solid using generalized edge dislocation theory.The solution for an elliptic cavity in an infinite two-dimensional MEE medium subjected to remotely applied uniform combined mechanical, electric, and magnetic loadings under permeable crack face boundary condition along the cavity of the surface had been obtained by Zhao et al. [10].Wang and Mai [11] discussed different electromagnetic boundary conditions on permeable and impermeable crack faces in 2 International Journal of Engineering Mathematics a MEE material.Ma et al. [12] addressed an antiplane problem of a functionally graded MEE strip containing an internal or edge permeable/impermeable crack lying transversely to the edges of the strip.A mixed boundary value problem was solved by Zhong and Li [13] for a crack in MEE solid.Kirilyuk [14] gave a method which enabled to find stress intensity factor (SIF) for a cracked MEE body directly from the analogous problem of elasticity.Zhao and Fan [15] proposed a strip electric-magnetic breakdown model for an electrically and magnetically impermeable crack in MEE media using extended Stroh formalism and the extended dislocation modelling of a crack.They have also proposed [16] a strip electric-magnetic polarization saturation model for an infinite and finite MEE strip.Using finite element method Krahulec et al. [17] discussed a crack problem for MEE solid under various electromagnetic boundary conditions on the crack rims.Recently, we, Bhargava and Verma [18], proposed a mechanical, electrical, and induction yield model (based on Dugdale [19] type model) for a cracked MEE 2D media.
In present paper, we have generalized the classical Dugdale model [19] and propose strip-yield-saturation-induction yield model for an unbounded cracked piezoelectromagnetic plate with electric and magnetic polarization in direction.Due to mechanical brittleness, it is assumed that developed mechanical yielding zone is the smallest zone.In the problem, two cases are considered: Case I is when developed saturation zone is bigger than induction zone and Case II is when developed saturation zone is smaller than induction zone.Fourier transform technique is employed to obtain the solution and derived closed form expressions for developed zone sizes, energy release rate, crack opening potential drop, and crack opening induction drop.To test the proposed model, a case study is presented for BaTiO 3 -CoFe 2 O 4 piezoelectromagnetic ceramic.Numerical results are presented graphically and these confirm that the proposed model is capable of crack arrest under small-scale electrical and magnetic yielding.

Fundamental Formulation and Solution Methodology
As are well known, the out-of-plane displacement and inplane electric and magnetic fields problems may be expressed as in -orthogonal coordinate system.For transversely isotropic magnetoelectroelastic medium which is poled along -direction, the constitutive equations for stress,   , electric-displacement,   , and magnetic induction,   ( = , ), components may be expressed as where  44 ,  15 , ℎ 15 ,  11 ,  11 , and  11 denote elastic constant, piezoelectric coefficient, piezomagnetic coefficient, magnetic permeability, and dielectric coefficient, respectively.A comma in subscript denotes the partial differentiation with respect to argument following it.
The equilibrium equations in the absence of body force, electric charge, and magnetic flux, respectively, are expressed as The gradient equations for strain components,   , magnetic field,   , and electric field,   , may be written as where ,  = , .
And the governing equations for electrically and magnetically poled piezoelectromagnetic ceramic under mode-III deformation may be written as where ) is two-dimensional Laplacian operator and  = [, , ]  .The superscript  denotes the transpose of matrix.
To obtain the desired potentials , , and , equation ( 5) is solved by taking Fourier integral transform.The solution of corresponding ordinary differential equation is given by where ∼ over the functions denotes its Fourier transform and superscripts + and − denote the value of function on the rims of crack as approached from  > 0 and  < 0 planes, respectively. and  are 3 × 1 column vectors and  is Fourier transform variable.For convenience of calculations, the boundary conditions of the problem are also written in terms of their Fourier transform.Fourier transform of constitutive equations (2) may be written as where ) .
Solution of equation ( 8) may be given as The continuity condition for traction on  = 0 and || <  yields Equations ( 10) and ( 11) together yield where The jump in displacement, electric-displacement, and magnetic induction are defined by And the dislocation function,  1 (), dipole density function,  2 (), and induction density function,  3 (), are defined as Taking Fourier transform of ( 14) one obtains Equations ( 12) and ( 15) may also be expressed as Inverse Fourier transform of the above equation and ( 11) and (16) yields a system of integral equations to determine f(x) as

The Problem
A piezoelectromagnetic ceramic plate occupies entire plane and is thick enough in -direction to allow antiplane deformations.The plate is poled along positive -direction both magnetically and electrically.The plate is cut along a hairline, quasi-stationary straight crack.The crack occupies the interval [−, ] on -axis.
Remote boundary of the plate is prescribed: (i) an antiplane shear stress: Case I.When developed saturation zone exceeds developed induction zone, ( < ).
To arrest the crack from further opening, the rims of saturation zone are subjected to normal cohesive in-plane saturation limit electrical displacement;   =   .And the rims of developed induction zone are subjected to in-plane normal cohesive magnetic load;   = (/)  , where   denotes saturation limit of magnetic induction and  is any point on induction zone.
Case II.When developed saturation zone is smaller than developed induction zone, ( > ).
For this case, developed induction zone rims are prescribed in-plane, normal cohesive saturation limit of magnetic induction,   =   , and developed saturation zone is subjected to in-plane normal cohesively linearly varying saturation limit electric displacement,   = (/)  , where   is saturation limit electrical displacement and  is any point on saturation zone.
For both considered cases, developed slide-yield zone is taken to be the smallest ( < &) and is prescribed cohesive yield-point shear stress,   =   .

Case I: Solution and Applications
Schematically, the configuration of the problem for Case I is depicted in Figure 1.
Consequently, the solution for  1 () may be written using (17) as provided that which on evaluation leads to a transcendental equation to determine : Hence the slide-yield zone is determined using | − |.Using (21), the solution of ( 19) may be written as where Analogously, the solution for  3 () is written using equation (16)  ) .
The solution of equation ( 25) is obtained under the condition Equation ( 27) enables to determine .The saturation zone length is then determined using | − |.

Applications for Case I 4.1.1. Crack Sliding Displacement (CSD), Crack Opening Induction Drop (COI), and Crack Opening Potential Drop (COP).
Quantities of interest, namely, CSD, COI, and COP, are obtained in closed form analytic expressions in this section.The crack sliding displacement, Δ I (), is obtained using the following formula: Substituting  1 () from ( 22) and evaluating, one obtains Crack opening induction drop (COI), Δ I (), across the developed induction zone rims is calculated using the definition (32)
The loads fixed for this study are As may be noted from Figure 2, CSD shows a linear increase as slide-yield zone length is increased for fixed crack length.But as volume fraction is increased the crack opening decreases in magnitude although the increasing trend is maintained.
Crack opening induction drop (COI) versus induction zone to half-crack length ratio is plotted in Figure 3. COI  shows a nonlinear parabolic increase as induction zone length is increased.COI stabilizes for bigger induction zone.However as the volume fraction is increased, the COI further drops although the increasing trend continues to stabilize.
Figure 4 depicts the variation of crack opening potential drop (COP) with respect to saturation zone length for a fixed crack length.COP shows a negative variation as the saturation zone length is increased.It may be noted that for higher volume fraction, there is substantial drop in COP.As the saturation zone length is increased, COP shows a continuous negative increase.Figure 5 depicts the variation of energy release rate (ERR) with respect to slide-yield zone for different volume fractions.For higher volume fraction, energy release rate is positive and increases as slide-yield zone length is increased.And for   > 0.7, it is observed that there is little change in ERR variation; in fact, it slightly decreases for bigger values of slide-yield zone length keeping crack length fixed.It may be noted that, for smaller value of volume fraction, the ERR is negative but increases continuously as slide-yield zone size is increased.
Figure 6 shows the variation of ERR with respect to increase in saturation zone size for different volume fractions.The ERR shows a uniform constant behaviour even when saturation zone is increased.For volume fraction value equal to 0.1 the ERR is negative; it makes no contribution to crack propagation.And for volume fraction value higher than 0.3, the ERR becomes positive.For volume fraction values equal and higher than 0.5 the ERR remains constant; that is, there is no effect of volume fraction increase.
Variation of ERR against induction zone for different values of volume fraction is plotted in Figure 7.For volume fraction equal to 0.1 the ERR continuously decreases parabolically and becomes negative as the induction zone length is increased.And for the values 0.5-0.9 of volume fraction the ERR decreases even when induction zone size is increased, indicating that the crack gets arrested for higher volume fraction values.

Case II: Solution and Applications
Schematically, the configuration of the problem is depicted in Figure 8.The traction boundary conditions  II  ( = 1, 2, 3) for Case II may be mathematically expressed as and for this case it is assumed that  <  <  < .
Equations to determine the desired potentials   () ( = 1, 2, 3) may be written using (16) as The dislocation function, under the constrain The above equation enables to determine , hence giving induction zone length using | − |.

Figure 1 :
Figure 1: The schematic representation of the problem for Case I.

Figure 2 :Figure 3 :
Figure 2: Normalized crack sliding displacement versus slide-yield zone to half-crack length ratio for Case I.

1 Figure 5 :
Figure 5: Normalized energy release rate versus slide-yield zone to half-crack length ratio for Case I.

Figure 7 :
Figure 7: Normalized energy release rate versus saturation zone to half-crack length ratio for Case I.

Figure 8 :
Figure 8: The schematic representation of the problem for Case II.

Table 1 :
Material constants of BaTiO 3 -CoFe 2 O 4 for different volume fraction of BaTiO 3 in composite material.