Anisotropic Bimaterials Research Articles

Energy variation and stress fields of spherical inclusions with eigenstrain in three-dimensional anisotropic bi-materials

In this study, expressions for the interior stress fields of a spherical inclusion with uniform eigenstrain embedded in an anisotropic bi-material featuring a planar interface are derived. These expressions involve surface integrals of the imaginary term of the first derivative of the Green tensor in an anisotropic bi-material which are well-suited for standard numerical integrations. Specific formulations are provided for cases where the inclusion belongs to either the same material or both materials. Additionally, expressions are presented for the equivalent Eshelby tensor. The interior stress fields and variations in elastic strain energy are computed using cubic elastic constants of Cu. Various inclusion positions relative to the interface, eigenstrain forms, and crystallographic orientations are considered. For instances involving dilatational eigenstrain, the elastic strain energy variation with the inclusion’s position may exhibit multiple extrema. The global minimum and maximum consistently occur when the inclusion spans both materials. In the context of symmetrical tilt boundaries, energy variations are perfectly symmetric, with a global minimum on the interface that decreases with the tilt angle. The significance of the observed energy variations for defects segregation is quantitatively assessed by comparing them with the interaction energy between an eigenstrain and a grain boundary stress field.

Kinked and forked crack arrays in anisotropic elastic bimaterials

The fracture problem of multiple branched crack arrays in anisotropic bimaterials is formulated by use of the Stroh formalism to the linear elasticity theory of dislocations. The general full-field solutions are obtained from the standard technique of continuously distributed dislocations along finite-sized cracks of arbitrary shapes, which are embedded in dissimilar anisotropic half-spaces under far-field stress loading conditions. The bimaterial boundary-value problem leads to a set of coupled integral equations of Cauchy-type that is numerically solved by using the Gauss–Chebyshev quadrature scheme with appropriate boundary conditions for kinked and forked crack arrays. The path-independent Jk-integrals as crack propagation criterion are therefore evaluated for equally-spaced cracks, while the limiting configuration of individual cracks is theoretically described by means of explicit expressions of the local stress intensity factors K for validation and comparison purposes on several crack geometries. The short-range interactions resulting from the idealized configurations of infinitely periodic cracks are investigated as well as various size- and heterogeneity-effects on the mixed-mode cracks in complex stress-state environments. The influences of anisotropic elasticity, elastic mismatch, applied stress direction, inter-crack spacings and crack length ratios on the predictions from the Jk- and K-based fracture criteria are examined in the light of different configurations from the single kinked crack case in homogeneous media to the network of closely-spaced forked cracks in presence of bimaterial interfaces.

Three-dimensional thermal Green's functions of quasi-steady-state motion in anisotropic bimaterials and some related problems

Three-dimensional thermal Green's functions of quasi-steady-state motion, as well as that of steady-state conduction in anisotropic bimaterials are derived based on two-dimensional Fourier transform, and are separated as a sum of a full-space Green's function and a complementary part. It is worth mentioning that Green's functions of steady-state case can be obtained directly by replacing the moving source by the static one and be expressed in a more concise form. Although the present paper aims to develop Green's functions in anisotropic bimaterials, the derived solutions can be reduced to simple cases, such as in isotropic or orthotropic materials, in half-space or full-space, and on the interface or surface. Numerical examples are presented to verify the validity of present solutions. Moreover, the comparison between steady-state case and quasi-steady-state cases of two different velocities, indicates high correlation with the properties of material and the distortion of thermal fields induced by inertial effect.

Three-dimensional Green’s functions for transient heat conduction problems in anisotropic bimaterial

In this paper, three-dimensional Green’s functions for transient heat conduction problems in general anisotropic bimaterial are obtained based on two-dimensional Fourier transform and Laplace transform, and are separated as a sum of a full-space Green’s function and a complementary part. We can get the bimaterial Green's functions in the transformed domain by boundary conditions, and then in the physical domain by the inverse Fourier and Laplace transform. Although the present paper aims to develop Green's function in anisotropic bimaterial, the derived solutions can be reduced to simple cases, such as in isotropic or orthotropic materials, and in half-space or full-space. Moreover, this method can be extended to derivation of new Green’s functions in multi-layer materials. Numerical examples are presented to verify the validity and applicability of present solutions. When the source is constant and time extends to infinity, the present transient solution approaches the steady one. Besides, the anisotropic solution indicates high correlation with the properties of material.

Fast Fourier transform-based micromechanics of interfacial line defects in crystalline materials

Spectral methods using Fast Fourier Transform (FFT) algorithms have recently seen a surge in interest in the mechanics community. The present contribution addresses the critical question of determining local mechanical fields using the FFT method in the presence of interfacial defects. Precisely, the present work introduces a numerical approach based on intrinsic discrete Fourier transforms for the simultaneous treatment of material discontinuities arising from the presence of disclinations, i.e., rotational discontinuities, and inhomogeneities. A centered finite difference scheme for differential rules are first used for numerically solving the Poisson-type equations in the Fourier space to get the incompatible elastic fields due to disclinations and dislocations. Second, centered finite differences on a rotated grid are chosen for the computation of the modified Fourier-Green’s operator in the Lippmann–Schwinger–Dyson type equation for heterogeneous media. Elastic fields of disclination dipole distributions interacting with inhomogeneities of varying stiffnesses, grain boundaries seen as DSUM (Disclination Structural Unit Model), grain boundary disconnection defects and phase boundary “terraces” in anisotropic bi-materials are numerically computed as applications of the method.

Singularity of subsonic and transonic crack propagations along interfaces of magnetoelectroelastic bimaterials

The unified method for addressing the propagation of subsonic and transonic cracks along the interfaces of anisotropic bimaterials in Shen and Nishioka (2000) is applied in this paper for the plain strain problem of subsonic and transonic crack propagation along the interfaces of anisotropic magnetoelectroelastic (MEE) bimaterials. Using a modified Eshelby–Stroh formalism and analytic continuation, the problem here leads to a Riemann–Hilbert problem. After solving the Riemann–Hilbert equation, near-tip asymptotic fields are obtained, which shows the properties of the subsonic and transonic crack propagation in an MEE bimaterial system. Finally, the numerical results related to different singularity powers of the crack-tip are presented and discussed for barium titanate-cobalt ferrite (BaTiO3CoFe2O4) composites. The effects of piezoelectricity and piezomagnetics are also examined.

Three-dimensional interaction and movements of various dislocations in anisotropic bicrystals with semicoherent interfaces

Lattice dislocation interactions with semicoherent interfaces are investigated by means of anisotropic field solutions in metallic homo- and hetero-structures. The present framework is based on the mathematically elegant and computationally powerful Stroh formalism, combining further with the Fourier integral and series transforms, which cover different shapes and dimensions of various extrinsic and intrinsic dislocations. Two-dimensional equi-spaced arrays of straight lattice dislocations and finite arrangements of piled-up dislocations as well as any polygonal and elliptical dislocation loops in three dimensions are considered using a superposition scheme. Self, image and Peach–Koehler forces are derived to compute the equilibrium dislocation positions in pile-ups, including the internal structures and energetics of the interfacial dislocation networks. For illustration, the effects due to the elastic and misfit mismatches are discussed in the pure misfit Au/Cu and heterophase Cu/Nb systems, while discrepancies resulting from the approximation of isotropic elasticity are clearly exhibited. These numerical examples not only feature and enhance the existing works in anisotropic bimaterials, but also promote a novel opportunity of analyzing the equilibrium shapes of planar glide dislocation loops at nanoscale.

Elastic Green’s Function in Anisotropic Bimaterials Considering Interfacial Elasticity

The two-dimensional elastic Green’s function is calculated for a general anisotropic elastic bimaterial containing a line dislocation and a concentrated force while accounting for the interfacial structure by means of a generalized interfacial elasticity paradigm. The introduction of the interface elasticity model gives rise to boundary conditions that are effectively equivalent to those of a weakly bounded interface. The equations of elastic equilibrium are solved by complex variable techniques and the method of analytical continuation. The solution is decomposed into the sum of the Green’s function corresponding to the perfectly bonded interface and a perturbation term corresponding to the complex coupling nature between the interface structure and a line dislocation/concentrated force. Such construct can be implemented into the boundary integral equations and the boundary element method for analysis of nano-layered structures and epitaxial systems where the interface structure plays an important role.

Interface Dislocation Core-Spreading Simulation and Corresponding Response

Dislocation can spread its core at an interface especially at a weak shear interface associated with shearing the interface. Such core-spreading dislocation can significantly reduce stress/strain concentration compared with the compact dislocation and thus trap the dislocation in the interface, correspondingly strengthening materials. Employing the Green’s function for a compact dislocation, we derived analytical expressions for the elastic fields of a dislocation with core spreading in anisotropic bimaterials. We proposed a conic model to mimic the spreading core of a dislocation at an interface. The accuracy and efficiency of the conic model are validated by the boundary conditions of both traction and displacement across the interface. Numerical simulation is calculated in the Cu/Nb biomaterial. The results of displacement and stress fields show that: (1) core-spreading dislocation can greatly reduce the stress intensity near the dislocation compared with the dislocation with a condensed core; (2) dislocation core spreading has a great influence on the elastic fields near the core region, while the influence can be negligible when the distance of a field point from the center of the dislocation core is greater than 1.51 times the width of the spreading core; (3) near the core region, Peach-Koehler force induced by the core spreading dislocation is larger than that of the compact dislocation.

Modeling core-spreading of interface dislocation and its elastic response in anisotropic bimaterial

Interfacial dislocation may have a spreading core corresponding to a weak shear resistance of interfaces. In this paper, a conic model is proposed to mimic the spreading core of interfacial dislocation in anisotropic bimaterials. By the Stroh formalism and Green’s function, the analytical expressions of the elastic fields are deduced for such a dislocation. Taking Cu/Nb bimaterial as an example, it is demonstrated that the accuracy and efficiency of the method are well validated by the interface conditions, a spreading core can greatly reduce the stress intensity near the interfacial dislocation compared with the compact core, and the elastic fields near the spreading core region are significantly different from the condensed core, while they are less sensitive to a field point that is 1.5 times the core width away from the center of the spreading core.

Integral identities for fracture along imperfectly joined anisotropic ceramic bimaterials

We study a crack lying along an imperfect interface in an anisotropic bimaterial. A method is devised where known weight functions for the perfect interface problem are used to obtain singular integral equations relating the tractions and displacements for both the in-plane and out-of-plane fields. The problem can be considered as modelling bimaterial ceramics which are joined with a thin soft adhesive substance. The integral equations for the out-of-plane problem are solved numerically for orthotropic bimaterials with differing orientations of anisotropy and for different extents of interfacial imperfection. These results are then compared with finite element computations.

Elastic fields of a core-spreading dislocation in anisotropic bimaterials

A dislocation at an interface especially at a weak shear interface spreads its core associated with shearing the interface. Such core-spreading significantly reduces stress/strain concentration of the dislocation at the interface and thus traps the dislocation in the interface, correspondingly strengthening materials. Employing the Green's function for a single dislocation, we derived analytical expressions for the elastic fields associated with a core-spreading dislocation in anisotropic bimaterials. We proposed three fractional dislocation models to mimic the spreading core of a dislocation at an interface, i.e. uniform distribution (UD), linear distribution (LD) and cosine distribution (CD). The accuracy and efficiency of the three fractional models are validated by the continuity of both traction and displacement across the interface. Numerical results of the stress and displacement fields of the dislocation in the Cu/Nb bimaterial show that: (1) such core-spreading greatly reduces the stress intensity near the dislocation compared with the dislocation with a condensed core; (2) the distribution of the Burgers vector associated with the core spreading determines the magnitude and patterns of the elastic fields; (3) The influence of the core-spreading on the elastic fields can be negligible when the distance of a field point from the center of the dislocation core is greater than 2.17 times the width of the spreading core; (4) The LD model is simple while it is able to capture the interaction force acting on an incoming dislocation. The findings offer insights into understanding interface roles in strengthening materials and designing interfaces-dominated composites.

Interface cracks with surface elasticity in anisotropic bimaterials

We consider the effect of surface elasticity on an interface crack between two dissimilar anisotropic elastic half-planes under generalized plane strain deformation. The surface mechanics is incorporated by using a modified anisotropic version of the continuum-based surface/interface model of Gurtin and Murdoch (1975). A system of first-order Cauchy singular integro-differential equations is derived by considering a distribution of line dislocations and line forces on the interface crack. The correctness of the obtained singular integro-differential equations is carefully verified by comparison with the existing results of Kim et al. (2010, 2011a,b,c), derived via the complex variable method. Our results show that generally the stresses exhibit only the weak logarithmic singularity at the tips of an interfacial crack when surface elasticity in anisotropic bimaterials is considered. Further, the crack-tip stresses in a homogeneous anisotropic solid exhibit both the stronger inverse square root and the weaker logarithmic singularities if the surface tension is ignored. The solution method for solving a system of singular integro-differential equations is also proposed.

Boundary integral equations and Green׳s functions for 2D thermoelectroelastic bimaterial

This paper presents a comprehensive study on the 2D boundary integral equations, Green׳s functions and boundary element method for thermoelectroelastic bimaterials containing cracks and thin inclusions. Based on the extended Stroh formalism, complex variable approach and the Cauchy integral formula, the paper derives integral formulae for the Stroh complex functions, and Somigliana type integral identities for 2D thermoelectroelastic bimaterial. The kernels arising in the integral formulae are obtained explicitly and in a closed-form. It is proved that these kernels are fundamental solutions for a line extended force and a line heat. The far-field mechanical, electric and thermal load and internal volume load are accounted for in the obtained integral formulae. The latter allow to derive boundary integral equations for a bimaterial containing holes, cracks and thin inclusions, and to develop the corresponding boundary element approach. Special tip boundary elements used in the analysis allow accurate determination of the stress and electric displacement intensity factors for cracks and thin deformable inclusions. Several numerical examples are considered that show the validity and efficiency of the developed boundary element approach in the analysis of defective thermoelectroelastic anisotropic bimaterials.

Elastic fields due to dislocation arrays in anisotropic bimaterials

Based on the single-dislocation Green’s function, analytical solutions of the elastic fields due to dislocation arrays in an anisotropic bimaterial system are derived by virtue of the Cottrell summation formula. The singularity in the Peach–Koehler (P–K) force is removed by both rigorous mathematical approach and physical energy consideration. Numerical results for dislocation arrays in the Cu/Nb bimaterial with Kurdjumov–Sachs (K–S) orientation show that: (1) the traction continuity and periodic condition are both satisfied; (2) the maximum magnitude of the traction at the interface due to a mixed dislocation array is smaller than that due to a single mixed dislocation. In other words, the traction at the interface could be suppressed by the corresponding array with a relatively high density (L<10nm); however, the shear stress on the glide plane increases with increasing dislocation density; (3) the Cu/Nb interface attracts the mixed dislocation array in copper and repels the screw one there. This implies that the P–K force depends not only on the material properties, but also on the crystal orientation and the type of Burgers vector, among others.

Electric field intensity factors for conducting paths in anisotropic dielectric bimaterials

Conducting paths in an anisotropic dielectric bimaterial as well as in a homogeneous anisotropic material subjected to electric loading are investigated. The conducting path problems are formulated by using a linear transformation method. Electric field intensity factors are obtained for conducting paths emanating from a surface electrode in an orthotropic material. The asymptotic problem of a kinked conducting path in dissimilar anisotropic dielectric materials is considered. The electric field intensity factor for the asymptotic problem is obtained in the infinite product form. To ascertain validity of the solution obtained from the linear transform method, numerical computations are carried out by using finite element method. The electric field intensity factor for a conducting path emanating from the vertex of a bimaterial wedge with a tilt boundary is also obtained in the closed form.

Two-Dimensional Stress Intensity Factor Analysis of Cracks in Anisotropic Bimaterial

This paper presents a 2D numerical technique based on the boundary element method (BEM) for the analysis of linear elastic fracture mechanics (LEFM) problems on stress intensity factors (SIFs) involving anisotropic bimaterials. The most outstanding feature of this analysis is that it is a singledomain method, yet it is very accurate, efficient, and versatile (i.e., the material properties of the medium can be anisotropic as well as isotropic). A computer program using the BEM formula translation (FORTRAN 90) code was developed to effectively calculate the stress intensity factors (SIFs) in an anisotropic bi-material. This BEM program has been verified and showed good accuracy compared with the previous studies. Numerical examples of stress intensity factor calculation for a straight crack with various locations in both finite and infinite bimaterials are presented. It was found that very accurate results can be obtained using the proposed method, even with relatively simple discretization. The results of the numerical analysis also show that material anisotropy can greatly affect the stress intensity factor.

Dislocation models of interfacial shearing induced by an approaching lattice glide dislocation

When a lattice glide dislocation approaches a bi-metal interface with relatively low shear strength, it causes the interface to shear. Interfacial shearing is accommodated by the nucleation and growth of interfacial dislocations, which have an attractive interaction with the incoming dislocation. Thus a critical length scale exists at which the net force on the incoming lattice glide dislocation can transition from being initially repulsive to attractive. In this paper, we develop dislocation-based interface shear models in order to represent this mechanism of interface/dislocation interaction at the continuum scale. Three versions are devised with different degrees of complexity and hence computational cost: the continuous shear model (CSM), simplified-CSM model (SCSM), and single dislocation shear model (SDSM). We simulate the interaction processes with these three models by means of a Green’s function method for an anisotropic bimaterial. All three models find that the critical length scale at which the dislocation becomes attracted to the interface increases as the interfacial shear resistance decreases. While the most complex model of the three, the CSM, performs the best, the SCSM and SDSM are more advantageous for implementation into higher-length scale dislocation dynamics models.

A Crack Emanating from a Wedge In Dissimilar Anisotropic Materials Under Antiplane Shear

A crack in a composite wedge consisting of two dissimilar anisotropic materials under concentrated antiplane loads is analyzed. The problem of a crack in an isotropic composite wedge is solved first by using the Mellin transform and the Wiener-Hopf method. Using a linear transformation of the original composite wedge into an isotropic composite wedge consisting of dissimilar isotropic materials, the antiplane displacement and stresses for the anisotropic composite wedge are obtained from the solution for the transformed wedge. The stress intensity factor for the crack in the original anisotropic composite wedge is obtained from the solution of the crack in the transformed wedge. Special attention is given to the asymptotic problem of a wedge crack in an anisotropic bimaterial. Numerical computations are carried out to obtain the energy release rate for various apex angles and anisotropic parameters as a function of crack angle.

Stress intensity factor analysis of a three-dimensional interface crack between dissimilar anisotropic materials under thermal stress

A numerical method for evaluating the stress intensity factors (SIFs) of a three-dimensional interface crack between dissimilar anisotropic materials subjected to thermal and mechanical stresses is proposed. The M1-integral method was applied to an interfacial crack between three-dimensional anisotropic bimaterials under thermal stress. The moving least square approximation was utilized to calculate the value of the M1-integral. The M1-integral in conjunction with the moving least square approximation can be used to calculate the SIFs from nodal displacements obtained by finite element analysis. SIF analyses were performed for double edge cracks in jointed dissimilar isotropic semi-infinite plates subjected to thermal load. Excellent agreement was achieved between the numerical results obtained by the present method and the exact solution. In addition, we computed the SIFs of an external circular interfacial crack in jointed dissimilar anisotropic solids under thermal stress and showed the distributions of SIFs along the crack front. The distribution of stress and the crack opening displacement obtained by the asymptotic solution using the computed SIFs were compared with those obtained by the finite element analysis with fine mesh. They were almost identical to each other, except for the minor component of SIFs that is much smaller than the major component of SIFs. These results indirectly demonstrate the accuracy of the obtained SIFs.