Bimaterial Case Research Articles

Concentrated sources on the interface of diffusive bimaterial media: The steady-state deformation

While the fundamental solutions of concentrated forces and dislocations in purely elastic media are available for various applications, solutions in the corresponding diffusive materials have seldom been derived due to the involved complexity. In this paper, in terms of the cylindrical system of vector functions and under the assumption of steady-state deformation, we derive analytically the elastic and diffusive fields induced by the concentrated forces and dislocations which are located on the interface of a transversely isotropic and diffusive bimaterial space. Solution of the concentrated diffusive source has been also derived. These solutions can further be reduced to the corresponding isotropic bimaterial case and the transversely isotropic full-space case. Based on the newly derived solutions, field quantities induced by different concentrated dislocation sources are presented numerically to demonstrate the effect of the diffusive coefficients, material heterogeneity, and dislocation types on the induced fields.

Elastic theory of dislocation loops in three-dimensional isotropic bi-materials

We develop a novel elastic Green's function of point force in three-dimensional isotropic bi-materials that resembles the classical Kelvin solution in the corresponding infinite media. Based upon such a simple bi-material Green's function, we then investigate dislocation loops of arbitrary shape embedded in isotropic bi-materials. The main contribution of this work is an elegant extension of the classical Burgers’ formula for displacements, Peach–Koehler's formula for stresses and Blin's formula for the interaction energy from the full-space case to the bi-material case. Most strikingly, our new formulae for bi-materials preserve the same simplicity as the classical expressions in the full space. The correctness of these extended formulae is verified by comparing them with available ones in the literature, and the importance of these formulae is on their direct application in the three-dimensional dislocation dynamics simulation involving interface or surface.

A novel method for the partition of mixed-mode fractures in 2D elastic laminated unidirectional composite beams

A powerful method for partitioning mixed-mode fractures on rigid interfaces in laminated unidirectional double cantilever beams (DCBs) is developed by taking 2D elasticity into consideration in a novel way. Pure modes based on 2D elasticity are obtained by introducing correction factors into the beam-theory-based mechanical conditions. These 2D-elasticity-based pure modes are then used to derive a 2D-elasticity-based partition theory for mixed-mode fractures. Excellent agreement is observed between the present partition theory and Suo and Hutchinson’s partition theory (Suo and Hutchinson, 1990) [1]. Furthermore, the method that is developed in this work has a stronger capability for solving more complex mixed-mode partition problems, for example, in the bimaterial case (Suo and Hutchinson, 1990) [1].

The Effects of Fault Stepovers on Ground Motion

Abstract Using 3D dynamic models, we investigate the effect of fault stepovers on near‐source ground motion. We use the finite‐element method to model the rupture, slip, and ground motion of two parallel strike‐slip faults with an unlinked overlapping stepover of variable width. We model this system as both an extensional and a compressional stepover and compare the results to those of single planar faults. We find that, overall, the presence of a stepover along the fault trace reduces the maximum ground motion when compared to the long planar fault. Whether the compressional or extensional stepover exhibits higher ground motion overall depends on the width of the separation between the faults. There is a region of reduced ground motion at the end of the first fault segment, when the faults are embedded in a homogeneous material. We also experiment with stress fields leading to supershear and subshear rupture velocities, and with different stress drops within those conditions. We find that subshear rupture produces stronger motions than supershear rupture, but supershear ruptures produce that maximum over a larger area than subshear areas, even though the overall area that experiences any shaking at all is not drastically different between the two cases. Lastly, we experiment with placing realistic materials along and around the faults, such as a sedimentary basin in an extensional stepover, a damage zone around the fault, and a soft rock layer on top of bedrock through the entire model area. These configurations alter the pattern of ground motion from the homogeneous case; the peaks in ground motion for the bimaterial cases depend on the materials in question. The results may have implications for ground‐motion prediction in future earthquakes on geometrically complex faults. Online Material: MPEG‐4 movies of models of dynamic rupture of fault stepovers embedded in heterogeneous material settings.

Energy-Release Rate and Mode Mixity of Face/Core Debonds in Sandwich Beams

Closed-form algebraic expressions for the energy-release rate and the mode mixity are obtained for a debonded sandwich (trimaterial). The most general case of an “asymmetric” sandwich is considered (i.e., the bottom face sheet not necessarily of the same material or thickness as the top face sheet). The energy-release rate is obtained by use of the J-integral, and the expression is derived in terms of the forces and moments at the debond section. Regarding the mode mixity, a closed-form expression is derived in terms of the geometry, material, and applied loading, and it is proven that, in the trimaterial case, just as in the bimaterial case, the mode mixity can be obtained in terms of a single scalar quantity , which is independent of loading; the value for a particular geometry and material can be extracted from a numerical solution for one loading combination. Thus, this analysis extends the existing formulas in the literature, which are for either a delamination in a homogeneous composite or an interface crack in a bimaterial. These new “trimaterial with a crack” formulas are also proven to yield the formulas for the limits of a bimaterial or for a homogeneous section with a crack.

The strengthening effect caused by an elastic contrast—part I: the bimaterial case

The final aim of the two parts of this study is to quantify the gain in strength of a layered heterogeneous structure caused by the elastic contrast between the layers, especially if no crack deflection is observed at the interface. In part I, the analysis focus on an example, the 3 point-bending of a bimaterial made of a compliant and a stiff layer; whereas part II is dedicated to the 3-point bending of a homogeneous beam embedding a thin stiff film. Baptized “step-over” and “jump-through”, two original mechanisms of crack crossing the interface are proposed herein. They rely on a coupled criterion involving both energy requirement and stress condition. It is established from asymptotic expansions and the theory of singularities. This criterion is able to respond to the paradox posed by the traditional tools of brittle fracture mechanics which might carelessly conclude to a quasi infinite strength enhancement. The subtitle of this first part might be: can a crack pass through an interface?

Analysis of plastic deformation in nanoscale metallic multilayers with coherent and incoherent interfaces

Nanoscale metallic multilayered (NMM) composites possess ultra high strength (order of GPa) and high ductility, and exhibit high fatigue resistance. Their mechanical behavior is governed mainly by interface properties (coherent and/or incoherent interfaces), dislocation mechanisms in small volume, and dislocation–interface interaction. In this work, we investigate these effects within a dislocation dynamics (DD) framework and analyze the mechanical behavior of two systems: (1) a bi-material system (CuNi) with coherent interface and (2) a newly developed tri-material system (CuNiNb) composed of both coherent and incoherent interfaces. For the bi-material case we analyze the influence of networks of interfacial dislocations whose nature and distribution are commensurate with the level of relaxation and loading of the structure. Misfit and pre-deposited interfacial dislocation arrays, as well as combinations of both, are studied and the dependence of strength on layer thickness is reported, along with observed dislocation mechanisms. It is shown that interfacial defect configurations significantly alter the strength and mechanical behavior of the material. Furthermore, it is shown that the implementation of penetrable interfaces in DD captures the strength dependence at layer thicknesses on the order of 3–7 nm. For the tri-material case we analyze the effects of coherent and incoherent interfaces in large-scale simulations. The results show that these materials have strong strength-size dependence but are limited by the strength of the incoherent (CuNb) interface which is weak in shear. The weak interface acts as a dislocation sink. This in turn induces an internal shear stress field that activates cross-slip in the adjacent CuNi interlace and thus causing softening. Moreover, it is shown that the yield stress of the CuNiNb system is controlled by the volume fraction of the Nb. Because Nb is the most compliant of the three materials, an increase in volume fraction of Nb decreases the overall yield strength of the material.

Asymptotic fields near an interface corner in orthotropic bi-materials

Based on the existing asymptotic solutions of the displacement and singular stress fields in the vicinity of a singular point in 2D orthotropic elastic materials, the two simple eigenequations are explicitly given for the symmetric and anti-symmetric deformation modes to determine the orders of the stress singularity at the interface corner in orthotropic bi-materials, respectively. The related displacement and singular stress fields near the interface corner are also explicitly established. The relevant stress intensity factors are defined as in the case of crack problems. The theoretical results have been confirmed by numerical, finite-element-based results in a special bi-material case. The solution obtained in this paper may be applied to the interface corner in the orthotropic/orthotropic, orthotropic/isotropic, and isotropic/isotropic bi-materials, and it will be very useful to evaluate the strength of the bonded orthotropic bi-materials with interface corners.

Analysis of slip‐weakening frictional laws with static restrengthening and their implications on the scaling, asymmetry, and mode of dynamic rupture on homogeneous and bimaterial interfaces

Dynamic simulations of homogeneous, heterogeneous and bimaterial fault rupture using modified slip‐weakening frictional laws with static restrengthening are presented giving rise to both crack‐like and pulse‐like rupture. We demonstrate that pulse‐like rupture is possible by making a modification of classical slip‐weakening friction to include static restrengthening. We employ various slip‐weakening frictional laws to examine their effect on the resulting earthquake rupture speed, size and mode. More complex rupture characteristics were produced with more strongly slip‐weakening frictional laws, and the degree of slip‐weakening had to be finely tuned to reproduce realistic earthquake rupture characteristics. Rupture propagation on a fault is controlled by the constitutive properties of the fault. We provide benchmark tests of our method against other reported solutions in the literature. We demonstrate the applicability of our elastoplastic fault model for modeling dynamic rupture and wave propagation in fault systems, and the rich array of dynamic properties produced by our elastoplastic finite element fault model. These are governed by a number of model parameters including: the spatial heterogeneity and material contrast across the fault, the fault strength, and not least of all the frictional law employed. Asymmetric bilateral fault rupture was produced for the bimaterial case, where the degree of material contrast influenced the rupture speed in the different propagation directions.

Electromagnetic fields induced by a point source in a uniaxial multiferroic full-space, half-space, and bimaterial space

Multiferroic magnetoelectrics are materials that are both ferroelectric and ferromagnetic in the same phase. In addition, electric and magnetic polarizations are strongly coupled in some magnetoelectric multiferroic materials. In this work, by virtue of the image method, exact closed-form Green’s functions are derived for a uniaxial multiferroic full-space, half-space, and bimaterial space. While for the bimaterial space case the interface is assumed to be perfect, for the half-space case four different sets of surface conditions are considered. The point source can be either an electric or a magnetic charge. Numerical results are presented to demonstrate the differences among the infinite-space, half-space, and bimaterial space Green’s functions.

Three-dimensional Green's functions of steady-state motion in anisotropic half-spaces and bimaterials

Three-dimensional Green's functions (GFs) of steady-state motion in linear anisotropic elastic half-space and bimaterials are derived within the framework of generalized Stroh formalism and two-dimensional Fourier transforms. The present study is limited to the subsonic case where the sextic equation has six complex eigenvalues. If the source and field points reside in the same material, the GF is expressed in two parts: a singular part that corresponds to the infinite-space GF, and a complementary part that corresponds to the reflective effects of the interface in the bimaterial case and of the free surface in the half-space case. The singular part in the physical domain is calculated analytically by applying the Radon transform and the residue theorem. If the source and field points reside in different materials (in the bimaterial case), the GF is a one-term solution. The physical counterparts of the complementary part in the half-space case and of the one-term solution in the bimaterial case are derived as a one-dimensional integral by analytically carrying out the integration along the radial direction in the Fourier-inverse transform. When the source and field points are both on the interface in the bimaterial case or on the surface in the half-space case, singularities appear in the Fourier-inverse transform of the GF. The singularities are treated explicitly using a method proposed recently by the authors. Numerical examples are presented to demonstrate the effects of wave velocity on the stress fields, which may be of interest in various engineering problems of steady-state motion. Furthermore, these GFs are required in the steady-state boundary-integral-equation formulation of anisotropic elasticity.

Weight functions for interface cracks in dissimilar anisotropic materials

Bueckner's work conjugate integral customarily adopted for linear elastic materials is established for an interface crack in dissimilar anisotropic materials. The difficulties in separating Stroh's six complex arguments involved in the integral for the dissimilar materials are overcome and then the explicit function representations of the integral are given and studied in detail. It is found that the pseudo-orthogonal properties of the eigenfunction expansion form (EEF) for a crack presented previously in isotropic elastic cases, in isotopic bimaterial cases, and in orthotropic cases are also valid in the present dissimilar arbitrary anisotropic cases. The relation between Bueckner's work conjugate integral and theJ-integral in these cases is obtained by introducing a complementary stress-displacement state. Finally, some useful path-independent integrals and weight functions are proposed for calculating the crack tip parameters such as the stress intensity factors.

The method of fundamental solutions for layered elastic materials

In this paper, we investigate the application of the method of fundamental solutions to two-dimensional elasticity problems in isotropic and anisotropic single materials and bimaterials. A domain decomposition technique is employed in the bimaterial case where the interface continuity conditions are approximated in the same manner as the boundary conditions. The method is tested on several test problems and its relative merits and disadvantages are discussed.

Three-dimensional Green’s functions in anisotropic bimaterials

In this paper, three-dimensional Green’s functions for anisotropic bimaterials are studied based on Stroh formalism and two-dimensional Fourier transforms. Although the Green’s functions can be expressed exactly in the Fourier transform domain, it is difficult to obtain the explicit expressions of the Green’s functions in the physical domain due to the general anisotropy of the material and a geometry plane involved. Utilizing Fourier inverse transform in the polar coordinate and combining with Mindlin’s superposition method, the physical-domain bimaterial Green’s functions are derived as a sum of a full-space Green’s function and a complementary part. While the full-space Green’s function is in an explicit form, the complementary part is expressed in terms of simple regular line-integrals over [0, 2 π] that are suitable for standard numerical integration. Furthermore, the present bimaterial Green’s functions can be reduced to the special cases such as half-space, surface, interfacial, and full-space Green’s functions. Numerical examples are given for both half-space and bimaterial cases with isotropic, transversely isotropic, and anisotropic material properties to verify the applicability of the technique. For the half-space case with isotropic or transversely isotropic material properties, the Green’s function solutions are in excellent agreement with the existing analytical solutions. For anisotropic half-space and bimaterial cases, numerical results show the strong dependence of the Green’s functions on the material properties.

Three-dimensional Green's functions in anisotropic trimaterials

In this paper, three-dimensional Green’s functions for anisotropic bimaterials are studied based on Stroh formalism and two-dimensional Fourier transforms. Although the Green’s functions can be expressed exactly in the Fourier transform domain, it is diAcult to obtain the explicit expressions of the Green’s functions in the physical domain due to the general anisotropy of the material and a geometry plane involved. Utilizing Fourier inverse transform in the polar coordinate and combining with Mindlin’s superposition method, the physical-domain bimaterial Green’s functions are derived as a sum of a full-space Green’s function and a complementary part. While the full-space Green’s function is in an explicit form, the complementary part is expressed in terms of simple regular line-integrals over [0, 2pa that are suitable for standard numerical integration. Furthermore, the present bimaterial Green’s functions can be reduced to the special cases such as half-space, surface, interfacial, and full-space Green’s functions. Numerical examples are given for both half-space and bimaterial cases with isotropic, transversely isotropic, and anisotropic material properties to verify the applicability of the technique. For the half-space case with isotropic or transversely isotropic material properties, the Green’s function solutions are in excellent agreement with the existing analytical solutions. For anisotropic half-space and bimaterial cases, numerical results show the strong dependence of the Green’s functions on the material properties. 7 2000 Elsevier Science Ltd. All rights reserved.

Three-dimensional Green’s functions in anisotropic piezoelectric bimaterials

In this paper, a recently proposed method by E. Pan and F.G. Yuan (Int. J. Solids Struct., 2000) for the calculation of the elastic bimaterial Green’s functions is extended to the analysis of three-dimensional Green’s functions for anisotropic piezoelectric bimaterials. The method is based on the Stroh formalism and two-dimensional Fourier transforms in combination with Mindlin’s superposition method. We first derive Green’s functions in exact form in the Fourier transform domain. When inverting the Fourier transform, a polar coordinate transform is introduced so that the radial integral from 0 to +∞ can be carried out exactly. Therefore, the bimaterial Green’s functions in the physical domain are derived as a sum of a full-space Green’s function and a complementary part. While the full-space Green’s function is in an explicit form, as derived recently by E. Pan and F. Tonon (Int. J. Solids Struct., 37 (2000): 943–958), the complementary part is expressed in terms of simple regular line integrals over [0, 2π] that are suitable for standard numerical integration. Furthermore, the present bimaterial Green’s functions can be reduced to the special cases such as half-space, surface, interfacial, and full-space Green’s functions. Uncoupled solutions for the purely elastic and purely electric case can also be simply obtained by setting the piezoelectric coefficients equal to zero. Numerical examples for Green’s functions are given for both half-space and bimaterial cases with transversely isotropic and anisotropic material properties to verify the applicability of the technique. Certain interesting features associated with these Green’s functions are observed and discussed, as related to the selected material properties.

A nonlinear finite element eigenanalysis of antiplane shear including higher order terms

A finite element formulation for the determination of higher order stress and displacement fields for antiplane shear loading of power-law hardening materials is presented. In this displacement based finite element formulation, the Newton method is coupled with singular value decomposition to solve the nonlinear system of equations. The first two asymptotic terms in the displacement field are included in the formulation, which gives the first three terms in the stress field. Examples are presented for the single material crack problem subjected to antisymmetric and arbitrary loading, the single material wedge, and the bimaterial case of a crack perpendicular to an interface.

Dynamic response of bimaterial and graded interface cracks under impact loading

The interfacial fracture in bimaterial and functionally graded material (FGM) under impact loading conditions is investigated using experimental and numerical techniques that are valid for both type of interfaces. Experiments are conducted on epoxy based specimens in three point bend configuration and the complex SIF is measured using an electrical strain gage mounted close to the crack-tip. A complementary two-dimensional finite element simulation is performed using tup force and support reactions as input tractions, and the SIF-time history is determined using a displacement extrapolation technique. The experimentally determined SIF-histories match closely with numerical simulation up to the time of fracture initiation. The test results show that the mode-mixity remains nearly constant through out the test in both the materials, and the mixity values correspond to their respective static counterparts. The general dynamic response of the bimaterial and FGM specimens in terms of impact load, support reaction and the magnitude of complex SIF are comparable, and the mode-mixity is the parameter that distinguishes the graded interface from the bimaterial case.

Order of singularity and singular stress field about an axisymmetric interface corner in three-dimensional isotropic elasticity

The spatial axisymmetric problem of an isotropic, elastic, homogeneous body containing an inclusion of a different material with an interface corner point (along a circular contour) and arbitrary joining angles is considered in this paper. It is found that the order of the stress singularity at the interface corner coincides with that of the corresponding plane strain problem, but the dependence of the singular stress field on the material properties depends on both the Poisson ratios (of the two isotropic materials) as well as on the ratio of their shear moduli. The theoretical results have been confirmed by numerical, finite-element-based results in a special bimaterial case.

On the contribution of subinterface microcracks near the tip of an interface macrocrack to the J-integral in bimaterial solids

The J-integral analysis is performed for the plane problems of multiple subinterface microcracks near the tip of an interface macrocrack in bimaterial solids. The analysis starts from a general solution based on the “pseudo-traction” method which has been addressed thoroughly in homogeneous cases. The contribution to the J-integral induced from the subinterface microcracks is shown in a consistent relation with those induced from an interface macrocrack tip and the remote stress field. A new technique is developed to evaluate the second component (being expressed by J2∗ in this paper) of the well-known Jk-vector of a subinterface crack for considering a contour enclosing the whole crack, which is necessary to evaluate the contribution to the J-integral arising from the subinterface microcracks. The consistency of the J-integral for numerical examples is proved, where two kinds of material combinations presented by Hutchinson et al. [ASME J. Appl. Mech., 1987, 54, 828–832] are considered. Some discussion and conclusions are then given which seem very useful in the investigation of microcrack shielding problems in bimaterial cases.