Showing papers in "Journal of The Mechanics and Physics of Solids in 1998"

Indentation size effects in crystalline materials: A law for strain gradient plasticity

Abstract: We show that the indentation size effect for crystalline materials can be accurately modeled using the concept of geometrically necessary dislocations. The model leads to the following characteristic form for the depth dependence of the hardness: H H 0 1+ h ∗ h where H is the hardness for a given depth of indentation, h, H0 is the hardness in the limit of infinite depth and h ∗ is a characteristic length that depends on the shape of the indenter, the shear modulus and H0. Indentation experiments on annealed (111) copper single crystals and cold worked polycrystalline copper show that this relation is well-obeyed. We also show that this relation describes the indentation size effect observed for single crystals of silver. We use this model to derive the following law for strain gradient plasticity: ( σ σ 0 ) 2 = 1 + l χ , where σ is the effective flow stress in the presence of a gradient, σ0 is the flow stress in the absence of a gradient, χ is the effective strain gradient and l a characteristic material length scale, which is, in turn, related to the flow stress of the material in the absence of a strain gradient, l ≈ b( μ σ 0 ) 2 . For materials characterized by the power law σ 0 = σ ref e 1 n , the above law can be recast in a form with a strain-independent material length scale l. ( builtσ σ ref ) 2 = e 2 n + l χ l = b( μ σ ref ) 2 = l ( σ 0 σ ref ) 2 . This law resembles the phenomenological law developed by Fleck and Hutchinson, with their phenomenological length scale interpreted in terms of measurable material parametersbl].

Revisiting brittle fracture as an energy minimization problem

Abstract: A variational model of quasistatic crack evolution is proposed. Although close in spirit to Griffith’s theory of brittle fracture, the proposed model however frees itself of the usual constraints of that theory : a preexisting crack and a well-defined crack path. In contrast, crack initiation as well as crack path can be quantified, as demonstrated on explicitly computable examples. Furthermore the model lends itself to numerical implementation in more complex settings.

Constitutive modeling of the large strain time-dependent behavior of elastomers

Abstract: The mechanical behavior of elastomeric materials is known to be rate-dependent and to exhibit hysteresis upon cyclic loading. Although these features of the rubbery constitutive response are well-recognized and important to its function, few models attempt to quantify these aspects of response perhaps due to the complex nature of the behavior and its apparent inconsistency with regard to current reasonably successful models of rubber elasticity. In this paper a detailed experimental investigation probing the material response of carbon black filled Chloroprene rubber subjected to different time-dependent strain histories is presented. Some of the key observations from the experiments are: (1) both filled and unfilled elastomers show significant amounts of hysteresis during cyclic loading; (2) the amount of carbon black particles does not strongly influence the normalized amount of hysteresis; (3) both filled and unfilled elastomers are strain-rate dependent and the rate dependence is higher during the uploading than during the unloading; (4) at fixed strain, the stress is observed to approach the same equilibrium level with relaxation time whether loading or unloading. Based on the experimental data a new constitutive model has been developed. The foundation of the model is that the mechanical behavior can be decomposed into two parts: an equilibrium network corresponding to the state that is approached in long time stress relaxation tests; and a second network capturing the non-linear rate-dependent deviation from the equilibrium state. The time-dependence of the second network is further assumed to be governed by the reptational motion of molecules having the ability to significantly change conformation and thereby relaxing the overall stress state. By comparing the predictions from the proposed three-dimensional constitutive model with experimental data for uniaxial compression and plane strain compression we conclude that the constitutive model predicts rate-dependence and relaxation behavior well.

Incorporation of deformation twinning in crystal plasticity models

Abstract: A new constitutive framework, together with an efficient time-integration scheme, is presented for incorporating the crystallography of deformation twinning in polycrystal plasticity models Previous approaches to this problem have required generation of new crystal orientations to reflect the orientations in the twinned regions or implementation of “volume fraction transfer” schemes, both of which require an update of the crystal orientations at the end of each time step in the simulation of the deformation process In the present formulation, all calculations are performed in a relaxed configuration in which the lattice orientation of the twinned and the untwinned regions are pre-defined based on the initial lattice orientation of the crystal The validity of the proposed constitutive framework and the time-integration procedures has been demonstrated through comparisons of predicted rolling textures in low stacking fault energy fcc metals and in hcp metals with the corresponding predictions from the earlier approaches as well as through qualitative comparisons with the measurements reported previously

The mechanics of size-dependent indentation

Abstract: Indentation tests at scales on the order of one micron have shown that measured hardness increases significantly with decreasing indent size, a trend at odds with the size-independence implied by conventional plasticity theory. In this paper, strain gradient plasticity theory is used to model materials undergoing small-scale indentations. Finite element implementation of the theory as it pertains to indentation modeling is briefly reviewed. Results are presented for frictionless conical indentations. A strong effect of including strain gradients in the constitutive description is found with hardness increasing by a factor of two or more over the relevant range of behavior. The results are used to investigate the role of the two primary constitutive length parameters in the strain gradient theory. The study indicates that indentation may be the most effective test for measuring one of the length parameters.

Numerical simulation of crack growth in an isotropic solid with randomized internal cohesive bonds

Abstract: A virtual internal bond (VIB) model with randomized cohesive interactions between material particles is proposed as an integration of continuum models with cohesive surfaces and atomistic models with interatomic bonding. This approach differs from an atomistic model in that a phenomenological “cohesive force law” is assumed to act between “material particles” which are not necessarily atoms; it also differs from a cohesive surface model in that, rather than imposing a cohesive law along a prescribed set of discrete surfaces, a randomized network of cohesive bonds is statistically incorporated into the constitutive law of the material via the Cauchy-Born rule, i.e., by equating the strain energy function on the continuum level to the potential energy stored in the cohesive bonds due to an imposed deformation. This work is motivated by the notion that materials exhibit multiscale cohesive behaviors ranging from interatomic bonding to macroscopic ductile failure. It is shown that the linear elastic behavior of the VIB model is isotropic and obeys the Cauchy relation; the instantaneous elastic properties under equibiaxial stretching are transversely isotropic, with all the in-plane components of the material tangent moduli vanishing at the cohesive stress limit; the instantaneous properties under equitriaxial stretching are isotropic with a finite strain modulus. We demonstrate through two preliminary numerical examples that the VIB model can be applied in direct simulation of crack growth without a presumed fracture criterion. The prospect of this type of approach in numerical simulations of fracture seems to be highly promising.

Adiabatic shear bands in a TI-6Al-4V titanium alloy

Abstract: Dynamic (γ≈103⧹sec) torsional experiments were performed to investigate the process of initiation and formation of adiabatic shear bands in Ti-6Al-4V alloy. In this study, thin-wall tubular specimens were deformed dynamically in a torsional Kolsky bar (torsional split Hopkinson bar) . Through high-speed photography of a grid pattern previously printed on the specimens outer surface, the local strain and the local strain rate were found to be in the range of 75%–350% and 8.0×104⧹sec, respectively. The width of the shear bands ranged from 12–55 μm. In addition, an array of infrared detectors was employed to measure the local temperature rise during the deformation process. A peak temperature of 440–550°C was found in the various tests. The fracture surface of the shear band material was characterized by (1) regions of elongated dimples within which no second phase particles were observed, and (2) regions with a relatively flat and smeared appearance. There was no clear evidence based either on the appearance of the shear band in SEM or the measured temperature rise to suggest that the material within the shear band had undergone a phase transformation.

Modelling of orthogonal cutting with a temperature dependent friction law

Abstract: In this paper the process of orthogonal cutting is studied by analytical means. A thermomechanical model of the primary shear zone is combined with a modelling of the contact problem at the tool-chip interface. A friction law is introduced that accounts for temperature effects. The effects of cutting conditions and material behaviour on the temperature distribution along the contact zone, on the mean friction and on the global cutting forces are evaluated. The experimental trends are shown to be well described by the proposed model. © 1998 Elsevier Science Ltd. All rights reserved.Keywords : A. cutting and forming, thermomechanical processes, B. viscoplastic material, friction.

Influence of wavy imperfections in cell walls on elastic stiffness of cellular solids

Abstract: The stiffness of open and closed cell low density cellular solids, or solid foams, is affected by “imperfections” such as non-uniform cell size (multi-dispersity), non-uniform cell wall thickness, wavy distortions of cell walls, etc. Metal foams generally have lower relative stiffnesses than, for example, expanded PVC based polymer foams, and a comparison of the morphologies suggests that the main difference between these cellular solids is wavy distortions of the cell walls of the metal foams. The influence of wavy distortions on stiffness is modeled in this paper. The concepts are introduced through application to open cell materials, for which closed form solutions are obtained, primarily to illustrate the phenomenon. Closed cell materials are analysed subsequently, and results that are considered to be in good agreement with experimental observations are obtained.

Fracture analysis of cellular materials: A strain gradient model

Abstract: A generalized continuum model is developed for cellular materials based on the equivalence of strain energy at the macro- and microscale. It is rather similar to the strain gradient theory, but has a well-defined characteristic length, namely, the cell size. The continuum model enables one to use powerful analytical methods to investigate fracture of cellular materials. The near-tip asymptotic fields and full-field solutions are obtained for cellular materials with hexagonal, triangular, or square lattice. Using the same strain-energy equivalence at the macro- and microscale, displacements and rotation of discrete cell walls are estimated from the continuum near-tip asymptotic fields. By postulating a maximum-tensile-stress failure criterion of cell walls, the fracture toughness of cellular materials is estimated to be proportional to the thickness h of cell walls and inversely proportional to √L, where L is the cell size. Moreover, the mixed-mode fracture toughness can be simply obtained from the fracture toughness in pure mode 1 and mode II, once the mode mixity is known. It is established that, with the same mass density, the hexagonal or triangular lattice in a cellular material can provide much higher fracture toughness than the square lattice.

Elasto-viscoplastic constitutive equations for polycrystalline metals: Application to tantalum

Abstract: Strain-rate and temperature-dependent constitutive equations for polycrystalline metals which are capable of modeling the initial and evolving anisotropy in ductile metallic materials owing to the evolution of crystallographic texture are reviewed and then specialized to reproduce the recently published stress-strain response of commercially pure b.c.c. tantalum for strains up to 60%, at strain rates from quasi-static to 30,000 s −1 , and temperatures from −200 to 525 °C (Hoge and Mukherjee, 1977; Vecchio, 1994; Nemat-Nasser and Isaacs, 1996). The constitutive equations have been implemented in a finite element program, and the computational capability is used to simulate the evolution of crystallographic texture in simple compression, plane-strain compression, and torsion under quasi-static conditions. A comparison of the predictions against corresponding experiments shows that the crystal plasticity-based model predicts the texture evolution and the macroscopic stress-strain curves satisfactorily. The computational capability is also used to simulate the dynamic Taylor rod-impact tests performed by Ting (1992) on pre-textured tantalum cylinders. The numerical simulations reasonably reproduce the final length and the ovalized macroscopic shape of the impact end of the cylinders observed in the experiments.

Onset of failure in aluminum honeycombs under general in-plane loading

Abstract: Of interest here is the theoretical prediction of the onset of failure in aluminum honeycombs under arbitrary macroscopic loading conditions. A failure surface is defined in macroscopic stress space by the onset of the first buckling-type instability encountered along proportional load paths, where each load path is defined by a fixed macroscopic load orientation and a fixed ratio of principal macroscopic stresses. The influence of specimen size (i.e., geometric scale effects), and the influence of geometric microstructural imperfections on these failure surfaces, are investigated through a combination of analytical (i.e., Bloch wave) and numerical (i.e., finite element) techniques. All of the analyses presented here are carried out for commercially available honeycombs, and the results show an extreme sensitivity of the onset of failure in these materials to the macroscopic load orientation and the principal macroscopic stress ratio. In addition, the failure surface for a perfectly periodic honeycomb of infinite extent, is found to be an upper bound for the failure surfaces of the corresponding finite honeycomb specimens with microstructural imperfections. Moreover, the construction of the failure surfaces for the imperfect specimens requires the numerical solution for large, multicell models, while the failure surface for the finite, perfectly periodic model is obtained with less computational effort, since calculations involving only the unit cell are required. The methodology proposed in this investigation, therefore, provides a useful predictive tool for the design of these materials.

Effective stiffness tensor of composite media : II. Applications to isotropic dispersions

Abstract: Accurate approximate relations for the effective elastic moduli of two- and three-dimensional isotropic dispersions are obtained by truncating, after third-order terms, an exact series expansion for the effective stiffness tensor of d-dimensional two-phase composites (obtained in the first paper) that perturbs about certain optimal dispersions. Our third-order approximations of the effective bulk modulus Ke and shear modulus Ge are compared to benchmark data, rigorous bounds and popular self-consistent approximations for a variety of macroscopically isotropic dispersions in both two and three dimensions, for a wide range of phase moduli and volume fractions. Generally, for the cases considered, the third-order approximations are in very good agreement with benchmark data, always lie within rigorous bounds, and are superior to popular self-consistent approximations.

A finite deformation continuum\discrete model for the description of fragmentation and damage in brittle materials

Abstract: A dynamic finite element analysis of large displacements, high strain rate deformation behavior of brittle materials is presented in total Lagrangian coordinates. A continuum\discrete damage model capable of capturing fragmentation at two size scales is derived by combining a continuum damage model and a discrete damage model for brittle failure. It is assumed that size and distribution of potential fragments are known a priori, through either experimental findings or materials properties, and that macrocracks can nucleate and propagate along the boundaries of these potential fragments. The finite deformation continuum multiple-plane microcracking damage model accounts for microcracks within fragments. Interface elements, with cohesive strength and reversible unloading before debonding, between potential fragments describe the initiation of macrocracks, their propagation, and coalescence leading to the formation of discrete fragments. A surface-defined multibody contact algorithm with velocity dependent friction is used to describe the interaction between fragments and large relative sliding between them. The finite element equations of motion are integrated explicitly using a variable time step. Outputs are taken at discrete time intervals to study material failure in detail. The continuum\discrete damage model and the discrete fragmentation model, employing interface elements alone, are used to simulate a ceramic rod on rod impact. Stress wave attenuation, fragmentation pattern, and overall failure behavior, obtained from the analyses using the two models, are compared with the experimental results and photographs of the failing rod. The results show that the continuum\discrete model captures the stress attenuation and rod pulverization in agreement with the experimental observations while the pure discrete model underpredicts stress attenuation when the same potential fragment size is utilized. Further analyses are carried out to study the effect of potential fragment size and friction between sliding fragments. It is found that compared with the continuum\discrete damage model, the discrete fragmentation model is more sensitive to the multi-body discretization.

Switch-toughening of ferroelectrics subjected to electric fields

Abstract: Electric fields can influence the fracture toughness of ferroelectrics. For example, poled ferroelectrics exhibit fracture toughness anisotropy: the material is tougher for a crack parallel to the poling direction but less tough for a crack perpendicular to it. When an electric field is applied to a poled sample, a positive field reduces its fracture toughness but a negative field enhances it. Previous investigations attribute these phenomena to polarization switching. This paper proposes a model of stress-assisted 90 polarization switching to quantify the toughening process. Small scale switching and uniform electric fields are assumed. An analytical solution is presented for a mono-domain ferroelectric crystal undergoing a confined polarization switch. This solution and the domain orientation pattern enable us to estimate the fracture resistance against the steady state crack growth in ferroelectrics by a Reuss-type multiple-domain assembly. A dimensionless group of material parameters and an electric field function emerge, and form the key ingredients of switch-toughening. The model is used to delineate several observations, including: poling-induced anisotropy of the fracture toughness, asymmetric variation of the fracture toughness under positive and negative electric fields of a poled specimen; upside-down butterfly loop for the fracture toughness response under cyclic electric loading.

A Multivariant model for single crystal shape memory alloy behavior

Abstract: A general 3-D multivariant model based on thermodynamics and micromechanics for single crystal shape memory alloy (SMA) behavior is presented. This model is based on the habit plane and transformation directions for the variants of martensite in a given material. From this information, the single crystal behavior of the material to temperature and mechanical loads is derived using the concept of a thermodynamic driving force. The Eshelby–Kroner approach is utilized to determine the interaction energy between the variants, where it is assumed that variants can be subdivided into several self-accommodating groups in which variants can grow together compatibly. This model is examined initially for a simple 2-variant case and then extended to the typical 24 variant case. The multivariant model is shown to exhibit appropriate responses for uniaxial results on single crystals : the transformations occur instantaneously when the critical stress\temperature is reached ; both pseudoelasticity and the shape memory effect are captured. The model is also examined for responses to multiaxial loadings and the distinction between perfectly compatible and imperfectly compatible variants (with nonzero volumetric transformation strain) is discussed.

Microstructural factors controlling the strength and ductility of particle-reinforced metal-matrix composites

Abstract: A micromechanical model is developed to simulate the mechanical response in tension of particle-reinforced metal-matrix composites. The microstructure of the composite is represented as a three-dimensional array of hexagonal prisms with one reinforcement at the centre of each prism. The shape, volume fraction and state (either intact or broken) of the reinforcement is independent for each cell, so the interaction among all these factors could be studied. The tensile response of the composite is determined from the behaviour of the intact and damaged cells, the fraction of damaged cells being calculated on the assumption that the reinforcement strength follows the Weibull statistics. The model is used to determine the microstructural factors which provide optimum behaviour from the point of view of the tensile strength and ductility. The analyses included the effect of the matrix and reinforcement properties, the reinforcement volume fraction, the interaction between reinforcements of different shape and the heterogeneous distribution of the reinforcements within the composite.

On the shock induced failure of brittle solids

Abstract: The response of brittle materials to uniaxial compressive shock loading has been the subject of much recent discussion. The physical interpretation of the yield point of brittle materials, the Hugoniot elastic limit (HEL), the dependence of this threshold on propagation distance and the effect of polycrystalline microstructure remain to be comprehensively explained. Evidence of failure occurring in glasses behind a travelling boundary that follows a shock front has been accumulated and verified in several laboratories. Such a boundary has been called a failure wave. The variations of properties across this front include complete loss of tensile strength, partial loss of shear strength, reduction in acoustic impedance, lowered sound speed and opacity to light. Recently we have reported a similar behaviour in the polycrystalline ceramics silicon carbide and alumina. It is the object of this work to present our observations of these phenomena and their relation to failure and the HEL in brittle materials.

A constitutive model for the non-shock ignition and mechanical response of high explosives

Abstract: An understanding of the non-shock ignition properties of energetic particulate composite materials, high explosives such as PBX-9501 is an important part of the safety assessments for conventional handling (transportation, storage, etc.) of weapons systems including assembly operations. This paper develops and demonstrates the use of a numerical constitutive model for PBX-9501 that includes viscoelastic response, statistical fracture mechanics, and an ignition hot-spot mechanism. The intent is that this model can be used in safety analyses involving accidents to prevent undesirable dispersion of Pu. The parameters have been determined that will predict the mechanical response and ignition:non-ignition of a set of experiments that have explored the non-shock properties of this material.

On the modeling of crush behavior of a closed-cell aluminum foam structure

Abstract: Crush behavior of a closed-cell aluminum foam is studied analytically and numerically. A new model of a truncated cube, which captures the basic folding mechanism of an array of cells, is developed. The model consists of a system of collapsing cruciform and pyramidal sections. Theoretical analysis is based on energy consideration in conjunction with the minimum principle in plasticity. The assumed kinematic model for the crushing mechanism of the truncated cube cell gives a good agreement with the deformation mechanism obtained from the numerical simulation. Analytical formulation for the crushing resistance of the truncated cube cell is shown to correlate very accurately with the numerical results. Closed form solutions for crushing resistance of closed-cell aluminum foam in terms of relative density are developed. The formulas are compared with the experimental results and give an excellent agreement.

Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning

Abstract: There have been considerable recent advances in the understanding of anisotropy due to crystallographic texturing, and a reasonably successful elasto-viscoplasticity theory for the deformation of face-centeredcubic (f.c.c.) single crystals and polycrystals with high stacking fault energies is now at hand. The high stacking fault energy f.c.c. materials (e.g. Cu, Al) deform predominantly by crystallographic slip. In contrast, for materials with low stacking energies, e.g. α-brass, in addition to crystallographic slip, deformation twinning plays an important role in maintaining generalized plastic flow. A direct manifestation of twinning is the different crystallographic texture that is observed in 70−30 brass as compared to pure copper. In this paper we formulate a rate-independent constitutive model which accounts for both slip and twinning. We have also developed a new scheme to determine the active systems and the shear increments on the active slip and twin systems. We have implemented our constitutive equations and computational procedures in the finite-element program ABAQUS/Explicit (1995). By using comparisons between model predictions and macroscopically-measured stress-strain curves and texture evolution we have deduced information about the values of the single-crystal parameters associated with slip and twin system deformation resistances and hardening due to slip and twinning. We show that our model is able to reproduce both the experimentally measured pole figures and the stress strain curves in plane strain compression for α-brass. With the model so calibrated, we show that the predictions for the texture and stress-strain curves from the model are also in reasonably good agreement with experiments in simple compression. We have also evaluated the applicability of a Taylor-type model for combined slip and twinning. Our calculations show that for the high-symmetry f.c.c. brass, a Taylor-type model for crystals deforming by combined slip and twinning is able to reasonably well predict the macroscopic stress-strain curves and crystallographic texture evolution. Our calculations show that in plane strain as well as simple compression, the crystallographic texture that develops is a result of lattice rotation due to both slip and twinning, and that as suggested by Wassermann (1963), in contrast to copper which does not twin under normal circumstances, it is twinning which is responsible for the brass-type texture that is observed in f.c.c. metals with low stacking fault energies.

Exact solutions for functionally graded and laminated elastic materials

Abstract: We consider isotropic linearly elastic materials in which, referred to a rectangular Cartesian coordinate system Oxyz, the Lame elastic moduli λ and μ depend in an arbitrary specified manner on the coordinate z. If this dependence is continuous the material may be regarded as a functionally graded elastic material ; the case in which it is discontinuous represents a laminate. A large class of exact solutions of the three-dimensional elasticity equations for materials of this type is established. It is shown that exact three-dimensional solutions for a thick plate are generated, in a simple manner, by solutions of the two-dimensional classical equations for stretching and bending of an equivalent plate. This is a hypothetical homogeneous plate with elastic moduli that are appropriate weighted averages of the moduli of the inhomogeneous plate. The formulation in cylindrical polar coordinates is also given, and the theory is illustrated by examining solutions with radial symmetry about the z axis.

On kink-band propagation in fiber composites

Abstract: Two kinds of kink-band propagation in the compression of aligned-fiber composites are studied analytically : band broadening, discovered experimentally by Moran, Liu and Shih, 1996 , in which a uniform kink band grows in the direction of loading at constant stress under increasing deformation; and transverse kink propagation, in which a kink band traverses a specimen quasi-statically under constant overall shortening. The analysis is based on a 1-D, geometrically nonlinear couple-stress theory of composite kinking that takes elastic fiber bending resistance into account together with idealized nonlinear stress–strain relations, but assumes non-breaking fibers. Simple results for the band-broadening and transverse propagation stresses are deduced, and their significance is discussed.

Crack front waves

Abstract: We present simulations of 3 D dynamic fracture which suggest that a persistent elastic wave is generated in response to a localized perturbation of a propagating crack front, e.g., by a local heterogeneity of critical fracture energy. The wave propagates along the moving crack front and spreads, relative to its origin point on the fractured surface, at a speed slightly below the Rayleigh speed. The simulations were done using the spectral elastodynamic methodology of Geubelle and Rice (1995). They model failure by a displacement-weakening cohesive model, which corresponds in the singular crack limit to crack growth at a critical fracture energy. Confirmation that crack front waves with properties like in our simulation do exist has been provided by Ramanathan and Fisher (1997). Through a derivation based on the linearized perturbation analysis of dynamic singular tensile crack growth by Willis and Movchan (1995), those authors found by numerical evaluation that a transfer function thereby introduced has a simple pole at a certain ω κ ratio, corresponding to a non-dispersive wave. Further, we show that as a consequence of these persistent waves, when a crack grows through a region of small random fluctuations in fracture energy, the variances of both the local propagation velocity and the deformed slope of the crack front increase, according to linearized perturbation theory, in direct proportion to distance of growth into the randomly heterogeneous region. That rate of disordering is more rapid than the growth of the variances with the logarithm of distance established by Perrin and Rice (1994) for a model elastodynamic fracture theory based on a scalar wave equation. That scalar case, which shows slowly decaying (as t − 1 2 ) rather than persistent crack front waves, is analyzed here too.

A general constitutive theory for linear and nonlinear particulate media with microstructure evolution

Abstract: This work is concerned with the development of a constitutive theory for composite materials with particulate microstructures, which is capable of predicting, approximately, the evolution of the microstructure and its influence on the effective response of composites under general three-dimensional finitestrain loading conditions, such as those present in metal-forming operations. In its present form, the theory is general enough to be used for linearly viscous, nonlinearly viscous and perfectly plastic composites with randomly oriented and distributed ellipsoidal inclusions (or pores), which, in the most general case, can change size, shape and orientation. In addition, the “shape” and “orientation” of their center-to-center statistical distribution functions can also evolve with the deformation. To illustrate the key features of the new theory in the context of a simple example, an application is carried out for plane-strain loading of two-phase systems consisting of random distributions of aligned rigid particles in a power-law matrix phase. The results show that the evolution of the relevant microstructural variables, as well as the effective response, depend in a complex fashion on the initial state of the microstructure, as well as on the specific boundary conditions. In particular, it is found that the changes in orientation of the particles provide a mechanism analogous to “geometric softening” in ductile single crystals, which can lead to significant changes in the instantaneous hardening rate of the composite. This is shown to have important consequences for the possible onset of shear localization in the composite.

Adhesion, slip, cohesive zones and energy fluxes for elastic spheres in contact

Abstract: The energy fluxes upon shrinkage of the contact area are calculated for a pair of spheres in adhesion. Various notions of energy release rate are introduced and analyzed for correlating the external work parameters and the work of adhesion. Decomposition of the energy release rate into reversible and irreversible parts shows that the reversible part is the work of adhesion and it can be described by the cohesive response purely at the contact zone-edge. This result justifies the use of local zone-edge quantities for modeling the interaction of adhesion and friction. For specific quantitative analysis, adhesion is represented by the Dugdale model, uniform cohesive traction up to a limited separation, as an approximation to more exact inter-surface forces. Exact results are given for the entirely reversible energy release rate to the edge of the contact and the energy release rate to the cohesive zone. The latter is named the strain energy release rate and found to depend on the path in loading parameter space, while the reversible energy release rate is independent of the loading path. The solution for shear of the contact is given as well. Energy released reversibly to be converted into surface energy is identified in contrast to energy released due to slip which will be partially or totally dissipated as heat. The relevance of the results for friction is discussed and contrasted with their significance for the mixed mode fracture of a circular joint.

Micromechanical models for graded composite materials : ii. thermomechanical loading

Abstract: Thermoelastic response of several discrete and homogenized models of unconstrained graded composite layers is examined for both uniform changes in temperature and steady-state heat conduction in the gradient direction. Detailed finite element studies of the overall response and local fields in the discrete models are conducted, using large plane-array domains containing simulated skeletal and particulate microstructures. Homogenized layered models, with the same composition gradient and effective properties derived from the Mori–Tanaka and\or self-consistent methods, are analyzed under identical boundary conditions. Comparisons of temperature distributions, and of overall and local stress and strain fields predicted by the discrete and homogenized models are made in the C\SiC composite system, with very different phase properties and relatively steep composition gradient, that was used in the first part of this study (T. Reiter, G. J. Dvorak and V. Tvergaard, J. Mech. Phys. Solids , Vol. 45, pp. 1281–1302, 1997). Homogenized models of combined microstructures which employ only a single averaging method do not provide reliable agreements with the discrete model predictions. However, close agreement with the discrete models is shown by homogenized models which derive effective properties estimates from several averaging methods : In those parts of the graded microstructure which have a well-defined continuous matrix and discontinuous reinforcement, the effective moduli, expansion coefficients and heat conductivities are approximated by the appropriate Mori–Tanaka estimates. In skeletal microstructures that often form transition zones between clearly defined matrix and reinforcement phases, the effective properties are approximated by the self-consistent estimates. The results do not support the proposition that nonlocal or new micromechanical theories are required for modeling of graded microstructures.

Intersonic crack propagation in bimaterial systems

Abstract: This paper describes experimental observations of various phenomena characteristic of dynamic intersonic decohesion of bimaterial interfaces. Two separate but complementary optical methods are used in conjunction with high-speed photography to explore the nature of the large-scale contact and mach wave formation at the vicinity of running cracks in two different bimaterial systems. Theoretical predictions of crack tip speed regimes, where large-scale contact is implied, are confirmed. Also, the theoretically predicted mach wave emanating from the intersonically propagating crack tip is observed. Direct visual evidence is also obtained for another traveling mach wave emanating from the end of the intersonically moving contact zone. Subsequently, a physical model for intersonic crack propagation along bimaterial interfaces is presented and ratified in view of recent experimental observations and theoretical developments. Finally, the paper presents very recent experimental evidence that shows crack tip speeds exceeding the intersonic regime and becoming clearly supersonic.

Quantification of constraint effects in elastic-plastic crack front fields

Abstract: In-plane and out-of-plane constraint effects on crack tip stress fields under both small-scale and large-scale yielding conditions are studied by means of three-dimensional numerical analyses of boundary layer models and of finite size specimens, M(T) and SE(B), respectively. It is shown that the ratio of the plastic zone size over the panel thickness, rpt, plays a key role in formation of the crack-tip fields, particularly the outof-plane stress components. For a vanishingly small plastic zone around the crack tip the stress fields are dominated by the plane strain solution. With increase of the applied loads, i.e. increasing the plastic zone size, the stress fields develop towards the plane stress state. Characterization of “constraint effects” in terms of Q-stress is investigated. The “second term” in the near tip stress field, which is defined as the difference between the full three-dimensional stress fields and the plane strain reference solution, appears to depend on the distance to the tip and to the free surface of the specimen. Hence, the whole three-dimensional crack front fields cannot be correctly described by a two-parameter formulation as the load increases. However, a unique linear relationship between Q and the hydrostatic stress was found in all three-dimensional crack front fields.

On the numerical evaluation of elastostatic fields in locally isotropic two-dimensional composites

Abstract: We present a fast algorithm for the calculation of elastostatic fields in locally isotropic composites. The method uses an integral equation approach due to Sherman, combined with the fast multipole method and an adaptive quadrature technique. Accurate solutions can be obtained with inclusions of arbitrary shape at a cost roughly proportional to the number of points needed to resolve the interface. Large-scale problems, with hundreds of thousands of interface points can be solved using modest computational resources.