The non-uniqueness of the atomistic stress tensor and its relationship to the generalized Beltrami representation

The non-uniqueness of the atomistic stress tensor is a well-known issue when defining continuum fields for atomistic systems. In this paper, we study the non-uniqueness of the atomistic stress tensor stemming from the non-uniqueness of the potential energy representation. In particular, we show using rigidity theory that the distribution associated with the potential part of the atomistic stress tensor can be decomposed into an irrotational part that is independent of the potential energy representation, and a traction-free solenoidal part. Therefore, we have identified for the atomistic stress tensor a discrete analog of the continuum generalized Beltrami representation (a version of the vector Helmholtz decomposition for symmetric tensors). We demonstrate the validity of these analogies using a numerical test. A program for performing the decomposition of the atomistic stress tensor called MDStressLab is available online at http://mdstresslab.org.     

Discrete element simulation of shock wave propagation in polycrystalline copper

The implementation of the characteristic of compressive plasticity into the Discrete Element Code, DM2, while maintaining its quasi-molecular scheme, is described. The code is used to simulate the shock compression of polycrystalline copper at 3.35 and 11.0 GPa. The model polycrystal has a normal distribution of grain sizes, with mean diameter 14 μm, and three distinct grain orientations are permitted with respect to the shock direction; 〈1 0 0〉, 〈1 1 0〉, and 〈1 1 1〉. Particle velocity dispersion (PVD) is present in the shock-induced flow, attaining its maximum magnitude at the plastic wave rise. PVD normalised to the average particle velocity of and are yielded for the 3.35 and 11.0 GPa shocks, respectively, and are of the same order as those seen in the experiment. Non-planar elastic and plastic wave fronts are present, the distribution in shock front position increasing with propagation distance. The rate of increase of the spread in shock front positions is found to be significantly smaller than that seen in probabilistic calculations on nickel polycrystals, and this difference is attributed, in the main, to grain interaction. Reflections at free surfaces yield a region of tension near to the target free surface. Due to the dispersive nature of the shock particle velocity and the non-planarity of the shock front, the tensile pressure is distributed. This may have implications for the spall strength, which are discussed. Simulations reveal a transient shear stress distribution behind the shock front. Such a distribution agrees with that put forward by Lipkin and Asay to explain the quasi-elastic reloading phenomenon. Simulation of reloading shocks show that the shear stress distribution can give rise to quasi-elastic reloading on the grain scale.     

Effect of pore concentration on localized plastic deformation in polycrystals

Presented is a mesomechanical model that simulates the behavior of localized plastic deformation in polycrystal subjected to uniaxial load. Edge effect and pore concentration are analyzed. Predicted results between the mechanical properties of polycrystal and pore concentration are inconclusive as porosity alone could not explain the increase or decrease of polycrystal strength. The model applies to low pore concentration of less than 5% and hence coalescence of pores is also neglected.     

A strong contrast homogenization formulation for multi-phase anisotropic materials

Homogenization relations, linking a material's properties at the mesoscale to those at the macroscale, are fundamental tools for design and analysis of microstructure. Recent advances in this field have successfully applied spectral techniques to Kroner-type perturbation expansions for polycrystalline and composite materials to provide efficient inverse relations for materials design. These expansions have been termed ‘weak-contrast’ expansions due to the conditionally convergent integrals, and the reliance upon only small perturbations from the reference property. In 1955, Brown suggested a different expansion for electrical conductivity that resulted in absolutely convergent integrals. Torquato subsequently applied the method to elasticity, with good results even for high-contrast materials; thus it is commonly referred to as a ‘strong contrast’ expansion. The methodology has been applied to elasticity for two phases of isotropic material, generally assuming macroscopic isotropy (with noted exceptions), thus resulting in a rather elegant form of the solution.

More recently, a multi-phase form of the solution was developed for conductivity. This paper builds upon this result to apply the method to elasticity of polycrystalline materials with both local and global anisotropy. New spectral formulations are subsequently developed for both the weak and strong contrast solutions. These form the basis for efficient microstructure analysis using these frameworks, and subsequently for inverse design applications. The process is taken through to demonstration of a property closure, which acts as the basis for materials design; the closure delineates the envelope of all physically realizable property combinations for the chosen properties, based on the particular homogenization relation being used.     

3-D simulation of spatial stress distribution in an AZ31 Mg alloy sheet under in-plane compression

A complete 3-D crystal plasticity finite element method (CPFEM) that considered both crystallographic slip and deformation twinning was applied to simulate the spatial distribution of the relative amount of slip and twin activities in a polycrystalline AZ31 Mg alloy during in-plane compression. A microstructure mapping technique that considered the grain size distribution and microtexture measured by electron backscatter diffraction (EBSD) technique was used to create a statistically representative 3-D microstructure for the initial configuration. Using a 3-D Monte Carlo method, a 3-D digital microstructure that matched the experimentally measured grain size distribution was constructed. Crystallographic orientations obtained from the EBSD data were assigned on the 3-D digital microstructure to match the experimentally measured misorientation distribution. CPFEM captured the heterogeneity of the stress concentration as well as the slip and twin activities of a polycrystalline AZ31 Mg alloy during in-plane compression.     

A two-level system with induced transitions as a model of stress relaxation

This paper outlines the basic features of a two-level cooperative model of the stress relaxation in solids. In contrast to similar models presented earlier, the transitions between the two energy levels can take place in both directions, the relaxation process thus assuming a quasi-equilibrium character. The kinetics of this model are the same as obtained in the model where the transitions take place from the higher to the lower level only.Dedicated to Prof. Dr. F. R. Schwarzl on the occasion of his 60th birthday     

Computationally efficient database and spectral interpolation for fully plastic Taylor-type crystal plasticity calculations of face-centered cubic polycrystals

A new computationally efficient database approach to fully plastic Taylor-type crystal plasticity calculations is presented in this paper. In particular, we explore strategies that circumvent the need to repeatedly solve sets of highly non-linear, extremely stiff, algebraic equations with poor convergence characteristics that are inherent to these calculations. The suggested strategies consist of computing only once all of the needed variables in crystal plasticity calculations, storing them, and retrieving the values of interest according to the need in any specific simulation. An algorithm is presented here that facilitates this approach, and involves local spectral interpolation using discrete fourier transform (DFT) methods. The approach described here results in major improvements in the computational time over both the conventional crystal plasticity calculations and our previously developed spectral approach using generalized spherical harmonics (GSH). Details of this new approach are described and validated in this paper through a few example case studies.     

FE analysis of stress and strain fields in finite random composite bodies

In this paper the mechanical behaviour of finite random heterogeneous bodies is considered. The analysis of non-local interactions between heterogeneities in microscopically heterogeneous materials is necessary when the spatial variation of the load or the dimensions of the body, relative to the scale of the microstructure, cannot be ignored. Microstructures can be periodic but generically they are random. In the first case, an exact calculation can be performed but in the second case recourse has to be made either to simulation or to some scheme of approximation. One such scheme is based on a stochastic variational principle. The novelty of the present work is that a stochastic variational principle is projected directly onto a finite-element basis so that all subsequent analysis is performed within a finite-element framework. The proposed formulation provides expressions for the local stress and strain fields in any realization of the medium, from which expressions for statistically-averaged quantities can be derived. Then an approximation of Hashin-Shtrikman type is developed, which generates a FE-based numerical procedure able to take account of interactions between random inclusions and boundary layer effects in finite composite structures. Finally, two examples are presented, namely a cylinder with square cross-section subjected to mixed boundary conditions of different types on different faces and a rectangular body containing a centre crack. The results show that in the vicinity of the boundary or close to the crack tip, the strain and the stress in the matrix and in the inclusions differ considerably from those obtained by the formal application of conventional homogenization.     

Irreducible structure, symmetry and average of Eshelby's tensor fields in isotropic elasticity

The strain field ?(x) in an infinitely large, homogenous, and isotropic elastic medium induced by a uniform eigenstrain ?⁰ in a domain ω depends linearly upon . It has been a long-standing conjecture that the Eshelby's tensor field S^ω(x) is uniform inside ω if and only if ω is ellipsoidally shaped. Because of the minor index symmetry , S^ω might have a maximum of 36 or nine independent components in three or two dimensions, respectively. In this paper, using the irreducible decomposition of S^ω, we show that the isotropic part S of S^ω vanishes outside ω and is uniform inside ω with the same value as the Eshelby's tensor S⁰ for 3D spherical or 2D circular domains. We further show that the anisotropic part A^ω=S^ω-S of S^ω is characterized by a second- and a fourth-order deviatoric tensors and therefore have at maximum 14 or four independent components as characteristics of ω's geometry. Remarkably, the above irreducible structure of S^ω is independent of ω's geometry (e.g., shape, orientation, connectedness, convexity, boundary smoothness, etc.). Interesting consequences have implication for a number of recently findings that, for example, both the values of S^ω at the center of a 2D C_n(n?3,n≠4)-symmetric or 3D icosahedral ω and the average value of S^ω over such a ω are equal to S⁰.     

Influence of anisotropic elasticity on the mechanical properties of fivefold twinned nanowires

Previous atomistic simulations and experiments have shown an increased Young's modulus and yield strength of fivefold twinned (FT) face-centered cubic metal nanowires (NWs) when compared to single crystalline (SC) NWs of the same orientation. Here we report the results of atomistic simulations of SC and FT Ag, Al, Au, Cu and Ni NWs with diameters between 2 and 50 nm under tension and compression. The simulations show that the differences in Young's modulus between SC and FT NWs are correlated with the elastic anisotropy of the metal, with Al showing a decreased Young's modulus. We develop a simple analytical model based on disclination theory and constraint anisotropic elasticity to explain the trend in the difference of Young's modulus between SC and FT NWs. Taking into account the role of surface stresses and the elastic properties of twin boundaries allows to account for the observed size effect in Young's modulus. The model furthermore explains the different relative yield strengths in tension and compression as well as the material and loading dependent failure mechanisms in FTNWs.     

Stress concentration of an ellipsoidal inclusion of revolution in a semi-infinite body under biaxial tension

Summary This paper deals with the stress concentration problem of an ellipsoidal inclusion of revolution in a semi-infinite body under biaxial tension. The problem is formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknowns are densities of body forces distributed in the r- and z-directions in semi-infinite bodies having the same elastic constants as the ones of the matrix and inclusion. In order to satisfy the boundary conditions along the ellipsoidal boundary, four fundamental density functions proposed in [24, 25] are used. The body-force densities are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield rapidly converging numerical results for stress distribution along the boundaries even when the inclusion is very close to the free boundary. The effect of the free surface on the stress concentration factor is discussed with varying the distance from the surface, the shape ratio and the elastic modulus ratio. The present results are compared with the ones of an ellipsoidal cavity in a semi-infinite body.accepted for publication 11 November 2003     

Boundary-layer corrections for stress and strain fields in randomly heterogeneous materials

Boundary-layer effects on the effective response of fibre-reinforced media are analysed. The distribution of the fibres is assumed random. A methodology is presented for obtaining non-local effective constitutive operators in the vicinity of a boundary. These relate ensemble averaged stress to ensemble averaged strain. Operators are also developed which re-construct the local fields from their ensemble averages. These require information on the local configuration of the medium. Complete information is likely not to be available, but averages of these operators conditional upon any given local information generate corresponding conditional averages of the fields. Explicit implementation is performed within the framework of an approximation of Hashin-Shtrikman type. Two types of geometry are considered in examples: a half-space and a crack in an infinite heterogeneous medium. These are representative, asymptotically, of the field in the vicinity of any smooth boundary, and in the vicinity of a crack tip, respectively. Results have been obtained for the case of anti-plane deformation, realized by the imposition of either Dirichlet or Neumann conditions on the boundary; those for the Neumann condition are presented and discussed explicitly. The stresses in both fibre and matrix adjacent to a crack tip are shown to differ substantially from the values that would be predicted by ordinary homogenization.     

On evolving deformation microstructures in non-convex partially damaged solids

The paper outlines a relaxation method based on a particular isotropic microstructure evolution and applies it to the model problem of rate independent, partially damaged solids. The method uses an incremental variational formulation for standard dissipative materials. In an incremental setting at finite time steps, the formulation defines a quasi-hyperelastic stress potential. The existence of this potential allows a typical incremental boundary value problem of damage mechanics to be expressed in terms of a principle of minimum incremental work. Mathematical existence theorems of minimizers then induce a definition of the material stability in terms of the sequential weak lower semicontinuity of the incremental functional. As a consequence, the incremental material stability of standard dissipative solids may be defined in terms of weak convexity notions of the stress potential. Furthermore, the variational setting opens up the possibility to analyze the development of deformation microstructures in the post-critical range of unstable inelastic materials based on energy relaxation methods. In partially damaged solids, accumulated damage may yield non-convex stress potentials which indicate instability and formation of fine-scale microstructures. These microstructures can be resolved by use of relaxation techniques associated with the construction of convex hulls. We propose a particular relaxation method for partially damaged solids and investigate it in one- and multi-dimensional settings. To this end, we introduce a new isotropic microstructure which provides a simple approximation of the multi-dimensional rank-one convex hull. The development of those isotropic microstructures is investigated for homogeneous and inhomogeneous numerical simulations.     

Stress relaxation of large amplitudes and long timescales in soft thermoplastic and filled elastomers

We propose a model for describing mesoscale relaxation mechanisms in soft thermoplastic elastomers and also in the high-temperature regime of filled rubbers. The model consists of hard spheres embedded in an elastic matrix. It is solved by dissipative particle dynamics. We study the response of the model to deformations of various amplitudes. We show that it displays slow relaxation processes of large amplitudes that are related to irreversible reorganizations at a mesoscopic scale. We characterize these reorganizations as buckling of instabilities that change the local environment of the hard inclusions. Paper presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

Flow birefringence study of sharkskin and stress relaxation in polybutadiene melts

In this paper we report rheo-optical and rheological observations made through a transparent slit die attached to a capillary rheometer. We find that the flow birefringence signal oscillates periodically near the die exit when sharkskin-like extrudate distortion is present. In contrast, steady behavior is observed in the die inland region. Specifically, the flow birefringence varies at the die exit with a period identical to that measured directly from the sharkskin extrudate. We also show that the exit flow instability leading to sharkskin can be observed directly through cross-polarizers in terms of the temporal change of the retardation order. We demonstrate that the same kind of interfacial flow instability can occur at a boundary discontinuity within the die land where the upper portion of a clean die wall meets the lower portion of a polysiloxane-coated die wall. Finally, stress relaxation upon the cessation of the slit die flow of two polybutadiene melts is studied through time-dependent flow birefringence measurements. The stress relaxation is then correlated with sharkskin time scales to describe the role of relaxation in sharkskin ridge formation. Received: 8 February 1999 Accepted: 28 July 1999     

Comparison of two methods for stress relaxation data presentation of solid foods

Published exponential relaxation equations, derived from Maxwellian models, were used to generate data for linear representation in the form ofP(0) ·t/(P(0) —P(t)) =k ₁ +k ₂t whereP(t) is the decaying parameter (force, stress or modulus),P(0) its initial value (att = 0) andk ₁ andk ₂ constants. The computer plots indicated that the fit of this normalized and linearized form was excellent for equations containing at least three exponential decay terms. The fit was not as good for some of the two-term exponential equations mainly due to the lack of accurate account for the initial stage of the relaxation process. In all the cases, however, the linear representation could clearly reveal the general rheological character of the analysed materials in terms of the relative degree of solidity.     

The Peierls stress in non-local elasticity

The effect of non-locality on the Peierls stress of a dislocation, predicted within the framework of the Peierls-Nabarro model, is investigated. Both the integral formulation of non-local elasticity and the gradient elasticity model are considered. A modification of the non-local kernel of the integral formulation is proposed and its effect on the dislocation core shape and size, and on the Peierls stress are discussed. The new kernel is longer ranged and physically meaningful, improving therefore upon the existing Gaussian-like non-locality kernels. As in the original Peierls-Nabarro model, lattice trapping cannot be captured in the purely continuum non-local formulation and therefore, a semi-discrete framework is used. The constitutive law of the elastic continuum and that of the glide plane are considered both local and non-local in separate models. The major effect is obtained upon rendering non-local the constitutive law of the continuum, while non-locality in the rebound force law of the glide plane has a marginal effect. The Peierls stress is seen to increase with increasing the intrinsic length scale of the non-local formulation, while the core size decreases accordingly. The solution becomes unstable at intrinsic length scales larger than a critical value. Modifications of the rebound force law entail significant changes in the core configuration and critical stress. The discussion provides insight into the issue of internal length scale selection in non-local elasticity models.     

Instability of an elastic film on a viscous layer

A flat, compressed elastic film on a viscous layer is unstable. The film can form wrinkles to reduce the elastic energy. In this paper, we are interested in the two-dimensional models for thin films bonded to a viscous layer and in particular we focus on generic instabilities evidenced in this context by Suo and coworkers [Huang, Z., Hong, W., Suo, Z., 2005. Non linear analyses of wrinkles in a film bonded to a compliant substrate. J. Mech. Phys. Solids 53, 2101–2118; Lo, Y.H., 1991. New approach to grow pseudomorphic structures over the critical thickness. Appl. Phys. Lett. 59, 2311–2320]. We present a rigorous linear perturbation analysis for anisotropic materials, that allows the prediction of both the orientation of the corrugations of the thin film, and the wavelength that maximize the growth velocity. Finally, we compare our theoretical estimates to experimental results for a In_0.65Ga_0.35As alloy constraint to InP.