Material microstructure optimization for linear elastodynamic energy wave management

We describe a systematic approach to design material microstructures to achieve desired energy propagation in a two-phase composite plate. To generate a well-posed topology optimization problem we use the relaxation approach which requires homogenization theory to relate the macroscopic material properties to the microstructure, here a sequentially ranked laminate. We introduce an algorithm whereby the laminate layer volume fractions and orientations are optimized at each material point. To resolve numerical instabilities associated with the dynamic simulation and constrained optimization problem, we filter the laminate parameters. This also has the effect of generating smoothly varying microstructures which are easier to manufacture. To demonstrate our algorithm we design microstructure layouts for tailored energy propagation, i.e. energy focus, energy redirection, energy dispersion and energy spread.     

Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity

We propose an approach to the definition and analysis of material instabilities in rate-independent standard dissipative solids at finite strains based on finite-step-sized incremental energy minimization principles. The point of departure is a recently developed constitutive minimization principle for standard dissipative materials that optimizes a generalized incremental work function with respect to the internal variables. In an incremental setting at finite time steps this variational problem defines a quasi-hyperelastic stress potential. The existence of this potential allows to be recast a typical incremental boundary-value problem of quasi-static inelasticity into a principle of minimum incremental energy for standard dissipative solids. Mathematical existence theorems for sufficiently regular minimizers then induce a definition of the material stability of the inelastic material response in terms of the sequentially weakly lower semicontinuity of the incremental variational functional. As a consequence, the incremental material stability of standard dissipative solids may be defined in terms of the quasi-convexity or the rank-one convexity of the incremental stress potential. This global definition includes the classical local Hadamard condition but is more general. Furthermore, the variational setting opens up the possibility to analyze the post-critical development of deformation microstructures in non-stable inelastic materials based on energy relaxation methods. We outline minimization principles of quasi- and rank-one convexifications of incremental non-convex stress potentials for standard dissipative solids. The general concepts are applied to the analysis of evolving deformation microstructures in single-slip plasticity. For this canonical model problem, we outline details of the constitutive variational formulation and develop numerical and semi-analytical solution methods for a first-level rank-one convexification. A set of representative numerical investigations analyze the development of deformation microstructures in the form of rank-one laminates in single slip plasticity for homogeneous macro-deformation modes as well as inhomogeneous macroscopic boundary-value problems. The well-posedness of the relaxed variational formulation is indicated by an independence of typical finite element solutions on the mesh-size.     

Strain gradient plasticity modeling of the cyclic behavior of laminate microstructures

Two recently proposed Helmholtz free energy potentials including the full dislocation density tensor as an argument within the framework of strain gradient plasticity are used to predict the cyclic elastoplastic response of periodic laminate microstructures. First, a rank-one defect energy is considered, allowing for a size-effect on the overall yield strength of micro-heterogeneous materials. As a second candidate, a logarithmic defect energy is investigated, which is motivated by the work of Groma et al. (2003). The properties of the back-stress arising from both energies are investigated in the case of a laminate microstructure for which analytical as well as numerical solutions are derived. In this context, a new regularization technique for the numerical treatment of the rank-one potential is presented based on an incremental potential involving Lagrange multipliers. The results illustrate the effect of the two energies on the macroscopic size-dependent stress–strain response in monotonic and cyclic shear loading, as well as the arising pile-up distributions. Under cyclic loading, stress–strain hysteresis loops with inflections are predicted by both models. The logarithmic potential is shown to provide a continuum formulation of Asaro's type III kinematic hardening model. Experimental evidence in the literature of such loops with inflections in two-phased FFC alloys is provided, showing that the proposed strain gradient models reflect the occurrence of reversible plasticity phenomena under reverse loading.     

Size effects in generalised continuum crystal plasticity for two-phase laminates

The solutions of a boundary value problem are explored for various classes of generalised crystal plasticity models including Cosserat, strain gradient and micromorphic crystal plasticity. The considered microstructure consists of a two-phase laminate containing a purely elastic and an elasto-plastic phase undergoing single or double slip. The local distributions of plastic slip, lattice rotation and stresses are derived when the microstructure is subjected to simple shear. The arising size effects are characterised by the overall extra back stress component resulting from the action of higher order stresses, a characteristic length l_c describing the size-dependent domain of material response, and by the corresponding scaling law lⁿ as a function of microstructural length scale, l. Explicit relations for these quantities are derived and compared for the different models. The conditions at the interface between the elastic and elasto-plastic phases are shown to play a major role in the solution. A range of material parameters is shown to exist for which the Cosserat and micromorphic approaches exhibit the same behaviour. The models display in general significantly different asymptotic regimes for small microstructural length scales. Scaling power laws with the exponent continuously ranging from 0 to −2 are obtained depending on the values of the material parameters. The unusual exponent value −2 is obtained for the strain gradient plasticity model, denoted “curl H^p” in this work. These results provide guidelines for the identification of higher order material parameters of crystal plasticity models from experimental data, such as precipitate size effects in precipitate strengthened alloys.     

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.     

The effect of microstructural representation on simulations of microplastic ratcheting

This paper assesses the sensitivity of cyclic plasticity to microstructure morphology by examining and comparing the microplastic ratcheting behavior of different idealized microstructures (square, hexagonal, tessellated, and digitized from experimental data). This analysis demonstrates the sensitivity of computational accuracy to the various approximations in microstructural representation. The methodology used to perform this study relies on a coupling between microstructural characterization, mechanical testing and numerical simulations to investigate the influence of the microstructure on the purely tensile uniaxial microplastic ratcheting behavior of pure nickel polycrystals. The morphology and deformation behavior of polycrystals were characterized using electron back-scatter diffraction (EBSD), while a finite element model (FEM) of crystal plasticity was used in a computational framework. The predicted cyclic behavior is compared to experimental results both at the macroscopic and microstructural scales. The stress–strain response is less sensitive to the details of the microstructural representation than might be expected with all representations displaying similar macroscopic constitutive response. However, the details of the plastic strain distribution at the microstructural scale and the related estimations of damage mechanics vary substantially from one microstructural representation to another.     

Phase-field slip-line theory of plasticity

A variational approach to determine the deformation of an ideally plastic substance is proposed by solving a sequence of energy minimization problems under proper conditions to account for the irreversible character of plasticity. The flow is driven by the local transformation of elastic strain energy into plastic work on slip surfaces, once that a certain energetic barrier for slip activation has been overcome. The distinction of the elastic strain energy into spherical and deviatoric parts is used to incorporate in the model the idea of von Mises plasticity and isochoric plastic strain. This is a “phase field model” because the matching condition at the slip interfaces is substituted by the evolution of an auxiliary phase field that, similar to a damage field, is unitary on the elastic phase and null on the yielded phase. The slip lines diffuse in bands, whose width depends upon a material length-scale parameter.Numerical experiments on representative problems in plane strain give solutions with noteworthy similarities with the results from classical slip-line field theory, but the proposed model is much richer because, accounting for elastic deformations, it can describe the formation of slip bands at the local level, which can nucleate, propagate, widen and diffuse by varying the boundary conditions. In particular, the solution for a long pipe under internal pressure is very different from the one obtainable from the classical macroscopic theory of plasticity. For this case, the location of the plastic bands may be an insight to explain the premature failures that are sometimes encountered during the manufacturing process. This practical example enhances the importance of this new theory based on the mathematical sciences.     

The Relaxation of Two-well Energies with Possibly Unequal Moduli

The elastic energy of a multiphase solid is a function of its microstructure. Determining the infimum of the energy of such a solid and characterizing the associated optimal microstructures is an important problem that arises in the modeling of the shape memory effect, microstructure evolution, and optimal design. Mathematically, the problem is to determine the relaxation under fixed phase fraction of a multiwell energy. This paper addresses two such problems in the geometrically linear setting. First, in two dimensions, we compute the relaxation under fixed phase fraction for a two-well elastic energy with arbitrary elastic moduli and transformation strains, and provide a characterization of the optimal microstructures and the associated strain. Second, in three dimensions, we compute the relaxation under fixed phase fraction for a two-well elastic energy when either (1) both elastic moduli are isotropic, or (2) the elastic moduli are well ordered and the smaller elastic modulus is isotropic. In both cases we impose no restrictions on the transformation strains. We provide a characterization of the optimal microstructures and the associated strain. We also compute a lower bound that is optimal except possibly in one regime when either (1) both elastic moduli are cubic, or (2) the elastic moduli are well ordered and the smaller elastic modulus is cubic; for moduli with arbitrary symmetry we obtain a lower bound that is sometimes optimal. In all these cases we impose no restrictions on the transformation strains and whenever the bound is optimal we provide a characterization of the optimal microstructures and the associated strain. In both two and three dimensions the quasiconvex envelope of the energy can be obtained by minimizing over the phase fraction. We also characterize optimal microstructures under applied stress.     

Heterogeneous crystal and poly-crystal plasticity modeling from a transformation field analysis within a regularized Schmid law

The regularized Schmid law (RSL) has recently been proposed as a plastic flow criterion for poly-crystals under the crude assumptions of either uniform stress or uniform strain. We first reconsider this law for application to heterogeneous intra-crystalline plasticity, with reference to a Homogeneous Equivalent Super-Crystal. We then extend the modeling to poly-crystals with the goal to account for both stress and strain heterogeneities within as well as between grains. The transformation field analysis (TFA) is used as the homogenization procedure. This TFA is known to be accurate for materials that can be described as assemblies of plastically homogeneous domains. Otherwise, the estimates of the material effective behavior that result from its application are too stiff. Because stress and strain fields are almost everywhere uniform in laminates, we consider crystal slip organizations into multi-laminate structures. It is demonstrated that laminate layers either parallel to slip planes or normal to slip directions do not contribute to the over-stiffness due to the TFA. Thus, hierarchical multi-laminate (HML) structures are introduced where the successive laminate orientations are taken parallel to the crystal slip planes. It is shown that a conveniently weighted superposition of all the possible plane hierarchies cancels out most of the undesirable TFA contributions to the overall stiffness estimates. A relevant extension to poly-crystal plasticity of this (RSL-TFA-HML) modeling is presented.     

A substructure mixtures model for the cyclic plasticity of single slip oriented nickel single crystal at low plastic strain amplitudes

The cyclic plasticity behavior of nickel single crystals oriented for single slip is characterized by uniaxial, symmetric, tension–compression, strain controlled tests carried out at constant plastic strain amplitudes ranging from 5(10⁻⁵) to 1(10⁻³). Annealed single crystals are cycled in this manner to post-cyclic saturation and microstructural characterizations, including transmission electron microscopy and optical micrographs of specimen surface replicas are used to verify and evaluate dislocation substructures. Stress–strain and microstructure data are used to construct a mixtures model that couples cyclic plasticity models for three substructures as well as a model for reverse magnetostriction (Villari effect) that is a significant component of inelastic strain at the lower plastic strain amplitudes. The model is used to correlate the stress–plastic strain hysteresis loop responses over the range of plastic strain amplitudes and from cumulative plastic strains from 0.3 to post-cyclic saturation. Complex evolution of substructure plastic strain amplitudes toward their so-called intrinsic values upon the formation of persistent slip bands is modeled. Additionally, bulk Young’s modulus is found to vary significantly with plastic strain amplitude and cumulative plastic strain. A correlation of this behavior is included.     

Single-Slip Elastoplastic Microstructures

We consider rate-independent crystal plasticity with constrained elasticity, and state the variational formulation of the incremental problem. For generic boundary data, even the first time increment does not admit a smooth solution, and fine structures are formed. By using the tools of quasiconvexity, we obtain an explicit relaxation of the first incremental problem for the case of a single slip system. Our construction shows that laminates between two different deformation gradients are formed. Plastic deformation concentrates in one of them, the other is a purely elastic strain. For the concrete case of a simple-shear test we also obtain a completely explicit solution.     

Interfacial Energies for Incoherent Inclusions

We study a variational problem describing an incoherent interface between a rigid inclusion and a linearly elastic matrix. The elastic material is allowed to slip relative to the inclusion along the interface, and the resulting mismatch is penalized by an interfacial energy term that depends on the surface gradient of the relative displacement. The competition between the elastic and interfacial energies induces a threshold effect when the interfacial energy density is non-smooth: small inclusions are coherent (no mismatch); sufficiently large inclusions are incoherent. We also show that the relaxation of the energy functional can be written as the sum of the bulk elastic energy functional and the tangential quasiconvex envelope of the interfacial energy functional.     

A cyclic plasticity model for single crystals

The Armstrong–Frederick type kinematic hardening rule was invoked to capture the Bauschinger effect of the cyclic plastic deformation of a single crystal. The yield criterion and flow rule were built on individual slip systems. Material memory was introduced to describe strain range dependent cyclic hardening. The experimental results of copper single crystals were used to evaluate the cyclic plasticity model. It was found that the model was able to accurately describe the cyclic plastic deformation and properly reflect the dislocation substructure evolution. The well-known three distinctive regimes in the cyclic stress–strain curve of the copper single crystals oriented for single slip can be reproduced by using the model. The model can predict the enhanced hardening for crystals oriented for multislip, showing the model's ability to describe anisotropic cyclic plasticity. For a given loading history, the model was able to capture not only the saturated stress–strain response but also the detailed transient stress–strain evolution. The model was used to predict the cyclic plasticity under a high–low loading sequence. Both the stress–strain responses and the microstructural evolution can be appropriately described through the slip system activation.     

Determination of Ti-6242 α and β slip properties using micro-pillar test and computational crystal plasticity

The properties and behaviour of an α−β colony Ti-6242 alloy have been investigated at 20 °C utilising coupled micro-pillar stress relaxation tests and computational crystal plasticity. The β-phase slip strength and intrinsic slip system strain rate sensitivity have been determined, and the β-phase shown to have stronger rate sensitivity than that for the α phase. Close agreement of experimental observations and crystal plasticity predictions of micro-pillar elastic-plastic response, stress relaxation, slip activation in both α and β-phases, and strain localisation within the α−β pillars with differing test strain rate, β morphology, and crystal orientations is achieved, supporting the validity of the properties extracted. The β-lath thickness is found to affect slip transfer across the α−β−α colony, but not to significantly change the nature of the slip localisation when compared to pure α-phase pillars with the same crystallographic orientation. These results are considered in relation to rate-dependent deformation, such as dwell fatigue, in complex multiphase titanium alloys.     

Cavity expansion of a gradient-dependent solid cylinder

This paper presents an elasto-plastic analysis for cavity expansion in a solid cylinder. The solid is modelled using a strain gradient plasticity model to account for the influence of microstructures on the macroscopic mechanical behaviour. A numerical shooting method, together with Broyden’s iteration procedure, is developed to solve the resulting fourth-order ordinary differential equation with two-point boundary conditions for the gradient-dependent problem. Fully elastic-plastic solutions to the cavity expansion are obtained and they are compared with conventional results for a number of examples. The effects of microstructure on macroscopic behaviour for the cavity expansion problem are analysed. It is demonstrated that, with consideration of microstructural effects, the deformation and stress distributions in the cylinder are highly inhomogeneous during both the initial loading and the subsequent elastic and plastic expansion stages. The gradient effects can result in a stiffer response in the elastic regime (as compared with the corresponding conventional prediction), but a weaker response in the plastic regime. As expected, the overall elasto-plastic behaviour of the gradient-dependent cylinder depends on the material parameters as well as the cylinder thickness. It is shown that the strain gradient theory solutions reduce to the conventional ones as a special case when the dimension of the microstructures is negligible compared with the cylinder size. The results in this paper can be used as a benchmark for further numerical investigations of the cavity expansion problem.     

Modelling microstructure evolution in engineering materials

In this paper we describe a general thermodynamically consistent variational principle for the rate of evolution of microstructure, which considers the competition between energy dissipation and the rate of change of Gibbs free energy of the system. We describe how numerical and approximate analytical procedures can be developed from the variational principle. Two examples are presented which demonstrate the utility of the approach: the kinetics of precipitate growth in an elastically strained body and the influence of an elastic strain on interdiffusion in a two-component system. Within these examples we pay particular attention to the effect of changes of elastic stored energy on the evolution process. The sensitivity of the morphology of growing phases to the ratio of the driving forces arising from elastic and chemical considerations is explored.     

Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem

A detailed variational formulation is provided for a simplified strain gradient elasticity theory by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the complete boundary conditions of the theory for the first time. To supplement the stress-based formulation, the coordinate-invariant displacement form of the simplified strain gradient elasticity theory is also derived anew. In view of the lack of a consistent and complete formulation, derivation details are included for the tutorial purpose. It is shown that both the stress and displacement forms of the simplified strain gradient elasticity theory obtained reduce to their counterparts in classical elasticity when the strain gradient effect (a measure of the underlying material microstructure) is not considered. As a direct application of the newly obtained displacement form of the theory, the problem of a pressurized thick-walled cylinder is analytically solved. The solution contains a material length scale parameter and can account for microstructural effects, which is qualitatively different from Lamé’s solution in classical elasticity. In the absence of the strain gradient effect, this strain gradient elasticity solution reduces to Lamé’s solution. The numerical results reveal that microstructural effects can be large and Lamé’s solution may not be accurate for materials exhibiting significant microstructure dependence.     

Inspection of free energy functions in gradient crystal plasticity

The dislocation density tensor computed as the cud of plastic distortion is regarded as a new constitutive variable in crystal plasticity. The dependence of the free energy function on the dislocation density tensor is explored starting from a quadratic ansatz. Rank one and logarithmic dependencies are then envisaged based on considerations from the statistical theory of dislocations. The rele- vance of the presented free energy potentials is evaluated from the corresponding analytical solutions of the periodic two-phase laminate problem under shear where one layer is a single crystal material undergoing single slip and the second one remains purely elastic.