In this work we analyze the relation between the multiplicative decomposition $\mathbf F=\mathbf F^{e}\mathbf F^{p}$ of the deformation gradient as a product of the elastic and plastic factors and the theory of uniform materials. We prove that postulating such a decomposition is equivalent to having a uniform material model with two configurations - total $ϕ$ and the inelastic $ϕ_{1}$. We introduce strain tensors characterizing different types of evolutions of the material and discuss the form of the internal energy and that of the dissipative potential. The evolution equations are obtained for the configurations $(ϕ,ϕ_{1})$ and the material metric $\mathbf g$.
Finally the dissipative inequality for the materials of this type is presented.It is shown that the conditions of positivity of the internal dissipation terms related to the processes of plastic and metric evolution provide the anisotropic yield criteria.
Elastoplastic lattice models for the response of solids to deformation typically incorporate structure only implicitly via a local yield strain that is assigned to each site. However, the local yield strain can change in response to a nearby or even distant plastic event in the system. This interplay is key to understanding phenomena such as avalanches in which one plastic event can trigger another, leading to a cascade of events, but typically is neglected in elastoplastic models. To include the interplay one could calculate the local yield strain for a given particulate system and follow its…
An effective temperature $T_{\rm eff}$ which differs from the bath temperature is believed to play an essential role in the theory of elasto-plasticity of amorphous solids. The definition of a measurable $T_{\rm eff}$ in the literature on sheared solids suffers however from being connected to a fluctuation-dissipation theorem which is correct only in equilibrium. Here we introduce a natural definition of $T_{\rm eff}$ based on measurable structural features without recourse to any questionable assumption. The value of $T_{\rm eff}$ is connected, using theory and scaling concepts, to the flow s…
Developing a macroscopic theory of elasto-plasticity in amorphous solids calls for (i) identifying the relevant macro state-variables and (ii) discriminating the different time-scales which characterize these variables. In current theories it is assumed that the stress reaches its elasto-plastic steady state value on the same time-scale as the configurational variables (be they the configurational energy, configurational entropy or the effective temperature). By examining numerical simulations in two and three dimensions we show that this is generally not the case, the configurational degrees …
Micro-plasticity theories and models are suitable to explain and predict mechanical response of devices on length scales where the influence of the carrier of plastic deformation - the dislocations - cannot be neglected or completely averaged out. To consider these effects without resolving each single dislocation a large variety of continuum descriptions has been developed, amongst which the higher-dimensional continuum dislocation dynamics (hdCDD) theory by Hochrainer et al. (Phil. Mag. 87, pp. 1261-1282) takes a different, statistical approach and contains information that are usually only …
The complex incremental behavior of granular materials is explored with multi-directional loading probes. An advanced discrete element model (DEM) was used to examine the reversible and irreversible strains for small loading probes, which follow an initial monotonic axisymmetric triaxial loading. The model used non-convex non-spherical particles and an exact implementation of the Hertz-like Cattaneo-Mindlin model for the contact interactions. Rectilinear true-triaxial probes were used in the study (i.e., no direct shear strain or principal stress rotation), with small strains increments of $2\…
This work presents a new modeling approach to macroscopic, polycrystalline elasto-plasticity starting from first principles and a few well-defined structural assumptions, incorporating the mildly rate-dependent (viscous) nature of plastic flow and the microscopic origins of plastic deformations. For the global dynamics, we start from a two-stage time-stepping scheme, expressing the fact that in most real materials plastic flow is much slower than elastic deformations, and then perform a detailed analysis of the slow-loading limit passage. In this limit, a rate-independent evolution can be expe…
We explore the nonlinear variational modelling of two-dimensional (2D) crystal plasticity based on strain energies which are invariant under the full symmetry group of 2D lattices. We use a natural parameterization of strain space via the upper complex Poincaré half-plane. This transparently displays the constraints imposed by lattice symmetry on the energy landscape. Quasi-static energy minimization naturally induces bursty plastic flow and shape change in the crystal due to the underlying coordinated basin-hopping local strain activity. This is mediated by the nucleation, interaction, and an…
The study is devoted to geometrically non-linear modelling of viscoplastic structures with residual stresses. We advocate and develop a special approach to residual stresses based on the transition between reference configurations. The finite strain kinematics of the viscoplastic material is modelled by the multiplicative decomposition of the deformation gradient tensor. Numerical algorithms originally developed for unstressed materials are extended to materials with pre-stresses. Owing to the weak invariance of constitutive equations, the incorporation of pre-stresses happens without addition…
The purpose of continuum plasticity models is to efficiently predict the behavior of structures beyond their elastic limits. The purpose of multiscale materials science models, among them crystal plasticity models, is to understand the material behavior and design the material for a given target. The current successful continuum hyperelastoplastic models are based in the multiplicative decomposition from crystal plasticity, but significant differences in the computational frameworks of both approaches remain, making comparisons not straightforward. In previous works we have presented a theory …
This study presents the applicability of conventional deep recurrent neural networks (RNN) to predict path-dependent plasticity associated with material heterogeneity and anisotropy. Although the architecture of RNN possesses inductive biases toward information over time, it is still challenging to learn the path-dependent material behavior as a function of the loading path considering the change from elastic to elastoplastic regimes. Our attempt is to develop a simple machine-learning-based model that can replicate elastoplastic behaviors considering material heterogeneity and anisotropy. The…
The mode I crack tip asymptotic response of a solid characterised by strain gradient plasticity is investigated. It is found that elastic strains dominate plastic strains near the crack tip, and thus the Cauchy stress and the strain state are given asymptotically by the elastic K-field. This crack tip elastic zone is embedded within an annular elasto-plastic zone. This feature is predicted by both a crack tip asymptotic analysis and a finite element computation. When small scale yielding applies, three distinct regimes exist: an outer elastic K field, an intermediate elasto-plastic field, and …
Necking instabilities, in which tensile (extensional) deformation localizes into a small spatial region, are generic failure modes in elasto-viscoplastic materials. Materials in this very broad class --- including amorphous, crystalline, polycrystalline and other materials --- feature a predominantly elastic response at small stresses, plasticity onset at a rather well-defined yield stress and rate-dependence. Necking instabilities involve a unique coupling between the system's geometry and its constitutive behavior. We consider generic elasto-viscoplastic constitutive relations involving an i…
In this paper we present a new general framework for anisotropic elastoplasticity at large strains. The new framework presents the following characteristics: (1) It is valid for non-moderate large strains, (2) it is valid for both elastic and plastic anisotropy, (3) its description in rate form is parallel to that of the infinitesimal formulation, (4) it is compatible with the multiplicative decomposition, (5) results in a similar framework in any stress-strain work-conjugate pair, (6) it is consistent with the principle of maximum plastic dissipation and (7) does not impose any restriction on…
We use considerations of energy balance and dissipation to derive a self-consistent version of the shear-transformation-zone (STZ) theory of plastic deformation in amorphous solids. The theory is generalized to include arbitrary spatial orientations of STZs. Continuum equations for elasto-plastic material and their energy balance properties are discussed.
We consider a thin soft adhesive interphase between dissimilar elastic media. The material of the intermediate layer is modelled by elasto-plastic pressure-sensitive constitutive law. An asymptotic procedure, together with a novel formulation of the deformation theory of plasticity for pressure-sensitive materials, is used in order to derive nonlinear transmission conditions for the corresponding imperfect zero-thickness interface. A FEM analysis of the original three-phase structure is performed to validate the transmission conditions for the simplified bimaterial structure.
In this paper the problem of a plane strain hydraulic fracture propagating in an elasto-plastic material is analyzed. A new stress redistribution model for the proximity of the fracture tip is formulated and a resulting plasticity-dependent crack propagation condition is introduced. A modified variant of the KGD problem that accounts for the plastic deformations in the near-tip zone only is proposed. It is demonstrated that this model can be a credible substitute for the full elasto-plastic hydraulic fracture problem in the case of moderate plastic deformation. The crack-tip shielding effect i…
We compare the macroscopic and the local plastic behavior of a model amorphous solid based on two radically different numerical descriptions. On the one hand, we simulate glass samples by atomistic simulations. On the other, we implement a mesoscale elasto-plastic model based on a solid-mechanics description. The latter is extended to consider the anisotropy of the yield surface via statistically distributed local and discrete weak planes on which shear transformations can be activated. To make the comparison as quantitative as possible, we consider the simple case of a quasistatically driven …
In situ elucidation of mechanisms governing crack transition to plasticity arrest
National Physical Laboratory（英国国家物理实验室）；Department of Materials, Royal School of Mines, Imperial College London（帝国理工学院皇家矿业学院材料系）；Department of Mechanical Engineering, Imperial College London（帝国理工学院机械工程系）
Finite plasticity theories are still a subject of controversy and lively discussions. Among the approaches to finite elastoplasticity two became especially popular. The first, implemented in the commercial finite element codes, is based on the introduction of a hypoelastic constitutive law and the additive elastic-plastic decomposition of the deformation rate tensor. Unfortunately, the use of hypoelasticity may lead to a nonphysical creation or dissipation of energy in a closed deformation cycle. In order to replace hypoelasticity with hyperelasticity the second popular approach based on the m…
We formulate a phenomenological elasto-plastic theory to describe a solid undergoing a structural transition from a square (p4mm) to an oblique (p2) lattice in two dimensions. Within our theory, the components of the strain may be decomposed additively into separate elastic and plastic contributions. The plastic strain, produced when the local stress crosses a threshold, is governed by a phenomenological equation of motion. We investigate the dynamics of shape of an initially square solid as it is cycled through a transformation protocol consisting of (1) a quench across the transition (2) def…
A number of current theories of plasticity in amorphous solids assume at their basis that plastic deformations are spatially localized. We present in this paper a series of numerical experiments to test the degree of locality of plastic deformation. These experiments increase in terms of the stringency of the removal of elastic contributions to the observed elasto-plastic deformations. It is concluded that for all our simulational protocols the plastic deformations are not localized, and their scaling is sub-extensive. We offer a number of measures of the magnitude of the plastic deformation, …
A system of partial differential equations (PDEs) is derived to compute the full-field stress from an observed kinematic field when the flow rule governing the plastic deformation is unknown. These equations generalize previously proposed equations that assume pure plastic behavior without elasticity. A method to numerically solve these equations is also presented. In addition to force balance, the equations are derived from the elastic-plastic decomposition of the deformation gradient, the assumption of isotropy, and the assumption that the function mapping the elastic strain to stress is kno…
Stressed dislocation pattern formation in crystal plasticity at finite deformation is demonstrated for the first time. Size effects are also demonstrated within the same mathematical model. The model involves two extra material parameters beyond the requirements of standard classical crystal plasticity theory. The dislocation microstructures shown are decoupled from deformation microstructures, and emerge without any consideration of latent hardening or constitutive assumptions related to cross-slip. Crystal orientation effects on the pattern formation and mechanical response are also demonstr…
The deformation and flow of disordered solids, such as metallic glasses and concentrated emulsions, involves swift localized rearrangements of particles that induce a long-range deformation field. To describe these heterogeneous processes, elastoplastic models handle the material as a collection of 'mesoscopic' blocks alternating between an elastic behavior and plastic relaxation, when they are too loaded. Plastic relaxation events redistribute stresses in the system in a very anisotropic way. We review not only the physical insight provided by these models into practical issues such as strain…
Fatigue crack advance induced by the application of cyclic quasistatic loads is investigated both numerically and analytically using a lattice spring model. The system has a quasi-one-dimensional geometry, and consists in two symmetrical chains that are pulled apart, thus breaking springs which connect them, and producing the advance of a crack. Quasistatic crack advance occurs as a consequence of the plasticity included in the springs which form the chains, and that implies a history dependent stress-strain curve for each spring. The continuous limit of the model allows a detailed analytical …
We propose an Eulerian thermodynamically compatible model for ideal plasticity of crystalline solids treated as a material flow through an adjustable crystal lattice space. The model is based on the additive splitting of the velocity gradient into the crystal lattice part and the plastic part. The approach extends a Gibbs-potential-based formulation developed by Rajagopal and Srinivasa for obtaining the response functions for elasto-visco-plastic crystals. The framework makes constitutive assumptions for two scalar functions: the Gibbs potential and the rate of dissipation. The constitutive eq…
We propose a theoretical framework for dealing with a transient polymer network undergoing small deformations, based on the rate of breaking and re-forming of network crosslinks and the evolving elastic reference state. In this framework, the characteristics of the deformed transient network at microscopic and macroscopic scales are naturally unified. Microscopically, the breakage rate of the crosslinks is affected by the local force acting on the chain. Macroscopically, we use the classical continuum model for rubber elasticity to describe the structure of the deformation energy, whose refere…
Plastic deformation in polycrystals is governed by the interplay between intra-granular slip and grain boundary-mediated plasticity. However, while the role played by bulk dislocations is relatively well-understood, the contribution of grain boundaries (GBs) has only recently begun to be studied. GB plasticity is known to play a key role along with bulk plasticity under a wide range of conditions, such as dynamic recovery, superplasticity, severe plastic deformation , etc., and developing models capable of simultaneously capturing GB and bulk plasticity has become a topic of high relevance. In…
Plastic yield of amorphous solids occurs by power law distributed slip avalanches whose universality is still debated. Determination of the power law exponents from experiments and molecular dynamics simulations is hampered by limited statistical sampling. On the other hand, while existing elasto-plastic depinning models give precise exponent values, these models to date have been limited to a scalar approximation of plasticity which is difficult to reconcile with the statistical isotropy of amorphous materials. Here we introduce for the first time a fully tensorial mesoscale model for the ela…
A new phenomenological model of cyclic creep is proposed which is suitable for applications involving finite creep deformations of the material. The model accounts for the the effect of the transient increase of the creep strain rate upon the load reversal. In order to extend the applicability range of the model, the creep process is fully coupled to the classical Kachanov-Rabonov damage evolution. As a result, the proposed model describes all the three stages of creep. Large strain kinematics is described in a geometrically exact manner using the assumption of a nested multiplicative split, o…
The plastic component of the deformation gradient plays a central role in finite kinematic models of plasticity. However, its characterization has been the source of extended debates in the literature and many important issues still remain unresolved. Some examples are the micromechanical understanding of F=FeFp with multiple active slip systems, the uniqueness of the decomposition, or the characterization of the plastic deformation without reference to the so-called intermediate configuration. In this paper, we shed some light to these issues via a two-dimensional kinematic analysis of the pl…
