Continuum Plasticity Theory Research Articles

A Transition State Theory-Based Continuum Plasticity Model Accounting for the Local Stress Fluctuation

Based on the transition state theory, a continuum plasticity theory is developed for metallic materials. Moreover, the nature of local stress fluctuation within a material point is considered by incorporating the probability distribution of the stresses. The model is applied to investigate the mechanical behaviors of 316 L stainless steel under various loading cases. The simulated results closely match the results obtained by the polycrystal plasticity model and experiments. The mechanical behaviors associated with strain rate sensitivity, temperature dependence, stress relaxation, and strain creep are correctly captured by the model. Furthermore, the proposed model successfully characterizes the Bauschinger effect, which is challenging to capture with a conventional continuum model without additional assumptions. The proposed model could be further employed in the design, manufacturing, and service of engineering components.

Criterion for void nucleation by particle debonding in metals obtained from molecular dynamics simulations

The critical conditions for void nucleation by particle debonding at a θ-particle in an aluminum matrix are identified through a comprehensive set of molecular dynamics simulations spanning a range of stress triaxialities and Lode parameter values. Specifically, it is determined that nucleation occurs when the local normal stress at the matrix–particle interface reaches a critical value of 8.14 GPa. When plasticity occurs in the matrix prior to nucleation, an additional plastic stress concentration is observed as a result of the plastic strain build up around the particle. Our results indicate that this plastic stress concentration factor increases roughly linearly with equivalent plastic strain, independent of triaxiality and Lode parameter. Comparing our results with the literature, we observe a much stronger influence of plastic strain than is predicted by continuum plasticity theory. We argue that this difference derives from the fact that nucleation is fundamentally driven by local stress hot spots resulting from the heterogeneity of plastic strain, while continuum theory is based on homogenized plastic strain fields. These results provide guidance for development of ductile fracture engineering models, while giving a warning regarding the use of continuum theory to predict damage initiation phenomena in metals.

Assessing continuum plasticity postulates with grain stress and local strain measurements in triaxially compressed sand

Critical state and continuum plasticity theories have been used in research and engineering practice in soil and rock mechanics for decades. These theories rely on postulated relationships between material stresses and strains. Some classical postulates include coaxiality between stress and strain rates, stress-dilatancy relationships, and kinematic assumptions in shear bands. Although numerical and experimental data have quantified the strains and grain kinematics in such experiments, little data quantifying grain stresses are available. Here, we report the first-known grain stress and local strain measurements in triaxial compression tests on synthetic quartz sands using synchrotron X-ray tomography and 3D X-ray diffraction. We use these data to examine the micromechanics of shear banding, with a focus on coaxiality, stress-dilatancy, and kinematics within bands. Our results indicate the following: 1) elevated deviatoric stress, strain, and stress ratios in shear bands throughout experiments; 2) coaxial principal compressive stresses and strains throughout samples; 3) significant contraction along shear bands; 4) vanishing volumetric strain but nonvanishing stress fluctuations throughout samples at all stages of deformation. Our results provide some of the first-known in situ stress and strain measurements able to aid in critically evaluating postulates employed in continuum plasticity and strain localization theories for sands.

Modeling dynamic formability of porous ductile sheets subjected to biaxial stretching: Actual porosity versus homogenized porosity

This paper investigates the effect of porous microstructure on the necking formability of ductile sheets subjected to dynamic in-plane stretching. We have developed an original approach in which finite element calculations which include actual void distributions obtained from additively manufactured materials are compared with simulations in which the specimen is modeled with the Gurson–Tvergaard continuum plasticity theory (Gurson, 1977; Tvergaard, 1982) which considers porosity as an internal state variable. A key point of this work is that in the calculations performed with the continuum model, the initial void volume fraction is spatially varied in the specimen according to the void distributions included in the simulations with the actual porous microstructure. The finite element computations have been carried out for different loading conditions, with biaxial strain ratios ranging from 0 (plane strain) to 0.75 (biaxial tension) and loading rates varying between 10000s−1 and 60000s−1. We have shown that for the specific porous microstructures considered, the necking forming limits obtained with the Gurson–Tvergaard continuum model are in qualitative agreement with the results obtained with the calculations which include the actual void distributions, the quantitative differences for the necking strains being generally less than ≈25% (the calculations with actual voids systematically predict greater necking strains). In addition, the spatial distribution of necks formed in the sheets at large strains is very similar for the actual porosity and the homogenized porosity models. The obtained results demonstrate that the voids promote plastic localization, acting as preferential sites for the nucleation of fast growing necks. Moreover, the simulations have provided individualized correlations between void volume fraction, maximum void size and necking formability, and highlighted the influence of the heterogeneity of the spatial distribution of porosity on plastic localization.

Elastoplastic source model for microseismicity and acoustic emission

SUMMARYThe microseismic events can often be characterized by a complex non-double couple source mechanism. Recent laboratory studies recording the acoustic emission during rock deformation help connecting the components of the seismic moment tensor with the failure process. In this complementary contribution, we offer a mathematical model which can further clarify these connections. We derive the seismic moment tensor based on classical continuum mechanics and plasticity theory. The moment tensor density can be represented by the product of elastic stiffness tensor and the plastic strain tensor. This representation of seismic sources has several useful properties: (i) it accounts for incipient faulting as a microseismicity source mechanism, (ii) it does not require a pre-defined fracture geometry, (iii) it accounts for both shear and volumetric source mechanisms, (iv) it is valid for general heterogeneous and anisotropic rocks and (v) it is consistent with elasto-plastic geomechanical simulators. We illustrate the new approach using 2-D numerical examples of seismicity associated with cylindrical openings, analogous to wellbore, tunnel or fluid-rich conduit and provide a simple analytic expression of the moment density tensor. We compare our simulation results with previously published data from laboratory and field experiments. We consider four special cases corresponding to ‘dry’ elastically homogeneous and elastically heterogeneous isotropic rocks, ‘dry’ transversely isotropic rocks and ‘wet’ isotropic rocks. The model highlights theoretical links between stress state, geomechanical parameters and conventional representations of the moment tensor such as Hudson source type parameters.

Stabilization of the response of cyclically loaded lattice spring models with plasticity

This paper develops an analytic framework to design both stress-controlled and displacement-controlledT-periodic loadings which make the quasistatic evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a functiont↦(e(t),p(t)), whereei(t) is the elastic elongation andpi(t) is the relaxed length of springi, defined on [t0,∞) by the initial condition (e(t0),p(t0)). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedronC(t) in a vector spaceEof dimensiond, it becomes natural to expect (based on a result by Krejci) that the elastic componentt↦e(t) always converges to aT-periodic function ast→∞. The achievement of this paper is in spotting a class of loadings where the Krejci’s limit doesn’t depend on the initial condition (e(t0),p(t0)) and so all the trajectories approach the sameT-periodic regime. The proposed class of sweeping processes is the one for which the normals of anyddifferent facets of the moving polyhedronC(t) are linearly independent. We further link this geometric condition to mechanical properties of the given network of springs. We discover that the normals of anyddifferent facets of the moving polyhedronC(t) are linearly independent, if the number of displacement-controlled loadings is two less the number of nodes of the given network of springs and when the magnitude of the stress-controlled loading is sufficiently large (but admissible). The result can be viewed as an analogue of the high-gain control method for elastoplastic systems. In continuum theory of plasticity, the respective result is known as Frederick-Armstrong theorem.

Spin and vorticity with vanishing rigid-body rotation during shear in continuum mechanics

A long-standing challenge in continuum mechanics has been how to separate shear deformation and corresponding shape changes from the rotation of a continuum. This can be obtained by a new decomposition of the spin tensor, i.e. of the skew part of the velocity gradient, into two parts, where one of them vanishes during shear flows. The same decomposition applies to the vorticity vector. In both cases, the two spin components are interpreted as generating plastic shear deformation and rigid body rotation. In continuum plasticity theories, the suggested rotational part of the spin tensor can be applied to avoid spurious behavior of the objective Lie derivatives of second order tensors, e.g. of the stress tensor. It provides a history-independent spin corresponding to a time-averaged angular velocity of the rotating line segments in a small homogeneous volume. In fluid mechanics, the new spin component can be used to quantify vortexes in shear flows and turbulent structures, and it provides a sound interpretation and generalization of the Δ and swirling-strength criteria for visualization of vortices.

Gradient Crystal Plasticity: A Grain Boundary Model for Slip Transmission.

Interaction between dislocations and grain boundaries (GBs) in the forms of dislocation absorption, emission, and slip transmission at GBs significantly affects size-dependent plasticity in fine-grained polycrystals. Thus, it is vital to consider those GB mechanisms in continuum plasticity theories. In the present paper, a new GB model is proposed by considering slip transmission at GBs within the framework of gradient polycrystal plasticity. The GB model consists of the GB kinematic relations and governing equations for slip transmission, by which the influence of geometric factors including the misorientation between the incoming and outgoing slip systems and GB orientation, GB defects, and stress state at GBs are captured. The model is numerically implemented to study a benchmark problem of a bicrystal thin film under plane constrained shear. It is found that GB parameters, grain size, grain misorientation, and GB orientation significantly affect slip transmission and plastic behaviors in fine-grained polycrystals. Model prediction qualitatively agrees with experimental observations and results of discrete dislocation dynamics simulations.

Dynamic rupture propagation on fault planes with explicit representation of short branches

An active fault zone is home to a plethora of complex structural and geometric features that are expected to affect earthquake rupture nucleation, propagation, and arrest, as well as interseismic deformation. Simulations of these complexities have been largely done using continuum plasticity or scalar damage theories. In this paper, we use a highly efficient novel hybrid finite element-spectral boundary integral equation scheme to investigate the dynamics of fault zones with small scale pre-existing branches as a first step towards explicit representation of anisotropic damage features in fault zones. The hybrid computational scheme enables exact near-field truncation of the elastodynamic field allowing us to use high resolution finite element discretization in a narrow region surrounding the fault zone that encompasses the small scale branches while remaining computationally efficient. Our results suggest that the small scale branches may influence the rupture in ways that may not be realizable in homogenized continuum models. Specifically, we show that these short secondary branches significantly affect the post event stress state on the main fault leading to strong heterogeneities in both normal and shear stresses and also contribute to the enhanced generation of high frequency radiation. The secondary branches also affect off-fault plastic strain distribution and suggest that co-seismic inelasticity is sensitive to pre-existing damage features. We discuss our results in the larger context of the need for modeling earthquake ruptures with high resolution fault zone physics.

Strain gradient continuum plasticity theories: Theoretical, numerical and experimental investigations

Over the last three decades, a number of researchers have used the gradient-enhanced plasticity theory to model the microstructural material behavior, which is not able to be captured by the conventional plasticity theory, within the continuum based framework. This paper aims to review the strain gradient continuum plasticity in the context of the theoretical, experimental and numerical investigations. A review of the theoretical developments by the several research groups and their applications to the finite element method is presented in this paper. In addition, a review of the various experimental methodologies to study the size effects is presented. Lastly, some open issues in the gradient-enhanced plasticity such as the physical nature of the strain gradients and the physical interpretation of the material length scales are discussed.

Small scale volume formulation based on coupled thermo-mechanical gradient enhanced plasticity theory

A coupled thermo-mechanical gradient enhanced continuum plasticity theory is developed within the thermodynamically consistent framework in this work and the corresponding two-dimensional finite element implementation is carried out to examine the micro-mechanical and thermal characteristics of small-scale metallic volumes. The proposed model is conceptually based on the dislocations interaction mechanisms and thermal activation energy. The thermodynamic conjugate microstresses are decomposed into dissipative and energetic components, correspondingly, the dissipative and energetic length scales are incorporated in the proposed model and an additional length scale related to the geometrically necessary dislocations-induced strengthening is also included. Not only the partial heat dissipation caused by the fast transient time, but also the distribution of temperature caused by the transition from the plastic work to the heat, are included into the coupled thermo-mechanical model by deriving a generalized heat equation. The derived constitutive framework and two-dimensional finite element model are validated through the comparison with the experimental observations conducted on micro-scale thin films. The proposed enhanced model is examined by solving the simple shear problem and the square plate problem respectively to explore the thermo-mechanical characteristics of small-scale metallic materials. Finally, some significant conclusions are presented.

System size effects on the mechanical response of cohesive-frictional granular ensembles

Shear resistance in granular ensembles is a result of interparticle interaction and friction. However, even the presence of small amounts of cohesion between the particles changes the landscape of the mechanical response considerably. Very often such cohesive frictional (c-ϕ ) granular ensembles are encountered in nature as well as while handling and storage of granular materials in the pharmaceutical, construction and mining industries. Modeling of these c-ϕ materials, especially in engineering applications have relied on the oft-made assumption of a “continua” and have utilized the popular tenets of continuum plasticity theory. We present an experimental investigation on the fundamental mechanics of c-ϕ materials specifically; we investigate if there exists a system size effect and any additional length scales beyond the continuum length scale on their mechanical response. For this purpose, we conduct a series of 1-D compression (UC) tests on cylindrical specimens reconstituted in the laboratory with a range of model particle–binder combinations such as sandcement, sand-epoxy, and glass ballotini-epoxy mixtures. Specimens are reconstituted to various diameters ranging from 10 mm to 150 mm (with an aspect ratio of 2) to a predefined packing fraction. In addition to the effect of the type of binder (cement, epoxy) and system size, the mean particle size is also varied from 0.5 to 2.5 mm. The peak strength of these materials is significant as it signals the initiation of the cohesive-bond breaking and onset of mobilization of the inter particle frictional resistance. For these model systems, the peak strength is a strong function of the system size of the ensemble as well as the mean particle size. This intriguing observation is counter to the traditional notion of a continuum plastic typical granular ensemble. Microstructure studies in a computed-tomograph have revealed the existence of a web patterned ‘entangled-chain’ like structure, we argue that this ushers an additional length scale as well as presents a system size effect.

A minimal gradient-enhancement of the classical continuum theory of crystal plasticity. Part I: The hardening law

A simple gradient-enhancement of the classical continuum theory of plasticity of single crystals deformed by multislip is proposed for incorporating size effects in a manner consistent with phenomenological laws established in materials science. Despite considerable efforts in developing gradient theories, there is no consensus regarding the minimal set of physically based assumptions needed to capture the slip-gradient effects in metal single crystals and to provide a benchmark for more refined approaches. In order to make a step towards such a reference model, the concept of the tensorial density of geometrically necessary dislocations generated by slip-rate gradients is combined with a generalized form of the classical Taylor formula for the flow stress. In the governing equations in the rate form, the derived internal length scale is expressed through the current flow stress and standard parameters so that no further assumption is needed to define a characteristic length. It is shown that this internal length scale is directly related to the mean free path of dislocations and possesses physical interpretation which is frequently missing in other gradient-plasticity models.

Strong discontinuity analysis of a class of anisotropic continuum damage constitutive models – Part I: Theoretical considerations

Several modeling techniques aiming at considering cracks as kinematics discontinuities have been proposed for the past years. Within this scope, the embedded finite element method (E-FEM) was introduced a couple of years ago. Among the features of this approach, it has been shown that a kinematic enhancement of the displacement field allows constructing a discrete model (expressed in terms of traction vector–displacement jump) from any continuous model (expressed in terms of stress–strain). This result has been rigorously established if the continuous model is formulated within the framework of either isotropic continuum damage or plasticity theories. The objectives of this study are (i) to extend this result in case where the continuous model belongs to a class of anisotropic continuum damage constitutive models and (ii) to show the main features of a specific traction/separation law derived from the aforementioned class of constitutive models through several numerical case-studies. In this paper, the light is put on the theoretical considerations which allow deriving discrete models in a consistent way.

Multipole expansion of continuum dislocations dynamics in terms of alignment tensors

The development of a dislocation-based continuum theory of plasticity remains one of the central challenges of applied physics and materials science. Developing a continuum theory of dislocations requires the solution of two long-standing problems: (i) to find a faithful representation of dislocation kinematics with a reasonable number of variables and (ii) to derive averaged descriptions of the dislocation dynamics (i.e. material laws) in terms of these variables. In the current paper, we solve the first problem, i.e. we develop tensorial conservation laws for distributions of oriented lines. This is achieved through a multipole expansion of the dislocation density in terms of so-called alignment tensors containing information on the directional distribution of dislocation density and dislocation curvature. A hierarchy of evolution equations of these tensors is derived from a higher dimensional dislocation density theory. Low-order closure approximations of this hierarchy lead to continuum dislocation dynamics models of plasticity with only few internal variables. Perspectives for more refined theories and current challenges in dislocation density modelling are discussed.

Strain gradient plasticity for amorphous and crystalline polymers with application to micro- and nano-scale deformation analysis

Microscale mechanical responses of semicrystalline glassy polymers are of utmost importance when these materials are deployed in microstructural applications, e.g. MEMS and NEMS. The classical continuum plasticity theories become ill-posed in the case of highly localized deformations and they result in mesh dependent Finite Element (FE) solutions. Then the continuum theories may not directly be applicable to study stress–strain responses in nano- and micro-scale structures. The non-local modeling approach is one solution to this deficiency in which the non-local effects, associated with the presence of highly localized plastic deformation fields, are integrated into the constitutive relations. The microstructural information together with the intrinsic material length scale are incorporated in these theories to take into account the non-local effects and to ensure the well-posedness of the governing equations. While non-local theories have been extensively investigated for Metals (class M materials), this work aims to develop a rate dependent plasticity theory with strain gradient effects for semicrystalline Polymers (class P materials).

A continuum model for dislocation dynamics incorporating Frank–Read sources and Hall–Petch relation in two dimensions

One of the main targets in the development of dislocation based continuum crystal plasticity theories is to establish continuum constitutive relations which approximately summarize the underlying discrete dislocation dynamics (DDD). However, rigorously transiting from discrete to continuum in describing the evolution of dislocation system is extremely challenging for complex networks of curved dislocations and their interactions at multiple length scales. To address this difficulty, a coarse-grained disregistry function (CGDF) was proposed to represent the continuous distributions of curved dislocations. In this paper, we present a dislocation based continuum model for crystal plasticity incorporating the Frank–Read sources, which serves as a crucial step towards systematically building a three-dimensional dislocation based continuum plasticity theory. The continuum model is derived accurately from the DDD model, and is validated by comparisons of the results with theoretical predictions and DDD simulations conducted under the same conditions. Furthermore by considering dislocation loop pileups within a rectangular grain, we derive analytical formulas which generalize the traditional Hall–Petch relation into two dimensions without any adjustable parameters. It is shown that the yield stress of a rectangular grain depends not only on the grain size, but also on the grain aspect ratio whose exact form is associated with the harmonic mean of the length and width of the rectangle. The derived formulas of the yield stress are shown in excellent agreement with the results by our continuum model and DDD simulations.

Elastoplastic modeling of polymeric composites containing randomly located nanoparticles with an interface effect

An elastoplastic constitutive model is proposed to predict the overall behavior of nanoparticle-reinforced polymeric composites. The effective elastic moduli of nanocomposites, composed of a polymer matrix and randomly dispersed nanoparticles, are constructed by incorporating the Eshelby tensor considering the interface effect into a micromechanics-based ensemble volume-averaged method. Micromechanical homogenization procedures are utilized to estimate an effective yield function in accordance with the continuum plasticity theory and are employed to predict the overall elastoplastic behavior. The effects of the particle size, interface moduli and the strengthening influence of the nanoparticles are investigated via numerical simulations. Finally, comparisons between theoretical predictions and the available experimental data are made to assess the predictive capability of the proposed framework.

Moving dislocations in finite plasticity: a topological approach

AbstractPlastic deformation of crystals is mostly mediated by the motion of dislocations. During the last two decades a lot of effort was directed towards including more knowledge about dislocations in continuum descriptions of plasticity. Promising approaches towards building continuum plasticity theories on averages of the behavior of many single dislocations have been formulated under the assumption of small deformations. In the current paper we derive the kinematics of single dislocations moving inside a dislocated crystal simultaneously deforming by the motion of other dislocations in the language of large deformation plasticity. The evolution equation of a single dislocation is connected to the formation of kinks and jogs due to cutting by other dislocations and is shown to parallel the evolution equation of the dislocation density tensor in finite deformation formulation. Implications for dislocation based modeling of plasticity are discussed.

Free-surface enhanced continuum model predicts size-effect for pillar compression at micro- and nano-scale

Abstract It has recently been observed that micro- and nano-sized pillars under uniaxial compression and tension attain a higher flow-stress than samples of macroscopic size. Here it is hypothesized that such a mechanical size-effect is driven by the interaction of dislocations with the free surfaces. An extended continuum plasticity theory is presented to account for such an effect by considering the free-surface imagestresses on the dislocation motion immediately following dislocation nucleation. The model employs only fundamental material parameters, Young's modulus, Poisson's ratio, bulk yield-strength, and the Burgers vector. The model predicts the magnitude of the size-effects observed in the compression of Au and Cu micro- and nano-pillars.