Simple Shear Problem Research Articles

A novel continuum dislocation density field-based crystal plasticity theory

In this work, a novel dislocation density field-based crystal plasticity formulation, that incorporates up-scaled continuum dislocation density fields to represent all possible characters of the dislocation density, is presented. The continuum dislocation field theory, formulated assuming large strain kinematics, is based on an all-dislocation concept, whereby individual dislocation density types are described as vector fields. The evolutionary behaviour of the dislocation density fields is defined in terms of a glide component derived from the classical general conservation law for dislocation density fields proposed originally by Kroner and Acharya, a rotation one given by the curl of the dislocation velocity vector as proposed originally by Ngan and co-workers, and a statistically stored dislocation source/sink component.The proposed formulation has been numerically implemented within the finite element method using a modified up-wind stabilisation approach, which results on eight extra independent nodal degrees of freedom per slip system, associated with the upscaled dislocation densities of pure edge and screw character. Furthermore, the required boundary conditions are simple: initial total dislocation density and dislocation flux or velocity, without the need for higher order boundary conditions. Full details of the numerical implementation of the crystal plasticity theory are given.The theory is then used to investigate classical boundary value problems involving 1D and 2D dislocation pile-ups, and the development of boundary layers in a deforming single crystal strip under constrained conditions. The main general understanding that emerges from the case studies is that, as either the dislocation pile-ups or the boundary layers develop with deformation, geometrically necessary dislocations are generated and lead to an overall behaviour that is size-dependent. Furthermore, it is shown that, in the absence of sources or sinks, which are statistically stored in nature, that total number of dislocations in the pile-up problems is preserved. The numerical predictions of the 1D pile-up and strip simple shear problems are in good agreement with published solutions.

Viscoplastic simple shear at finite strains

The equations governing the simple shear deformation of an incompressible inelastic material undergoing finite strain are derived in this paper. The constitutive assumptions are kept in their most general form to allow the incorporation of widely used viscoplastic or viscoelastic models from the literature. It is shown that, while for a hyperelastic material the simple shear problem is completely determined by a single parameter, the amount of shear, in the viscoplastic case, the elastic deformation is the superposition of a triaxial stretch and a simple shear, whose determination requires the solution of three coupled nonlinear evolution equations. We evaluate such a solution for different material models and compare it with three-dimensional finite element simulations to assess its accuracy. We further assess the performance of these models using experimental data from filled rubber, focusing on their ability to capture the observed behaviour, such as the well-known Payne effect. Additionally, we extend our simple shear solution to address torsion and the extension of thin-walled cylinders. These derivations and analyses offer valuable insights for experimentalists engaged in the mechanical characterization of soft materials.

Non-Quadratic Strain Gradient Plasticity Theory and Size Effects in Constrained Shear

Abstract A previously proposed strain gradient plasticity theory is extended to incorporate a non-quadratic power law function of the plastic strain gradient in the free energy expression with an exponent of N + 1. The values of N are taken to vary from N = 1 to N = 0. A simple shear problem of a metal layer between rigid boundaries is analyzed. Two stages of plastic deformation are considered. In stage I, the plastic strain is taken to be zero at the boundaries. Stage I ends when a specified magnitude of the plastic strain gradient is attained at the boundaries. In stage II, the magnitude of the plastic strain gradient at the boundaries is fixed at the specified value. With N = 0, a critical plastic strain gradient cannot be specified at the boundaries because the plastic strain gradient is infinite at the boundaries. The theory with N = 0 predicts a constant plateau stress immediately after initial yield, and the dependence of the plateau stress on the layer thickness can fit experimentally observed plateau stress values. However, with N = 0, a stress gap occurs between the initial yield stress and the plateau stress. The theory with 0 < N ≤ 1 and with stage II also can reproduce the experimentally observed dependence of the plateau stress on the layer thickness for any value of N in that range, with an appropriate value of critical plastic strain gradient at the boundaries. The solution for 0 < N ≤ 1 includes that for N = 0 as a limiting case.

An Eulerian thermodynamical formulation of size-dependent plasticity

The objective of this work is to develop a finite-deformation, thermodynamically-consistent theory to model the size-dependent irreversible behavior of metals at the micron-scale. In contrast with other approaches, the proposed Eulerian formulation introduces an evolution equation directly for an elastic distortional deformation measure without the need for defining total or plastic deformation measures. Motivated both by experimental observations and by the literature on generalized continua, the proposed theory assumes tensorial fields separately describing macro-plasticity and micro-plasticity. An additional evolution equation defines a Nye–Kröner-like tensor that depends on the curl of the non-symmetric micro-plasticity distortional rate. By drawing inspiration from higher-order strain gradient plasticity, the micro-plasticity distortional rate is determined by a micro-plasticity balance equation with associated boundary conditions. A quadratic function of the Nye–Kröner-like tensor contributes to the Helmholtz free energy, which leads to an energetic stress entering the micro-plasticity balance that controls the material size-dependent response. The theory is designed to allow the onset of micro-plasticity to occur at a stress level below that triggering macro-plasticity and, regarding size-effects, the response characterized by macro-plasticity alone is the strongest one. In addition, the formulation uses a smooth elastic–plastic transition model for rate-independent response. The theory is specialized to the case of small strains and rotations and applied to the constrained simple shear problem, which allows the derivation of an analytical solution able to demonstrate the predictive capabilities ensuing from the rich interplay between micro- and macro-plasticity.

Constitutive modeling of magneto-viscoelastic polymers, demagnetization correction, and field-induced Poynting effect

Magneto-active polymers are polymer-based composites consisting of magnetizable micro-particles embedded in an elastomeric matrix material. The presence of magnetic particles provides strong tunability to the stiffness and damping properties of the polymeric composites under the magnetic field. We propose here a novel free energy-based phenomenological modeling for magneto-active polymers. Coupled mechanical and magnetization constitutive equations are derived by considering magneto-viscoelasticity in a finite deformation framework. The model calibration takes into account the demagnetization effect, which depends on the specimen size, shape, and its own magnetization. The model prediction is verified with the available experimental data. Finally, a coupled simple shear problem is solved to demonstrate the influence of magnetic field on the Poynting effect. Classical nonlinear soft elastic material shows positive Poynting effect, i.e., shearing planes try to expand. We found the possibilities of field-induced reverse or negative Poynting effect in shear for magneto-active polymers. Such a negative Poynting effect could potentially affect a very diverse range of applications from the performances of miniature sensors and actuators to controlled drug delivery.

Analytical solutions of the simple shear problem for micromorphic models and other generalized continua

To draw conclusions as regards the stability and modelling limits of the investigated continuum, we consider a family of infinitesimal isotropic generalized continuum models (Mindlin–Eringen micromorphic, relaxed micromorphic continuum, Cosserat, micropolar, microstretch, microstrain, microvoid, indeterminate couple stress, second gradient elasticity, etc.) and solve analytically the simple shear problem of an infinite stripe. A qualitative measure characterizing the different generalized continuum moduli is given by the shear stiffness $$\mu ^{*}$$ . This stiffness is in general length-scale dependent. Interesting limit cases are highlighted, which allow to interpret some of the appearing material parameter of the investigated continua.

Family of Continuous Strain-Consistent Convective Tensor Rates and Its Application in Hooke-Like Isotropic Hypoelasticity

This paper presents a new family of objective (Lagrangian and Eulerian) continuous strain-consistent convective (corotational and non-corotational) tensor rates. Objective strain-consistent convective tensor rates are defined as having the property that there exist objective (Lagrangian and Eulerian) strain tensors from the Hill family such that the considered rates of these strain tensors are equal to the rotated/standard (Lagrangian/Eulerian) stretching (strain rate) tensors. The family of such Eulerian strain-consistent convective tensor rates was introduced by Bruhns et al. (Proc. R. Soc. Lond. A 460:909–928, 2004). On the one hand, the family of continuous strain-consistent convective tensor rates presented in this paper extends the Bruhns et al. family by including Lagrangian tensor rates and, on the other hand, it is narrower than the Bruhns et al. family due to the continuity requirement imposed on tensor rates, which is necessary for applications. The theorem that any strain tensors only from the Doyle–Ericksen family (which is a subfamily of the Hill family) provide sufficient continuity conditions for strain-consistent convective tensor rates was formulated and proved. The expressions obtained by proving this theorem for convector tensors show that each tensor from the Doyle–Ericksen family is associated with a single strain-consistent convective tensor rate (in the Bruhns et al. family, any strain tensor from the Hill family generates infinitely many convector tensors associated with infinitely many strain-consistent convective tensor rates). In addition, a new family of Hooke-like isotropic hypoelastic material models based on objective continuous strain-consistent convective rates of the rotated/standard (Lagrangian and Eulerian) Kirchhoff stress tensors was constructed. The theorem that any material model from this family is self-consistent provided that the Lame parameter $\lambda =0$ and/or the deformation of the body is isochoric was formulated and proved. By self-consistent hypoelastic material models are meant those models for which constitutive hypoelastic relations are counterparts of constitutive relations for Cauchy/Green elasticity. Some material models from the new family were tested by solving the simple shear problem. Both new and well-known solutions of this problem for material models were obtained using an approach that takes into account the self-consistency property of material models from the new family.

On instability of constitutive models for isotropic elastic-perfectly plastic material behaviour at finite deformations

The main goal of this work is to test the possibility of a newly introduced constitutive law to model the behaviour of the isotropic elastic-perfectly plastic material which is exposed to large elastoplastic deformations. The proposed constitutive relation is based on the hypo-elastic relation and the inelastic INTERATOM model. The verification of the model is done by its implementation into the commercial software ABAQUS/Standard via the user subroutine UMAT. For that purpose, the large simple shear problem is studied where selected objective corotational rates, i.e. the logarithmic rate, the Jaumann rate and the Green-Naghdi rate, are individually implemented in the aforementioned constitutive relations. The obtained results are compared mutually and with the relevant literature. The proposed constitutive model is also used to test the behaviour of the part of a real engineering structure, i.e. a seismic isolator, in order to obtain the correct input data for further analysis of superstructure behaviour due to seismic excitation.

Size Scaling of Plastic Deformation in Simple Shear: Fractional Strain-Gradient Plasticity and Boundary Effects in Conventional Strain-Gradient Plasticity

Abstract A recently developed model based on fractional derivatives of plastic strain is compared with conventional strain-gradient plasticity (SGP) models. Specifically, the experimental data and observed model discrepancies in the study by Mu et al. (2016, “Dependence of Confined Plastic Flow of Polycrystalline Cu Thin Films on Microstructure,” MRS Com. Res. Let. 20, pp. 1–6) are considered by solving the constrained simple shear problem. Solutions are presented both for a conventional SGP model and a model extension introducing an energetic interface. The interface allows us to relax the Dirichlet boundary condition usually assumed to prevail when solving this problem with the SGP model. We show that the particular form of a relaxed boundary condition does not change the underlying size scaling of the yield stress and consequently does not resolve the scaling issue. Furthermore, we show that the fractional strain-gradient plasticity model predicts a yield stress with a scaling exponent that is equal to the fractional order of differentiation.

Numerical investigations of strain-gradient plasticity with reference to non-homogeneous deformations

In this work, a higher-order irrotational strain gradient plasticity theory is studied in the small strain regime. A detailed numerical study is based on the problem of simple shear of a non-homogeneous block comprising an elastic-plastic material with a stiff elastic inclusion. Combinations of micro-hard and micro-free boundary conditions are used. The strengthening and hardening behaviour is explored in relation to the dissipative and energetic length scales. There is a strong dependence on length scale with the imposition of micro-hard boundary conditions. For micro-free conditions there is marked dependence on dissipative length scale of initial yield, though the differences are small in the post-yield regime. In the case of hardening behaviour, the variation with respect to energetic length scale is negligible. A further phenomenon studied numerically relates to the global nature of the yield function for the dissipative problem; this function is given as the least upper bound of a function of plastic strain increment, and cannot be determined analytically. The accuracy of an upper-bound approximation to the yield function is explored, and found to be reasonably sharp in its prediction of initial yield.

ФЕНОМЕНОЛОГИЧЕСКАЯ МОДЕЛЬ ГИПЕРВЯЗКОУПРУГОСТИ ЭЛАСТОМЕРОВ ПРИ НЕИЗМЕННОЙ ОРИЕНТАЦИИ ГЛАВНЫХ ОСЕЙ НАПРЯЖЕНИЙ В СОПУТСТВУЮЩЕЙ СИСТЕМЕ ОТСЧЕТА

The Bergstrom-Boyce hyper-viscoelasticity model, which is based on considerations of the microstructure of rubber-like materials and uses the multiplicative decomposition of the total strain gradient, is considered and analyzed. Special attention is paid to the choice of the single-valuedness condition, which determines the rotation of the intermediate (relaxed) configuration and ensures the uniqueness of the multiplicative decomposition of the gradient total strain to the gradient of elastic deformations and the gradient of viscous deformations. To verify the validity of the statement that this choice is not essential, we found a solution of the test problem of simple shear according to the Bergstrom-Boyce model is obtained for the three most frequently used single-valuedness conditions. The results of numerical calculations showed a significant discrepancy for dynamic stresses and a less significant discrepancy for total stresses. To separate the permissible single-valuedness conditions from the physically unacceptable single-valuedness conditions, it is proposed to use the principle of the material frame-indifference. For this purpose, of we studied the transformation of the elastic deformations and viscous deformations gradients when replacing the reference system, to which there is no single point of view in the scientific literature. For the most frequently proposed and used single-valuedness conditions, it is shown which of them do not depend on the reference system choice. Since the list of such kind of allowable ratios can be expanded many times, the choice a single-valuedness condition should be solved in the same way as it is done, e.g. for elastic potentials: a proper formulation and conducting experimental studies, or a theoretical study of the material microstructure. A phenomenological model of hyper-viscoelasticity is proposed, based on a one-dimensional Kelvin-Poynting model and limited to the case when the main axes of stresses and strains (full, elastic and viscous ones) coincide and do not change their orientation relative to the material lines (fibers). This ensures the uniqueness of the corresponding multiplicative decomposition. In order to expand the range of strain rates when describing experimental data, the dependence of the viscosity coefficient on the second invariant of the right Cauchy-Green viscous strain measure in the power law of the apparent viscosity of the Reiner-Rivlin model is taken into account, which generalizes the corresponding dependence of the Bergstrom-Boyce model. The developed mathematical model of hyper-viscoelasticity of rubber-like materials is intended for the stress-strain state calculation of highly elastic shells of rotation under symmetric loading.

Objective Symmetrically Physical Strain Tensors, Conjugate Stress Tensors, and Hill’s Linear Isotropic Hyperelastic Material Models

We introduce a new family of strain tensors—a family of symmetrically physical (SP) strain tensors—which is also a subfamily of the well-known Hill family of strain tensors. For the further analysis, five scale functions are chosen which generate strain tensors belonging to the families of strain tensors previously introduced by other authors (i.e., the Doyle–Ericksen, Curnier–Rakotomanana, Curnier–Zysset, Itskov, and Darijani–Naghdabadi families) and to the new family of SP strain tensors. In particular, these five scale functions include the scale function generating the Lagrangian and Eulerian Hencky strain tensors. We introduce the family of SPH models of isotropic hyperelastic materials (with Hill’s linear relations) which are generated by SP strain tensors and work-conjugate stress tensors based on Hill’s natural generalization of Hooke’s law. Five SPH models of isotropic hyperelastic materials are generated on the basis of chosen SP strain tensors and work-conjugate stress tensors. These models are tested by solving two problems with homogeneous strain and stress tensors fields: the simple elongation and simple shear problems. Analysis of these solutions shows that the solutions of both problems for the Hencky isotropic hyperelastic material model (one of the five generated SPH models of isotropic hyperelastic materials) are qualitatively different from the solutions for the remaining four material models. That is, the solutions using the Hencky isotropic hyperelastic material model are of yielding nature typical of inelastic deformation of metals whereas the solutions for the other four material models reproduce strain diagrams typical of rubber.

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.

Use of a one-parameter family of Gordon–Schowalter objective derivatives to describe finite strains of viscoelastic bodies

A constitutive relation is considered for viscoelastic materials at finite strains. This relation is obtained using a one-parameter family of Gordon–Schowalter objective derivatives and generalizes the elementary Maxwell model. It is shown that, in the problem of simple shear of an incompressible viscoelastic material, this constitutive relation allows one to obtain the Poynting effect for any parameters of the model.

Strain Gradient Plasticity: An Application to Plastic Flow Localization Analysis

In this paper, strain gradient plasticity theory is extended to include the corner-like effect that is inherent in crystal plasticity. The predictive feature of the extended theory is examined via finite element analysis of a constrained simple shear problem and a plane-strain tension problem involving plastic flow localization. Numerical issues with respect to finite element formulations are also discussed.

A Natural Generalization of Linear Isotropic Relations with Seth-Hill Strain Tensors to Transversely Isotropic Materials at Finite Strains

Hooke’s law was naturally generalized to finite strains by Hill in 1978, by introducing the Seth-Hill strain and its conjugate stress. This paper presents the transversely isotropic relations, which are not only a natural extension of Hill’s theory from isotropic materials to transversely isotropic materials, but also the natural generalization of the transversely isotropic Hooke’s law from infinitesimal strains to moderate strains. This generalization introduces a class of transversely isotropic hyperelastic models, which are adopted to investigate the uniaxial stretch and the simple shear problems. Results show that the material responses for different constitutive equations are significantly different; the stiffening or softening behaviors of materials at moderate deformations can be described by the appropriate model with proper material parameters.

A strain-gradient plasticity theory with a corner-like effect: a thermodynamics-based extension

A higher-order strain gradient plasticity theory with a corner-like effect, which was recently proposed by the author, is updated to a form suitable for its thermodynamics-based extensions. It is assumed that the free energy is augmented by a defect energy that has a non-quadratic dependence on the plastic strain gradients. Predictive feature of the extended theory is examined via finite element analyses of a constrained simple shear problem and a plane strain tension problem involving plastic flow localization.

A higher-order strain gradient plasticity theory with a corner-like effect

A corner-like plasticity model originally proposed as a size-independent theory is extended to include a size effect resulting from plastic strain gradients. A method of solving boundary value problems at finite strains is also presented. The efficiency of the new theory is demonstrated through two typical numerical examples: a constrained simple shear problem in which an infinitely long strip bounded by two hard materials is subjected to large shear under plane strain conditions, and a problem of shear band formation in plane strain tension.

A numerical basis for strain-gradient plasticity theory: Rate-independent and rate-dependent formulations

A numerical model formulation of the higher order flow theory (rate-independent) by Fleck and Willis [2009. A mathematical basis for strain-gradient plasticity theory – part II: tensorial plastic multiplier. Journal of the Mechanics and Physics of Solids 57, 1045-1057.], that allows for elastic–plastic loading/unloading and the interaction of multiple plastic zones, is proposed. The predicted model response is compared to the corresponding rate-dependent version of visco-plastic origin, and coinciding results are obtained in the limit of small strain-rate sensitivity. First, (i) the evolution of a single plastic zone is analyzed to illustrate the agreement with earlier published results, whereafter examples of (ii) multiple plastic zone interaction, and (iii) elastic–plastic loading/unloading are presented. Here, the simple shear problem of an infinite slab constrained between rigid plates is considered, and the effect of strain gradients, strain hardening and rate sensitivity is brought out. For clarity of results, a 1D model is constructed following a procedure suitable for generalization to 2D and 3D.

Failure in shear bands for granular materials: thermo-hydro-chemo-mechanical effects

The failure of geomaterials in localised shear bands is one of the most common features in geomechanics. Early studies have provided the necessary criteria for the conditions of localisation, inclination angle with respect to the loading axes and thickness of the shear bands, when loaded at room temperature. This work extends these criteria for the problem of simple shear of a chemically active granular cohesionless material at higher temperatures, where thermal or chemical pressurisation may set in and influence the response of the material. It is deduced that failure occurs at higher positive values of the hardening modulus as temperature increases, that the shear band thickness also depends on the chemical reaction characteristics and that micro-inertia due to grain translations and rotations introduce the necessary rate dependency to regularise the system.