Extension of a multi-surface plasticity model to two-phase materials

The work reported in this paper is part of the ongoing research on the development of suitable elastic–plastic constitutive models for multiphase materials. This paper is concerned with the application of an elastic–plastic constitutive model based on the Mróz-multi-surface kinematic hardening rule to particulate metal matrix composites (PMMCs). Details of the Mróz-based elastic–plastic constitutive model for PMMCs and its explicit implementation are presented to enhance the applicability of the model for a stress controlled simulation. Comparison between numerical predictions and experimental results is also presented for uniaxial loading and biaxial proportional and non-proportional loading paths. For the load paths tested, reasonable agreement is observed between the numerical and the experimental results.     

Effective field method in ideal plasticity theory for composite materials

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 149–155, May–June, 1989.     

A plasticity model for cellular materials with open-celled structure

Based on a rigid-plastic material model that obeys the von Mises yield criterion, the plastic behavior of foams with an open-celled structure is studied in this paper using a single unit cell. An approximate continuum plasticity model is developed within the framework of the upper bound theorem of plasticity to describe the yield behavior of foams. The microscopic velocity fields are derived for the unit cell, which satisfy the incompressibility and the kinematic boundary conditions, and expressed in macroscopic rate of deformation. From the microscopic velocity fields, a macroscopic yield function is developed for foams under multi-axial stresses and includes the effects of the hydrostatic stress due to the void presence and growth. The dependency of the derived yield surfaces of foams on their relative densities is studied. The plastic behavior of foams is also studied numerically using the finite element method. The newly developed plasticity model is compared with the finite element analysis results and other available foam models and then correlated with the finite element results.     

桁架板等效刚度分析

桁架材料的连续介质等效模型的研究已有相当基础,而工程中桁架材料往往以类板结构形式出现,其变形表现出明显的弯曲特征。将类板桁架材料采用弯曲板模型模拟,研究合理的方法确定等效板模型的刚度具有重要意义。本文在基于Kirchhoff假定的小挠度薄板弹性理论框架下,研究了类板桁架材料的等效弯曲薄板模型,提出了确定薄板模型等效刚度的基于Dirichlet位移边界条件的代表体元法,给出了确定各刚度系数所对应的代表体元的边界位移形式。具体计算了几种典型形式桁架板的等效刚度,并采用有限元离散模型和实验技术分析了桁架板在一定的边界约束和荷载作用下的响应,并与等效板模型的分析结果进行了对比。结果表明,在响应分析中,具有等效刚度的薄板模型可准确模拟类板桁架材料;连续介质板等效刚度计算的积分法不能给出准确的桁架板等效刚度,而基于Dirichlet位移边界条件的代表体元法获得的等效板的刚度具有很高的精度。     

Mechanism-based strain gradient plasticity in C0 axisymmetric element

Non-uniform plastic deformation of materials exhibits a strong size dependence when the material and deformation length scales are of the same order at micro- and nano-metre levels. Recent progresses in testing equipment and computational facilities enhancing further the study on material characterization at these levels confirmed the size effect phenomenon. It has been shown that at this length scale, the material constitutive condition involves not only the state of strain but also the strain gradient plasticity. In this study, C⁰ axisymmetric element incorporating the mechanism-based strain gradient plasticity is developed. Classical continuum plasticity approach taking into consideration Taylor dislocation model is adopted. As the length scale and strain gradient affect only the constitutive relation, it is unnecessary to introduce either additional model variables or higher order stress components. This results in the ease and convenience in the implementation. Additional computational efforts and resources required of the proposed approach as compared with conventional finite element analyses are minimal. Numerical results on indentation tests at micron and submicron levels confirm the necessity of including the mechanism-based strain gradient plasticity with appropriate inherent material length scale. It is also interesting to note that the material is hardened under Berkovich compared to conical indenters when plastic strain gradient is considered but softened otherwise.     

Analysis and calibration of a three-invariant plasticity model for granular materials

Summary In this paper we briefly review issues related to the characterization of properties of granular materials subjected to micro-gravity and one-gravity conditions at very low effective stress levels. We describe the development of a three-invariant plasticity model that resembles the model devised by Lade. An inverseidentification scheme where the analysis tools are used to extract constitutive model parameters from experiments is also discussed.

---

Untersuchung und Kalibrierung eines mit drei Spannungsinvarianten formulierten Plastizitätsmodells für granulare Stoffe

Zusammenfassung Übersicht: Über Resultate, die mit der Charakterisierung der Eigenschaften von granularen Stoffen in schwachen Schwerefeldern bzw. dem der Erde bei sehr kleinen Spannungen zusammenhängen, wird ein kurzer Überblick gegeben. Beschrieben wird ein mit drei Spannungsinvarianten formuliertes plastisches Stoffgesetz, welches dem von Lade entwickelten Modell ähnlich ist. Weiterhin wird ein inverses Identifikationsschema diskutiert, bei dem mit analytische Methoden benutzt werden, um die Parameter des Stoffmodells zu gewinnen.

---

---

Presented at the workshop on Limit Analysis and Bifurcation Theory, held at the University of Karlsruhe (FRG), February 22–25, 1988     

On the implementation of a multi-surface kinematic hardening plasticity and its applications

This paper is concerned with an application of the multi-surface plasticity in solid mechanics and geotechnical problems. The model is of a von-Mises type with associated flow rule, originally proposed by Montans. The Mroz translation rule is implemented to the movements of the yield surfaces and the fully implicit scheme with radial mapping method is applied in numerical computations. Algorithmic consistent tangent modulus with numerical integration algorithm of constitutive equations is extracted. The model is developed in the class of kinematic hardening models, so the ‘Masing’ rule is preserved. The model is able to consider the plastic strain accumulation in constant axial stress state, such as ratcheting. The implementation is validated by means of a simple deformation path of combined extension and compression test, a pure shear test with pseudo-random loading, a test which demonstrates the capabilities of the model in simulation of cyclic loading and ratcheting, a cyclic shear test in saturated undrained sand and finally, the analysis of a plate with holes, which presents the shear band using the multi-surface plasticity model.     

Bifurcation conditions for ideal fibre-reinforced materials

A bifurcation of an equilibrium state for ideal fibre-reinforced material is discussed. It is assumed that the material is elastic, locally transversely isotropic, incompressible and inextensible in the direction of fibres. On a finite state of strain an arbitrary field of small displacements is superposed and a set of governing equations for the perturbed state is derived.As an example a stability problem of a rectangular block. Objected to a finite, homogeneous deformation is considered. A discussion of the results is focused on the influence on the stability of the pressure applied in the direction of fibres.Due to the assumption of inextensibility this pressure has no influence on the state of strain, but it is shown that it may cause a loss of stability.     

On relations for general two-dimensional ideal plasticity problems under the full plasticity condition

Relations for two-dimensional ideal plasticity problems under the full plasticity condition are determined with material anisotropy, inhomogeneity, and compressibility properties taken into account. These properties are determined by the direction cosines of the principal stress, the coordinates of points in space, and the mean stress.For the yield strength we take a function of the form k = k(σ, n ₁, n ₂, n ₃, x, y, z). The desired relations are determined for the general plane ideal plasticity problem. The relations thus obtained are generalized to the cases of axisymmetric and spherical plasticity problems.     

Well-posedness of a model of strain gradient plasticity for plastically irrotational materials

The initial boundary value problem corresponding to a model of strain gradient plasticity due to [Gurtin, M., Anand, L., 2005. A theory of strain gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. J. Mech. Phys. Solids 53, 1624–1649] is formulated as a variational inequality, and analysed. The formulation is a primal one, in that the unknown variables are the displacement, plastic strain, and the hardening parameter. The focus of the analysis is on those properties of the problem that would ensure existence of a unique solution. It is shown that this is the case when hardening takes place. A similar property does not hold for the case of softening. The model is therefore extended by adding to it terms involving the divergence of plastic strain. For this extended model the desired property of coercivity holds, albeit only on the boundary of the set of admissible functions.     

Reproduction of solutions of bidimensional ideal plasticity

The symmetries of a system of differential equations allowed the transformation of its solutions to a solution of this system. New analytical exact solutions of a system of two-dimensional ideal plasticity equations were constructed from two well-known solutions, that for a circular cavity stressed by normal pressure, and Prandtl's solution for a block compressed between perfectly rough plates, for the case where the thickness of the block was rather small. A mechanical sense of new solutions was discussed.     

Nonlinear dynamics and stability of microstructures in a lattice model for ferroelastic materials

With the view of understanding how precise macroscopic properties of a material emerge from the underlying physics of homogeneous microstructures, a lattice model which can describe complex non-linear patterns made of elastic domains and interfaces is proposed. On the basis of a two-dimensional lattice model involving non-linear and competing interactions the dynamics of microstructure formation is examined. The emphasis is placed especially on an instability mechanism of a strain band producing localized domains. The influence of applied forces and dissipative effects on the dynamics of two perpendicular strain bands is studied. The results are interpreted as a microtwinning in crystalline alloys. The physical conjectures are checked by means of numerical simulations performed directly on the microscopic system.

---

Sommario Si propone un modello reticolare che può descrivere complessi arrangiamenti fatti di domini elastici ed interfacce. Sulla base di un modello bidimensionale in cui sono presenti interazioni contrastanti e nonlineari si esamina la dinamica della formazione di microstrutture. L'accento è posto sui meccani'smi di instabilità che determinano bande di deformazione localizzata. Si studia l'influenza delle forze applicate e degli effetti dissipativi sulla dinamica di due bande perpendicolari e si interpretano i risultati come un microtwinning in leghe cristalline. Si verificano le congetture fisiche per mezzo di simulazioni numeriche del modello microscopico.

     

On real and ideal materials

Various plasticity theories are discussed as models for the behavior of real materials. The simple rigid/perfectly plastic model is shown to be a reasonable first approximation. Once the theory for this model has been developed, it is easily modified to account for strain hardening and elastic effects. It is shown in various examples that, for loads less than the rigid/perfectly plastic limit load, strain hardening has a negligible effect, whereas for greater loads elastic strains can reasonably be neglected.     

A crack-mechanics based model for damage and plasticity of brittle materials under dynamic loading

A rate-dependent model for damage and plastic deformation of brittle materials under dynamic loading is presented. The model improves upon a recently developed micromechanical damage model (Zuo et al., 2006) by incorporating plastic deformation of the material. The distribution of the microcracks in the material is assumed to remain isotropic, and the damage evolution is through the growth of the average crack size. Plasticity is considered through an additive decomposition of the total strain rate, and a rate-independent, von Mises model is used. The model was applied to simulate the response of a model material (SiC) under uniaxial strain loading. To further examine the behavior of the model, cyclic loading and large-strain compressive loading were considered. Numerical results of the model predictions are presented, and comparisons with those from a previous model are provided.     

Non linear constitutive models for lattice materials

We use a computational homogenisation approach to derive a non linear constitutive model for lattice materials. A representative volume element (RVE) of the lattice is modelled by means of discrete structural elements, and macroscopic stress–strain relationships are numerically evaluated after applying appropriate periodic boundary conditions to the RVE. The influence of the choice of the RVE on the predictions of the model is discussed. The model has been used for the analysis of the hexagonal and the triangulated lattices subjected to large strains. The fidelity of the model has been demonstrated by analysing a plate with a central hole under prescribed in plane compressive and tensile loads, and then comparing the results from the discrete and the homogenised models.     

A model for extreme plasticity

We present a mathematical model for elastoplasticity in the regime where the applied stress greatly exceeds the yield stress. This scenario is typically found in violent impact testing, where millimetre thick metal samples are subjected to pressures on the order of 10–10² GPa, while the yield stress can be as low as 10⁻² GPa. In such regimes the metal can be treated as a barotropic compressible fluid in which the strength, measured by the ratio of the yield stress to the applied stress, is negligible to lowest order. Our approach is to exploit the smallness of this ratio by treating the effects of strength as a small perturbation to a leading order barotropic model. We find that for uniaxial deformations, these additional effects give rise to features in the response of the material which differ significantly from the predictions of barotropic flow.