Plastic Yield Criterion Research Articles

Shakedown analysis of shape memory alloy structures

Phase transformational shakedown of a structure refers to a status that plastic strains cease developing after a finite number of loading cycles, and subsequently the structure undergoes only elastic deformation and alternating phase transformations with limited magnitudes. Due to the intrinsic complexity in the constitutive relations of shape memory alloys (SMA), there is as yet a lack of effective methods for modeling the mechanical responses of SMA structures, especially when they develop both phase transformation and plastic deformation. This paper is devoted to present an algorithm for analyzing shakedown of SMA structures subjected to cyclic or varying loads within specified domains. Based on the phase transformation and plastic yield criteria of von Mises-type and their associated flow rules, a simplified three-dimensional phenomenological constitutive model is first formulated accounting for different regimes of elastic–plastic deformation and phase transformation. Different responses possible for SMA bodies exposed to varying loads are discussed. The classical Melan shakedown theorem is extended to determine a lower bound of loads for transformational shakedown of SMA bodies without necessity of a step-by-step analysis along the loading history. Finally, a simple example is given to illustrate the application of the present theory as well as some basic features of shakedown of SMA structures. It is interesting to find that phase transformation may either increase or decrease the load-bearing capacity of a structure, depending upon its constitutive relations, geometries and the loading mode.

Plastic deformation and yield criterion for compressible sintered powder materials

This paper analyses the basic aspects of plastic deformation behaviours of compressible sintered powder materials, such as plastic volume change, mass constancy, small Poisson's ratio and the effect of hydrostatic pressure on yield. Based on the uniaxial compression tests using sintered powder copper as a model material, the true stress–strain equation, a generalized yield criterion and some plastic constitutive equations for compressible sintered powder materials are proposed. The proposed generalized yield criterion can depict both the initial yield and successive yield of compressible sintered powder materials in the form of unified yield function, and reflect comprehensively the effects of initial density and deformation strengthening on yield of compressible sintered powder materials as well. Experiments show that the generalized yield criterion can be applied to materials with various densities from loose powder to fully dense metal.

A Mixed Finite Element Implementation of a Gradient-enhanced Coupled Damage—Plasticity Model

The numerical implementation of a gradient-enhanced continuum coupled damage—plasticity model as a constitutive framework to model the nonlocal response of materials is presented. By the introduction of “nonlocal, ” gradient-enhanced measures in the plasticity potential function and yield criterion and in the damage potential function and damage criterion, the proposed model introduces microstructural characteristic material length scales, which allow us to predict the size of localized zones based on material constants, as opposed to local models where the loss of ellipticity causes localized zones to be mesh dependent. As numerical methods are used to compute the gradients of the hardening terms, no additional gradient internal state variables are introduced into the Helmholtz free energy. As opposed to previous theories that only incorporate linear hardening, the gradient model proposed here uses coupled nonassociative plasticity — nonassociative damage and can account for a wide range of material models because of the consistently expanded Laplacian evolution equations. However, due to the nonassociative material model, anisotropic hardening, and the corresponding gradients in plasticity, the model becomes highly complex as additional equations have to be introduced for the evolution of the gradients of plastic normal, plastic strains, damage normal, and damage tensor. The numerical implementation uses a small deformation finite element formulation and includes the displacements, plastic multiplier, and damage multiplier as nodal degrees of freedom, thus allowing the three fields to have different interpolation functions. Higher-order elements are used for the plastic multiplier and damage multiplier to enforce continuity of the second gradients, which requires the introduction of additional boundary conditions.

An anisotropic damage-based plastic yield criterion and its application to analysis of metal forming process

Based on the continuum damage mechanics (CDM), both the total damage growth force and the damage equivalent stress has been firstly derived. Material and damage are considered to be anisotropic in this theoretical model. A new damage-based plastic yield criterion, where the damage equivalent stress is taken as the yield function, and the corresponding damage evolution equation has been established. A relation between the damage variable and the plastic deformation has been proposed. This relation can be applied to obtain the damage history in terms of Moiré photo-mechanics technology in this work. In addition, from the anisotropic damage-based plastic yield criterion presented, the large-scale finite element codes ABAQUS/Explicit in conjunction with the damage experimental results has been then used to calculate some mechanical variable fields in metal forming process. It has finally been verified that the position corresponding to the maximum value in damage equivalent stress field accurately coincides with the initial place of macro-crack given by Erichsen cupping tests carried out by authors.

Atomistic basis for the plastic yield criterion of metallic glass.

Because of their disordered atomic structure, amorphous metals (termed metallic glasses) have fundamentally different deformation mechanisms compared with polycrystalline metals. These different mechanisms give metallic glasses high strength, but the extent to which they affect other macroscopic deformation properties is uncertain. For example, the nature of the plastic-yield criterion is a point of contention, with some studies reporting yield behaviour roughly in line with that of polycrystalline metals, and others indicating strong fundamental differences. In particular, it is unclear whether pressure- or normal stress-dependence needs to be included in the plastic-yield criterion of metallic glasses, and how such a dependence could arise from their disordered structure. In this work we provide an atomic-level explanation for pressure-dependent yield in amorphous metals, based on an elementary unit of deformation. This simple model compares favourably with new atomistic simulations of metallic glasses, as well as existing experimental data.

Effects of plastic anisotropy and yield criteria on prediction of forming limit curves

The effects of yield criteria on predictions of the right-hand side of forming limit diagrams (FLDs) are investigated. Predictions of limit strains are determined from an initial imperfection model based on the early work of Marciniak and Kuczynski (1967). Particular attention is placed on the effect of normal plastic anisotropy on limit strains during biaxial sheet stretching. The anisotropic yield criteria investigated in this paper include Hill’s (1948) quadratic criterion, Hosford’s (1979) higher-order criterion, and case 4 of Hill’s (1979) non-quadratic criterion. Several important characteristics of the yield surface shape are discussed and a new parameter that quantifies some of these aspects is introduced. Similar to the work of Barlat (1987), this parameter is based on the relative position of plane strain on the yield surface and can be used to predict the various effects of yield criteria on limit strains. Results indicate that predictions of FLD are very sensitive to selection of yield criteria.

Poroelastic-plastic consolidation — analytical solution

Consolidation of a poroelastic material that yields according to Drucker–Prager or Mohr–Coulomb criteria leads to a Stefan problem for time-dependent pore fluid pressure. The solution to the Stefan problem for a column of infinite depth is known and is adapted to poroelastic/plastic consolidation of a weightless material under a uniform surface load applied instantaneously and subsequently maintained constant. In this approach, the plastic potential and yield criterion need not be the same. If yielding occurs concurrently with application of load, then collapse is instantaneous. Otherwise, yielding may occur during the consolidation period. If so, then the elastic–plastic zone first appears at the surface and subsequently moves down the column. Depth to the elastic–plastic boundary is given by the simple expression Z = 2β√t where β is a constant determined from continuity conditions at the elastic–plastic boundary. Time-dependent surface displacement that occurs during consolidation is directly proportional to Z. There is little difference between elastic–plastic and purely elastic results in a numerical example because there is little difference in the respective consolidation coefficients. Elastic–plastic finite element results obtained from a column of finite depth are in close agreement with analytical results as long as the pore pressure at the bottom of the column does not change significantly from the value induced by application of the surface load. The analytical solution provides for: (1) efficient evaluation of material properties effects on consolidation, including strength and fluid compressibility, and (2) an accurate way of validating poroelastic/plastic computer codes that are based on Drucker–Prager and Mohr–Coulomb criteria. Copyright © 1999 John Wiley & Sons, Ltd.

Unconfined flow of granular avalanches along a partly curved surface. I. Theory

This paper deals with three-dimensional gravity driven free surface flows of piles of granular materials along bottom profiles that are weakly curved downward and plane laterally. We present in detail a three-dimensional extension of the two-dimensional Savage-Hutter model for such granular avalanches. In this extended model, the avalanche is described as a three-dimensional incompressible continuum obeying a Coulomb dry friction law at the base and a Mohr-Coulomb plastic yield criterion in the interior. Based on this, the balance laws of mass and linear momentum and kinematic and stress boundary conditions at the free surface and the base are used to derive depth-averaged dynamic equations that describe the temporal evolution of the height and the depth-averaged horizontal velocity components as functions of position and time. A computation is performed for a pile of granular material with an initial spherical cap geometry moving down an inclined plane.

Couplled Geomechanical-thermal Simulation For Deforming Heavy-oil Reservoirs

Abstract The paper describes a novel numerical model for the solution of poro-elasto-plasticity and multiphase, thermal flow in unconsolidated heavy-oil and oil-sand reservoirs. The elasto-plastic deformation is calculated using a finite-element incremental plasticity model with Mohr-Coulomb and Drucker-Prager as the yield criteria. This model is coupled with CMG's thermal simulator STARS which is capable of handling many advanced thermal recovery processes. The geomechanical calculation is in fact an effective stress analysis which solves a set of force balance (equilibrium) equations based on total stresses. These are augmented by the elastic material behaviors, plastic yield criteria, plastic flow rule, work-hardening law, and the strain-displacement relationships. Besides volumetric deformation, the solution gives the displacements in each direction, stresses, and strain distribution in the reservoir. The behaviors of reservoir multiphase flow and heat transfer processes are most significantly affected by the volume change and the associated permeability increase. The volume change is calculated by the plasticity model, whereas the permeability increase is related to the volume change via tabular data. The prediction of shear failure region, the rotation of principal stress axis, and the reduction of effective stresses as a result of pore pressure increase, are all important phenomena affecting the placement of fluid and hear in the formation. Introduction Considerable experiences on the in-situ recovery of heavy oil and bitumen from unconsolidated media have been accumulated both in field operations and in laboratory research over the last two decades. The significance of geomechanical responses on the recovery of these reservoirs is now generally recognized. In order to understand the recovery processes, we must consider the complex interaction of multiphase fluid flow, heat transfer, changes in stress state, and formation deformation as a result of injection and production operations. Previous work on the coupling of stress modeling with reservoir simulation(1,2) uses linear elasticity. However, extensive experimental data on oil sands indicate extremely complex constitutive behaviors at elevated pore pressure and temperature(3–6). Oil sands are dense frictional materials which tend to dilate under shear. This dilatancy is the result of an increase in deviatory stresses which leads to the slippage and loss of grain-to-grain contacts. Since the original sand fabric is disturbed, strains thus accumulated are irrecoverable. More recent contributions to this class of coupled simulations(7–11) treat the nonlinearity and plasticity behaviors of oil sands with varying degrees or sophistication. The stresses and strains are related by the material constructive model which must be satisfied for the geomechanical calculation. In this work, an elastoplastic formulation is chosen such that plastic deformation as a result of shear dilatancy can he modeled. During cyclic steam stimulation, the sand matrix goes through several loading and unloading cycles. This leads to complex deformation history which must be properly accounted for. As a reservoir grid block is loaded in the elastic regime, elastic strains accumulate which are recoverable upon unloading. The stress space is delimited by the yield surface. As the stress state reaches the yield surface, additional loading will cause the material to yield plastically.

Degenerated isoparametric finite elements in nonlinear analysis of 2D-problems

Geometrically nonlinear elastic-plastic analysis of 2D-problems of structural mechanics, based on the application of degenerated isoparametric finite elements, is carried out. The investigations concentrate on the derivation of nonlinear equilibrium equations using the total Lagrangian formulation. Large rotation increments and a displacement dependent pressure loading are taken into account, developing, in consequence, additional components of the corresponding stiffness matrix. For the elastic-plastic behaviour the Prandtl-Reuss associated flow rule and the von Mises plastic yield criterion are adopted. The Riks-Wempner constant arc length method, together with Newton-Raphson iterations, are employed to solve the incremental nonlinear equations in question. Several numerical sample problems are presented and compared with analytical reference solutions. A proof of the good accuracy of the presented method is given.

The geometric nature of plasticity laws

This paper concerns the plasticity constitutive laws in small strain. In the thermodynamic approach developed here, the key concept is that of internal variables. The differential nature of plasticity law has been pointed out for a long time. If we unite the invariance condition of these laws in a state variable transformation, this involves, ultimately, that the natural mathematic frame of plasticity theory is Differential Geometry. The system state is defined as a point of a differentiable manifold. The state variable are the local coordinates of this point in a chart. The internal stresses are the components of a covariant vector of the cotangent bundle to internal state manifold and the elastic domain is a convex part of cotangent vector space. The plastic yield criteria such as von Mises condition define a Riemannian structure over the manifold. The metric element is identified with the internal dissipation element. Constitutive laws link the covariant derivatives of the thermodynamic stress with the state variable. Hardening modulus splits up in two parts, kinematic hardening and metric hardening. This last is defined by Christoffel connection coefficients. Applied to von Mises isotropic yield condition, the metric hardening is identified with isotropic hardening. The Baltov-Sawczuk model is also analysed. The use of appropriate polar coordinates simplifies significantly the computations. Generalization to a significant category of non-differentiable yield criteria, such as Tresca condition, is considered by introducting a metric tensor family. The adaptation of Drucker's postulate to the proposed model requires the introduction of parallel transport of the internal stress covector. Generally, this transport is different over distinctive paths joining two points. This fact expresses internal state manifold curvature. The Riemann-Christoffel tensor is computed for von Mises, Baltov-Sawczuk and Tresca models.

The geometric nature of plasticity laws

This paper concerns the plasticity constitutive laws in small strains. In the thermodynamic approach developed here, the key concept is that of internal variables. The differential nature of plasticity law has been discussed for a long time. If we unite the invariance condition of these laws in a state variable transformation, this involves, ultimately, the natural mathematical frame of plasticity theory which is differential geometry. The system state is defined as a point of a differentiable manifold. The state variables are the local co-ordinates of this point in a chart. The plastic yield criteria such as von Mises condition define a Riemannian structure over the manifold. The metric element is identified with the internal dissipation element. Constitutive laws link the covariant derivatives of the thermodynamic stresses with the state variable. The adaptation of Drucker's postulate to the proposed model requires the introduction of parallel transport of the internal stress covector.

A physical model for the inelasticity of concrete

A physical model for the inelasticity of concrete is proposed in this paper. The main effects are attributed to microcracking and softening elastoplastic coupling. The composite nature of concrete is seen to influence decisively the process of microcracking, resulting, for instance, in stable crack growth under uniaxial compression, but unstable crack growth under uniaxial tension. A model is proposed that relates the degradation of the elastic compliances of the material to the extent of microcracking, as described by a set of internal variables that represent the sizes of the microcracks oriented along some selected directions. A thermodynamic approach to the inelasticity of concrete is then presented that proves advantageous in characterizing the coupling between the plasticity of the material and its elastic degradation, i. e. elastoplastic coupling. It is shown that a softening elastoplastic coupling may result in lack of normality and in unstable behaviour, in violation of Drucker’s postulates of classical plasticity. A suitable thermodynamic plastic stability criterion is proposed that generalizes Drucker’s second postulate in the presence of elastoplastic coupling allowing for unstable plastic behaviour. A specific model of elastoplastic coupling is then proposed that allows for the explicit generalization of the classical yield criteria when elastoplastic coupling is present. It is shown, with the aid of this model, how a general anisotropic distribution of microcracks renders the plastic yield criterion anisotropic. It is also shown that the plastic strain rates depart from isochoricity in the presence of microcracking, even if the uncracked material exhibits isochoric plasticity. The proposed thermodynamic criterion for the extension of microcracks is seen to generalize the Griffith criterion in the presence of elastoplastic coupling. It is shown that tensile plastic strains in the direction normal to a microcrack tend to decrease the critical stress for the extension of the microcrack, and that compressive plastic strains tend to increase it. Finally, the proposed model is seen to lead to rate-independent stress─strain incremental relations with symmetric tangent stiffness compliances.

Unified plastic yield criterion for ductile solids.

Introduction T basic hypothesis in the theory of plasticity is that there exists a scalar function of stresses which characterizes the yielding of materials. This scalar function, also called the yield function,/(crfj.) generates a closed surface in the stress space by a relation /(o^) = 0. When a material is subjected to increasing forces and torques, it describes a path in the stress space. When this path intersects the yield surface /(afj-) = 0, the material yields and becomes plastic. All the classical failure theories, e.g., those of Tresca, Von Mises, and Reuss, give conflicting predictions of the yield stresses of metals even for such simple cases of pure torsion or elongation. The reason for the conflicts in these theories is that these theories essentially form one-parameter yield surfaces. So fitting these criteria on tensile tests alone make these incompatible with the torsion test data, and vice versa. These theories also do not predict the failure under hydrostatic compression. Detailed references on these theories can be obtained in Ref. 1. The proposed theory, on the other hand, forms a multiparameter yield surface. Here a start is made from the basic yield point data from both tensile and torsion test experiments, as well as the hydrostatic compression test. Thus the present theory can predict yielding under hydrostatic pressure while posing no conflict between theory and, at least, the basic tension and torsion experiments. The formulation is flexible enough to satisfy a large number of the available experimental data on the yield point and subsequent flow for ductile materials, and includes both ideally plastic and strain-hardening solids. The yield surface generated is invariant with respect to rotation of coordinate axes, a necessary requirement for all such criteria.

Analysis of laboratory test data to derive soil constitutive properties: J.G. Jackson, Jr. 39 pp (Apr. 1969).

Abstract : Computer codes used in the solution of free-field ground shock problems are based on mathematically defined constitutive models. Quantitative input for these constitutive models is primarily based on laboratory test data, but extensive analysis and numerous assumptions are required to convert these data to a form suitable for actual code input. This report presents a detailed illustration of the analyses involved in deriving the soil constitutive properties required for a specific code formulation using laboratory test data from just one stratum of a single site. Data are available from static and dynamic uniaxial strain and triaxial shear tests; code property requirements are for mathematical expressions relating mean pressure to volumetric strain, Poisson's ratio to mean pressure, and a plastic yield criterion to mean pressure. The illustration indicates that progress is being made in developing mathematical constitutive models that are realistic in terms of actual physical behavior, but that if many of the assumptions presently being made in soil property analyses are to be eliminated, additional soil tests and measurements must be developed.