Classical Plasticity Theory Research Articles

A ductile growth model for elasto-plastic material

Abstract In bulk forming processes, the soundness of a manufactured part can be related to the final mechanical properties of its material, which can be locally deteriorated by an induced porosity increase. In order to predict such damage evolution, a void growth model, based upon the Rice and Tracey analysis and suited for thermo-elasto-plastic finite-element modelling is discussed and compared to existing models. The workpiece material is considered as a metallic matrix, following the classical plasticity theory, with microscopic spherical voids. Its slightly compressible macroscopic behaviour is predicted by the Rice and Tracey analysis, the results of a numerical study of a unit cell with or without an incompressible inclusion filling the void showing that the plastic volume changes come mainly from geometric effects. Comparisons between existing growth models and this new model are carried out with the same finite-element software for axisymmetric thermo-mechanical closed-die modelling of a forward extrusion and the collar test.

A more sophisticated hardening law for the theory of plasticity

A new hardening law is introduced into the classical incremental theory of plasticity for initially isotropic solids allowing us not only to rigidly translated, transversely isotropically deform and rotate, but also to distort the yield surface. The law, on the whole, can satisfactorily describe modifications of the surface experimentally obtained by other authors in complex biaxial tensiontorsion loading of thin-walled tubular specimens of pure aluminum.

A model for coupled mechanical and hydraulic behaviour of a rock joint

SUMMARY Constitutive laws for rock joints should be able to reproduce the fundamental mechanical behaviour of real joints, such as dilation under shear and strain softening due to surface asperity degradation. In this work, we extend the model of Plesha1 to include hydraulic behaviour. During shearing, the joint can experience dilation, leading to an initial increase in its permeability. Experiments have shown that the rate of increase of the permeability slows down as shearing proceeds, and, at later stages, the permeability could decrease again. The above behaviour is attributed to gouge production. The stress—strain relationship of the joint is formulated by appeal to classical theories of interface plasticity. It is shown that the parameters of the model can be estimated from the Barton—Bandis empirical coeƒcients; the Joint Roughness Coeƒcient (JRC) and the Joint Compresive strength (JSC). We further assume that gouge production is also related to the plastic work of the shear stresses, which enables the derivation of a relationship between the permeability of the joint and its mechanical aperture. The model is implemented in a finite element code (FRACON) developed by the authors for the simulation of the coupled thermal—hydraulic—mechanical behaviour of jointed rock masses. Typical laboratory experiments are simulated with the FRACON code in order to illustrate the trends predicted in the proposed model. ( 1998 by John Wiley & Sons. Ltd.

Effective methods for the analysis of steel structures with strain-softening behaviour

Abstract In this paper two methods are presented for the solution of steel structures taking into account softening phenomena. This behaviour results from the coupling of local buckling and lateral-torsional buckling phenomena and leads to a nonconvex function of the potential energy of the structure. The problem is formulated on the rigorous mathematical background of hemivariational inequalities. Both methods reduce the nonconvex minimization problem to a number of convex minimization subproblems which are treated numerically with the methods used for the solution of the classical plasticity problems. In this sense, both approaches proposed here can be understood as extensions of the classical plasticity theory which is familiar to structural engineers.

Numerical modelling of masonry: A material model accounting for damage effects and plastic strains

The paper deals with a model based upon notions taken from plasticity and damage mechanics. Plane stress states only are considered, and a yield surface is introduced in the stress space. Inelastic strain increments are supposed to occur only when the stress point lies along that surface. The increments are considered to be partially nonreversible and are partially related to a decrease of stiffness. By following concepts typical of the classical theory of plasticity, inelastic strain increments are assumed normal to the yield surface. After implementing the model into a finite element code a suitable procedure is utilized in order to limit mesh-dependence effects. It is also shown that the material model falls within a class of softening models for which convergence can be proved (under convenient conditions) when dynamic actions and the backward difference time-integration scheme are utilized. Finally comparisons between experimental tests on masonry panels and the relevant numerical results are presented.

ELASTO-PLASTIC INTERFACE MODEL FOR 3D-FRICTIONAL ORTHOTROPIC CONTACT PROBLEMS

The present study deals with the solution of the fully three-dimensional contact/friction problem taking into account microstructural characteristics of the surfaces. An incremental non-associated hardening friction law model analogous to the classical theory of plasticity is used. Two different non-linear friction functions in the orthogonal directions are used to account for the orthotropic properties of the contacting bodies. A frontal solver processing unsymmetric matrices is adopted. Two numerical examples have been selected to show applicability of the method proposed. © 1997 by John Wiley & Sons, Ltd.

A Cosserat theory for elastoviscoplastic single crystals at finite deformation

IN THIS WORK, displacement and lattice rotation are regarded as independent degrees of freedom. They are connected only on the constitutive level and by the balance equations. The description of plastic deformation is based on the slip theory. Elastic lattice curvature and torsion are associated with couple-stresses. The continuum theory of dislocations has been revisited to derive the kinematics of plastic lattice torsion-curvature. Explicit constitutive equations and hardening rules are proposed to close the theory in the case of elastoviscoplasticity. The thermodynamical formulation of the model involves internal variables which are similar to the densities of statistically stored dislocations and the densities of geometrically necessary dislocations. Accordingly, the proposed Cosserat theory can be regarded, on the one hand, as the classical crystal plasticity theory complemented by lattice curvature and torsion variables and, on the other hand, as the continuum theory of dislocations closed by the missing hardening variables and constitutive equations within the appropriate micropolar framework. A generalization of Mandel's elastoviscoplastic decomposition of strain is used especially for the torsion-curvature measure at finite deformation.

Inelastic Thermal Response of Gr/Cu with Nonuniform Fiber Distribution

An analytical model for the thermomechanical response of metal matrix composites was developed based on the method of cells micromechanics model and classical lamination theory. The inelastic behavior of the matrix material is modeled using either the classical incremental plasticity theory or the Freed-Walker viscoplasticity theory, which is a more recent development designed for use with copper and copper alloys. The composite model was subsequently used to investigate the effect of the matrix constitutive model and the effect of nonuniform fiber distribution, on the thermal expansion of unidirectional graphite/copper (Gr/Cu) composites. Results illustrate that when the viscoplastic model is used, stress relaxation decreases the ability of the copper matrix to support load, and the predicted longitudinal thermal response of the composite resembles that of the graphite fiber. The effect of nonuniform fiber distribution on the longitudinal thermal expansion is more pronounced when the plastic constitutive model is used. In addition, the effect of the nonuniformity is more pronounced for the longitudinal thermal expansion of Gr/Cu than the transverse thermal expansion. The model predictions were compared with experimental thermal expansion data for Gr/Cu, and the use of the viscoplastic constitutive model was found to provide better correlation than the plastic constitutive model.

FE-formulation of a nonlocal plasticity theory

A nonlocal continuum plasticity theory is presented. The nonlocal field introduced here is defined as a certain weighted average of the corresponding local field, taken over all the material points in the body. Hereby, a quantity with the dimension of length occurs as a material parameter. When this so-called internal length is equal to zero, the local classical plasticity theory is regained. In the present model, the yield function will depend on a nonlocal field. The consistency condition and the integration algorithm result in integral equations for determination of the field of plastic multipliers. The integral equations are classified as Fredholm equations of the second kind and the existence of a solution will be commented upon. After discretization, a matrix equation is obtained, and an algorithm for finding the solution is proposed. For a generalized von Mises material, a plane boundary value problem is solved with a FE-method. Since the nonlocal quantities are integrals, C0-continuous elements are sufficient. The solution strategy is split into a displacement estimate for equilibrium and the integration of constitutive equations. In the numerical simulations shear band formation is analysed and the results display mesh insensitivity.

Stability of a frictional, cohesive layer on a viscous substratum: Variational formulation and asymptotic solution

This contribution is concerned with the stability of a frictional, cohesive material layer, called the overburden, resting on a viscous, incompressible layer of lower density referred to as the substratum. The viscous layer is either perfectly bonded or free to slip with no friction on a rigid basement. The in situ stress is assigned a realistic gradient in the overburden and is assumed to be purely hydrostatic in the substratum. A general variational formulation of the linearized stability problem for the stratified system is obtained in which the viscous response of the substratum provides the characteristic time. To a given state of stress and perturbation corresponds a rate of growth or decay which is calculated by an asymptotic method for small overburden thickness compared to the perturbation wavelength. It is found that the substratum's thickness influences the rate of growth of the instability if its product by the perturbation wavenumber is smaller than 3 or 4, depending on the type of boundary condition at the basement. However, these conditions and the substratum thickness have no influence on neutral stability. Furthermore, the conditions for stability depend on the tectonic stress distribution in the overburden and not solely on the density contrast, as is the case for viscoelastic systems. It is also shown that the classical flow theory of plasticity adopted for the overburden, which successfully captures the onset of folding, fails to predict initiation of faulting for similar stress conditions.

A hygro-thermo-mechanical constitutive model for multiphase composite materials

The paper describes a hygro-thermo-mechanical constitutive model appropriate for numerical simulation of multiphase composite material behaviour. The theory is based on the mixture of the basic substances of the composite and allows to evaluate the inter-dependent behaviour between the different compounding constitutive models. The initial anisotropy of each compounding can be taken into account by means of a mapped isotropic plastic formulation. This is a generalization of classic isotropic plasticity theory applied to orthotropic or anisotropic materials. The basic assumption is the existence of a stress and strain real anisotropic spaces and the respective fictitious isotropic spaces where a mapped fictitious problem is solved. Those spaces are related by means of two fourth order transformation tensor. The combination of mixing theory and the mapped isotropic plastic approach provides a powerful constitutive framework for the numerical treatment of bulk-fiber composite materials.

Coupled plastic-damaged model

Abstract A constitutive model that couples plasticity and damage is presented. The model is thermodynamically consistent and comes from a generalization of classical plasticity theory and isotropic damage theory of Kachanov. Coupling between plasticity and damage is achieved through a simultaneous solution of the plastic and the damage problem. After a description of the model, a numerical algorithm for the integration of the resulting constitutive equations is presented. It is an Euler Backward type of algorithm that is particularly suitable to solve plain stress non-linear problems with a 2D finite element program. The consistent stiffness matrix is also derived. The paper is completed with some application examples that show that the model presented accurately reproduces the behaviour of elastic-plastic-damaged materials.

Bearing Capacity of Highly Frictional Material

Abstract Highly angular dense soils are known to possess secant friction angles well in excess of commonly encountered sands. These soils also exhibit a significant variation of the friction angle over a common range of mean normal stress confinement. These characteristics raise questions regarding the ability of conventional solutions to predict bearing capacity and how linear strength parameters are to be determined for a nonlinear material. To examine these issues, centrifuge experiments were performed on shallow foundations at three embedment depths. Linear strength parameters required for three conventional solutions were obtained from the nonlinear envelope first by a simple averaging technique. To indirectly account for the material nonlinearity, tangent and secant linear strength parameters were estimated once an appropriate level of mean normal stress was determined. A relationship between mean normal stress and the resulting footing bearing capacity was developed using classical plasticity theory. The latter approach offered significantly better predictions than the averaging method, yet the results indicate the inability of conventional theories to model experimentally observed behavior.

Thermo-inelastic response of functionally graded composites

Abstract A recently developed micromechanical theory for the thermo-elastic response of functionally graded composites is further extended to include the inelastic and temperature-dependent response of the constituent phases. In contrast to currently employed micromechanical approaches applied to this newly emerging class of materials, which decouple the local and global effects by assuming the existence of a representative volume element at every point within the composite, the new theory explicitly couples the local and global effects. Previous thermo-elastic analysis has demonstrated that such coupling is necessary when: the temperature gradient is large with respect to the dimension of the inclusion phase; the characteristic dimension of the inclusion phase is large relative to the global dimensions of the composite; and the number of inclusions is small. In these circumstances, the concept of the representative volume element is no longer applicable and the standard micromechanical analyses based on this concept produce questionable results. Examples of composite materials that fall into this category include large-diameter fiber composites such as SiC/Ti and B/A1. Herein, we extend this new approach to include the inelastic and temperature-dependent response of the constituent phases in order to be able to realistically model functionally graded metal matrix composites in the presence of large temperature gradients. The inelastic behavior of the matrix phase is modeled using two inelastic models, namely the Bodner-Partom unified viscoplasticity theory and the classical incremental plasticity theory. Results are presented that illustrate the differences between elastic and inelastic analyses, defining under what circumstances the inclusion of inelastic effects is important. Application of the theory to composites with thermal barrier coatings demonstrates the utility of the concept of internal temperature management through functional grading of the microstructure using differently-distributed particulate inclusions.

An inelastic analysis of a welded aluminum joint

Butt weld joints are most commonly designed into pressure vessels by using weld material properties that are determined from a tensile test. These properties are provided to the stress analyst in the form of a stressvs strain diagram. Variations in properties through the thickness of the weld and along the width of the weld have been suspect but not explored because of inaccessibility and cost. The purpose of this study is to investigate analytical and computational methods used for analysis of multiple pass aluminum 2219-T87 butt welds. The weld specimens are analyzed using classical plasticity theory to provide a basis for modeling the inelastic properties in a finite element solution. The results of the analysis are compared to experimental data to determine the weld behavior and the accuracy of currently available numerical prediction methods.

A generalized elastoplastic plate theory and its algorithmic implementation

AbstractIn continuum mechanics within specific classes of problems, one‐ or two‐dimensional theories are often simpler to apply than the more complete three‐dimensional one. This is, for example, the case of thin bodies, such as plates or shells, which may be studied using appropriate two‐dimensionai theories.Within this approach, the reduction of the dimension is traded for a loss of information relative to the motion in the transverse direction. For example, in the case of non‐linear material behaviour, classical plasticity plate theories are usually not able to model the effects related to the spreading of plasticity through the cross‐section.In the present paper we discuss a generalized plasticity plate model, which can be used to reproduce some of the three‐dimensional effects in a two‐dimensional setting. We present the continuous and the discrete time model, including both isotropic and kinematic hardening mechanisms; moreover, the form of the tangent matrix consistent with the discrete model is addressed.Finally, some examples (cantilever beam, clamped circular plate and clamped square plate under monotonic and cyclic loading) are studied numerically using a three‐dimensional classical plasticity theory, a classical plasticity plate theory and the proposed plate theory. The generalized plasticity plate model matches the three‐dimensional response with greater accuracy, than the classical plasticity plate model.

Strain gradients and size effects in nonhomogeneous plastic deformation

Phenomenological constitutive equation for plastic deformation in metallic alloys are usually developed with the basic assumption of uniformity of the deformation field, or by the homogenization of the field over a macroscopic representative element. This, in turn, coupled with simple nondimensional analysis within the classical continuum theory of plasticity, leads to the development of homogeneous constitutive equations that relate the flow stress to some internal variables, such as strain hardening, strain rate, and volume fraction of second phase particles in metal matrix composites. However, there are circumstances where the size of the microstructure significantly influences the overall mechanical properties of the material. This becomes even more crucial when the size of the plastic deformation zone and the magnitude of the strain gradient within it become comparable to the size of the underlying dislocation structure. For example, predicting the width of the shear band in metallic alloys and its scaling with the size of the microstructure is an important issue from a fundamental point of view. A constitutive model that can capture this phenomenon can also provide an explanation of other observed phenomena such as the size of the plastic zone near a crack tip, the dependence of flow stress onmore » particle size in metal matrix composite, the effect of specimen size on the stress-strain curve in pure copper, and the significant increase in flow stress observed in indentation tests when the size of the indenter is in the nanometer range. The objective of this paper is to provide a possible explanation of how a local microscopic strain inhomogeneity resulting from a heterogeneous microstructure can generate a strain gradient effect when the deformation field is averaged over a macroscopic representative element, leading to a specific predictive relation for the strain gradient coefficient c in terms of the materials elastic-plastic moduli and grain size.« less

Analysis of Plastic Behavior to Cyclically Uniaxial Tests Using an Endochronic Approach

This paper is devoted to the modeling of metallic materials behaving path dependent strain/loading hardening using endochronic theory. Characteristics of the model include the description of the strain path dependent rate of kinematic hardening and of the strain path dependent distortion of the yield surface. Comparisons between theoretical and experimental results are presented to proof the theory. It is shown that the present model can reasonably predict stress response to uniaxially cyclic loading under a variety of test conditions. In particular, the celebrated Bauschinger effect, which is well-known to be inadequately described by the classical plasticity theories, can be quantitatively predicted well by the model.

Experimental evidence of non-homogeneous strain distribution in aluminium polycrystals

The relationship between local non-homogeneous distribution of structural strain and the macroscopic state of loading has already received considerable attention [1, 2]. Knowledge of this relationship promises to contribute to the solution of the problems associated with the initiation and development of yield and the breakdown of material. Three types of strain can be distinguished, taking into account different structural levels: (i) those that are dealt with by the classical theory of elasticity and plasticity, which fit well with the idea of material continuity; (ii) structural strains, which represent strain distribution over structural elements, e.g. grains in polycrystals; and (iii) those that correspond to the local strain field in the neighbourhood of dislocations, vacancies and so on. Deformation inhomogeneity is closely connected with structural strains. These strains characterize the deformed state of each grain. The aim of this work was to prove experimentally the non-homogeneous nature of the plastic strain distribution in polycrystalline atuminium. The correlation between the direction of the principal strains, slip bands and the crystallographic orientation of particular grains was verified by the measured data. Several methods can be used for this purpose. A well-known method is that of Moir6 patterns [3], created by interference of two systems of lines. Artificial marks made on the surface of the specimen are commonly used. We have adopted another method, developed by Berka and Rtlf~ek [4]. This method, called time-based microphotogrammetry, uses elements of the material's structure as natural marks. They can be grain boundaries, natural flaws, etc. The principle of the method consists of a comparison of a stereoscopic pair of micrographs. The first is made in the predeformed state of the material, the second after the well-defined overall deformation. The two micrographs differ slightly. Parallaxes, indirectly expressing displacements, are measured using a stereocomparator. The local displacements in a preselected set of points can then be evaluated quantitatively. The quantitative data obtained can be used for further computations. The measurements were carried out on highpurity aluminium (99.85%) in the form of strip specimens. The grain structure was revealed on their surface in a circular region of diameter 5 mm (Fig. 1)

A Plastic Potential Function Suitable for Anisotropic Fiber Composites

A 3-D plastic potential function was proposed to describe nonlinear be havior in anisotropic fiber composites following classical plasticity theory. The assump tion that uniform dilatational strains do not contribute to plastic deformation was used. Further simplification was obtained by ignoring the plastic deformation along the fiber direction which is an acceptable assumption for most advanced fiber composites. The va lidity of this plastic potential was examined by employing finite element micromechanical analysis.