Incremental Stress-strain Relations Research Articles

Internal Variable Representation of Microcrack Induced Inelasticity in Brittle Materials

For brittle materials strained in compression, nonlinear constitutive equations are developed within the thermodynamic framework with microstructural internal variables. The sliding crack dissipative micromechanism, activated on preexisting dilutely distributed flaws, is assumed to govern the macroscopic deformation. The losses of energy resulting from frictional sliding on closed preexisting flaws, wing crack propagation, and wing crack rotation are included in the analysis. Two models (displacement-driven vs traction-driven) of the basic deformational micromechanism are considered in detail. The obtained incremental stress-strain relations are compared with the corresponding kinematic solutions available in the literature.

Tangent modulus matrix for finite element analysis of hyperelastic materials

In this study, using the VEC operator [1], compact expressions are formulated for the tangent modulus matrix of hyperelastic materials, in particular elastomers, using Lagrangian coordinates. Compressible, incompressible, and near-compressible materials are considered. Expressions are obtained for the corresponding finite element tangent stiffness matrices. It is observed that the incremental stress-strain relations should be considered anisotropic. Numerical procedures based on Newton iteration are sketched. The limiting case of small strain is developed. Finally, the tangent modulus matrix is presented for the Mooney-Rivlin material, with application to the rubber rod element.

A generalised D matrix for anisotropic elastic granular media

A matrix relating stress and elastic strain tensors for anisotropic particulate materials has been derived. The magnitude of the matrix depends on the state of the material anisotropy. Anisotropy in granular materials depends on strain because normal and tangential particle contact forces, as well as the spatial distribution of the contacts, vary with stress and strain. However, the rotation tensor and the strain tensor cannot be independent; they must satisfy certain constraints to meet the requirement for macroscopic stress tensor symmetry. These conditions and constraints lead to the derivation of the matrix presented in this article. The principal directions of the stress tensor and strain tensor are generally not coincident, and the values of deformation parameters, Young's modulus and Poisson's ratio, are direction dependent; these two aspects are also discussed in this paper. Whereas this matrix can be used in static numerical analyses for elastic problems, we note that this relationship can also be used as a basis upon which to derive a fully incremental stress–strain relationship for anisotropic granular materials in the plastic state, where the anisotropy is evolving with strain.

A constitutive model and FEM analysis of jointed rock masses

In the present study, a constitutive model of jointed rock masses is presented which reflects the size, density, orientation and connectivity of joints as well as their mechanical properties. Following the continuum approach, the incremental stress-strain relation of the jointed rock mass is formulated by taking the volume average of stress and strain inside a representative volume element where the evaluation of the relative displacement across the joints is required. Employing an elasto-plastic constitutive model of the joint behavior, the relative displacement across the joint can be obtained once the stress acting on the joint is known. The fundamental difficulty in the constitutive modeling of jointed rock masses is due to the fact that the stress acting on a joint is different from the average stress since a joint does not cut through the rock mass but terminates within the rock mass, possibly connecting with other joints. The lower the stiffness of the surrounding matrix is, due to the existence of other joints or the connection of joints, the higher the stress acting on the joint will be. In the present study, the stress concentration tensor, which gives a relation between average stress and the stress acting on the joint, is introduced and a simple method to evaluate it is developed. The interaction effect between joints and the effect of joint connection are properly considered in the model. Some simple examples are solved by the proposed constitutive model. The results are in agreement with experimental data showing the characteristic features of the behavior of jointed rock masses. The proposed constitutive model for jointed rock masses is implemental into a finite element analysis program with a 3-D isoparametric element to analyze actual engineering problems. As an example, the program is used to analyze a plate-loading test problem and the results of the 3-D finite element analysis of the problem are compared with the test data. For highly jointed rock masses, the continuum model offers a powerful analytical method.

Plasticity models for the biaxial behaviour of concrete at elevated temperatures, Part II: Implementation and simulation tests

The thermo-mechanical behaviour of concrete at elevated temperatures under multiaxial stress is extremely nonlinear. In Part I we developed a general analytical failure criterion based on the available biaxial test data. Here, the loading surfaces are used to derive the elastic-plastic incremental stress-strain relationships for a plasticity approach, and the model is used to predict experimental strain measurements under biaxial stress. Agreement is shown to be very good. The predictions are also used to suggest possible variations in material parameters such as Poisson's ratio, for which scant data exist.

The crack tip zone shielding effect for ductile particle reinforced brittle materials

The crack tip zone shielding effect for the ductile particle reinforced brittle materials is analyzed by using a micromechanics constitutive theory. The theory is developed here to determine the elastoplastic constitutive behavior of the composite. The elastoplastic particles, with isotropic or kinematical hardening, are uniformly dispersed in the brittle elastic matrix. The method proposed is based on the Mori-Tanaka's concept of average stress in the composite. The macroscopic yielding condition and the incremental stress strain relation of the composite during plastic deformation are explicity given in terms of the macroscopioc applied stress and the microstructural parameters of the composite such as the volume fraction and yield stress of ductile particles, elastic constants of the two phases, etc. Finally, the contribution of the plastic deformation in the particles near a crack tip to the toughening of the composite is evaluated.

A micromechanics constitutive model of transformation plasticity with shear and dilatation effect

B ased on micromechanics, thermodynamics and microscale t → m transformation mechanism considerations a micromechanics constitutive model which takes into account both the dilatation and shear effects of the transformation is proposed to describe the plastic, pseudoelastic and shape memory behaviors of structural ceramics during transformation under different temperatures. In the derivation, a constitutive element (representative material sample) was used which contains many of the transformed m-ZrO 2 grains or precipitates as the second phase inclusions embedded in an elastic matrix. Under some basic assumptions, analytic expressions for the Helmholtz and complementary free energy of the constitutive element are derived in a self-consistent manner by using the Mori-Tanaka method which takes into account the interaction between the transformed inclusions. The derived free energy is a function of externally applied macroscopic stress (or strain), temperature, volume fraction of transformed phase and the averaged stressfree transformation strain (eigenstrain) of all the transformed inclusions in the constitutive element, the latter two quantities being considered to be the internal variables describing the micro-structural rearrangement in the constitutive element. In the framework of the Hill-Rice internal variable constitutive theory, the transformation yield function and incremental stress strain relations, in analogy to the theory of metal plasticity, for proportional and non-proportional loading histories are derived, respectively. The theoretical predictions are compared with the available experimental data of Mg-PSZ and Ce-TZP polycrystalline toughening ceramics.

A micromechanics constitutive theory for forward transformation plasticity with shear and dilatation effect: I, nonproportional loading history

A micromechanics constitutive theory which takes into account both the dilatation and shear ef fects of the transformation is proposed to describe the macroscopic plastic behavior of structure ceramics during forward transformation under different temperatures. Under some basic assumptions, the analytic expressions of the Helmholtz and complementary free energy of the constitutive element are derived in a self-consistent manner by using the Mori-Tanaka's method which takes into account the interaction between the transformed inclusions. In the framework of Hill-Rice's internal variable constitutive theory, the forward transformation yield function and incremental stress strain relations, in analogy to the theory of metal plasticity, for non-proportional loading histories are obtained.

Incremental stress-strain relationships for regular packings made of multi-sized particles : Chang, C S; Misra, A; Xue, J H Int J Solids StructV25, N6, 1989, P665–681

Incremental stress-strain relationships for regular packings made of multi-sized particles : Chang, C S; Misra, A; Xue, J H Int J Solids StructV25, N6, 1989, P665–681

A Stochastic Nonlinear Constitutive Law for Concrete

An incremental three-dimensional stress-strain relationship for concrete with induced anisotropy has been developed. The nonlinearity and path-dependency are modeled by expressing the elastic moduli at each increment as function of the octahedral and deviatoric strains, based on a uniaxial stochastic model developed earlier. Predictions of multiaxial response under proportional and nonproportional loading are in good agreement with experimental results.

Incremental stress-strain relationships for regular packings made of multi-sized particles

The mechanical behavior of a granular system is greatly influenced by its packing geometry. In this work, the concept of “Voronoi polyhedron” is utilized to characterize granular packings and it is shown that polyhedral tessellations can be used to represent packings of multisized particles. A stress strain theory, based on micro-mechanical considerations, tor small deformations of such packings is then described. The derived stress strain relationship is compared with experimental results.

Mechanical sublayer model for elastic-plastic analyses

Strain-hardening behavior for plane stress problems is modeled by a panel with n layers, the first (n - 1) layers are elastic-perfectly-plastic under Mises-Hencky condition, each with different yield stress, and the n-th layer is elastic. Equivalent incremental stress-strain relations for the panel can be obtained. The resulting uniaxial stress-strain curve contains n segments. Those segments in the plastic range are not straight lines.

柱の変動軸力を考慮したRC造骨組の弾塑性解析 : その1 解析法と芯筋柱の構造実験解析

This paper presents the elastic-plastic analytical method of the reinforced concrete plane frame considering the fluctuation of axial forces on columns and results of application of this method to the lateral loading tests of columns with core rebars in their centers against large axial forces during earthquake. In this analytical method, two surfaces of crack and yield are introduced on the force plane of bending moment and axial force (M-N force plane) at the ends of column as the yield criteria. The yield surface is assumed as the parabolic curves approximated from M-N values at the ultimate strengths which are obtained by summation of M-N strength curves of concrete and reinforcing bars. The crack surface is assumed as the similar parabolic curves to the yield surface. Three states of elastic, crack and yield are assumed at column ends. Introducing the yield criteria, the nomality flow rule regarding to the incremental plastic strain, and the hardening rule, the elastic-plastic equilibrium equations regarding to the incremental stress-strain relationship of column are derived. From the result of application of this method to the analysis of columns with core rebars in their centers, good agreements are shown between the two load-deflection skeleton curves which are obtained from the test and the analysis. From this results it is clarified that this analytical method is adequate and effective in order to investigate the elastic-plastic behavior of the reinforced concrete plane frame under the fluctuating axial forces of columns.

Theoretical framework for modelling the behaviour of frictional materials

A constitutive theory is proposed, which possesses the possibilities of modelling all the important features or the behaviour or frictional materials such as: influence of all three stress invariants, coupling between deviatoric and volumetric response, dilatancy, softening, and different behaviour in loading and unloading. The basic constitutive assumptions are relations between properly defined stress and strain rate invariants, from which the component equations are derived by means of a suitable reformulation. After the incremental stress-strain relations have been derived, they are augmented by consistent loading/unloading criteria. Emphasis is given to a fundamental discussion of the general properties of the theory proposed and it is shown to fulfil all the formal requirements (causality, determinism, admissibility, form-invariance, continuity) that a properly formulated constitutive theory must obey. Moreover, the theory contains a surprisingly large number of classical as well as nonclassical theories as special cases. In particular, it contains formulations ranging from nonassociated plasticity theory, associated plasticity theory, hypoelasticity to elasticfracturing theory.

A plastic-fracture model for concrete

The paper summarizes recent efforts in formulating an elastic-plastic-fracture model for the finite-element analysis of concrete structures. Based on the geometrical considerations, a four-parameter fracture (or yielding) criterion was proposed which embraces some of the simpler existing models. Isotropic elastic and anisotropic elastic behaviors were proposed for the initial loading and the post-failure behaviors. A plastic model displaying mixed hardening was proposed to describe material behaviors between the initial yielding and the fracture failure. Incremental stress-strain relationships were derived based on the associated flow rule and Ziegler's kinematic hardening rule. Three different types of failure modes were considered. A simple crushing coefficient was defined based on a dual criterion to identify the crushing type, the cracking type and the mixed type of failure. Material parameters required for each element of the plastic-fracture model were determined. An important feature of the paper is that matrix formulations for all the constitutive equations were derived and are available for finite-element implementations.

A constitutive model for structural analysis of fusion magnets

A constitutive model to simulate the elastic-plastic and slip actions of fusion magnets under operating or abnormal loads is outlined. To represent the elastic-plastic responses a unified material homogenization procedure based on the existing composite technology was applied to obtain an effective incremental stress-strain relationship for the heterogeneous, laminated magnets. The inter-layer slip behavior of the magnets was represented by a friction-type model from which slip deformation can be calculated for the situation where inter-layer shear stresses exceed the bonding strength of the adhesives. Numerical results of three example problems are presented to demonstrate the utility of the proposed model.

Strain-rate dependent plasticity in thermo-mechanical transient analysis

The thermo-mechanical transient behavior of fuel element cladding and other reactor components is generally governed by the strain-rate properties of the material. Relevant constitutive modeling requires extensive material data in the form of strain-rate response as function of true-stress, temperature, time and environmental conditions, which can then be fitted within a theoretical framework of an inelastic constitutive model. In this paper, we present a constitutive formulation that deals continuously with the entire strain-rate range and has the desirable advantage of utilizing existing material data. The derivation makes use of strain-rate sensitive stress-strain curve and strain-rate dependent yield surface. By postulating a strain-rate dependent von Mises yield function and a strain-rate dependent kinematic hardening rule, we are able to derive incremental stress-strain relations that describe the strain-rate behavior in the entire deformation range spanning high strain-rate plasticity and creep. The model is sufficiently general as to apply to any materials and loading histories for which data is available.

A Finite-Element Model for Residual Stresses and Deflections in Girth-Butt Welded Pipes

Computational models for predicting transient temperature distributions, residual stresses, and residual deflections for girth-butt welds are described. Comparisons of predicted and measured temperatures for a two-pass welded pipe show agreement to within 9 percent and 17 percent of the measured values for passes one and two, respectively, the model for predicting residual stresses and residual deflections is based on a finite-element representation recognizing individual passes, temperature dependent elastic-plastic constitutive behavior, elastic unloading for material in the nonlinear stress-strain range, and changes in geometry due to the deformation of each weld pass. Load incrementation and incremental stress-strain relations are also used. Results for a two-pass girth-butt welded pipe show good correlation between residual stresses and residual deflections obtained from the computational model and data obtained from a welded 304 stainless steel pipe.

Numerical implementation of a transverse-isotropic inelastic, work-hardening constitutive model

This paper documents the numerical implementation of a transverse-isotropic inelastic, work-hardening plastic constitutive model. A brief review of the model is presented first to facilitate the understanding of its numerical implementation. This model is formulated in terms of ‘pseudo’ stress invariants, so that the incremental stress-strain relationship can be readily incorporated into existing finite-difference or finite-element computer codes. The anisotropic model reduces to its isotropic counterpart without any changes in the mathematical formulation or in the numerical implementation (algorithm) of the model. The input quantities to the algorithm are the initial stress components obtained at the end of the previous strain increment, and the new strain increments; the output quantities are the new values of the stress components. In the first step of the algorithm a set of elastic trial stresses computed, which are then tested with respect to the failure criteria. If they do not violate the failure criteria, the behavior of the material is truly elastic and these stresses are the correct stresses for the given strain increments. On the other hand, if the elastic trial stresses violate the failure criteria, they are corrected through an iteration scheme using an appropriate flow rule. A typical example of the model and its behavior in uniaxial strain and triaxial compression is presented.

Nonlinear analysis of reinforced concrete structures

For the prediction of yield and failure of concrete under combined stress, a generalization of the Mohr-Coulomb behavior is made in terms of the principal stress invariants. The generalized yield and failure criteria are developed to account for the two major sources of nonlinearity: the progressive cracking of concrete in tension, and the nonlinear response of concrete under multiaxial compression. Using these criteria, incremental stress-strain relationships are established in suitable form for the nonlinear finite element analysis. For the analysis of reinforced concrete members by finite elements, a method is introduced by which the effect of reinforcement is directly included. With this approach, the stress-strain laws for the constituent materials of reinforced concrete are uncoupled permitting efficient and convenient implementation of a finite element program. The applicability of the method is shown on sample reinforced concrete analysis problems.