Surface In Stress Space Research Articles

Calculation of Life Characteristics for Structural Alloys under the Mutual Influence of Fatigue and Long-Term Strength

The processes of degradation of the initial strength properties of polycrystalline structural alloys are considered under mechanisms that combine low-cycle fatigue and long-term strength of the material. From the point of view of mechanics of damaged medium (MDM) and fracture mechanics (FM), a mathematical model that describes the processes of cyclic viscoplastic deformation and damage accumulation in structural alloys under multiaxial disproportionate modes of combined thermomechanical loading has been developed. The model consists of three interrelated components: relations that determine the cyclic viscoplastic behavior of the material by taking into account the dependence on the fracture process; evolutionary equations describing the kinetics of damage accumulation; criterion of the strength of the damaged material. The viscoplasticity model is based on the idea of the existence of plasticity and creep surfaces in stress space and the principle of gradient vectors of plastic and creep strain rates to the corresponding surface at the loading point. This form of the equations of state reflects the main effects of cyclic viscoplastic deformation of the material for arbitrary complex loading trajectories. The form of kinetic equations for damage accumulation is based on the introduction of a scalar damage parameter and on energy principles. It also takes into account the main effects of the formation, growth and merging of microdefects under arbitrary complex modes of combined thermomechanical loading. A joint form of the evolutionary equation for damage accumulation in the areas of low-cycle fatigue and long-term strength of the material is proposed. As a criterion for the strength of a damaged material, the condition for reaching a critical value is used. The material parameters and scalar functions included in the constitutive relations of the MDM mathematical model are obtained. The results of numerical simulation of the processes of deformation and damage accumulation in structural alloys under the mutual influence of low-cycle fatigue and long-term strength of the material are presented. The results of comparison of calculated and experimental data show that the proposed MDM model qualitatively and with the accuracy necessary for practical calculations quantitatively describes the durability of materials under the mutual influence of low-cycle fatigue and long-term strength of the material.

Split stress rate plasticity formulation

A novel general constitutive formulation framework of rate independent plasticity is presented where the stress rate tensor is split in two components, and plastic loading is affected separately by each one of these two components in relation to a common loading direction normal to a single smooth yield surface in stress space. This split-stress rate separate effect provides greater flexibility for simulating the material response under combined stress rate components loading. The difficulty arises when loading and unloading events may occur interchangeably between the two stress rate components and special attention is required to avoid irrational results and guarantee continuity of response upon rotation of the total stress rate direction at a loading stress point. The formulation can acquire two different analytical schemes, one more complicated but incrementally linear, while the other is simpler but incrementally nonlinear. The key constitutive ingredient of split stress rate definition is a multifaceted choice, and three options are presented. One option which is of particular interest to granular mechanics, splits the stress rate into a component that changes the stress principal values at fixed principal axes, while the other component does exactly the opposite, i.e., rotates the stress principal axes at fixed principal values. For this option various constitutive ingredients and new concepts are introduced and extensively elaborated, with emphasis on dilatancy, understanding that these elaborations are of an exploratory nature intending to prompt further research along the proposed guidelines.

A computational multiscale model for anisotropic failure of sheet molding compound composites

We present a holistic multiscale approach for constructing anisotropic criteria describing the macroscopic failure of sheet molding compound composites based on full-field simulations of microscale damage evolution. We use an anisotropic damage model on the microscale that directly operates on the compliance tensor, captures matrix and bundle damage via dedicated stress-based damage criteria and allows for a comparison of simulation and experimental results. To identify the damage material-parameters used in the non-linear and time-consuming full-field simulations, we utilize Bayesian optimization with Gaussian processes. We validate the full-field predictions on the microscale and subsequently identify macroscopic failure criteria based on distributions taken from experimental findings. We propose failure surfaces in stress space and stiffness-reduction triggered failure surfaces to cover both a structural analysis and a design process perspective.

Dilatancy-triggering surface for advanced constitutive modelling of sand

This paper presents, for sand, a formulation for a surface in stress space defining the onset of dilatancy based on the volumetric threshold strain concept, the expression for the small-strain shear modulus in terms of soil state and an expression for secant shear modulus degradation with increasing shear strain. The size of the surface depends on both density (void ratio) and mean effective stress. The use of the proposed surface is demonstrated for Toyoura sand. The paper also presents a multiaxial formulation for the surface that can include kinematic hardening. The proposed surface can be useful in constitutive modelling, especially to capture seismic or cyclic loading conditions.

Effect of the plasticity model on the yield surface evolution after abrupt strain-path changes

Abrupt strain path changes without elastic unloading have been used in the literature to investigate the yield surface of sheet metals, both experimentally and theoretically. Such pioneering studies emphasized an apparent non-normality of the plastic strain rate tensor with respect to the trace of the yield surface in stress space, following such a strain-path change. They inspired numerous subsequent developments of plasticity models including non-associated flow rules. In this paper, this type of abrupt strain-path changes is investigated using state-of-the-art plasticity models. The aim is to emphasize the respective contributions of elasticity, isotropic-kinematic hardening, and rate sensitivity, to the apparent violation of the normality condition. The results show that these classical ingredients of plasticity models significantly contribute to the apparent vertex and loss of normality. These effects are quantified for typical sheet metals subject to biaxial-to-shear orthogonal strain path change.

Unified Expression of Rotational Hardening in Clay Plasticity

AbstractRotational hardening is the constitutive ingredient of clay plasticity models that allows rotation of the yield surface in stress space to simulate, more realistically than isotropic models, the stress-strain response under different loading conditions. Based on well-known rotational hardening rules and extensive experimental results, a new unified expression of rotational hardening is proposed in this work, in which the parameters of the unified expression can be determined directly. The proposed rotational hardening expression was imported into a clay model and adopted to simulate the stress-strain behavior of different clays. The simulated results and the experimental results were compared to analyze the influence of the equilibrium values on the stress-strain behaviors. The comparisons show that the constitutive model with the proposed rotational hardening rule had good predictive capability for triaxial tests under different loading paths.

Rotational hardening with and without anisotropic fabric at critical state

The question of zero or non-zero anisotropic fabric at critical state for clays is a debatable issue. The present note addresses a way to model either eventuality by proposing a novel rotational hardening rule for clay plasticity constitutive modelling. Various conditions are derived on the model constants such that some basic requirements are satisfied, among them the uniqueness of the critical state line and prevention of excessive rotation of the plastic potential and yield surfaces in stress space. Schematic and real data simulations illustrate the performance of the new rotational hardening rule.

Anatomy of rotational hardening in clay plasticity

Rotational hardening is a constitutive feature of anisotropic clay plasticity models that allows rotation of the yield and plastic potential surfaces in stress space in order to simulate, more realistically than isotropic models, the material response under various loading conditions. The anatomy of some well-known rotational hardening rules, as well as a novel proposition with certain advantages, is performed in relation to the following characteristics: (a) the conditions under which rotational hardening yields a unique critical state line in e–p space; (b) the restrictions on model parameters necessary to prevent excessive and unrealistic rotation of the yield and plastic potential surfaces; and (c) the analytical description of the response of rotational hardening to loading at constant stress and strain-rate ratios, including K0 consolidation, and to undrained shearing. The anatomy will reveal strong and weak features, and corrections for the latter are proposed. Special attention is given to calibration of the rotational hardening rules parameters based on easily available experimental data, such as the K0 value. Simulations of experimental data under drained and undrained loading after isotropic and K0 consolidation at various OCRs for associative and non-associative flow rules illustrate the simulative capabilities of the rotational hardening rules considered.

Determination of an image point on a surface based on a $$\pi $$ π plane-based algorithm

Constitutive models of particulate materials often rely on distances between the current stress state in stress space and various surfaces. Examples of these surfaces include the bounding surface and the dilatancy surface. This paper proposes a rigorous method for determination of distance to a surface in stress space. It starts by examining operations on stress variables defined in the $$\pi $$ plane. Algorithms for determination of an image point on a surface are then presented as a function of the location of the current stress state with respect to the surface. For points within the surface, the bisection method is used; otherwise, the secant method is used. The paper shows that implementation of the proposed algorithm locates the image point on a surface in stress space with accuracy and rigor, providing an accurate measure of the distance to the surface that can be used in hardening or flow rules.

On the effects of deformation induced anisotropy in isotropic materials

SUMMARYEffects of recoverable deformation induced anisotropy in the elastic stiffness of isotropic materials are described. In isotropic materials, thermodynamics predicts coupling of hydrostatic and deviatoric responses. It is shown that the coupling of the two responses is more significant than previously recognized in the literature. Properly accounting for the coupling of hydrostatic and deviatoric responses requires re‐evaluating elastic materials characterization data, allowing for the coupled response. The result is an apparent decrease in the pressure sensitivity of the elastic shear modulus. The decrease in the pressure sensitivity of the shear modulus leads to stress paths that are more tangential to the yield surface in stress space, resulting in an increase in predicted elastic strain at each step of an elastic–plastic stress update. Consequently, predicted plastic strains and, in particular, volumetric plastic strains, are smaller than if recoverable deformation induced anisotropy had been neglected. The result is an associated plasticity model, which appears to be non‐associated. Copyright © 2012 John Wiley & Sons, Ltd.

Cyclic plasticity modeling and finite element analyzes of a circumferentially notched round bar under combined axial and torsion loadings

A cyclic plasticity model suitable for fatigue damage modeling of metallic structures is presented and its finite element (FE) implementation is described within the small strain plasticity framework. The model uses the von Mises yield surface in stress space whose evaluation follows an Armstrong–Frederick type of nonlinear kinematic hardening rule, and the normality hypothesis in conjunction with the associative flow rule is assumed. An incremental implicit-iterative algorithm was employed for the numerical solution of resulting stress–strain equations, and the continuum tangent obtained from plasticity model was used in FE implementation. The developed FE computational model is applied in the cyclic deformation analysis of a circumferentially notched specimen in combined axial force-torsion loading tests. The computed notch root deformations were compared with measured notch root strain histories. An assessment of model predictions showed that non-proportional loading tests have been simulated with a good accuracy. The computed strain loops were in accord with experimental data and matched qualitatively with measured shear – axial strain histories irrespective of loading path of the test. In proportional balanced torsion-axial loading, the nonlinear shear strain – axial strain loops were also simulated properly. The errors in notch root strains were dependent on the loading path shape, and compared to axial strains, the shear strain errors were relatively greater. The computer solution times were also acceptable.

Elastic-growing tissues: A growth rate law that satisfies the dissipation inequality

A general framework including deformation, generalized diffusion, mass transfer and growth in a multi-species mixture context has been exposed in Loret and Simões (2005). Embedding the analysis in the thermodynamics of open multi-phase media helped to segregate the physical phenomena, and structure interactions. A growth rate law was derived, assuming the internal entropy production to be quadratic in terms of the dissipation variables. Here, a stress-driven growth rate law is exposed in the simpler context of a single solid in a system that is open to mass only. The solid is elastic-growing, in the sense that the deformation gradient is multiplicatively decomposed into a growth transformation and an elastic transformation. The growth rate law is based on the concept of a convex homeostatic surface in stress space. Depending on the position of the stress state with respect to the homeostatic surface, growth, equilibrium or resorption may take place. The dissipation inequality is the central tool that motivates the rate laws for the growth transformation and for the structural variables that define an evolving homeostatic surface. The constitutive equations are used in two simple boundary value problems, that allow to test the material responses in terms of aggregated/resorbed mass and mechanical characteristics, stress and strain, under various representative loading configurations. While the loads corresponding to non homeostatic states are fixed in time, the adaption of the tissue allows finite growth to take place.

Simple Yield Surface Expressions Appropriate for Soil Plasticity

The objective of this paper is to present a number of simple, practical, and useful analytical expressions of a yield surface for geomaterials. In classical plasticity, the analytical expression of a yield surface defines the locus of points in stress space at which plastic flow initiates, and the corresponding function must depend on direct and mixed invariants of stress and tensor-valued internal variables. One single function describes a yield surface in order to avoid singularities and computational difficulties arising from the use of multiple functions representing intersecting surfaces in stress space that are often used for cap-type models in soil plasticity. The presented functions are conveniently subdivided in three main categories depending on the type of analytical expression used, and they all describe properly closed yield surfaces which are continuous and convex. The internal variables in these functions can be used in order to address classical plasticity features such as isotropic and kinematic hardening, the latter in the form of rotational hardening. The effects of parameters on the shape of yield surfaces are clearly demonstrated and illustrated for all functions in triaxial stress space. The generalization of these functions to the multiaxial stress space is presented using a consistent method such that if one applies triaxial loading conditions on the multiaxial expressions, the triaxial ones are retrieved. Finally, the appropriateness of the yield functions in regards to the soil type is discussed.

On the Algorithmic Implementation of a Material Model Accounting for the Effects of Martensitic Transformation

The formulation and numerical implementation of a material model with as little phenomenological input as possible requires a thorough investigation of the physical relationships relevant for the material under consideration. In the present case the focus is on the correct description of the TRIP (Transformation Induced Plasticity) effect and its interpretation in terms of a transformation surface concept similar to the well‐known yield surface approach used in classical plasticity. Some general aspects of the thermodynamics of martensitic transformation along with a general idea of the topology of a transformation surface in stress space are presented and cast into equations suitable for the implementation into a finite element program. A normality rule for the transformation strain rate can be derived in analogy to the well established procedures of flow plasticity. The notion of a transformation hardening is introduced, collectively characterizing the mechanisms inhibiting transformation. They can be quantified based on the data from dilatometer tests. In contrast to earlier models of this group of authors [1], the algorithm at hand allows to keep track of the evolution of each single variant thus providing detailed insights into the variant selection mechanisms for various load levels and loading directions.

A less hypothetical perspective on rate-independent continuum theory of metal plasticity

In the framework of rate-independent theory of metal plasticity the hypothesis of maximum plastic dissipation is commonly applied to derive the associated flow-rule and to prove convexity of the yield surface in stress space. Alternatively, Drucker’s postulate of material stability may be used to prove convexity. Both hypotheses appear reasonable, but it is appealing to derive the basic framework for rate-independent metal plasticity without involving additional hypotheses. In the present article it will be shown that indeed both hypotheses can be dropped and an alternative way is established to derive the associated flow-rule and to prove convexity of the yield surface. It turns out that the associated flow-rule as well as the convexity of the yield surface are intrinsic features of the proposed theoretical framework.

A numerical model for Transformation Induced Plasticity (TRIP)

The peculiar behavior of steels exhibiting a TRIP (Transformation Induced Plasticity) effect, especially for the case of non-proportional loading paths, has given rise to a wide variety of different material descriptions ranging from purely phenomenological models to more rigorous approaches based on thermodynamic principles as proposed by this group of authors [1]. While this latter model gives reasonable results for the multigrain compound, its validity has limitations at the integration poit level since it selects only one variant within each time increment. Alternatively, this paper suggests an implicit formulation capable of reflecting the simultaneous evolution of multiple variants during transformation and suitable for an implementation into a finite element program. Particular attention has to be paid to the mechanisms inhibiting transformation, collectively characterized as hardening terms. To this end the well known yield surface concept of classical plasticity is extended by the notion of an additional phase transformation surface in stress space. The algorithm allows to identify the active variants and to monitor their contribution to the transformation kinetics. The overall strain component in loading direction is worked out and discussed in context with the orientation effect as the dominating mechanism for the accommodation of the new phase.

Yield criteria for quasibrittle and frictional materials

A new yield/damage function is proposed for modelling the inelastic behaviour of a broad class of pressure-sensitive, frictional, ductile and brittle-cohesive materials. The yield function allows the possibility of describing a transition between the shape of a yield surface typical of a class of materials to that typical of another class of materials. This is a fundamental key to model the behaviour of materials which become cohesive during hardening (so that the shape of the yield surface evolves from that typical of a granular material to that typical of a dense material), or which decrease cohesion due to damage accumulation. The proposed yield function is shown to agree with a variety of experimental data relative to soil, concrete, rock, metallic and composite powders, metallic foams, porous metals, and polymers. The yield function represents a single, convex and smooth surface in stress space approaching as limit situations well-known criteria and the extreme limits of convexity in the deviatoric plane. The yield function is therefore a generalization of several criteria, including von Mises, Drucker–Prager, Tresca, modified Tresca, Coulomb–Mohr, modified Cam-clay, and––concerning the deviatoric section––Rankine and Ottosen. Convexity of the function is proved by developing two general propositions relating convexity of the yield surface to convexity of the corresponding function. These propositions are general and therefore may be employed to generate other convex yield functions.

Linear and nonlinear modulus surfaces in stress space, from stress-strain measurements on Berea sandstone

Abstract. The elastic response of many rocks to quasistatic stress changes is highly nonlinear and hysteretic, displaying discrete memory. Rocks also display unusual nonlinear response to dynamic stress changes. A model to describe the elastic behavior of rocks and other consolidated materials is called the Preisach-Mayergoyz (PM) space model. In contrast to the traditional analytic approach to stress-strain, the PM space picture establishes a relationship between the quasistatic data and a number density of hysteretic mesoscopic elastic elements in the rock. The number density allows us to make quantitative predictions of dynamic elastic properties. Using the PM space model, we analyze a complex suite of quasistatic stress-strain data taken on Berea sandstone. We predict a dynamic bulk modulus and a dynamic shear modulus surface as a function of mean stress and shear stress. Our predictions for the dynamic moduli compare favorably to moduli derived from time of flight measurements. We derive a set of nonlinear elastic constants and a set of constants that describe the hysteretic behavior of the sandstone.

Transformation Surfaces of a Textured Pseudoelastic Polycrystalline Cu-Zn-Al Shape Memory Alloy

Pseudoelasticity is a useful regime of behavior for industrial applications of shape memory alloys. The complexity of the physical phenomena of pseudoelastic shape memory behavior requires micromechanical methods of scale transition to model at higher length scales using crystallographic information. Using a self-consistent model to transition from single crystal to polycrystal behavior, we present results for computed surfaces of constant effective transformation strain for random, drawn, tension, compression and rolled textures of a polycrystalline Cu-Zn-Al shape memory alloy in the pseudoelastic regime. These results are relevant to the development of macroscopic transformation surfaces in stress space. A feature of the results is that normality of transformation strain rate to surfaces of constant macroscopic transformation strain is observed for the most part, apart from regions of high curvature that bridge different segments of the transformation surface.

Finite Element Modelling Damage in Brittle Matrix Composites

A finite element model of the deformation and failure of brittle matrix composites during matrix cracking and delamination has been developed. The model is underpinned by the micro-mechanics developed by Aveston, Cooper and Kelly, represented through a damage mechanics approach, in which the effect of individual cracks and fibres is smeared into a continuum response. The model allows matrix micro-cracking, shear, and tensile delamination to be modelled by intersecting failure surfaces in stress space. Comparison of the predictions by the finite element model with experiments shows promise for the analysis of critical engineering structures and components.