Incremental Stress-strain Relations Research Articles

A Micromechanical Model for Deformation Behavior of Nanocrystalline Copper

Molecular dynamics simulations have show that nanocrystalline (NC) materials can be treated as composite materials consisting of two phases of grain and grain boundary. In this paper, the incremental stress-strain relation is derived based on deformation mechanism of NC materials and internal variable theory from micromechanics point of view. The developed model is exemplified by the pure copper subjected to uniaxial tension. Implicated iteration algorithm is then employed to obtain the stress-strain relation. Moreover, the effects of grain shape and statistical distribution of grain sizes are also discussed, and predicted results are compared with experimental values to verify the model.

A variational formulation for finite deformation wrinkling analysis of inelastic membranes

This paper is concerned with a novel, fully variational formulation for finite deformation analysis of inelastic membranes with wrinkling. In contrast to conventional approaches, every aspect of the physical problem derives from minimization of suitable energy functionals. A variational formulation of finite strain plasticity theory, which leads to a minimization problem for the constitutive updates, serves as the starting point for the derivations. In order to take into account the kinematics induced by wrinkles and slacks, a relaxed version of the finite strain functional is postulated. In effect, the local incremental stress–strain relations are established via differentiation of the relaxed energy functional with respect to the strains. Hence, the presented formulation is fully analogous to that of hyperelasticity with the sole exception that the aforementioned functional depends on history variables and, accordingly, it is path dependent. The advantages associated with the developed variational method are manifold. From a practical point of view, the possibility of applying standard optimization algorithms to solve the minimization problem describing inelastic membranes is remunerative. From a mathematical point of view, on the other hand, the energy of the system induces some sort of natural metric representing an essential requirement for error estimation and thus, for adaptive finite element methods. The presented derivation of the model allows to consider possible material symmetries in the elastic as well as plastic response of the material. As a prototype, a von Mises-type model is implemented. The efficiency and performance of the resulting algorithm are demonstrated by means of numerical examples.

Unsaturated soils: From constitutive modelling to numerical algorithms

This paper presents an overview of constitutive modelling of unsaturated soils and the numerical algorithms for solving the associated boundary value problems. It first discusses alternative stress and strain variables that can be used in constitutive models for unsaturated soils. The paper then discusses the key issues in unsaturated soil modelling and how these issues can be incorporated into an existing model for saturated soils. These key issues include (1) volumetric behaviour associated with saturation or suction changes; (2) strength behaviour associated with saturation and suction changes, and (3) hydraulic behaviour associated with saturation or suction changes. The paper also shows how hysteresis in soil–water characteristics can be incorporated into the elasto-plastic framework, leading to coupled hydro-mechanical models. Finally, the paper demonstrates the derivation of the incremental stress–strain relations for unsaturated soils and discusses briefly the new challenges in implementing these relations into the finite element method.

Incremental stress-strain relation from granular elasticity: Comparison to experiments

Granular media are reversible and elastic if the stress increments are small enough. An elastic stress-strain relation, employed previously to determine static stress distributions, in this paper is compared to experiments by Kuwano and Jardine [Geotechnique 52, 727 (2002)] on incremental stress-strain relations, and shown to yield satisfactory agreement. In addition, the yield condition is given a firmer footing.

Nonlinear Stress-Strain Relationship of Soil Reinforced with Flexible Geofibers

A nonlinear stress-strain relationship of soil reinforced with flexible geofibers under static loading is derived based on a nonlinear elastic stress-strain relationship of soil and a linear elastic stress-strain relationship of geofibers in the paper. This investigation includes the following aspects: First, the homogenization technique is introduced to find the volume average stress tensor and volume average strain tensor and further an elastic incremental stress-strain relation is introduced to describe deviatoric shear stress-axial strain relationship of equivalent homogeneous geofiber-reinforced soil. Second, the relation of geofiber numbers, content, mechanical behavior, distribution, and geometrical features to shear modulus of geofiber-reinforced soil is expressed and assessed by employing an elastic energy method. Third, the deviatoric shear stress and axial strain curves of geofiber-reinforced soil are calibrated by laboratory testing data of geofiber reinforced soil samples. Finally, the theoretical computational curves of geofiber reinforced soil are compared with the curves calibrated by testing data of geofiber reinforced soil. The model prediction has a good agreement with experimental results.

An analytical model for estimating deformation in laser forming

Laser forming of metal sheets offers the advantages of requiring no external forces and thus reduces cost and increases flexibility. This paper presents an analytical model to estimate the angle bent during the laser forming of a sheet. Plastic deformation is considered during both heating and cooling and is calculated based on a history-dependent incremental stress–strain relationship. On the basis of the proposed model with known temperature distributions, the bending angle induced by laser can be calculated. Comparison of the present model with experiment data is provided to demonstrate the accuracy of the present model under both TGM and BM.

Quasi-static crush behavior of aluminum honeycomb specimens under non-proportional compression-dominant combined loads

The quasi-static crush behavior of aluminum 5052-H38 honeycomb specimens under non-proportional compression-dominant combined loads is investigated by experiments. A test fixture was designed such that dominant compressive and shear loads with respect to the strongest material symmetry direction can be controlled and applied independently. Compression-dominant combined loads and pure compressive loads were applied in different sequences to induce non-proportional combined loads. The experimental results show that the normal crush and shear strengths in combined loading regions and the normal crush strengths in pure compressive loading regions of the non-proportional combined loads are quite consistent with the existing phenomenological yield criterion based on the experimental normal crush and shear strengths under proportional combined loads. The experimental results also indicate that the sequence of loading paths for the non-proportional combined loads does not affect the crush strengths when compared with those obtained under proportional combined loads. In addition, the experimental results indicate that the non-normality plastic flow behavior of honeycomb specimens under non-proportional combined loads is consistent with that under proportional combined loads. Finally, specimens crushed under non-proportional combined loads show developments of different stacking patterns of folds in different loading regions of the non-proportional combined loads. The experimental results suggest that the incremental stress–strain relation for the transition loading path within the so-called yield surface from one yield state to another may be related to the displacement increments that correspond to the change of microscopic folding mechanisms instead of the usual elastic relation according to the classical plasticity theory.

A constitutive model for unsaturated soils: thermomechanical and computational aspects

This paper first presents a complete formulation of a constitutive model that deals with the irreversible behaviour of unsaturated soils under various loading and drying/wetting conditions. A standard form of incremental stress-strain relations is derived. The constitutive model is then cast into the thermodynamical theories and verified using the thermomechanical principles. It is shown that hydraulic hysteresis does not contribute to the plastic dissipation, though it contributes to the plastic work. All plastic work associated with a plastic increment of the degree of saturation is stored and can be recovered in a reversed plastic increment of saturation. The incremental constitutive equations are also reformulated for implementation in finite element codes where displacements and pore pressures are primary unknowns. Qualitative predictions of the constitutive model show that incorporating two suction related yield surfaces and non-associated flow rules into the Barcelona Basic Model opens a full range of possibilities in modelling unsaturated soil behaviour.

Mechanics of Solids and Shells: Theories and Approximations

Mechanics of Solids and Shells: Theories and Approximations

Anisotropic damage model under continuum slip crystal plasticity theory for single crystals

Modern concepts in the safe and economical design of components and structures need modern material models, which describe the material behavior more correctly. In high temperature design, the study of creep damage accumulation and its influence on the deformation behavior is very important. For ductile materials large deformation takes place at the level of damage appearance. Damage is anisotropic in nature. In this paper an anisotropic damage mechanics model based on a continuum damage mechanics (CDM) has been developed to model creep behavior of single crystal superalloys. Using the theory of CDM, the slip system model coupling with anisotropic damage model is developed. The model is formulated in the context of irreversible thermodynamics and the internal state variable theory. The distinguishing characteristic of the proposed constitutive model is that, by construction, the corresponding incremental stress–strain relations including damage evolution equation derive from a thermodynamical framework. The finite element program ABAQUS has been used and the slip system model is written using a user material subroutine. The numerical simulations show that the developed damage crystal model can reflect the microstructure such as the lattice orientation, self- and latent-hardening and rate sensitivity has great influence on the creep and damage development.

Calculation of the incremental stress-strain relation of a polygonal packing.

The constitutive relation of the quasistatic deformation on two-dimensional packed samples of polygons is calculated using molecular dynamics simulations. The stress values at which the system remains stable are bounded by a failure surface, which shows a power law dependence on the pressure. Below the failure surface, nonlinear elasticity and plastic deformation are obtained, which are evaluated in the framework of the incremental linear theory. The results show that the stiffness tensor can be directly related to the microcontact rearrangements. The plasticity obeys a nonassociated flow rule with a plastic limit surface that does not agree with the failure surface.

Poromechanics: from Linear to Nonlinear Poroelasticity and Poroviscoelasticity

Due to the impact on productivity and oil in place estimates, reliable modeling of rock behavior is essential in reservoir engineering. This paper examines several aspects of rock poroelastic behavior within the framework of Biot's mechanics of fluid saturated porous solids. Constitutive laws of linear and nonlinear poroelasticity are first determined from a fundamental stress decomposition, which allows to clearly connect linear and nonlinear models. Concept of effective stress and rock compressibility are considered. Linear incremental stress-strain relations are derived from the proposed nonlinear constitutive law by defining tangent elastic properties. These characteristics are naturally functions of strains and pore pressure, but explicit expressions as functions of stresses and pore pressure are established herein. Experiments performed on a reservoir sandstone illustrate these points. A constitutive law of poroviscoelasticity is finally presented and applied to experimental data obtained on clay.

Constitutive modeling of aluminum matrix NiTi fiber-reinforced smart composite

The purpose of this study is to provide a numerical method to model the constitutive behavior of aluminum matrix NiTi fiber-reinforced smart composite along the fiber direction. In the present approach, pseudo-elasticity, shape-memory, and strain-hardening effects have been taken into account. During phase transformation, the strain–temperature relation of NiTi fibers has been assumed to be linear. Hence, the incremental stress–strain relationship of the fibers has been derived for each temperature range where phase transformation takes place. A bi-linear and isotropic hardening elasto-plastic model has been employed to describe the stress–strain relation of the aluminum matrix. With the consideration of the fabrication procedure, the stress–strain behavior of the composite has been computed.

Particle spin in anisotropic granular materials

We adopt the simplest theory for the description of the mechanical behavior of a random array of elastic spheres and show that it is necessary that the average rotation of the spheres about their centers be distinguished from the average rotation of the aggregate. Without such a distinction, the stress in an anisotropic aggregate, subjected to a deformation that involves an average rotation, would not be symmetric. We illustrate this by calculating the incremental stress–strain relation for an orthotropic aggregate.

A Unified Micromechanical Model for the Mechanical Properties of Two Constituent Composite Materials. Part IV: Rubber-Elastic Behavior

This series of papers reports a new, general, and unified micromechanical model for estimating the three-dimensional mechanical properties of a composite made from two constituent materials, i.e., continuous fiber and matrix. The present paper addresses the application of the model to rubber-based composites, with a focus on prediction of the entire stress-strain response characteristic of a composite having an elastomer matrix constituent material. For this purpose, an accurate constitutive theory has been developed for any incompressible rubber-like material. Its incremental stress-strain relationship at any given level is simply correlated by a load-dependent compliance matrix, similar to Hooke's law. The theory is consistent in that when the involved material parameters, i.e., the load-dependent Young's modulus and Poisson's ratio, are determined using a single type of test data such as uniaxial tension data, it is then able to predict precisely the material responses to other types of deformation conditions. Combining the incremental theory with the unified model developed in Parts I and II, an analysis procedure for the overall properties of elastomer-based composites follows. As an application, an interlock weft-knitted polyester-fiber fabric-reinforced polyurethane elastomer composite is investigated. Favorable correlation has been found between the theoretical and experimental results.

A Unified Micromechanical Model for the Mechanical Properties of Two Constituent Composite Materials. Part IV: Rubber-Elastic Behavior

This series of papers reports a new, general, and unified micromechanical model for estimating the three-dimensional mechanical properties of a composite made from two constituent materials, i.e., continuous fiber and matrix. The present paper addresses the application of the model to rubber-based composites, with a focus on prediction of the entire stress-strain response characteristic of a composite having an elastomer matrix constituent material. For this purpose, an accurate constitutive theory has been developed for any incompressible rubber-like material. Its incremental stress-strain relationship at any given level is simply correlated by a load-dependent compliance matrix, similar to Hooke's law. The theory is consistent in that when the involved material parameters, i.e., the load-dependent Young's modulus and Poisson's ratio, are determined using a single type of test data such as uniaxial tension data, it is then able to predict precisely the material responses to other types of deformation conditions. Combining the incremental theory with the unified model developed in Parts I and II, an analysis procedure for the overall properties of elastomer-based composites follows. As an application, an interlock weft-knitted polyester-fiber fabric-reinforced polyurethane elastomer composite is investigated. Favorable correlation has been found between the theoretical and experimental results.

The variational formulation of viscoplastic constitutive updates

We present a class of constitutive updates for general viscoplastic solids including such aspects of material behavior as finite elastic and plastic deformations, non-Newtonian viscosity, rate-sensitivity and arbitrary flow and hardening rules. The distinguishing characteristic of the proposed constitutive updates is that, by construction, the corresponding incremental stress—strain relations derive from a pseudo-elastic strain-energy density. This, in turn, confers the incremental boundary value problem a variational structure. In particular, the incremental deformation mapping follows from a minimum principle. This minimum principle may conveniently be taken as a basis for error estimation and mesh adaption. The accuracy and robustness of the variational constitutive updates is demonstrated with the aid of convergence tests involving the finitely-deforming Mises solid and ductile single crystals. The ability of the updates to resolve the complex patterns of slip activity which arise in the latter application is particularly noteworthy.

Energy flux and dissipation in pre-stressed solids

The modelling of a pre-stressed anisotropic viscoelastic solid is briefly reviewed and a general thermodynamic restriction is derived for a linearized costitutive equation around the prestressed state. Next the thermodynamic restriction is shown to imply the negative definiteness of the divergence of the energy flux. Indeed, the divergence proves to be unaffected by the pre-stress. The pre-stress instead affects the wave modes as is shown explicitly in the particular case when the pre-stress and the incremental stress-strain relation are isotropic.

Triaxial Composite Model for Basic Creep of Concrete

This paper shows how the mechanics of elastic composite materials can be adapted to predict the basic creep of concrete with aging caused by hydration. The prediction is made on the basis of the given composition of concrete, the elastic constants of the aggregate, and the aging viscoelastic properties of the portland cement mortar. The triaxial action of the composite is approximated by Dvorak's transformation field analysis. To convert the aging creep problem to an elastic problem of a composite material with inelastic strains, Granger and Bazant's approach is used. This approach relies on Bazant's age-adjusted effective modulus method that reduces the integral-type stress-strain relation for linear aging creep with nonconvolution kernel to a quasi-elastic incremental stress-strain relation with inelastic incremental strain. Explicit expressions for the aging creep properties of concrete as a composite are deduced. The model is calibrated and verified by Ward and colleagues and Counto's test data. The predictions obtained are almost as close as those recently obtained by Granger and Bazant's model. While the present model has the advantage of describing the triaxial composite action in a rational manner, it does not yet capture the effect of the deviation of the aggregate configuration from the case of a continguous aggregate skeleton of maximum possible compactness. Further refinements in this respect are needed. Another refinement might be needed to take into account possible enhancement of creep in the interface layers between the mortar and the aggregate.

The sliding crack model of brittle deformation: An internal variable approach

Rice's internal variable theory including a micro-to-macro transition is employed to formulate a micromechanical, two-dimensional damage model of brittle deformation in compression. The sliding crack model is selected as a basic dissipative mechanism underlying macroscopic inelastic deformation. Microfluxes and conjugated thermodynamic forces are identified and incremental stress-strain relations are derived in loading and unloading. Comparison with the model of Nemat-Nasser and Obata (Nemat-Nasser, S. and Obata, M. (1988) A microcrack model of dilatancy in brittle materials. Journal of Applied Mechanics55, 24–35, in which the sliding crack mechanism was analyzed from a kinematic point of view, is presented. An illustrative example is worked out showing the capability of the present model to predict experimentally observed response of a compact rock.