Showing papers on "Constitutive equation published in 2004"

Constitutive laws for the matrix-logarithm of the conformation tensor

Abstract: We show how to transform a large class of differential constitutive models into an equation for the (matrix) logarithm of the conformation tensor. Under this transformation, the extensional components of the deformation field act additively, rather than multiplicatively. This transformation is motivated by numerical evidence that the high Weissenberg number problem may be caused by the failure of polynomial-based approximations to properly represent exponential profiles developed by the conformation tensor. The potential merits of the new formulation are demonstrated for a finitely-extensible fluid in a two-dimensional lid-driven cavity at Weissenberg number W i = 5 .

Accumulated creep strain and energy density based thermal fatigue life prediction models for SnAgCu solder joints

Abstract: Pb free solder is fast becoming a reality in electronic manufacturing due to marketing and legislative pressures. The industry has pretty much concluded that various versions of SnAgCu solder alloy offer the best alternative for eutectic Sn/Pb solder currently in use. With the current trend of cheaper, faster, and better electronic equipment, it has become increasingly important to evaluate the package and system performance very early in the design cycle using simulation tools. This requires life prediction models for new solder alloy systems so that the package-to-board interconnect reliability can be predicted for various environmental and field conditions. This paper describes in detail the life prediction models for SnAgCu solder joints. The models are based on published constitutive equations for this alloy and thermal cycle fatigue data on actual components. The approach uses advanced finite element modeling and analysis techniques and is based on mechanics of deformation. Both accumulated creep strain and creep strain energy density based models are developed. The model has been correlated with a number of data points and predicts life within 25% in most cases. The framework of modeling and prediction methodology described here is fully compatible with the framework used for SnPb solder previously.

A simple technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces

Abstract: Numerical simulations of crack initiation which use a cohesive zone law to model a weak interface in the solid are often limited by the occurrence of an elastic snap-back instability. At the point of instability, quasi-static finite element computations are unable to converge to an equilibrium solution, which usually terminates the calculation and makes it impossible to follow the post-instability behaviour. In this paper, we show that such numerical difficulties can easily be avoided by introducing a small viscosity in the constitutive equations for the cohesive interface. Simple boundary value problems are used to develop guidelines for selecting appropriate values of viscosity in numerical simulations involving crack nucleation and growth. As a representative application, we model crack nucleation at the interface between an elastic thin film and an elastic–plastic substrate, which is subjected to contact loading.

Comparison of a multi-layer structural model for arterial walls with a fung-type model, and issues of material stability.

Abstract: The goals of this paper are (i) to re-examine the constitutive law for the description of the (passive) highly nonlinear and anisotropic response of healthy elastic arteries introduced recently by the authors, (ii) to show how the mechanical response of a carotid artery under inflation and extension predicted by the structural model compares with that for a three-dimensional form of Fung-type strain-energy function, (iii) to provide a new set of material parameters that can be used in a finite element program, and (iv) to show that the model has certain mathematical features that are important from the point of view of material and numerical stability.

A constitutive model for unsaturated soils: thermomechanical and computational aspects

Abstract: 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.

Peridynamic modeling of membranes and fibers.

Abstract: The peridynamic theory of continuum mechanics allows damage, fracture, and long-range forces to be treated as natural components of the deformation of a material. In this paper, the peridynamic approach is applied to small thickness two- and one-dimensional structures. For membranes, a constitutive model is described appropriate for rubbery sheets that can form cracks. This model is used to perform numerical simulations of the stretching and dynamic tearing of membranes. A similar approach is applied to one-dimensional string like structures that undergrow stretching, bending, and failure. Long-range forces similar to van der Waals interactions at the nanoscale influence the equilibrium configurations of these structures, how they deform, and possibly self-assembly.

Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling

Abstract: The dynamic response of an initially spherical capsule subject to different externally imposed flows is examined. The neo-Hookean and Skalak et al. (Biophys. J., vol. 13 (1973), pp. 245–264) constitutive laws are used for the description of the membrane mechanics, assuming negligible bending resistance. The viscosity ratio between the interior and exterior fluids of the capsule is taken to be unity and creeping-flow conditions are assumed to prevail. The capillary number , beyond the interval of stability, the membrane has two tips along the direction of elongation where the deformation is most severe, and no equilibrium shapes could be identified. For both regions outside the interval of stability, the membrane model is not appropriate and bending resistance is essential to obtain realistic capsule shapes. This pattern persists for the two constitutive laws that were used, with the Skalak et al. law producing a wider stability interval than the neo-Hookean law owing to its strain hardening nature.

Sand Plasticity Model Accounting for Inherent Fabric Anisotropy

Abstract: A sand plasticity constitutive model is presented herein, which accounts for the effect of inherent fabric anisotropy on the mechanical response. The anisotropy associated with particles’ orientation distribution, is represented by a second-order symmetric fabric tensor, and its effect is quantified via a scalar-valued anisotropic state variable, \iA. \iA is defined as the first joint isotropic invariant of the fabric tensor and a properly defined loading direction tensor, scaled by a function of a corresponding Lode angle. The hardening plastic modulus and the location of the critical state line in the void ratio—mean effective stress space, on which the dilatancy depends, are made functions of \iA. The incorporation of this dependence on \iA in a pre-existing stress-ratio driven, bounding surface plasticity constitutive model, achieves successful simulations of test results on sand for a wide variation of densities, pressures, loading manners, and directions. In particular, the drastic difference in material response observed experimentally for different directions of the principal stress axes with respect to the anisotropy axes, is well simulated by the model. The proposed definition and use of \iA has generic value, and can be incorporated in a large number of other constitutive models in order to account for inherent fabric anisotropy effects.

Numerical implementation of a neural network based material model in finite element analysis

Abstract: Neural network (NN) based constitutive models can capture non-linear material behaviour. These models are versatile and have the capacity to continuously learn as additional material response data becomes available. NN constitutive models are increasingly used within the finite element (FE) method for the solution of boundary value problems. NN constitutive models, unlike commonly used plasticity models, do not require special integration procedures for implementation in FE analysis. NN constitutive model formulation does not use a material stiffness matrix concept in contrast to the elasto-plastic matrix central to conventional plasticity based models. This paper addresses numerical implementation issues related to the use of NN constitutive models in FE analysis. A consistent material stiffness matrix is derived for the NN constitutive model that leads to efficient convergence of the FE Newton iterations. The proposed stiffness matrix is general and valid regardless of the material behaviour represented by the NN constitutive model. Two examples demonstrate the performance of the proposed NN constitutive model implementation.

Finite element formulation of viscoelastic sandwich beams using fractional derivative operators

Abstract: This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler– Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grunwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects.

Numerical finite element formulation of the Schapery non-linear viscoelastic material model

Abstract: This study presents a numerical integration method for the non-linear viscoelastic behaviour of isotropic materials and structures. The Schapery's three-dimensional (3D) non-linear viscoelastic material model is integrated within a displacement-based finite element (FE) environment. The deviatoric and volumetric responses are decoupled and the strain vector is decomposed into instantaneous and hereditary parts. The hereditary strains are updated at the end of each time increment using a recursive formulation. The constitutive equations are expressed in an incremental form for each time step, assuming a constant incremental strain rate. A new iterative procedure with predictor–corrector type steps is combined with the recursive integration method. A general polynomial form for the parameters of the non-linear Schapery model is proposed. The consistent algorithmic tangent stiffness matrix is realized and used to enhance convergence and help achieve a correct convergent state. Verifications of the proposed numerical formulation are performed and compared with a previous work using experimental data for a glassy amorphous polymer PMMA. Copyright © 2003 John Wiley & Sons, Ltd.

Constitutive equations for ionic transport in porous shales

Abstract: [1] The constitutive coupled equations describing ionic transport in a porous shale are obtained at the scale of a representative elementary volume by volume averaging the local Nernst-Planck and Stokes equations. The final relationships check the Onsager reciprocity to the first order of perturbation of the state variables with respect to the thermostatic state. This state is characterized by a modified version of the Donnan equilibrium model, which accounts for the partition of the counterions between the Stern and diffuse Gouy-Chapman layers. After upscaling the local equations the material properties entering the macroscopic constitutive equations are explicitly related to the porosity of the shale, its cation exchange capacity, and some textural properties such as the electrical cementation exponent entering Archie's law. This new model is then applied to predict the salt filtering and electrodiffusion efficiencies of a shale layer.

Testing and Modeling of Soil-Structure Interface

Abstract: An accurate modeling of soil-structure interfaces is very important in order to obtain realistic solutions of many soil-structure interaction problems. To study the mechanical characteristics of soil-structure interface, a series of direct shear tests were performed. A charged-coupled-device camera was used to observe the sand particle movements near the interface. It is shown that two different failure modes exist during interface shearing. Elastic perfect-plastic failure mode occurs along the smooth interface, while strain localization occurs in a rough interface accompanied with strong strain-softening and bulk dilatancy. To describe the behavior of the rough interface, this paper proposes a damage constitutive model with ten parameters. The parameters are identified using data from laboratory interface shear tests. The proposed model is capable of capturing most of the important characteristics of interface behavior, such as hardening, softening, and dilative response. The interface behaviors under direct and simple shear tests have been well predicted by the model. Furthermore, the present model has been implemented in a finite element procedure correctly and calculation results are satisfactory.

Constitutive model for stretch-induced softening of the stress?stretch behavior of elastomeric materials

Abstract: Elastomeric materials experience stretch-induced softening as evidenced by a pre-stretched material exhibiting a significantly more compliant response than that of the virgin material. In this paper, we propose a fully three-dimensional constitutive model for the observed softening of the stress–strain behavior. The model adopts the Mullins and Tobin concept of an evolution in the underlying hard and soft domain microstructure whereby the effective volume fraction of the soft domain increases with stretch. The concept of amplified strain is then utilized in a mapping of the macroscopic deformation to the deformation experienced by the soft domain. The strain energy density function of the material is then determined from the strain energy of the soft domain and thus evolves as the volume fraction of soft domain evolves with deformation. Comparisons of model results for cyclic simple extension with the experimental data of Mullins and Tobin show the efficacy of the model and suggest that an evolution in the underlying soft/hard domain microstructure of the elastomer captures the fundamental features of stretch-induced softening. Model simulations of the cyclic stress–strain behavior and corresponding evolution in structure with strain for uniaxial tension, biaxial tension and plane strain tension are also presented and demonstrate three-dimensional features of the constitutive model.

Thermoelastic models of continua

Abstract: Introduction. 1: Thermoelastic Materials with Voids. 1.1. Preliminaries. The Laws of Thermodynamics. 1.2. Constitutive Equations. Thermodynamic Restrictions. 1.3. Boundary-Initial-Value Problems. 1.4. Continuous Data Dependence and Uniqueness Results. 1.5. The Linear Theory. 2: Dynamic Theory. 2.1. Uniqueness Results. 2.2. Reciprocal Theorem. Applications. 2.3. Homogeneous and Isotropic Bodies. 2.4. Acceleration Waves. 2.5. Harmonic Waves. 2.6. Concentrated Loads. 2.7. Radiation Conditions. 2.8. Potentials. 3: Equilibrium Theory. 3.1. Thermoelastic States. 3.2. The Thermoelastic Plane Strain. 3.3. Exercises. 4: Prestressed Thermoelastic Bodies. 4.1. Equations of Perturbed Motion. 4.2. An Existence Result. 5: Thermoelastic Cosserat Continua. 5.1. The Linear Theory of Micropolar Thermoelasticity. 5.2. Thermoelastic Processes. Boundary-Initial-Value Problem. 5.3. Reciprocity. Existence and Uniqueness Results. 5.4. Variational Theorem. 5.5. Homogeneous and Isotropic Solids. Plane Waves. 5.6. A Representation of Galerkin Type. Fundamental Solutions. 5.7. Transient Waves. 5.8. Plane Strain Problem. 5.9. Bending of Micropolar Thermoelastic Plates. 5.10. A Generalized Theory of Micropolar Thermoelasticity. 6: Thermoelastostatics of Micropolar Bodies. 6.1. Boundary Value Problems. Basic Theorems. 6.2. Special Results for Homogeneous and Isotropic Bodies. 6.3. The Equilibrium Plane Problems. 6.4. Exercises. 6.5. Thermal Stresses in Beams. 6.6. Cylinders Composed of Different Materials. 7: Nonsimple Materials. 7.1. The Nonlinear Thermoelasticity. 7.2. Uniqueness and Continuous Dependence results. 7.3. Linear Theories. 7.4. Isotropic Bodies. 7.5. A Grade Consistent Theory of Micropolar Thermoelasticity. References. Index.

Nonlinear magnetoelastic deformations

Abstract: In this paper we first summarize, in a simple form, the equilibrium equations for a solid material capable of large magnetoelastic deformations. Such equations are needed for the analysis of boundary-value problems for elastomers endowed with magnetic properties by the embedding of distributions of ferrous particles. The general constitutive law for an isotropic material in the presence of a magnetic field is described and expressed in a compact form, with either the magnetic field or the magnetic induction as the independent magnetic variable. The equations are applied, in the case of an incompressible material, to the solution of representative problems involving circular cylindrical geometry, specifically the helical shear of a circular cylindrical tube, its specializations to axial and azimuthal shear, and the problem of extension and torsion of a solid circular cylinder. For each problem a general formulation is afforded without specialization of the (isotropic) constitutive law, and then specific results are discussed briefly for special choices of such laws. It is noted, in particular, that certain restrictions may be placed on the class of constitutive laws for a considered combination of deformation and magnetic field to be admitted.

Nonlinear magnetoelastic deformations of elastomers

Abstract: Magneto-sensitive (MS) elastomers are “smart materials” whose mechanical properties may be changed rapidly by the application of a magnetic field Such materials typically consist of micron-sized ferrous particles dispersed within an elastomeric matrix The equations governing deformations of these materials were discussed in a recent paper by the present authors and applied in a particular specialization of the constitutive model to the problem of axial shear of a circular cylindrical tube subject to a radial magnetic field In the present paper we develop the governing equations for a more general form of constitutive model and provide alternative forms of the equations, including a Lagrangian formulation To illustrate the theory the problem of azimuthal shear of a circular cylindrical tube is formulated and then solved for a specific constitutive law with a magnetic field that is initially radial The results, which show the stiffening of the azimuthal shear stress/strain response with increasing magnetic field strength, are illustrated graphically

A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin

Abstract: This study develops a gradient theory of small-deformation viscoplasticity based on: a system of microforces consistent with its peculiar balance; a mechanical version of the second law that includes, via the microforces, work performed during viscoplastic flow; a constitutive theory that accounts for the Burgers vector through a free energy dependent on curl H p , with Hp the plastic part of the elastic–plastic decomposition of the displacement gradient. The microforce balance and the constitutive equations, restricted by the second law, are shown to be together equivalent to a nonlocal flow rule in the form of a coupled pair of second-order partial differential equations. The first of these is an equation for the plastic strain-rate E p in which the stress T plays a basic role; the second, which is independent of T, is an equation for the plastic spin W p . A consequence of this second equation is that the plastic spin vanishes identically when the free energy is independent of curl H p , but not generally otherwise. A formal discussion based on experience with other gradient theories suggests that sufficiently far from boundaries solutions should not differ appreciably from classical solutions, but close to microscopically hard boundaries, boundary layers characterized by a large Burgers vector and large plastic spin should form. Because of the nonlocal nature of the flow rule, the classical macroscopic boundary conditions need be supplemented by nonstandard boundary conditions associated with viscoplastic flow. As an aid to solution, a variational formulation of the flow rule is derived. Finally, we sketch a generalization of the theory that allows for isotropic hardening resulting from dissipative constitutive dependences on ∇ E p .

Mastering Calculations in Linear and Nonlinear Mechanics

Abstract: The notion of quality of a finite element solution.- The constitutive relation error method for linear problems.- Other methods for linear problems.- Principles of the comparison of the various estimators in the linear case.- Mesh adaptation for linear problems.- The constitutive relation error method for nonlinear evolution problems.- The constitutive relation error method in dynamics.- Techniques for constructing admissible fields.- Estimation of local errors.

Prediction of shape distortions Part I. FE-implementation of a path dependent constitutive model

Abstract: There is a great interest, especially from the aircraft industry, to increase the ability to understand and predict development of shape distortions and residual stresses during manufacture of polymer composite components. An increased ability to predict shape distortions will result in more cost efficient development, improved performance and optimised manufacturing of composites. To be able to predict residual stresses and shape distortions a model is needed that accounts for all important mechanisms involved. In a previous work by the authors, it was demonstrated that such—models must account for thermal expansion (different in glassy and rubbery state), chemical shrinkage due to the crosslinking reaction and finally frozen-in deformations. The present paper presents a simple mechanical constitutive model that accounts for the mechanisms mentioned above. The model is a limiting case of linear visco-elasticity that permits us to replace the rate dependence by a path dependence on the state variables: strain, degree of cure and temperature. This means significant savings in computational time, memory requirements and costs for material characterisation as compared to conventional visco-elastic models. This is the first of two papers, the second paper deals with experimental validation and analysis of mechanical boundary conditions during prediction of shape distortion.

Plasticity: A Treatise on Finite Deformation of Heterogeneous Inelastic Materials

Abstract: Preface 1. Geometry 2. Kinematics 3. Stress and stress-rate measures and balance relations 4. Continuum theories of elastoplasticity 5. Integration of continuum constitutive equations 6. Finite elastoplastic deformation of single crystals 7. Finite plastic deformation of granular materials 8. Average quantities and homogenisation models 9. Special experimental techniques Cited authors Subject index.