Showing papers on "Constitutive equation published in 2011"

A finite strain model of stress, diffusion, plastic flow, and electrochemical reactions in a lithium-ion half-cell

Abstract: We formulate the continuum field equations and constitutive equations that govern deformation, stress, and electric current flow in a Li-ion half-cell. The model considers mass transport through the system, deformation and stress in the anode and cathode, electrostatic fields, as well as the electrochemical reactions at the electrode/electrolyte interfaces. It extends existing analyses by accounting for the effects of finite strains and plastic flow in the electrodes, and by exploring in detail the role of stress in the electrochemical reactions at the electrode–electrolyte interfaces. In particular, we find that that stress directly influences the rest potential at the interface, so that a term involving stress must be added to the Nernst equation if the stress in the solid is significant. The model is used to predict the variation of stress and electric potential in a model 1-D half-cell, consisting of a thin film of Si on a rigid substrate, a fluid electrolyte layer, and a solid Li cathode. The predicted cycles of stress and potential are shown to be in good agreement with experimental observations.

Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method

Abstract: The deformation of an initially spherical capsule, freely suspended in simple shear flow, can be computed analytically in the limit of small deformations [D. Barthes-Biesel, J.M. Rallison, The time-dependent deformation of a capsule freely suspended in a linear shear flow, J. Fluid Mech. 113 (1981) 251-267]. Those analytic approximations are used to study the influence of the mesh tessellation method, the spatial resolution, and the discrete delta function of the immersed boundary method on the numerical results obtained by a coupled immersed boundary lattice Boltzmann finite element method. For the description of the capsule membrane, a finite element method and the Skalak constitutive model [R. Skalak, A. Tozeren, R.P. Zarda, S. Chien, Strain energy function of red blood cell membranes, Biophys. J. 13 (1973) 245-264] have been employed. Our primary goal is the investigation of the presented model for small resolutions to provide a sound basis for efficient but accurate simulations of multiple deformable particles immersed in a fluid. We come to the conclusion that details of the membrane mesh, as tessellation method and resolution, play only a minor role. The hydrodynamic resolution, i.e., the width of the discrete delta function, can significantly influence the accuracy of the simulations. The discretization of the delta function introduces an artificial length scale, which effectively changes the radius and the deformability of the capsule. We discuss possibilities of reducing the computing time of simulations of deformable objects immersed in a fluid while maintaining high accuracy.

Understanding and predicting viscous, elastic, plastic flows

Abstract: Foams, gels, emulsions, polymer solutions, pastes and even cell assemblies display both liquid and solid mechanical properties. On a local scale, such “soft glassy” systems are disordered assemblies of deformable rearranging units, the complexity of which gives rise to their striking flow behaviour. On a global scale, experiments show that their mechanical behaviour depends on the orientation of their elastic deformation with respect to the flow direction, thus requiring a description by tensorial equations for continuous materials. However, due to their strong non-linearities, the numerous candidate models have not yet been solved in a general multi-dimensional geometry to provide stringent tests of their validity. We compute the first solutions of a continuous model for a discriminant benchmark, namely the flow around an obstacle. We compare it with experiments of a foam flow and find an excellent agreement with the spatial distribution of all important features: we accurately predict the experimental fields of velocity, elastic deformation, and plastic deformation rate in terms of magnitude, direction, and anisotropy. We analyse the role of each parameter, and demonstrate that the yield strain is the main dimensionless parameter required to characterize the materials. We evidence the dominant effect of elasticity, which explains why the stress does not depend simply on the shear rate. Our results demonstrate that the behaviour of soft glassy materials cannot be reduced to an intermediate between that of a solid and that of a liquid: the viscous, the elastic and the plastic contributions to the flow, as well as their couplings, must be treated simultaneously. Our approach opens the way to the realistic multi-dimensional prediction of complex flows encountered in geophysical, industrial and biological applications, and to the understanding of the link between structure and rheology of soft glassy systems.

A constitutive model with microstructure evolution for flow of rate-independent granular materials

Abstract: A constitutive model is developed for the complex rheology of rate-independent granular materials. The closures for the pressure and the macroscopic friction coefficient are linked to microstructure through evolution equations for coordination number and fabric. The material constants in the model are functions of particle-level properties and are calibrated using data generated through simulations of steady and unsteady simple shear using the discrete element method (DEM). This model is verified against DEM simulations at complex loading conditions.

Homogenization-based constitutive models for magnetorheological elastomers at finite strain

Abstract: This paper proposes a new homogenization framework for magnetoelastic composites accounting for the effect of magnetic dipole interactions, as well as finite strains. In addition, it provides an application for magnetorheological elastomers via a “partial decoupling” approximation splitting the magnetoelastic energy into a purely mechanical component, together with a magnetostatic component evaluated in the deformed configuration of the composite, as estimated by means of the purely mechanical solution of the problem. It is argued that the resulting constitutive model for the material, which can account for the initial volume fraction, average shape, orientation and distribution of the magnetically anisotropic, non-spherical particles, should be quite accurate at least for perfectly aligned magnetic and mechanical loadings. The theory predicts the existence of certain “extra” stresses—arising in the composite beyond the purely mechanical and magnetic (Maxwell) stresses—which can be directly linked to deformation-induced changes in the microstructure. For the special case of isotropic distributions of magnetically isotropic, spherical particles, the extra stresses are due to changes in the particle two-point distribution function with the deformation, and are of order volume fraction squared, while the corresponding extra stresses for the case of aligned, ellipsoidal particles can be of order volume fraction, when changes are induced by the deformation in the orientation of the particles. The theory is capable of handling the strongly nonlinear effects associated with finite strains and magnetic saturation of the particles at sufficiently high deformations and magnetic fields, respectively.

A Brief Review of Elasticity and Viscoelasticity for Solids

Abstract: There are a number of interesting applications where modeling elastic and/or viscoelastic materials is fundamental, including uses in civil engineering, the food industry, land mine detection and ultrasonic imaging. Here we provide an overview of the subject for both elastic and viscoelastic materials in order to understand the behavior of these materials. We begin with a brief introduction of some basic terminology and relationships in continuum mechanics, and a review of equations of motion in a continuum in both Lagrangian and Eulerian forms. To complete the set of equations, we then proceed to present and discuss a number of specific forms for the constitutive relationships between stress and strain pro- posed in the literature for both elastic and viscoelastic materials. In addition, we discuss some applications for these constitutive equations. Finally, we give a com- putational example describing the motion of soil experiencing dynamic loading by incorporating a specific form of constitutive equation into the equation of motion.

A comparative study on Arrhenius-type constitutive model and artificial neural network model to predict high-temperature deformation behaviour in Aermet100 steel

Abstract: For predicting high-temperature deformation behaviour in Aermet100 steel, the experimental stress–strain data from isothermal hot compression tests on a Gleeble-3800 thermo-mechanical simulator, in a wide range of temperatures (1073–1473 K) and strain rates (0.01–50 s−1), were employed to develop the Arrhenius-type constitutive model and artificial neural network (ANN) model, and their predictability for high-temperature deformation behaviour of Aermet100 steel was further evaluated. The predictability of two models was quantified in terms of correlation coefficient (R) and average absolute relative error (AARE). The R and AARE for the Arrhenius-type constitutive model were found to be 0.9861 and 7.62% respectively, while the R and AARE for the feed-forward back-propagation ANN model are 0.9995 and 2.58% respectively. The breakdown of the Arrhenius-type constitutive model at the instability regimes (i.e. at 1073 K and 1173 K in 0.1, 1, 10 and 50 s−1, and at 1373 K in 50 s−1) is possibly due to that physical mechanisms in the instability regimes, where microstructure exhibits cracking, shear bands and twin kink bands, are far different from that of the stability regimes where dynamic recovery and recrystallization occur. But the feed-forward back-propagation ANN model can accurately track the experimental data across the whole hot working domain, which indicates it has good capacity to model the complex high-temperature deformation behaviour of materials associated with various interconnecting metallurgical phenomena like work hardening, dynamic recovery, dynamic recrystallization, flow instability, etc.

Thixotropic elasto-viscoplastic model for structured fluids

Abstract: A constitutive model for structured fluids is presented. Its predictive capability includes thixotropy, viscoelasticity and yielding behavior. It is composed by two differential equations, one for the stress and the other for the structure parameter—a scalar quantity that represents the structuring level of the fluid. The equation for stress is obtained in accordance with a simple mechanical analog composed by a structuring-level-dependent Maxwell element in parallel with a Newtonian element, leading to an equation of the same form as the Jeffreys (or Oldroyd-B) equation. The relaxation and retardation times that arise are functions of the structure parameter. The ideas found in de Souza Mendes, J. Non-Newtonian Fluid Mech., 2009, 164, 66 are employed for the structure parameter equation as well as for the dependencies on the structure parameter of the structural viscosity and structural shear modulus. The model is employed in constant-rate, constant-stress, and oscillatory shear flows, and its predictive capability is shown to be excellent for all cases.

A fractional calculus approach to nonlocal elasticity

Abstract: If the attenuation function of strain is expressed as a power law, the formalism of fractional calculus may be used to handle Eringen nonlocal elastic model. Aim of the present paper is to provide a mechanical interpretation to this nonlocal fractional elastic model by showing that it is equivalent to a discrete, point-spring model. A one-dimensional geometry is considered; the static, kinematic and constitutive equations are presented and the governing fractional differential equation highlighted. Two numerical procedures to solve the fractional equation are finally implemented and applied to study the strain field in a finite bar under given edge displacements.

Computational modeling of passive myocardium

Abstract: This work deals with the computational modeling of passive myocardial tissue within the framework of mixed, non-linear finite element methods. We consider a recently proposed, convex, anisotropic hyperelastic model that accounts for the locally orthotropic micro-structure of cardiac muscle. A coordinate-free representation of anisotropy is incorporated through physically relevant invariants of the Cauchy–Green deformation tensors and structural tensors of the corresponding material symmetry group. This model, which has originally been designed for exactly incompressible deformations, is extended towards entirely three-dimensional inhomogeneous deformations by additively decoupling the strain energy function into volumetric and isochoric parts along with the multiplicative split of the deformation gradient. This decoupled constitutive structure is then embedded in a mixed finite element formulation through a three-field Hu–Washizu functional whose simultaneous variation with respect to the independent pressure, dilatation, and placement fields results in the associated Euler–Lagrange equations, thereby minimizing the potential energy. This weak form is then consistently linearized for uniform-pressure elements within the framework of an implicit finite element method. To demonstrate the performance of the proposed approach, we present a three-dimensional finite element analysis of a generic biventricular heart model, subjected to physiological ventricular pressure. The parameters employed in the numerical analysis are identified by solving an optimization problem based on six simple shear experiments on explanted cardiac tissue. Copyright © 2010 John Wiley & Sons, Ltd.

Hot working behavior of 2205 austenite–ferrite duplex stainless steel characterized by constitutive equations and processing maps

Abstract: The hot deformation characteristics of the 2205 duplex stainless steel were analyzed using constitutive equations and processing maps. The hot compression tests were performed at temperature range of 950–1200 °C and strain rate of 0.001–1 s−1. Flow stress was modeled by the constitutive equation of hyperbolic sine function. However, the stress exponent and strain rate sensitivity were different at low and high deformation temperatures where austenite and ferrite are dominant, respectively. It was recognized that strain at the peak point of flow curve increases with the Zener–Hollomon parameter, Z, at low temperature deformation while at high temperature deformation it actually decreases with Z. The power dissipation map, instability map and processing map were developed for the typical strain of 0.3. It was realized that dynamic restoration mechanisms could efficiently hinder the occurrence of flow instability at low and medium strain rates. Otherwise, the increase in strain rate at low and high temperatures could increase the risk of flow instability.