Showing papers on "Constitutive equation published in 2003"

Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release

Abstract: A refined version of the Doi and Edwards tube model for entangled polymer liquids is presented. The model is intended to cover linear chains in the full range of deformation rates from linear to strongly nonlinear flows. The effects of reptation, chain stretch, and convective constraint release are derived from a microscopic stochastic partial differential equation that describes the dynamics of the chain contour down to the length scale of the tube diameter. Contour length fluctuations are also included in an approximate manner. Predictions of mechanical stresses as well as the single chain structure factor under flow are shown. A comparison with experimental data is made in which all model parameters are fixed at universal values or are obtained from linear oscillatory shear measurements. With no parameter modification the model produces good agreement over a wide range of rheological data for entangled polymer solutions, including both nonlinear shear and extension.

On Implicit Constitutive Theories

Abstract: In classical constitutive models such as the Navier-Stokes fluid model, and the Hookean or neo-Hookean solid models, the stress is given explicitly in terms of kinematical quantities. Models for viscoelastic and inelastic responses on the other hand are usually implicit relationships between the stress and the kinematical quantities. Another class of problems wherein it would be natural to develop implicit constitutive theories, though seldom resorted to, are models for bodies that are constrained. In general, for such materials the material moduli that characterize the extra stress could depend on the constraint reaction. (E.g., in an incompressible fluid, the viscosity could depend on the constraint reaction associated with the constraint of incompressibility. In the linear case, this would be the pressure.) Here we discuss such implicit constitutive theories. We also discuss a class of bodies described by an implicit constitutive relation for the specific Helmholtz potential that depends on both the stress and strain, and which does not dissipate in any admissible process. The stress in such a material is not derivable from a potential, i.e., the body is not hyperelastic (Green elastic).

Incorporation of experimentally-derived fiber orientation into a structural constitutive model for planar collagenous tissues

Abstract: Structural constitutive models integrate information on tissue composition and structure, avoiding ambiguities in material characterization. However, critical structural information (such as fiber orientation) must be modeled using assumed statistical distributions, with the distribution parameters estimated from fits to the mechanical test data. Thus, full realization of structural approaches continues to be limited without direct quantitative structural information for direct implementation or to validate model predictions. In the present study, fiber orientation information obtained using small angle light scattering (SALS) was directly incorporated into a structural constitutive model based on work by Lanir (J. Biomech., v. 16, pp. 1-12, 1983). Demonstration of the model was performed using existing biaxial mechanical and fiber orientation data for native bovine pericardium (Sacks and Chuong, ABME, v.26, pp. 892-902, 1998). The structural constitutive model accurately predicted the complete measured biaxial mechanical response. An important aspect of this approach is that only a single equibiaxial test to determine the effective fiber stress-strain response and the SALS-derived fiber orientation distribution were required to determine the complete planar biaxial mechanical response. Changes in collagen fiber crimp under equibiaxial strain suggest that, at the meso-scale, fiber deformations follow the global tissue strains. This result supports the assumption of affine strain to estimate the fiber strains. However, future evaluations will have to be performed for tissue subjected to a wider range of strain to more fully validate the current approach.

Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation

Abstract: Recently we developed a theory for fast flows of entangled polymer melts which includes the processes of reptation, convective and reptation-driven constraint release, chain stretch and contour length fluctuations The theory is derived from a stochastic microscopic equation of motion of the chain inside the tube and of the tube itself As a result we obtain a partial differential equation for the tube tangent correlation function, the solution of which requires quite intensive calculations At the same time the application of this theory to realistic flows (which is anything other than the laboratory rheometer) requires a simple and less computationally intensive set of equations for the stress tensor similar to the Giesekus, PTT, Larson or Pom–Pom equations In particular, the last was derived from molecular theory for a generic type of branched polymer In this paper we demonstrate that molecular tube theory can also provide a route to constructing a family of very simple differential constitutive equations for linear polymers They capture the full model quite well and therefore can be used in flow solving software to model spatially inhomogeneous flows We present a comparison of the proposed equations with our full model and with experimental data

Load–displacement behavior during sharp indentation of viscous–elastic–plastic materials

Abstract: A model is developed that describes the sharp indentation behavior of time-dependent materials. The model constitutive equation is constructed from a series of quadratic mechanical elements, with independent viscous (dashpot), elastic (spring), and plastic (slider) responses. Solutions to this equation describe features observed under load-controlled indentation of polymers, including creep, negative unloading tangents, and loading-rate dependence. The model describes a full range of viscous–elastic–plastic responses and includes as bounding behaviors time-independent elastic–plastic indentation (appropriate to metals and ceramics) and time-dependent viscous–elastic indentation (appropriate to elastomers). Experimental indentation traces for a range of olymers with different material properties (elastic modulus, hardness, viscosity) are econvoluted and ranked by calculated time constant. Material properties for these polymers, deconvoluted from single load–unload cycles, are used to predict the indentation load–displacement behavior at loading rates three times slower and faster, as well as the steady-state creep rate under fixed load.

A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates

Abstract: The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. A generalized Maxwell model with the fractional calculus was considered. Exact solutions of some unsteady flows of a viscoelastic fluid between two parallel plates are obtained by using the theory of Laplace transform and Fourier transform for fractional calculus. The flows generated by impulsively started motions of one of the plates are examined. The flows generated by periodic oscillations of one of the plates are also studied.

A constitutive scaling law and a unified comprehension for frictional slip failure, shear fracture of intact rock, and earthquake rupture

Abstract: [1] It is widely recognized that some of the physical quantities inherent in a rupture are scale-dependent, and the scale dependence is one of the most no facts and features of rupture phenomena. The paper addresses how such scale-dependent shear rupture of a broad range from laboratory-scale frictional slip failure and shear fracture of intact rock to field-scale rupture as an earthquake source can be unified by a single constitutive law. Noting that the earthquake rupture is a mixed process between frictional slip failure and the shear fracture of intact rock, it is concluded that the constitutive law for the earthquake rupture be formulated as a unifying law that governs both frictional slip failure and shear fracture of intact rock. It is demonstrated that the slip-dependent constitutive law is such a unifying law, and a constitutive scaling law is derived from laboratory data on both frictional slip failure and shear fracture of intact rock. This constitutive scaling law enables one to provide a consistent and unified comprehension for scale-dependent physical quantities inherent in the rupture, over a broad range from small-scale frictional slip failure and shear fracture in the laboratory to large-scale earthquake rupture in the field.

Elastoviscoplastic constitutive frameworks for generalized continua

Abstract: A unifying thermomechanical constitutive framework for generalized continua including additional degrees of freedom or/and the second gradient of displacement is presented. Based on the analysis of the dissipation, state laws, flow rules and evolution equations are proposed for Cosserat, strain gradient and micromorphic continua. The case of the gradient of internal variable approach is also incorporated by regarding the nonlocal internal variable as an actual additional degree of freedom. The consistency of the continuum thermodynamical framework is ensured by the introduction of a viscoplastic pseudo–potential of dissipation, thus extending the classical class of so–called standard material models to generalized continua.

Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers

Abstract: A micromechanically based constitutive model for the elasto-viscoplastic deformation and texture evolution of semi-crystalline polymers is developed. The model idealizes the microstructure to consist of an aggregate of two-phase layered composite inclusions. A new framework for the composite inclusion model is formulated to facilitate the use of finite deformation elasto-viscoplastic constitutive models for each constituent phase. The crystalline lamellae are modeled as anisotropic elastic with plastic flow occurring via crystallographic slip. The amorphous phase is modeled as isotropic elastic with plastic flow being a rate-dependent process with strain hardening resulting from molecular orientation. The volume-averaged deformation and stress within the inclusions are related to the macroscopic fields by a hybrid interaction model. The uniaxial compression of initially isotropic high density polyethylene (HDPE) is taken as a case study. The ability of the model to capture the elasto-plastic stress–strain behavior of HDPE during monotonic and cyclic loading, the evolution of anisotropy, and the effect of crystallinity on initial modulus, yield stress, post-yield behavior and unloading–reloading cycles are presented.

A model of large-strain cyclic plasticity and its application to springback simulation

Abstract: This paper presents a framework of constitutive modeling of large-strain cyclic plasticity which describes both the deformation- and texture-induced anisotropies of materials. In this model, the transient Bauschinger effect is described accurately by a new equation of the backstress evolution, and the strain-range and mean-strain dependencies of cyclic strain hardening are expressed by a model of workhardening stagnation. For the description of texture-induced anisotropy, any of widely accepted yield criteria of orthotropic anisotropy is easily incorporated into this model. This paper demonstrates the advantage of the present model in the springback analysis over other classical models by comparing the FE simulations of draw bending with the corresponding experimental results.

Finite element formulation and algorithms for unsaturated soils. Part I: theory

Abstract: This paper presents a complete finite-element treatment for unsaturated soil problems A new formulation of general constitutive equations for unsaturated soils is first presented In the incremental stress–strain equations, the suction or the pore water pressure is treated as a strain variable instead of a stress variable The global governing equations are derived in terms of displacement and pore water pressure The discretized governing equations are then solved using an adaptive time-stepping scheme which automatically adjusts the time-step size so that the integration error in the displacements and pore pressures lies close to a specified tolerance The non-linearity caused by suction-dependent plastic yielding, suction-dependent degree of saturation, and saturation-dependent permeability is treated in a similar way to the elastoplasticity An explicit stress integration scheme is used to solve the constitutive stress–strain equations at the Gauss point level The elastoplastic stiffness matrix in the Euler solution is evaluated using the suction as well as the stresses and hardening parameters at the start of the subincrement, while the elastoplastic matrix in the modified Euler solution is evaluated using the suction at the end of the subincrement In addition, when applying subincrementation, the same rate is applied to all strain components including the suction Copyright © 2003 John Wiley & Sons, Ltd

A non-orthogonal constitutive model for characterizing woven composites

Abstract: Thermoforming of woven fabric reinforced composites usually results in significant in-plane shear deformation in materials, and induces additional anisotropy into the composite. In this paper, a new constitutive model for characterizing the non-orthogonal material behavior under large deformation is proposed. On the basis of stress and strain analysis in the orthogonal and non-orthogonal coordinates and the rigid body rotation matrices, the relationship between the stresses and strains in the global coordinates is obtained. The equivalent material properties are then determined by fitting the numerical load vs. displacement curves to experimental results under biaxial tension and pure shear conditions. This model can be used to efficiently predict material responses under various loading paths for woven composites with different weave architectures. The geometrical non-linearity and the material non-linearity, as well as the complex redistribution and reorientation of the warp and weft yarns during deformation are taken into account. To demonstrate the performance of this model, numerical simulations using a commercial finite element package (ABAQUS/Standard) incorporated with our material model are conducted for various loading cases. Numerical results are in excellent agreement with experimental data.

Thermal effects in the superelasticity of crystalline shape-memory materials

Abstract: This paper develops a three-dimensional theory for the superelastic response of single-crystal shape-memory materials. Since energetic considerations play a major role in the phase transformations associated with the superelastic response, we have developed the theory within a framework that accounts for the laws of thermodynamics. We have implemented a special set of constitutive equations resulting from the general theory in a finite-element computer program, and using this program have simulated the superelastic response of a single crystal Ti–Ni shape-memory alloy under both isothermal and thermo-mechanically coupled situations. Both manifestations of superelasticity—stress–strain response at fixed temperature and strain–temperature response at fixed stress—are explored. The single-crystal constitutive-model is also used to discuss the superelastic response of a polycrystalline aggregate with a random initial crystallographic texture. The overall features of the results from the numerical simulations are found to be qualitatively similar to existing experimental results on Ti–Ni.

Nonlinear Elasticity, Anisotropy, Material Stability and Residual Stresses in Soft Tissue

Abstract: In this chapter the basic equations of nonlinear elasticity theory needed for the analysis of the elastic behaviour of soft tissues are summarized. Particular attention is paid to characterizing the material symmetries associated with the anisotropy that arises in soft tissue from its fibrous constituents (collagens) that endow the material with preferred directions. The importance of the issue of convexity in the construction of constitutive laws (strain-energy functions) for soft tissues is emphasized with reference to material stability. The problem of extension and inflation of a thick-walled circular cylindrical tube is used throughout as an example that is closely associated with arterial wall mechanics. This is discussed first for isotropic materials, then for cylindrically orthotropic materials. Since residual stresses have a very important role in, in particular, arterial wall mechanics these are examined in some detail. Specifically, for the tube extension/inflation problem the residual stresses arising from the assumption that the circumferential stress is uniform under typical physiological conditions are calculated for a representative constitutive law and compared with those calculated using the ‘opening angle’ method.