Rate-type Constitutive Equations Research Articles

Rheological model for rock-type materials under large deformations

In this paper the behaviour of rock-type materials is described with respect to the current configuration through rate-type constitutive equations, within the finite deformation framework, using the relative motion. The viscoplastic part of the constitutive equations concerns the long-time effects, i.e. the deformation by creep, while the instantaneous elastic response characterizes the short-time effect. The instantaneous elastic response expresses an appropriate objective time derivative of Cauchy stress tensor via the rate of strain, with the aid of fourth-order elastic stiffness tensor. Using mathematical arguments it follows that our new model extends, to finite strains, Cristescu’s rheological model Cristescu (1989)[2]. To solve boundary value problems for rock-type materials described by the proposed model, the appropriate variational equality for the velocity field, at a time t, is derived starting from the weak formulation of the incremental equilibrium problem. The rate form of the constitutive model is coupled with the variational equality, and serves to update the current state of the material.

Comparative assessment of the contending force and placement methods for weightless sagging cables

Sagging elastic cables are inherently nonlinear structures due to the presence of self-weight apart from the presence of conventional physical and geometrical nonlinearities. Recently, authors have proposed a force method of weightless sagging elastic cables with linear tension–extension relations and undergoing small elastic displacements to focus explicitly on their characteristic ‘configurational’ nonlinearity. This force method is based upon the rate-type constitutive equations and third-order differential equations of motion. Existing displacement method is reconstructed here for the physically linear weightless sagging elastic cables undergoing finite elastic displacements. In place of nodal forces in force method, deformed state nodal placements are the primary variables in this placement method. Rate-type constitutive equations for two-node weightless planar sagging elastic cables are derived. The objective of the present paper is to compare and contrast these two contending methods in reference to their underlying hypotheses, analytical reach, and quantitative predictions of structural response. Possible equivalence of the proposed third-order and the popular incremental second-order differential equation of motion are explored. Theoretical significance of the present paper is critically evaluated.

Lubrication approximation of flows of a special class of non-Newtonian fluids defined by rate type constitutive equations

• We investigate the flow of a fluid defined by a rate-type implicit constitutive equation. • The model takes into account viscoelastic effects. • We study the channel flow with non-flat walls. • The pressure satisfies an integro-differential equation. • We plot the pressure and velocity distribution for different wall profiles. We consider the flow of a rate-type fluid defined by an implicit constitutive equation in a channel with non flat walls. We also assume that the channel characteristic width is small in comparison to the channel length, so that the lubrication approximation can be applied. The model developed is mainly motivated by the evidence that many lubricants seem to be well described by implicit rate type fluid models. The mathematical problem is reduced to an integro-differential equation for the pressure that is solved numerically for several channel profiles.

Thermodynamically-consistent constitutive modeling of aligned Silk fibroin sponges: Theory and application to uniaxial compression

Microstructurally-aligned Silk fibroin sponges have emerged as a viable candidate for replicating the anisotropic fibrous microarchitecture that is encountered in several functional tissues in vivo. But for the material to reach its full potential in terms of its application in biomimetic soft tissue constructs or scaffolds, a complete understanding and quantification of its nonlinear viscoelastic response in the finite strain regime is needed. To address this gap from the mechanical modeling perspective, an anisotropic, nonlinear, viscoelastic model is developed within a thermodynamic framework in this study, to capture the macroscopic response of these sponges in a phenomenological manner. The rate-type constitutive equations that are developed in the process are subsequently applied for the case of uniaxial compression, and satisfactorily corroborated against the results from finite strain viscoelastic characterization experiments done on hydrated Silk fibroin sponges under uniaxial compression for different constituent material concentrations of the sponges ranging from 1 w/v % - 4 w/v %.

Mathematical study of a single leukocyte in microchannel flow

The recruitment of leukocytes and subsequent rolling, activation, adhesion and transmigration are essential stages of an inflammatory response. Chronic inflammation may entail atherosclerosis, one of the most devastating cardiovascular diseases. Understanding this mechanism is of crucial importance in immunology and in the development of anti-inflammatory drugs. Micropipette aspiration experiments show that leukocytes behave as viscoelastic drops during suction. The flow of non-Newtonian viscoelastic fluids can be described by differential, integral and rate-type constitutive equations. In this study, the rate-type Oldroyd-B model is used to capture the viscoelasticity of the leukocyte which is considered as a drop. Our main goal is to analyze a mathematical model describing the deformation and flow of an individual leukocyte in a microchannel flow. In this model we consider a coupled problem between a simplified Oldroyd-B system and a transport equation which describes the density considered as non constant in the Navier–Stokes equations. First we present the mathematical model and we prove the existence of solution, then we describe its numerical approximation using the level set method. Through the numerical simulations we analyze the hemodynamic effects of three inlet velocity values. We note that the hydrodynamic forces pushing the cell become higher with increasing inlet velocities.

Rate type constitutive equations for fiber reinforced nonlinearly vicoelastic solids using spectral invariants

In this paper we are interested in developing constitutive equations for fiber-reinforced nonlinearly viscoelastic solids. It has been shown that constitutive equations for such bodies can be expressed in terms of a complete minimal set of 18 classical invariants associated with deformation and fiber orientation. In this paper, we give an alternative formulation using a set of spectral invariants. It is shown via the use of spectral invariants that only 11 of the 18 classical invariants are independent. We analyze the spectral invariants for two illustrative deformation gradients: (i) simple tension, and (ii) simple shear.

Static and dynamic analysis of cable-suspended concrete beams

A new theory of weightless sagging planer elasto-flexible cables under point loads is developed earlier by the authors and used for predicting the nonlinear dynamic response of cable-suspended linear elastic beams. However, this theory is not valid for nonlinear elastic cracked concrete beams possessing different positive and negative flexural rigidity. In the present paper, the flexural response of simply supported cracked concrete beams suspended from cables by two hangers is presented. Following a procedure established earlier, rate-type constitutive equations and third order nonlinear differential equations of motion for the structures undergoing small elastic displacements are derived. Upon general quasi-static loading, negative nodal forces, moments and support reactions may be introduced in the cable-suspended concrete beams and linear modal frequencies may abruptly change. Subharmonic resonances are predicted under harmonic loading. Uncoupling of the nodal response is proposed as a more general criterion of crossover phenomenon. Significance of the bilinearity ratio of the concrete beam and elasto-configurational displacements of the cable for the structural response is brought out. The relevance of the proposed theory for the analysis and the design of the cable-suspended bridges is critically evaluated.

A comparative study of hyperelastic and hypoelastic material models with constant elastic moduli for large deformation problems

Many finite element programs including commercial codes for large deformation analysis employ incremental formulations of rate-type constitutive equations which are based on hyperelastic or hypoelastic material models with constant elastic moduli. In this paper, a comparative study is carried out for hyperelastic and hypoelastic material models with constant elastic moduli of a face-centered cubic single crystal of copper. A strain energy function from the inter-atomic potential for single-crystal copper is also considered for the hyperelastic material model to obtain physically based elastic deformations. Numerical results show that constant elastic moduli of hypoelastic material models can cause considerable errors in stress and strain increments when the changes in volume and cross-sectional area of a material are not negligible.

Some remarks about the use of Gordon–Schowalter time derivative in rate-type viscoelastic constitutive equations

The present paper is concerned with the analysis of the thermodynamic consistency of rate-type constitutive equations for viscoelastic fluids that adopt the Gordon–Schowalter time derivative. The choice of the objective time derivative is important not only to assure a mathematically correct modelling and physically realistic description of the viscoelastic behaviour, but also to perform a thermodynamically consistent modelling. It is shown that, for some very popular constitutive equations that adopt this particular family of objective time derivatives (such as the upper-convected Oldroyd fluids), it is impossible to automatically assure that the second law of thermodynamics is satisfied in any process.

A class of models to predict the normal force and torque under torsional loading of a viscoelastic liquid

Viscoelastic liquids exhibit diverse mechanical behavior and offers a tremendous challenge in modeling its nonlinear response. This work is concerned with the development of a class of constitutive equations for viscoelastic liquids, which can capture the nonlinear response, especially, under torsional loading. By extending a universal solution developed by Rivlin (1948) for elastic bodies under torsional loading to viscoelastic bodies, one can arrive at a constitutive equation that can simultaneously predict torque and normal force with a reasonable accuracy. In order to develop such a constitutive equation, a suitable thermodynamical framework developed by Rajagopal and Srinivasa (2000) is chosen because of the fact that it can be used to exploit the mentioned result of Rivlin. Consequently, a suitable rate-type constitutive equation is also derived in Section 5. The efficacy of the developed model is checked by comparing the predictions of the model with that of the experimental data for torsional deformation of asphalt. It is found that the predictions of the model agree reasonably with that of the experimental data.

The structure of shock and interphase layers for a heat conducting Maxwellian rate-type approach to solid–solid phase transitions

We consider a thermoelastic model for phase transforming materials which can adequately describe the evolution with respect to the temperature of the hysteresis loop both in compression and tension tests. The specificity of this model is that the Gruneisen coefficient changes its sign. The model is augmented by considering a dissipative mechanism governed by a Maxwellian rate-type constitutive equation that can describe stress relaxation phenomena toward equilibrium between phases. Existence and uniqueness of traveling wave solutions are investigated. One derives that the admissibility condition induced by the Maxwellian rate-type approach, coupled or not with Fourier heat conduction law is related to the chord criterion with respect to the Hugoniot locus. We investigate the structure of profile layers, and we focus on their thermodynamic properties. The influence of the exothermic or endothermic character of phase transitions on the inner structure of interphase layers is captured. A phenomenon of temperature overshoot/undershoot with respect to the front state temperature and Hugoniot back state temperature inside an interphase layer is revealed.

Thermodynamic restrictions and wave features of a non-linear Maxwell model

A non-linear rate-type constitutive equation, established by Rajagopal, provides a generalization of the Maxwell fluid. This note embodies such a constitutive equation within the scheme of materials with internal variables thus allowing also for solids with both dissipative and thermoelastic mechanisms. The compatibility with the second law of thermodynamics, expressed by the Clausius–Duhem inequality, is examined and the restrictions on the evolution equations are determined. Next the propagation condition of discontinuity waves is derived, for shock waves and acceleration waves, by regarding the body as a definite conductor. Infinitesimal shock waves and acceleration waves show similar effects. The effective acoustic tensor proves to be the sum of a thermoelastic tensor and a tensor arising from the rate-type equation.

Rate boundary value problems and variational inequalities in rate-independent finite elasto-plasticity

The paper deals with variational inequalities related to the rate quasistatic boundary value problem and associated with a generic stage of the process in our approach to finite elasto-plastic materials. The Eulerian setting of the models can be associated with the various models, firstly postulated with respect to the plastically deformed configurations, by the pushing forward procedure, as well as the Lagragian formalism of the models can be adopted by pulled back to the reference configuration procedure. Only when the constitutive and the evolution functions which describe the behaviour of the elasto-plastic material are compatible with the restrictions imposed by the dissipation postulate, the variational inequalities are characterized by symmetric bilinear forms. In the present formulation of the variational inequalities we used an appropriate procedure that allows us to reformulate pointwise the consistency condition in the form of an inequality. We formulate the variational inequalities to be solved for the unknowns, namely the velocity field and the plastic factor, in the deformed configuration at time t, considered as a reference one. The coefficients of the variational inequalities are dependent on the current values of the stress, the elastic and plastic deformations, as well as the internal state variables. We propose an update algorithm which allows us to find the current state of the process at the time, by integrating the rate type constitutive equations, which characterizes the current state of the elasto-plastic materials through the objective derivative. We prove an existence and uniqueness result concerning the solution, namely the spatial velocity and plastic factor, for the symmetric variational inequality of the first kind and formulate an incrementally objective algorithm in order to update the state of the process. By solving the appropriate variational inequality at a fixed time t, we avoid the application of the Return-Mapping Algorithm proposed by Simo (1998).

A model for anisotropic viscoelastic damage in composites

A model for continuous damage combined with viscoelasticity is proposed. The starting point is the formulation connecting the elastic properties to the tensor of damage variables. A hardening law associated with the damage process is identified from available experimental information and the rate-type constitutive equations are derived. This elastic damage formulation is used to formulate an internal variable approximation to viscoelastic damage in the form of a non-linear Kelvin chain. Elastic and viscoelastic equations are implemented into a finite element procedure. The code is verified by comparison with closed-form solutions in simplified configurations, and validated by fitting results of experimental creep tests.

A simplified approach to continuous damage of composite materials and micromechanical analysis

A comprehensive approach to the continuous damage of materials is formulated within the framework of thermodynamic reasoning. Particular emphasis is put on the application to composites. The starting point lies in the assumption that damage evolution can be described at the macroscopic scale by means of a tensor variable. The thermodynamic force associated with the evolution of the damage tensor is deduced from the expression of the intrinsic dissipation. A phenomenological criterion for damage yielding is then proposed and a hardening law associated with the damage process is identified from available experiment results. For completeness, the rate-type constitutive equations are derived. This damage model is therefore implemented within a finite element procedure. The numerical tool is then applied to analyze the damage induced by mechanical loading in several examples of composite materials. The numerical predictions are compared to available closed-form solutions or experimental results. The last part of the paper is dedicated to a micromechanical analysis of damage in the composite material. A simplified macro-to-micro framework is formulated for the evaluation of damage and stress concentrations within the constituents of the composite from the knowledge of the composite morphology and from the stress and damage at the macroscopic scale.

Explicit Runge–Kutta methods for the integration of rate-type constitutive equations

Modern constitutive models have the potential to improve the quality of engineering calculations involving non-linear anisotropic materials. The adoption of complex models in practice, however, depends on the availability of reliable and accurate solution methods for the stress point integration problem. This paper presents a modular implementation of explicit Runge–Kutta methods with error control, that is suitable for use, without change, with any rate-type constitutive model. The paper also shows how the complications caused by the algebraic constraint of conventional plasticity are resolved through a simple subloading modification. With this modification any rate-independent model can be implemented without difficulty, using the integration module as an accurate and robust standard procedure. The effectiveness and efficiency of the method are demonstrated through a comparative evaluation of second and fifth-order formulas, applied to a complex constitutive model for natural clay, full details of which are given.

A unified stress update algorithm for explicit transient shell dynamics with combined isotropic-kinematic hardening in Eulerian rate-type phenomenological finite elasto-plasticity models

This paper discusses applications of the objective stress rates to a unified stress update algorithm for transient shell dynamic analysis within the context of explicit time integration. We propose a unified stress update algorithm via establishing the Eulerian rate-type constitutive equations of hypoelastic–plastic materials. The constitutive equations are often given on the additive decomposition of the rate of deformation tensor under the assumption that the elastic part of the rate of deformation tensor may be characterized to be hypoelastic with grade zero, i.e., the fourth-rank constant isotropic tensor. Several objective stress rates are derived through the Lie derivative of the Cauchy or Kirchhoff stress tensor. Those stress rates cover, for instance, the Jaumann, Green–Naghdi, Truesdell, Oldroyd, and Cotter–Rivlin stress rates. Further, the unified stress update algorithm is implemented along with the so-called radial return method and the phenomenological plasticity theory with combined isotropic/kinematic hardenings. Among the objective stress rates employed here, we emphasize especially the stress update procedure of the Green–Naghdi stress rate for transient shell dynamics, because this algorithm developed in this work is not shown even in the representative commercial codes. In the implementation of the unified stress update algorithm, we applied our algorithms to the Belytschko–Lin–Tsay shell theory to validate its accuracy and effectiveness. Moreover, a slightly modified secant algorithm for the zero transverse normal stress of the shell is also developed. The ultimate objectivity of this work is to provide a guideline for the best choice of an appropriate objective stress rate for phenomenological elastoplastic models. Several numerical examples are demonstrated, including non-contact transient shell dynamics examples and contact–impact examples such as a ball-plate impact problem, by which the accuracy and effectiveness of the unified stress update algorithm, developed in this work, are addressed.

New formulations, positivity preserving discretizations and stability analysis for non-Newtonian flow models

We propose a class of new discretization schemes for solving the rate-type non-Newtonian constitutive equations. The so-called conformation tensor has been known to be symmetric and positive definite in a large class of constitutive equations. Preserving such a positivity property on the discrete level is believed to be crucially important but difficult. High Weissenberg number problems on numerical instabilities have been often associated with this issue. In this paper, we present various discretization schemes that preserve the positive-definiteness of the conformation tensor regardless of the time and spatial resolutions. Moreover, the robustness of the algorithm has been also demonstrated by the stability analysis using the discrete analogue of energy estimates. New schemes presented in this paper are constructed based upon the newly discovered relationship between the rate-type constitutive equations and the symmetric matrix Riccati differential equations.

Comparison of Jiles–Atherton and Preisach models extended to stress dependence in magnetoelastic behaviors of a ferromagnetic material

The Jiles–Atherton and the Preisach models are extended and employed in the rate-type constitutive equations of magnetoelastic materials. These models have individual advantages in applicable materials, suitability of numeric calculations, easiness of parameter identification, etc. From the comparison of the induced stress magnetization effects, it is proved that the constitutive equations maintain the original magnetization characteristics and can unifiedly describe the various magnetoelastic behaviors. In particular, it is confirmed that the constitutive equations with the extended Preisach model can express the stress magnetization curve of nickel more accurately than the one with the extended Jiles–Atherton model.

Reynolds stress model involving the mean spin tensor

In this work, we develop a Reynolds stress model along the line of the approach presented by Commun. Nonlinear Sci. Numer. Simul. 9, 543 (2004)]], aiming to assess the role and contribution of the mean spin tensor in turbulence modeling. Here, the constitutive functional for the Reynolds stress depends on the mean spin tensor as well as the mean stretching tensor and its Jaumann derivative, the turbulent kinetic energy K , and the turbulent dissipation rate epsilon , which is at the complexity level of p=1,m=1 , and n=0 of a rate-type constitutive equation for the Reynolds stress proposed in the aforementioned paper. The explicit form for the Reynolds stress is obtained with recourse to the representation theorem and the theory of invariants developed in modern rational continuum mechanics, and, as an approximation, a nonlinear cubic K-epsilon model is worked out in which the model coefficients are analytically identified based on the experimental results of Tavoularis and Corrsin [J. Fluid Mech. 104, 311 (1981)]]. In addition, numerical results based on this model, in the forms of employing the Jaumann derivative and the Oldroyd derivative, respectively, for homogeneous turbulent shear flow and fully developed turbulent flow over a backward-facing step, are presented in comparison with those obtained based on a few previously proposed linear and nonlinear K-epsilon models, showing reasonably good agreement with the experimental results and the DNS data concerned and a better performance than the previously developed quadratic models.