Class Of Constitutive Equations Research Articles

On a class of nonlinear isotropic thermo-elastic body that is non-Green elastic: Applications to large elastic deformations

Summary A new class of constitutive equation is proposed for isotropic thermoelastic solids, wherein the Hencky strain tensor is assumed to be a function of the Cauchy stress tensor, via a Gibbs potential. The solid is assumed to be incompressible in the referential state, but the volume can change due to differences in the temperature relative to a reference temperature. The change in volume only depends on temperature. Some restrictions are found for the Gibbs potential, resulting in a constitutive equation for isotropic solids, wherein the volume depends on temperature. Using the resulting constitutive equation, some boundary value problems are studied, considering some relatively simple distributions for the temperature, deformations and stresses.

Rheological synergistic thermal conductivity of HEC-based silicon dioxide nanofluids in shear flow fields

The classical constitutive equations for heat transfer of non-Newtonian fluids in shear flow fields neglect viscous dissipation and the field synergistic properties of rheology and heat transfer. In this paper, a novel constitutive Eq. (6) for rheological synergy thermal conductivity was derived by considering viscous dissipation. Based on this, we established an experimental platform to investigate the heat transfer properties of HEC-based silicon dioxide nanofluids (different nanoparticle sizes and volume fractions) in shear flow field for two cases: neglecting viscous dissipation and considering viscous dissipation. The results show that viscous dissipation have very important effects on heat transfer behavior at different shear rates and rheological properties. The contribution of viscous dissipation to the thermal conductivity increases with shear rate and particle volume fraction, but decreases with temperature. Among all the measured results, the maximum contribution of viscous dissipation to the thermal conductivity reached 19.86%. This confirms that the viscous dissipation is not negligible and the field synergy between rheology and heat transfer of non-Newtonian fluids must be considered in the shear flow fields.

A bond-based correspondence model and its application in dynamic plastic fracture analysis for quasi-brittle materials

This work presents a generalized bond-based peridynamic model named bond-based correspondence model to simulate the fracture. This model combines the advantages of bond-based peridynamics and the correspondence model (non-ordinary state-based peridynamics). The peridynamic pair-wise force is calculated by classical constitutive equations through the relation between peridynamic pair-wise force and stress. With specific function types of relation between peridynamic pair-wise force and stress, the bond-based correspondence model can degenerate to the existing bond-based peridynamic models, such as the extended peridynamic model with bond rotation and the micropolar peridynamics. Besides, this model is easy to be extended to plasticity, in this study, the small deformation von Mises elastoplastic model is developed. And the effects of plastic parameters on dynamic fracture are analyzed for quasi-brittle materials. Numerical examples illustrate the model’s capacity in solving the fixed Poisson’s ratio and simulating dynamic crack propagation. And the numerical results show that plastic parameters have a great impact on fracture propagation, including crack initiation and the speed of propagation.

A spectral approach for nonlinear transversely isotropic elastic bodies, for a new class of constitutive equation: Applications to rock mechanics

A constitutive equation is provided for nonlinear transversely isotropic elastic solids, wherein the linearized strain tensor is assumed to be a function of the Cauchy stress tensor; this elastic constitutive equation belongs to a subclass of a more general set of implicit constitutive relations proposed in the recent years. The proposed constitutive equation is valid for both compressible and incompressible bodies and can be simply modified, to exclude the mechanical influence of compressed fibres and to model inextensible fibres. A crude specific constitutive model is proposed to compare with a uniaxial experimental data on Marcellus shale. Some simple boundary value problems are analyzed.

Magnetohydrodynamics free convection flow of Carbon nanotubes viscous nanofluids over an infinite plate with Newtonian heating and fractional derivative

The problem of hydromagnetic (MHD) free convection flow of viscous nanofluids over a moving infinite vertical plate with carbon nanotubes (CNTs) and Newtonian heating on the boundary is considered. The Caputo time‐fractional derivative is introduced to classical constitutive equations by using the generalized constitutive shear stress equation and generalized Fourier law. Analytical solution for the temperature distribution and semi‐analytical solution for the velocity profile are found by means of Laplace transform. The solution of temperature distribution is presented in terms of Wright and Gauss complementary error functions. The inverse Laplace transform of velocity profile is approximated by Stehfest algorithm, and for validation, Tzou algorithm is used. Finally, for illustration, three special cases are considered, and the influence of physical parameters on some fundamental motions are graphically underlined and discussed. The fractional derivative and volume faction parameter has a good influence on temperature distribution and velocity field.

Fractional model for simulating the viscoelastic behavior of artificial fracture in shale gas

With the development of geothermal energy and unconventional reservoirs, shales have become the popular target formation. This kind of organic-rich or clay-rich geomaterial has significant viscoelastic behavior which may lead to severe accidents, due to the creep rupture and the relaxation of pre-stress. In order to understand the viscoelastic behavior and simulate it quickly and accurately, we have developed a novel procedure for solving viscoelastic problems based on fractional calculus. Fractional viscoelastic constitutive equations were built by introducing a springpot into the classical component models. Fractional and classical constitutive equations were compared by the fitting results of triaxial experimental data of shale samples. Fitting efficiency of fractional constitutive equation is enhanced if compared to classical equations. Besides, we also found that fractional order is directly correlated to the amount of total volume content of clay and kerogen. This implies that the physical meaning of fractional model is clearer and its extrapolation is more accurate. Fracture width creep model for unpropped fractures and closure stress relaxation model for well propped fractures were developed and analyzed. Their simulation results in terms of long-term and short-term scales differ tremendously and hence should be studied individually. The influence of lateral compliances of real shale samples on fracture width creep or closure stress relaxation is not significant and may be ignored in long-term simulations. Simulation results of closure stress relaxation could explain the abnormal proppant production, which occurs several days later than the start of natural flow-back.

Natural convection of bio-nanofluid between two vertical parallel plates with damped shear and thermal flux

In this article, we studied the nature (free) convection flow of bio-nanofluid between two vertical plates. We developed a fractional model in which the classical constitutive equations are generalized by using the constitutive shear stress equation and the generalized Fourier’s law. The solutions of fractional differential equations are found by means of integral transforms (Laplace and sine-Fourier transforms). Solutions are expressed in terms of generalized G-function of Lorenzo and Hartley and Mittag-Leffler function. We compare our results with numerical methods. The effects of fractional and physical parameters are graphically discussed.

Spatial behavior of the dual-phase-lag deformable conductors

The purpose of the present article is to investigate about the spatial behavior question for a thermoelastic model describing the intrinsic coupling between the high-order effects of thermal lagging and the ultrafast deformation. The main achievements of this study consist of four types of results concerning the spatial behavior, namely: I) when the orders of approximation in the constitutive equation for the heat flux differ by one and (i) when short elapsed times are considered we establish a result describing the domain of influence that characterizes the wave like-behavior, while (ii) when long elapsed times are considered then we establish in the inside of the influence domain, some appropriate exponential decay estimates with uniform in time decay rate and II) when approximations of the same order are considered and (iii) when short elapsed times are considered we establish some exponential decaying estimates of Saint Venant type with decay rate depending on time, while (iv) when long elapsed times are considered then we establish some appropriate Saint Venant exponential estimates with uniform in time decaying rate or Phragmén-Lindelöf alternative (for a semi-infinite cylinder). The last two results characterize the diffusion like-behavior of the model. The reported spatial behavior is established for a large class of constitutive equations for the heat flux under mild assumptions upon the characteristics of the high-order thermal lagging and for both finite and semi-infinite cylinders.

On high-order approximations for describing the lagging behavior of heat conduction

This paper is dedicated to high-order effects of thermal lagging in correlation with heat transfer models in micro- or nanoscale, relating the number of energy carriers and the associated resonance phenomenon under high-frequency excitations. Thus, a class of constitutive equations is considered for the heat flux describing high-order effects in the lagging behavior of heat transport. Tzou’s model, which is based on time-differential dual-phase-lag approximations of heat conduction, is generalized, incorporating the microstructural interaction effect in the fast-transient process of heat transport. More precisely, polynomial approximations of order n for the heat flux vector and of order m for the gradient of the temperature variation are considered. Further, well-posedness is established for solutions of the specific initial boundary value problems for the mathematical model when: (i) ; (ii) ; and (iii) . This means that uniqueness and continuous dependences of solutions are established with respect to the given prescribed data. With this aim, some modified initial boundary value problems related to the operators of the constitutive equation are introduced and suitable time-weighted measures of the solution are presented, and are used to establish the uniqueness theorems and to generate some estimates describing the continuous dependence of solutions with respect to the given data for each of the three cases specified. Moreover, the spatial behavior of the transient solutions is studied and an influence domain result is established for , while for some exponential decay estimates of Saint-Venant type are established. All results are established without restrictions on the characteristic parameters of the constitutive equation other than that the product of the coefficients of the time derivatives of greatest order of the two operators involved in the constitutive equation is positive.

Comparison among Different Constitutive Equations on Investigating Tensile Plastic Behavior and Microstructure in Austempered Ductile Iron

The capabilities of different constitutive equations of approximating the tensile flow curves and correlating plastic behavior with the microstructure were investigated in austempered ductile iron ADI 1050. In a previous paper, the microstructure evolution of ADI 1050 during austempering was investigated through quenching the ADI 1050 after 14 increasing austempering times to room temperature. The 14 samples were tensile tested and two classes of constitutive equations were examined in the present paper. The Hollomon-type constitutive equations approximated all of the tensile flow curves of ADI 1050 very well but failed in correlating the plastic behavior with microstructure evolution. Voce-type constitutive equations approximated the tensile flow curves only at high stresses very well but could correlate the plastic behavior with the microstructure evolution of ADI 1050 during austempering excellently. The reason of this success was rationalized in terms of the physical basis of Voce-type equations, while Hollomon-type equations are empirical.

Two-dimensional nonlinear stress and displacement waves for a new class of constitutive equations

The propagation of displacement waves and stress waves for implicit constitutive equations is investigated. This new class of constitutive equations contain Cauchy elastic and hyperelastic bodies as subclasses. We consider a particular subclass where the strain is expressed in terms of a non-invertible function of the stress. This subclass of constitutive equations describe elastic responses where the stress and linearised strain are nonlinearly related. Such a phenomenon cannot be captured in the classical theory. Two constitutive equations are studied. The first constitutive equation is analogous to the constitutive equation for a power-law fluid with an exponent n in the expression for the stress and the second constitutive equation can describe elastic bodies which exhibit limiting strain. The special semi-inverse solution gives a system of nonlinear hyperbolic partial differential equations. These systems can be written as single partial differential equations which describe nonlinear shear stress waves. Solitary wave solutions for each constitutive equation are derived. Perturbation solutions for the system of partial differential equations for displacement and stress are considered and travelling wave and standing wave solutions are found. Although the solutions for both the travelling waves and standing waves contain a secular term, the perturbation expansions break down outside the range of interest. The speed of the solitary wave for both constitutive equations was obtained. The solitary waves develop a shock front and estimates for the time that this will occur are derived. The shock front develops at the back of the wave for the power-law constitutive equation and at the front of the wave for the strain-limiting constitutive equation. For the travelling waves the stress is non-zero at the wave front and the stress waves propagate as shock waves. The speed of propagation and the amplitude of the travelling waves and the period of oscillation of the standing waves are compared for the two constitutive equations.

Finite element implementation of slightly compressible and incompressible first invariant-based hyperelasticity: theory, coding, exemplary problems

The present study is concerned with the finite element (FE) implementation of slightly compressible hyperelastic material models. A class of constitutive equations is considered where the isochoric potential functions are based on the first invariant of the right Cauchy-Green (C-G) deformation tensor. Special attention is paid to the most recently developed model formulations. The incremental form of hyperelasticity and its numerical implementation into both commercial and non-commercial FE software are discussed. A Fortran 77 UMAT code is attached which allows for a simple implementation of arbitrary first invariant-based constitutive models into Abaqus and Salome-Meca FE packages. Several exemplary problems are considered.

Existence theory for the equations of inelastic material behavior of metals - Transformation of interior variables and energy estimates

The system of equations, which we study, consists of linear partial differential equations and of nonlinear ordinary differential equations for internal variables. The existence theory for such systems was studied first by the french mathematicians G. Duvaut and J.L. Lions [1]. Next we can find in the literature a work of C. Johnson [2] on a quasi-static problem for a special model. Then in the nineties we can find more works in the domain. This work consists of two parts. In the first part we will classify constitutive equations and therefore we define constitutive equations of monotone type. Moreover by transformation of internal variables we will enlarge the class of constitutive equations, for which we can prove a. global in time existence theorem for large initial data. But there exist models, which are not of monotone type and which we can not transform to monotone type. Therefore we must study such models with other methods. This is the second part, of the work. We write about the energy method for the model of Bodner-Partom.

Bulging bifurcation of inflated circular cylinders of doubly fiber-reinforced hyperelastic material under axial loading and swelling

In this paper, we examine the influence of swelling on the bulging bifurcation of inflated thin-walled cylinders under axial loading. We provide the bifurcation criteria for a membrane cylinder subjected to combined axial loading, internal pressure and swelling. We focus here on orthotropic materials with two preferred directions which are mechanically equivalent and are symmetrically disposed. Arterial wall tissue is modeled with this class of constitutive equation and the onset of bulging is considered to give aneurysm formation. It is shown that swelling may lead to compressive hoop stresses near the inner radius of the tube, which could have a potential benefit for preventing aneurysm formation. The effects of the axial stretch, the strength of the fiber reinforcement and the fiber winding angle on the onset of bifurcation are investigated. Finally, a boundary value problem is studied to show the robustness of the results.

Homogenization of periodic hexa- and tetrachiral cellular solids

The homogenization of periodic hexachiral and tetrachiral honeycombs is dealt with two different techniques. The first is based on a micropolar homogenization. The second approach, developed to analyse two-dimensional periodic cells consisting of deformable portions such as the ring, the ligaments and possibly a filling material, is based on a second gradient homogenization developed by the authors. The obtained elastic moduli depend on the parameter of chirality, namely the angle of inclination of the ligaments with respect to the grid of lines connecting the centers of the rings. For hexachiral cells the auxetic property of the lattice together with the elastic coupling modulus between the normal and the asymmetric strains is obtained; a property that has been confirmed here for the tetrachiral lattice. Unlike the hexagonal lattice, the classical constitutive equations of the tetragonal lattice turns out to be characterized by the coupling between the normal and shear strains through an elastic modulus that is an odd function of the parameter of chirality. Moreover, this lattice is found to exhibit a remarkable variability of the Young’s modulus and of the Poisson’s ratio with the direction of the applied uniaxial stress. Finally, a simulation of experimental results is carried out.

Analysis of Shear Flexible Layered Isotropic and Composite Shells by ‘EPSA’

We present a simple and efficient method for the analysis of shear flexible isotropic and orthotropic composite shells. Classical thin shell constitutive equations used in the explicit finite element code EPSA to model homogenous isotropic shells using "through-the-thickness-integration" and layered orthotropic composite shells [1–3,5] are modified to account for transverse shear deformation. This effect is important in the analysis of thick plates and shells as well as composite laminates, where interlaminar effects matter. Transverse shear stresses are calculated using a linear normal strain distribution, where first the shear forces are calculated and then the stresses are calculated by means of the generalized section properties, i.e., first and second moments of area. The formulation is a generalization of the analytical method of analyzing composite beams. It is simple and computationally inexpensive, and it yields accurate results without employing higher order displacement interpolations. In the case of isotropic shells, the transverse shear stresses are distributed parabolically, based on the assumption of linear normal strain distribution through the thickness and on application of the quadratic shape function to transverse shear strains. The transverse shear stresses are included in the elastic-perfectly plastic yield function of the Huber-Mises-Hencky type.

First-order gradient damage theory

Taking the strain tensor, the scalar damage variable, and the damage gradient as the state variables of the Helmholtz free energy, the general expressions of the firstorder gradient damage constitutive equations are derived directly from the basic law of irreversible thermodynamics with the constitutive functional expansion method at the natural state. When the damage variable is equal to zero, the expressions can be simplified to the linear elastic constitutive equations. When the damage gradient vanishes, the expressions can be simplified to the classical damage constitutive equations based on the strain equivalence hypothesis. A one-dimensional problem is presented to indicate that the damage field changes from the non-periodic solutions to the spatial periodic-like solutions with stress increment. The peak value region develops a localization band. The onset mechanism of strain localization is proposed. Damage localization emerges after damage occurs for a short time. The width of the localization band is proportional to the internal characteristic length.

An enhanced single-layer variational formulation for the effect of transverse shear on laminated orthotropic plates

A novel mixed formulation is derived by means of Reissner's variational approach-based on Castigliano's principle of least work in conjunction with a Lagrange multiplier method for the calculus of variations. The governing equations present an alternative theory for modeling the important three-dimensional structural aspects of plates in a two-dimensional form. By integrating the classical Cauchy's equilibrium equations with respect to the thickness co-ordinate, and enforcing continuity of shear and normal stresses at each ply interface, condenses the effect of the thickness. A reduced system of partial differential equations of sixth-order in one variable, is also proposed, which contains differential correction factors that formally modify the classical constitutive equations for composite laminates. The theory degenerates to classical composite plate analysis for thin configurations. Significant deviations from classical plate theory are observed when the thickness becomes comparable with the in-plane dimensions. A variety of case studies are presented and solutions are compared with other models available in the literature and with finite element analysis.

Consistency for two multi-mechanism models in isothermal plasticity

A new class of constitutive equations (the multi-mechanism models) devoted to the prediction of ratcheting and of complex material behavior, for instance of steel, has developed during the last years. The main difference between these models and the “Chaboche” type models is the representation of different mechanisms of plastic strain with a possible interaction between them. This sophistication has a cost related to the complexity to understand well how the various parameters act. The identification process thus becomes more difficult as it will be difficult to give reasonable bounds. Two multi-mechanism models have been proposed during the last decade, see Cailletaud and Sai [Cailletaud, G., Sai, K., 1995. Study of plastic/viscoplastic models with various inelastic mechanisms. International Journal of Plasticity 11, 991], Taleb et al. [Taleb, L., Cailletaud, G., Blaj, L., 2006. Numerical simulation of complex ratcheting tests with a multi-mechanism model type. International Journal of Plasticity, 22, 724], and Sai and Cailletaud [Sai, K., Cailletaud, G., 2007. Multi-mechanism models for the description of ratcheting: effect of the scale transition rule and of the coupling between hardening variables. International Journal of Plasticity 23, 1589]. In this paper, we would like to ensure a kind of “after-sales service” by exploring the thermodynamic way in order to see if there exist any restrictions on the parameter values. We prove thermodynamic consistency of some two-mechanism models under suitable conditions. As a result, we obtain conditions for the parameters which ensure the non negativity of the free energy and the fulfilling of the Clausis–Duhem inequality. Besides this, we perform numerical simulations, considering classical histories (monotonic tension, cyclic strain controlled, uniaxial ratcheting, biaxial ratcheting). These simulations show that the violation of the established restrictions may lead to anomalous predictions.

On the thermodynamics of fluids defined by implicit constitutive relations

In this paper, we develop a thermodynamically consistent theory for describing the response of nonlinear viscous fluids whose constitutive equations are of the form f (T, D) = 0. We show that such constitutive equations which include classical constitutive equations wherein the stress is expressed explicitly in terms of the kinematical quantities, provide a rich class of physically meaningful fluid response functions which allows us to describe a wider range of material behavior, including that of a general class of incompressible fluids, incompressible fluids with pressure dependent viscosity, and Bingham (or pseudoplastic) materials.