Rate Constitutive Equations Research Articles

Computational determination of in-plane shear mechanical behaviour of textile composite reinforcements

The knowledge of the mechanical behaviour of woven fabrics is necessary in many applications in particular for the simulation of textile composite forming. This mechanical behaviour is very specific due to the possible motions between the fibres and the yarns. In this paper, the in-plane shear behaviour is analysed from virtual tests on the Representative Unit Cell. The in-plane shear strains can be very large (up to 50°) in case of draping on a double curved surface. These virtual tests avoid performing tricky experimental tests. The presented 3D finite element analyses involve two main specific aspects. Firstly the boundary conditions have to render the periodicity at large deformations and, in some cases, the evolution of contacts between neighbouring yarns during the motion. Secondly the yarn that is made of thousand of fibres is modelled as a continuous medium but its constitutive law has to take its fibrous nature into account. For that reason a rate constitutive equation using a specific objective stress rate is used. It is based on the rotation of the fibre. The analysis is performed for two unit cells. Both results are in good agreement with the experiments, but the use of one of the cells turns out to be much easier.

Intermediate strain rate constitutive equation of 304L stainless steel from modified notched specimen torsion test

A new experimental torsion test is developed to achieve intermediate strain rates. The gauge length of the classical torsion bar (27 mm) is reduced within the range 0 to 5 mm. Analytical predictions with simple constitutive equation show that strain rates achieve the magnitude of 250 s -1 . A finite difference analysis is proposed to simulate the test with more evolved constitutive equations. A comparison between experimental and numerical results shows a good agreement.

Laminated metal composites—High temperature deformation behavior

A constitutive model for deformation of a novel laminated metal composite (LMC) which is comprised of 21 alternating layers of Al 5182 alloy and Al 6090/SiC/25p metal matrix composite (MMC) has been proposed. The LMC as well as the constituent or neat structures have been deformed in uniaxial tension within a broad range of strain-rates (i.e. 10 −6 to 10 +1 s −1) and moderate to high homologous temperatures (i.e. 0.85 ≥ 0.95 T m). The stress exponent, n, of the Al 5182 layers increases from 3 to 5 with increasing strain-rate (i.e. >10 −3 s −1). The MMC layer's apparent n value decreases from 7 to 5 with increasing temperature. Furthermore, at low strain-rates (i.e. <10 −3 s −1) an apparent threshold stress dominates the MMC's observed mechanical behavior. The LMC structure exhibits a behavior somewhat closer to the MMC layers. The results of these experiments have lead to a characterization of the neat layer's mechanical behavior and a subsequent semi-empirical constitutive rate equation for both the Al 5182 and Al 6090/SiC/25p. These predictive relations for the neat layers have been coupled with a proposed model, which takes into account the dynamic load sharing between the elastically stiffer and softer layers when loaded axially during isostrain deformation of the LMC. This deformation model has led to the development of a constitutive relationship between flow stress and applied strain-rate for the laminated structure, which has been compared with experimental data.

A return mapping algorithm for cyclic viscoplastic constitutive models

This paper presents a return mapping algorithm for cyclic viscoplastic constitutive models that include material memory effects. The constitutive model is based on multi-component forms of kinematic and isotropic hardening variables in conjunction with von Mises yield criterion. Armstrong–Frederick (A–F) type rules are used to describe the nonlinear evolution of each of the multi-component kinematic hardening variables. A saturation type (exponential) rule is used to describe the nonlinear evolution of each of the isotropic hardening variables. The concept of memory surface is used to describe the strain range dependent material memory effects that are induced by the prior strain histories. In this paper, the above class of cyclic viscoplastic constitutive models is formulated within a consistent thermodynamic framework that encompasses the standard generalized materials framework. Furthermore, a complete algorithmic treatment of the above rate-dependent constitutive model is also presented for any desired stress or strain constrained configuration subspace. A generalized midpoint algorithm is used to integrate the rate constitutive equations. The consistent tangent operator is obtained by linearizing the return mapping algorithm, and is found to be unsymmetric due to the presence of nonlinear evolution rules for the kinematic hardening variables. Several numerical examples representing the cyclic hardening and softening behavior, transient and stabilized hysteresis behavior, and the non-fading memory effects of the material are presented. These examples demonstrate the accuracy and robustness of the present algorithmic framework for modeling the cyclic viscoplastic behavior of the material.

Mixed finite element method for coupled thermo-hydro-mechanical process in poro-elasto-plastic media at large strains

A mixed finite element for coupled thermo-hydro-mechanical (THM) analysis in unsaturated porous media is proposed. Displacements, strains, the net stresses for the solid phase; pressures, pressure gradients, Darcy velocities for pore water and pore air phases; temperature, temperature gradients, the total heat flux are interpolated as independent variables. The weak form of the governing equations of coupled THM problems in porous media within the element is given on the basis of the Hu–Washizu three-filed variational principle. The proposed mixed finite element formulation is derived. The non-linear version of the element formulation is further derived with particular consideration of the THM constitutive model for unsaturated porous media based on the CAP model. The return mapping algorithm for the integration of the rate constitutive equation, the consistent elasto-plastic tangent modulus matrix and the element tangent stiffness matrix are developed. For geometrical non-linearity, the co-rotational formulation approach is utilized. Numerical results demonstrate the capability and the performance of the proposed element in modelling progressive failure characterized by strain localization and the softening behaviours caused by thermal and chemical effects. Copyright © 2005 John Wiley & Sons, Ltd.

A constitutive model for finite deformation of endochronic plasticity in powder forming processes

In this paper, an endochronic constitutive model is developed for cold powder compaction processes. As large deformations are observed in powder forming processes, the endochronic constitutive model based on large strain plasticity is presented and an integration scheme for the rate constitutive equations is established. Although the concept of yield surface has not been explicitly assumed in the endochronic theory, it is demonstrated that the cone-cap yield surface can be derived as a special case of the proposed endochronic model. The plastic flow rule and the dilatancy condition of the model are derived and a procedure for calibration of material parameters of the constitutive model is proposed. The material parameters in the endochronic model are calibrated for MH-100 iron powder to reproduce experimental data and the predictive capability of the model is assessed by simulating a powder compaction process.

A hypoelasto-plastic finite strain simulation of powder compaction processes with density-dependent endochronic model

In this paper, a new approach is developed based on an endochronic density-dependent plasticity model for describing the isothermal deformation behavior of metal powder at low homologous temperature. As large deformations are observed in powder compaction processes, the endochronic constitutive model is presented based on large strain plasticity and an integration scheme is established for the rate constitutive equations. Endochronic constitutive equations are established based on coupling between deviatoric and hydrostatic behavior. The elastic response is stated in term of hypoelastic model and endochronic constitutive equations are stated in unrotated frame of reference. Finally, the algorithmic modulus consistent with numerical integration algorithm of constitutive equations is extracted. Although the concept of yield surface has not been explicitly assumed in endochronic theory, it is demonstrated that the cone-cap yield surface can be derived as a special case of the proposed endochronic model. The material parameters in endochronic model are calibrated for two samples of metal powder by fitting the model to reproduce data from true-triaxial compression experiments. The numerical schemes are examined for efficiency and accuracy in the modeling of three powder compaction components.

A density-dependent endochronic plasticity for powder compaction processes

This paper is concerned with the numerical modeling of powder cold compaction process using a density-dependent endochronic plasticity model. Endochronic plasticity theory is developed based on a large strain plasticity to describe the nonlinear behavior of powder material. The elastic response is stated in terms of hypoelastic model and endochronic plasticity constitutive equations are stated in unrotated frame of reference. A trivially incrementally objective integration scheme for rate constitutive equations is established. Algorithmic modulus consistent with numerical integration algorithm of constitutive equations is extracted. It is shown how the endochronic plasticity describes the behavior of powder material from the initial stage of compaction to final stage, in which material behaves as solid metals. It is also shown that some commonly used plasticity models for powder material can be derived as special cases of the proposed endochronic theory. Finally, the numerical schemes are examined for efficiency in the modeling of a plain bush, a rotational-flanged and a shaped tablet powder compaction component.

THE K-VERSION OF FINITE ELEMENT METHOD FOR NONLINEAR OPERATORS IN BVP

In the companion papers [1,2], authors introduced the concepts of k-version of finite element method and k, hk, pk, hkp-processes of the finite element method for boundary value problems described by self-adjoint and non-self adjoint operators using Ĥk,p(Ω) spaces with specific details including numerical studies for weak forms and least square processes. It was demonstrated that a variationally consistent (VC) weak form is possible when the differential operator is self-adjoint, however, in case of non-self-adjoint operators the weak forms are variationally inconsistent (VIC) which lead to degenerate computational processes that can produce spurious oscillations in the computed solutions. In this paper we demonstrate that when the boundary value problems are described by non-linear differential operators, Galerkin processes and weak forms can never be variationally consistent and hence result in degenerate computational processes and suffer from same problems as in the case of non-self-adjoint operators ...

THE k-VERSION OF FINITE ELEMENT METHOD FOR NON-SELF-ADJOINT OPERATORS IN BVP

THE k-VERSION OF FINITE ELEMENT METHOD FOR NON-SELF-ADJOINT OPERATORS IN BVP

Analytical stress solutions of a closed deformation path with stretching and shearing using the hypoelastic formulations

In the present work a set of hypoelastic analytical stress solutions of a closed deformation path, consisting of four deformation phases with stretching and shearing, is developed. The rate constitutive equations, based on the Oldroyd, Truesdell, Cotter–Rivlin, Jaumann, Green–Naghdi and logarithmic stress rates, are exploited. According to a representation principle of the material spin tensors, issued in Xiao et al. (1998a, 1998b), a set of new and concise representation equations of the spin tensors for the four deformation phases is derived. The derived spin equations play a crucial role for formulating the hypoelastic closed-form corotational-rate-based stress solutions of the specific deformation path. The analytical stress solutions as functions of the deformation phases are illustrated and complete comparisons between the residual stresses at the end of the deformation path are performed. These analytical stress solutions, as a set of living examples of applying the hypoelastic equations, support the conclusion that in all spatial hypoelastic rate equations only the one based on the logarithmic stress rate is consistent with elasticity.

An implicit algorithm for hypoelasto-plastic and hypoelasto-viscoplastic endochronic theory in finite strain isotropic–kinematic-hardening model

This paper is concerned with objective stress update algorithm for elasto-plastic and elasto-viscoplastic endochronic theory within the framework of additive plasticity. The elastic response is stated in terms of hypoelastic model and endochronic constitutive equations are stated in unrotated frame of reference. A trivially incrementally objective integration scheme for rate constitutive equations is established. Algorithmic modulus consistent with numerical integration algorithm of constitutive equations is extracted. The implementation is validated by means of a set of simple deformation paths (simple shear, extension and rotation), two benchmark test in nonlinear mechanics (the necking of a circular bar and expansion of a thick-walled cylinder), a test which demonstrates the capabilities of the proposed model in simulation of cyclic loading and ratcheting in finite strain case (cyclically loaded notched bar) and finally, the analysis of a tensile test, which presents a shear band with a finite thickness independent of the finite element mesh using endochronic viscoplastic constitutive model.

Mixed finite element method for saturated poroelastoplastic media at large strains

AbstractA mixed finite element for hydro‐dynamic analysis in saturated porous media in the frame of the Biot theory is proposed. Displacements, effective stresses, strains for the solid phase and pressure, pressure gradients, and Darcy velocities for the fluid phase are interpolated as independent variables. The weak form of the governing equations of coupled hydro‐dynamic problems in saturated porous media within the element are given on the basis of the Hu–Washizu three‐field variational principle. In light of the stabilized one point quadrature super‐convergent element developed in solid continuum, the interpolation approximation modes for the primary unknowns and their spatial derivatives of the solid and the fluid phases within the element are assumed independently. The proposed mixed finite element formulation is derived. The non‐linear version of the element formulation is further derived with particular consideration of pressure‐dependent non‐associated plasticity. The return mapping algorithm for the integration of the rate constitutive equation, the consistent elastoplastic tangent modulus matrix and the element tangent stiffness matrix are developed. For geometrical non‐linearity, the co‐rotational formulation approach is used. Numerical results demonstrate the capability and the performance of the proposed element in modelling progressive failure characterized by strain localization due to strain softening in poroelastoplastic media subjected to dynamic loading at large strain. Copyright © 2003 John Wiley &amp; Sons, Ltd.

Hypoelasticity‐based analytical stress solutions in the simple shearing process

AbstractIn the present work a set of hypoelasticity‐based analytical stress solutions of simple shearing is formulated. The rate constitutive equations based on the Jaumann, Green‐Naghdi, Oldroyd, Truesdell, Cotter‐Rivlin, logarithmic and OE‐ and OL‐based stress rates are exploited. From a representation principle of the material spins, a concise formula of the spin tensors for simple shearing is derived. Based on the derived spin equation a novel analytical stress solution related to the OL‐based stress rates is formulated. For completeness the issued stress solutions in conjunction with the other stress rates are also reformulated. All analytical stress solutions are illustrated and compared.

Consistent tangent operators for constitutive rate equations

AbstractA general approach for obtaining the consistent tangent operator for constitutive rate equations is presented. The rate equations can be solved numerically by the user's favourite time integrator. In order to obtain reliable results, the substepping in integration should be based on a control of the local error. The main ingredient of the consistent tangent operator, namely the derivative of the stress with respect to the strain increment must be computed simultaneously with the same integrator, applied to a numerical approximation of the variational equations. This information enables finite‐element packages to assemble a consistent tangent operator and thus guarantees quadratic convergence of the equilibrium iterations. Several numerical examples with a hypoplastic constitutive law are given. As numerical integrator we used a second‐order extrapolated Euler method. Quadratic convergence of the equilibrium iteration is shown. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Numerical study of consistency of rate constitutive equations with elasticity at finite deformation

AbstractThe present work is concerned with the numerical study of the elasticity consistency of the spatial rate equations using the conventional Oldroyd, Truesdell, Cotter–Rivlin, Jaumann and Green–Naghdi rates and the three novel co‐rotational ΩE‐ and Ω¯L‐based, logarithmic rates, and of the rotated material rate equation describing the relationship between the material time derivatives of the rotated Kirchhoff stress and material logarithmic strain. To this end, three integration procedures for updating stress are presented. The stress responses of several typical deformation processes are simulated. According to the numerical results we know that among the spatial rate equations only the logarithmic rate equation is consistent with elasticity under constant material parameters. Integrating the other spatial rate equations will provide path‐dependent stress response. These numerical conclusions support the arguments in H. Xiao et al. (Acta Mechanica 1999; 138:31–50). The reasons leading to elasticity inconsistency of spatial rate equations are analysed. If the material parameters are assumed to be strain‐dependent, the logarithmic rate equation loses also its elasticity‐consistent property. The numerical results prove also that the spatial logarithmic and rotated material rate equations are equivalent to each other. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Modelling of laminated microstructures in stress-induced martensitic transformations

This paper is concerned with micromechanical modelling of stress-induced martensitic transformations in crystalline solids, with the focus on distinct elastic anisotropy of the phases and the associated redistribution of internal stresses. Micro–macro transition in stresses and strains is analysed for a laminated microstructure of austenite and martensite phases. Propagation of a phase transformation front is governed by a time-independent thermodynamic criterion. Plasticity-like macroscopic constitutive rate equations are derived in which the transformed volume fraction is incrementally related to the overall strain or stress. As an application, numerical simulations are performed for cubic β 1 (austenite) to orthorhombic γ 1′ (martensite) phase transformation in a single crystal of Cu–Al–Ni shape memory alloy. The pseudoelasticity effect in tension and compression is investigated along with the corresponding evolution of internal stresses and microstructure.

THE k-VERSION OF FINITE ELEMENT METHOD FOR SELF-ADJOINT OPERATORS IN BVP

In this paper a new mathematical and computational framework for boundary value problems described by self-adjoint differential operators is presented. In this framework, numerically computed solutions, when converged, possess the same degree of global smoothness in terms of differentiability up to any desired order as the theoretical solutions. This is accomplished using spaces Ĥk,p that contain basis functions of degree p and order k - 1 (or the order of the space k). It is shown that the order of space k is an intrinsically important independent parameter in all finite element computational processes in addition to the discretization characteristic length h and the degree of basis functions p when the theoretical solutions are analytic. Thus, in all finite element computations, all quantities of interest (e.g., quadratic functional, error or residual functional, norms and seminorms, error norms, etc.) are dependent on h, p as well as k. Therefore, for fixed h and p, convergence of the finite element process can also be investigated by changing k, hence k-convergence and thus the k-version of finite element method. With h, p, and k as three independent parameters influencing all finite element processes, we now have k, hk, pk, and hpk versions of finite element methods. The issue of minimally conforming finite element spaces is reexamined and it is demonstrated that the definition of currently believed minimally conforming space which permit weak convergence of the highest-order derivatives of the dependent variables appearing in the bilinear form is not justifiable mathematically or from physics view point. A new criterion is proposed for establishing the minimally conforming spaces which is more in agreement with the physics and mathematics of the BVP. Significant features and merits of the proposed mathematical and computational framework are presented, discussed, illustrated, and substantiated mathematically as well as numerically with the Galerkin and least-squares finite element formulations for self-adjoint boundary-value problems.

On the construction of extended problems and related functionals for general nonlinear equations

Starting from existing methods for the symmetrisation of general nonlinear, nonpotential operators (Tonti, Int. J. Engng. Sci. 22 (11–12) (1984) 1343–1371; Auchmuty, Nonlinear Anal. Theory Methods Appl. 12 (5) (1988) 531–564) this work discusses some alternative formulations and illustrates some significant implications of such methods, which should make them more suited to practical application. Further, a new class of the so-called “extended” functionals is proposed, much simpler to construct than the preceding ones. Even if the definition of the functionals requires the doubling of the unknown functions, the new unknowns have a precise physical meaning in the solution of the problem, which may help in the actual solution process. The application of the new method is illustrated by means of two examples in the field of continuum mechanics: the nonassociated elastic–plastic rate constitutive equations, and the nonlinear continuum dynamics equations with initial conditions.

An implicit integration algorithm for plane stress damage coupled elastoplasticity

Boundary value problems based on Continuum Damage Mechanics(CDM) exhibiting damage with plasticity are highly nonlinear and usually, finite element method is employed for solution. The heart of a nonlinear finite element analysis is the integration of constitutive rate equations. An efficient and accurate integration scheme is required to integrate the rate equations. The basic aim of this paper is to present an efficient algorithm to integrate damage coupled elastoplasticity for plane stress case. It has been shown that the present integration algorithm is reduced to the linearLzation of three equations to solve them by Newton Raphson method unlike Doghri's algorithm[2] where fifteen equations are required to be linearized. Also, difficulties associated with extending the integration scheme of Simo and Taylor[9] to damage coupled elastoplasticity in plane stress case have been studied. Backward Euler scheme has been employed for the integration. An example has been presented to illustrate the efficiency of the algorithm.