Showing papers on "Viscoplasticity published in 1997"

Nonlinear Viscoelastic and Viscoplastic Constitutive Equations Based on Thermodynamics

Abstract: An approach to modeling the mechanical behavior of fiber reinforced and unreinforced plastics with an evolving internal state is described. Intrinsic nonlinear viscoelastic and viscoplastic behavior of the resin matrix is taken into account along with growth of damage. The thermodynamic framework of the method is discussed first. The Gibbs free energy is expressed in terms of stresses, internal state variables (ISVs), temperatureand moisture content. Simplifications are introduced based on physical models for evolution of the ISVs and on experimental observations of thedependence of strain state on stress state and its history. These simplifications include use of master creep functions that account for multiaxial stresses, environmental factors and aging in a reduced time and other scalars. An explicit representation of the strains follows, which isthen specialized to provide three-dimensional homogenized constitutiveequations for transversely isotropic, fiber composites. Experimentalsupport for these equations is briefly reviewed. Finally, physicalinterpretation of some of the constitutive functions is discussed usingresults from a microcracking model as well as molecular rate process andfree volume theories. It is shown that the present thermodynamicformulation leads to a generalized rate process theory that accounts for abroad distribution of thermally activated transformations in polymers.

Viscoplasticity for instabilities due to strain softening and strain-rate softening

Abstract: Three viscoplastic approaches are examined in this paper. First, the overstress viscoplastic models (i.e. the Perzyna model and the Duvaut—Lions model) are outlined. Next, a consistency viscoplastic approach is presented. In the consistency model a rate-dependent yield surface is employed while the standard Kuhn—Tucker conditions for loading and unloading remain valid. For this reason, the yield surface can expand and shrink not only by softening or hardening e⁄ects, but also by softening/hardening rate e⁄ects. A full algorithmic treatment is presented for each of the three models including the derivation of a consistent tangential sti⁄ness matrix. Based on a limited numerical experience it seems that the consistency model shows a faster global convergence than the overstress approaches. For softening problems all three approaches have a regularising e⁄ect in the sense that the initial-value problem remains well-posed. The width of the shear band is determined by the material parameters and, if present, by the size of an imperfection. A relation between the length scales of the three models is given. Furthermore, it is shown that the consistency model can properly simulate the so-called S-type instabilities, which are associated with the occurrence of travelling Portevin-Le Chatelier bands. ( 1997 John Wiley & Sons, Ltd.

A physically based method to represent the thermo-mechanical behaviour of elastomers

Abstract: A geometric nonlinear and thermodynamical consistent constitutive theory is proposed, which allows the representation of the thermomechanical behaviour of carbon black filled rubber. In a recent paper [1] it was shown that the mechanical behaviour of this material is mainly influenced by nonlinear elasticity coupled with some inelastic effects, in particular the Mullins-effect, nonlinear rate dependence and a weak equilibrium hysteresis. In the present paper, the Mullins-effect is not taken into consideration. At first we discuss a uniaxial approach, based on a simple spring dashpot system of viscoplasticity. The essential feature of this model is a decomposition of the total stress into a rate independent equilibrium stress and a nonlinear rate dependent overstress. The equilibrium stress is decomposed into a sum of two terms as well: The first term, the elastic part of the equilibrium stress, is a nonlinear function of the total strain, and the second term, the so-called hysteretic part, depends in a rate independent manner on the strain history. Both the overstress and the hysteretic part of the equilibrium stress are determined by nonlinear elasticity relations which depend on internal variables. These internal variables are inelastic strains, and the corresponding evolution equations are developed in consideration of the second law of thermodynamics. Accordingly, we demonstrate that the principle of non-negative dissipation is satisfied for arbitrary deformation processes. In a further step, we transfer the structure of this model to the three-dimensional and geometric nonlinear case. In a certain sense similar to finite deformation elasto-plasticity, we introduce two multiplicative decompositions of the deformation gradient into elastic and inelastic parts. The first decomposition is defined with respect to the overstress and the second one with respect to the hysteretic part of the equilibrium stress. Consequently, two intermediate configurations are induced, which lead two different decompositions of the Green's strain tensor into elastic and inelastic parts. The latter are the internal variables of the model. For physical reasons, we define the corresponding stress tensors and derivatives in the sense of the concept of dual variables [7], [39]. Theconstitutive equations for the overstress and for the hysteretic part of the equilibrium stress are specified by nonlinear elasticity relations, formulated with respect to the different intermediate configurations. In order to facilitate a separate description of inelastic bulk and distortional effects, we introduce kinematic decompositions of the deformation gradient into volumetric and distortional parts. Numerical simulations demonstrate that the developed theory represents the mechanical behaviour of a tread compound at room temperature very well. Thermomechanical heating effects, which are caused by inelastic deformations are also described by the theory. The method proposed in this paper can be utilised to generalise uniaxial rheological models to three-dimensional finite strain theories, which are admissible in the sense of the second law of thermodynamics.

Nonlinear mechanical response of high density polyethylene. Part II: Uniaxial constitutive modeling

Abstract: Based on the experimental data presented in Part I, two uniaxial constitutive models are constructed. The first, a nonlinear viscoelastic (NVE) model, is formulated using the mechanical analogy consisting of one independent spring and six Kelvin elements in series. Creep data are used to determine the model parameters. The second model, a viscoplastic (VP) formulation, is developed using the viscoplastic theory proposed by Bodner to characterize the uniaxial viscoplastic behavior of metals. Inelastic strain rate is introduced into the state variable in addition to inelastic work to depict the strong rate dependent behavior of HDPE. Experimental data from constant strain rate tests are employed to construct the material functions of the model. Limitations in the application of each model are discussed in conjunction with possibilities for future work.

Nonlinear mechanical response of high density polyethylene. Part I: Experimental investigation and model evaluation

Abstract: The nonlinear behavior of high density polyethylene (HDPE) is investigated for samples cut from thick-walled HDPE pipe. Extensive experimental work has been performed to characterize the non-linear time-dependent response of the material tested under uniaxial compression. Tests were conducted under conditions of constant strain rate, creep, stress relaxation, constant loading rate, abrupt change of strain rate, creep-recovery, cyclic strain rate, and various combinations of these loading conditions. Creep and stress relaxation response after strain reversal and the effect of the transient response on the following stress-strain behavior is examined. Permanent strains for the test specimens and their dependence on loading histories are investigated. Specimens cut at various orientations from the pipe are used to quantify the small amounts of local anisotropy in the pipe specimen. The experimental work has been used to develop both nonlinear viscoelastic (NVE) and viscoplastic (VP) constitutive models in a companion paper. Both the test results and the corresponding model predictions are reported in this paper. It is found that the VP model reproduces the nonlinear viscoelastic-viscoplastic behavior of HDPE very well provided that the current strain is not below the maximum strain imposed (there is no strain reversal). The NVE model predicts the material behavior reasonably well for some loading conditions, but inadequately for others.

Collective behavior and spacing of adiabatic shear bands

Abstract: The failure of metals subjected to high strain rates is frequently related to the collective development of adiabatic shear bands which results in a patterning depending on the material properties and the loading conditions. In this paper, the spacing between adiabatic shear bands is characterized by analytical means in a one-dimensional formulation. Using a perturbation analysis, a dominant instability mode can be characterized, whose wavelength is related to the shear-band spacing. Explicit solutions are found for materials with no strain hardening. Asymptotic developments are used to obtain results that account for strain hardening. Comparisons are made with experimental results and other existing models.

Void growth in glassy polymers

Abstract: This paper deals with a study of voids in amorphous glassy polymers that exhibit elastic-viscoplasticity with rate dependent yield, intrinsic softening and progressive strain hardening at large strains. The study is motivated by the plastic deformation in voided polymer-rubber blends caused by cavitation of the rubber particles, and thus attempts to contribute to the understanding of the toughening mechanisms in blends. Axisymmetric cell analyses are presented to study the plastic deformation around initially spherical voids and their resulting growth in terms of size and shape up to large overall strains. This void growth is demonstrated to inherit particular properties from the typical features of plasticity in glassy polymers, viz. small strain softening and large strain hardening. The role of strain localization into shear bands and their subsequent propagation in controlling void growth is highlighted. Furthermore, an approximate constitutive model is presented for the description of the macroscopic overall behaviour of porous glassy polymers. This model includes a modification of existing porous plasticity models to account for elasticity effects on the initiation of overall plasticity, which are important in polymers because of their relatively high yield strain. Its predictions are compared with the results from the numerical cell analyses.

A finite element analysis of the squeeze flow of an elasto-viscoplastic paste material

Abstract: A finite element analysis of squeeze flow has been implemented for a material that exhibits elasto-viscoplasticity. The formulation is based upon the assumption that linear elastic deformation occurs prior to yield and that the yield surface is strain rate hardening as defined by an associated viscoplastic flow rule. Both no-slip and lubricated wall boundary conditions are considered. The numerical simulation results are compared with experimental measurements involving a model elasto-viscoplastic material for which the material parameters were derived from tensile and ram extrusion measurements. Satisfactory agreement was obtained for the compressive forces as a function of displacement, the radial displacement fields and the wall normal and shear stress distributions.

A Cosserat theory for elastoviscoplastic single crystals at finite deformation

Abstract: IN THIS WORK, displacement and lattice rotation are regarded as independent degrees of freedom. They are connected only on the constitutive level and by the balance equations. The description of plastic deformation is based on the slip theory. Elastic lattice curvature and torsion are associated with couple-stresses. The continuum theory of dislocations has been revisited to derive the kinematics of plastic lattice torsion-curvature. Explicit constitutive equations and hardening rules are proposed to close the theory in the case of elastoviscoplasticity. The thermodynamical formulation of the model involves internal variables which are similar to the densities of statistically stored dislocations and the densities of geometrically necessary dislocations. Accordingly, the proposed Cosserat theory can be regarded, on the one hand, as the classical crystal plasticity theory complemented by lattice curvature and torsion variables and, on the other hand, as the continuum theory of dislocations closed by the missing hardening variables and constitutive equations within the appropriate micropolar framework. A generalization of Mandel's elastoviscoplastic decomposition of strain is used especially for the torsion-curvature measure at finite deformation.

An efficient stress algorithm with applications in viscoplasticity and plasticity

Abstract: This paper deals with two main topics. The first one concerns the equivalence of stress algorithms, based on a Backward-Euler-step applied on viscoplastic models of Chaboche-type, and their elastoplastic counterpart. Generally, the stress algorithm yields a system of non-linear algebraic equations and the corresponding consistent tangent operator, occurring in the principle of virtual displacements, leads to a system of linear equations. This procedure can be obtained utilizing only numerical methods. The second topic concerns a special constitutive relation based on a kinematic hardening model using a sum of Armstrong/Frederick terms, which is equivalent to a multi-surface plasticity model. Applying this model a so-called problem-adapted stress algorithm is derived, where only one non-linear equation must be solved. This result is independent of the number of terms in the hardening model. Furthermore, only the viscoplastic algorithm must be implemented, since it includes the elastoplastic constitutive model as a special case. © 1997 by John Wiley & Sons, Ltd.

Effective properties of porous ideally plastic or viscoplastic materials containing rigid particles

Abstract: This study deals with the overall yield surface of porous materials containing rigid inclusions and governed by a generalized Gurson model. The dissipation potential of these voided materials can be obtained in a closed form. An estimate for the effective dissipation potential of the composite is derived in terms of the strain energy of a linear comparison solid. The corresponding yield surface is described by a generalized Gurson criterion. These results are extended to voided viscous matrices. The accuracy of the analytical estimates is assessed by comparison with cell calculations performed by the finite element method.

Dynamic crack growth across an interface

Abstract: The growth of a crack first in an elastic solid, then across an interface and into an elastic-viscoplastic solid is analyzed numerically. The analyses are carried out within a framework where the continuum is characterized by two constitutive relations; one that relates stress and strain in the bulk material, the other relates the traction and separation across a specified set of cohesive surfaces. Crack initiation, crack growth and crack arrest emerge naturally as outcomes of the imposed loading, without any ad hoc assumptions concerning crack growth or crack path selection criteria. Full transient analyses are carried out using two characterizations of strain rate hardening for the viscoplastic solid; power law strain rate hardening and a combined power law-exponential relation that gives rise to enhanced strain rate hardening at high strain rates. Results are presented for two values of interface strength. For the higher strength interface the crack grows straight through the interface into the elastic–viscoplastic solid, while for the lower strength interface the crack deflects into the interface.

Microstructurally Based Finite Element Simulation on Solder Joint Behaviour

Abstract: The most commonly used solder for electrical interconnects in electronic packages is the near eutectic 60Sn‐40Pb alloy. This alloy has a number of processing advantages(suitable melting point of 183°C and good wetting behaviour). However, under conditions of cyclic strain and temperature (thermomechanical fatigue) the microstructure of this alloy undergoes a heterogeneous coarsening and failure process that makes the prediction of solder joint lifetime complex. A finite element simulation methodology to predict solder joint mechanical behaviour, that includes microstructural evolution, has been developed. The mechanical constitutive behaviour was incorporated into the time‐dependent internal state variable viscoplastic model through experimental creep tests. The microstructural evolution is incorporated through a series of mathematical relations that describe mass flow in a temperature/strain environment. The model has been found to simulate observed thermomechanical fatigue behaviour in solder joints.

A secant-viscosity approach to the time-dependent creep of an elastic viscoplastic composite

Abstract: At present no homogenization theory seems to exist to estimate the time-dependent creep behavior of a two-phase viscoplastic composite. In this paper an approach introducing a linear viscoelastic comparison composite in conjunction with the secant viscosity is proposed for this purpose. The method makes use of a Maxwell matrix in the viscoelastic composite, and sets its shear viscosity equal to the secant viscosity of the viscoplastic matrix at every stage of deformation. The property of the viscoelastic composite is in turn determined from its elastic comparison composite, and the results are cast in the rate forms to reflect the continuously changing secant viscosity of the viscoplastic matrix. This new approach is neither restricted to rigid particles nor to a viscoplastic matrix with rigid elasticity, and it is capable of accounting for the strain-hardening of the matrix phase. The theory is developed fully for the particle-reinforced composite and, when applied to a TiC/Al system, it leads to a satisfactory agreement with available experimental data. Both concepts of viscoelastic comparison composite and secant viscosity are believed to be new in a homogenization scheme, and, as the secant moduli in the rate-independent plasticity, the secant viscosity now can play a useful role in the determination of the effective behaviour of an elastic-viscoplastic composite.

Viscoplastic behaviour of soft clay

Abstract: Based on an extensive experimental programme, this paper presents the stress—strain—time behaviour of a natural soft clay. The clay was part of the foundation soil of Le Flumet dam, whose delayed settlements were very large. The main aim of this study was to explain, to quantify and to anticipate that settlement. Stiffness and strength of clays are both functions of time. Studying the relation (q—p—t) through conventional laboratory tests showed their dependence upon the loading rate in a stress—strain plane. The effect of time was also observed during creep tests (triaxial and oedometer) or during stress—relaxation tests. A constitutive model based on the critical state concept and viscoplasticity theory was used to predict the viscous behaviour. The model was based on Hujeux's elastoplastic model and it was focused on the plastic flow function which was modified. The model parameters were determined by carrying out different simulations. The model was validated using all the experimental test res...