Molecular simulation guided constitutive modeling on finite strain viscoelasticity of elastomers

Viscoelasticity characterizes the most important mechanical behavior of elastomers. Understanding the viscoelasticity, especially finite strain viscoelasticity, of elastomers is the key for continuation of their dedicated use in industrial applications. In this work, we present a mechanistic and physics-based constitutive model to describe and design the finite strain viscoelastic behavior of elastomers. Mathematically, the viscoelasticity of elastomers has been decomposed into hyperelastic and viscous parts, which are attributed to the nonlinear deformation of the cross-linked polymer network and the diffusion of free chains, respectively. The hyperelastic deformation of a cross-linked polymer network is governed by the cross-linking density, the molecular weight of the polymer strands between cross-linkages, and the amount of entanglements between different chains, which we observe through large scale molecular dynamics (MD) simulations. Moreover, a recently developed non-affine network model (Davidson and Goulbourne, 2013) is confirmed in the current work to be able to capture these key physical mechanisms using MD simulation. The energy dissipation during a loading and unloading process of elastomers is governed by the diffusion of free chains, which can be understood through their reptation dynamics. The viscous stress can be formulated using the classical tube model (Doi and Edwards, 1986); however, it cannot be used to capture the energy dissipation during finite deformation. By considering the tube deformation during this process, as observed from the MD simulations, we propose a modified tube model to account for the finite deformation behavior of free chains. Combing the non-affine network model for hyperelasticity and modified tube model for viscosity, both understood by molecular simulations, we develop a mechanism-based constitutive model for finite strain viscoelasticity of elastomers. All the parameters in the proposed constitutive model have physical meanings, which are signatures of polymer chemistry, physics or dynamics. Therefore, parametric materials design concepts can be easily gleaned from the model, which is also demonstrated in this study. The finite strain viscoelasticity obtained from our simulations agrees qualitatively with experimental data on both un-vulcanized and vulcanized rubbers, which captures the effects of cross-linking density, the molecular weight of the polymer chain and the strain rate.     

A micro-mechanical model for the response of filled elastomers at finite strains

Constitutive equations are developed for the isothermal response of particle-reinforced elastomers at finite strains. A rubbery polymer is treated as a network of chains bridged by junctions. A strand between two junctions is thought of as a series of inextensible segments linked by bonds. Two stable conformations are ascribed to a bond: flexed and extended. Deformation of a specimen induces transition of bonds from their flexed conformation to the extended conformation. A concept of trapped entanglements is adopted, according to which not all junctions are active in the stress-free state. Under straining, some entanglements are transformed from their passive (dangling) state to the active state, which results in a decrease in the average length of a strand. Stress–strain relations for an elastomer and kinetic equations for the rate of transition of bonds from their flexed conformation to the extended conformation are derived by using the laws of thermodynamics. Simple phenomenological equations are suggested for the evolution of the number of active entanglements. The model is determined by five adjustable parameters which are found by fitting experimental data in uniaxial tensile tests. Fair agreement is demonstrated between the results of numerical simulation and observations for a polysulfide elastomer reinforced with polystyrene particles and two natural rubber vulcanizates with different cross-linkers.     

A viscoelastic constitutive model of rubber under small oscillatory load superimposed on large static deformation

Summary  A viscoelastic constitutive equation of rubber that is under small oscillatory load superimposed on large static deformation is proposed. The model is derived through linearization of Simo's nonlinear viscoelastic constitutive model and reference configuration transformation. Most importantly, in this model, static deformation correction factor is introduced to consider the influence of pre-strain on the relaxation function. Natural statically pre-deformed state is served as reference configuration. The proposed constitutive equation is extended to a generalized viscoelastic constitutive equation that includes widely used Morman's model as a special case using objective stress increment. The proposed constitutive model is tested for dynamic behavior of rubber specimens with different carbon black content. It is concluded from the test that the assumption that the effects of static deformation can be separated from time effects, which is the basis of Morman's model, is only applicable to unfilled rubber. The viscoelastic constitutive equation for filled rubber must include, therefore, the influence of the static deformation on the time effects. The suggested constitutive equation with static deformation correction factor shows good agreement with test values. Received 4 January 2001; accepted for publication 13 June 2001     

A thermo-viscoelastic model for elastomeric behaviour and its numerical application

Summary  This paper presents a model of thermo-mechanical behaviour of viscoelastic elastomers under large strain. A formulation is proposed with a generalisation to large strain of the Poynting–Thomson rheological model. A finite element formulation is then exposed taking the incompressibility constraint for mechanical equilibrium into account. On the thermomechanical coupling aspect, an algorithm of time discretisation is proposed with two time scales corresponding respectively to mechanical and thermal behaviours. Finally, an application for the simulation of a double-shearing test is presented with an analysis of parameters' influence and a comparison between numerical and experimental results. Received 22 November 2000; accepted for publication 26 June 2001     

The influence of pre-loading and thermal recovery on the viscoelastic response of filled elastomers

Experimental data are reported in tensile relaxation tests on carbon black-filled natural rubber at strains up to 200%. Constitutive equations are derived for the time-dependent response of a particle-reinforced elastomer at finite strains. Adjustable parameters in the model are found by fitting observations. The effects on mechanical pre-loading and thermal recovery are analyzed on the material constants.     

Bending of an elastoplastic strip with isotropic and kinematic hardening

Summary  We propose an exact analysis for finite bending of a compressible elastoplastic strip with combined hardening at a given stretch normal to the bending plane. We apply the self-consistent eulerian rate-type elastoplastic model based on the logarithmic rate, in conjunction with a Tresca-type loading function. Utilizing the maximum circumferential stretch at the outer surface as an independent parameter, we derive exact analytic expressions for the bending angle, the bending moment, the outer and inner radii, the radii of the two elastic-plastic interfaces and the circumferential stretches at these two interfaces, as well as the stress distributions in every current cross section. In particular, we establish an explicit relation between the two circumferential stretches at the two elastic-plastic interfaces, and we show that this relation is universal for all cases of hardening. We show also that the maximum and minimum circumferential stretches at the outer and inner surfaces obey a reciprocal relation in the course of both elastic and elastic-plastic deformations. Received 4 June 2002; accepted for publication 14 November 2002 This research was carried out under financial support from the German Science Foundation (DFG) (Contract No.: Br. 580/26-2) and from Alexander von Humboldt Foundation. We wish to express our sincere gratitude to these Foundations.     

Effects of dynamic viscoelastic properties on acoustic diffraction by a solid sphere submerged in a viscous fluid

Summary  This study provides a general analysis for scattering of a planar monochromatic compressional sound wave by a homogeneous, isotropic, viscoelastic, solid sphere immersed in an unbounded viscous, heat-nonconducting, compressible fluid. The dynamic viscoelastic properties of the spherical scatterer and the viscosity of the surrounding fluid are rigorously taken into account in the solution of the acoustic-scattering problem. Havriliak–Negami model for viscoelastic material behaviour along with the appropriate wave-harmonic field expansions and the pertinent boundary conditions are employed to develop a closed-form solution in form of infinite series. Subsequently, the associated acoustic quantities such as the scattered far-field pressure directivity pattern, scattered intensity distribution, differential scattering cross section, and the acoustic radiation force are evaluated for given sets of viscoelastic material properties. Numerical results clearly indicate that, in addition to the traditional fluid viscosity-related mechanisms, the dynamic viscoelastic properties of the solid obstacle can be of major significance in sound scattering. Limiting cases are examined and fair agreements with well-known solutions are established. Received 15 January 2002; accepted for publication 2 July 2002 The authors wish to sincerely thank professors Daniel Levesque, Roderic Lakes, Yves Berthelot, S. Temkin, and Andrei Dukhin for valuable and productive consultations on dynamic theory of viscoelasticity and acoustics of (thermo)viscous media.     

Numerical analysis of the stress concentration near holes originating in previously loaded viscoelastic bodies at finite strains

The state of stress and strain of previously loaded viscoelastic bodies with holes originating in them, successively or simultaneously, is analyzed under finite plane deformations. The problem statement and solution are based on the theory of repeatedly superimposed large deformations. The material mechanical properties are described using integral relations of the convolution type over time with a weakly singular kernel. The problem solving is based on the finite-element method. To calculate the integral of the convolution type, a recurrence formula is used that can be obtained by approximating the initial kernel with a linear combination of exponential functions (the truncated Prony’s series). The nonlinear effects and the effect of the interaction between holes on the stress concentration are analyzed. For the dynamic problems, the results for incompressible and weakly compressible materials are compared.     

Computational micro-to-macro transitions of discretized microstructures undergoing small strains

Summary  The paper investigates algorithms for the computation of homogenized stresses and overall tangent moduli of microstructures undergoing small strains. Typically, these microstructures define representative volumes of nonlinear heterogeneous materials such as inelastic composites, polycrystalline aggregates or particle assemblies. We consider a priori given discretized microstructures, without focusing on details of specific discretization techniques in space and time. The key contribution of the paper is the construction of a family of algorithms and matrix representations of the overall properties of discretized microstructures. It is shown that the overall stresses and tangent moduli of a typical microstructure may exclusively be defined in terms of discrete forces and stiffness properties on the boundary. We focus on deformation-driven microstructures, where the overall macroscopic deformation is controlled. In this context, three classical types of boundary conditions are investigated: (i) linear displacements, (ii) constant tractions and (iii) periodic displacements and antiperiodic tractions. Incorporated by the Lagrangian multiplier method, these constraints generate three classes of algorithms for the computation of equilibrium states and the overall properties of microstructures. The proposed algorithms and matrix representations of the overall properties are formally independent of the interior spatial structure and the local constitutive response of the microstructure and are therefore applicable to a broad class of model problems. We demonstrate their performance for some representative model problems including elastic–plastic deformations of composite materials. Received 29 May 2001; accepted for publication 2 January 2002     

Accurate modelling of piezoelectric plates: single-layered plate

Summary  The paper presents an efficient two-dimensional approach to piezoelectric plates in the framework of linear theory of piezoelectricity. The approximation of the through-the-thickness variations accounts for the shear effects and a refinement of the electric potential. Using a variational formalism, electromechanically coupled plate equations are obtained for the generalized stress resultants as well as for the generalized electric inductions. The latter are deduced from the conservative electric charge equation, which plays a crucial role in the present model. Emphasis is placed on the boundary conditions at the plate faces. The model is used to examine some problems for cylindrical bending of a single simply supported plate. Number of situations are examined for a piezoelectric plate subject to (i) an applied electric potential, (ii) a surface density of force, and (iii) a surface density of electric charge. The through-thickness distributions of electromechanical quantities (displacements, stresses, electric potential and displacement) are obtained, and compared with results provided by finite element simulations and by a simplified plate model without shear effects. A good agreement is observed between the results coming from the present plate model and finite element computations, which ascertains the effectiveness of the proposed approach to piezoelectric plates. Received 17 July 2000; accepted for publication 26 September 2000     

Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals

The paper presents new continuous and discrete variational formulations for the homogenization analysis of inelastic solid materials undergoing finite strains. The point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro-structure as a generalized standard medium in terms of an energy storage and a dissipation function. Consistent with this type of finite inelasticity we develop a new incremental variational formulation of the local constitutive response, where a quasi-hyperelastic micro-stress potential is obtained from a local minimization problem with respect to the internal variables. It is shown that this local minimization problem determines the internal state of the material for finite increments of time. We specify the local variational formulation for a distinct setting of multi-surface inelasticity and develop a numerical solution technique based on a time discretization of the internal variables. The existence of the quasi-hyperelastic stress potential allows the extension of homogenization approaches of finite elasticity to the incremental setting of finite inelasticity. Focussing on macro-deformation-driven micro-structures, we develop a new incremental variational formulation of the global homogenization problem for generalized standard materials at finite strains, where a quasi-hyperelastic macro-stress potential is obtained from a global minimization problem with respect to the fine-scale displacement fluctuation field. It is shown that this global minimization problem determines the state of the micro-structure for finite increments of time. We consider three different settings of the global variational problem for prescribed displacements, non-trivial periodic displacements and prescribed stresses on the boundary of the micro-structure and develop numerical solution methods based on a spatial discretization of the fine-scale displacement fluctuation field. Representative applications of the proposed minimization principles are demonstrated for a constitutive model of crystal plasticity and the homogenization problem of texture analysis in polycrystalline aggregates.     

Crack initiation at the rectangular step notch surface under combined loading

Summary  Under external forces acting on the face of a notch, cracks originate at corners, and the system is liable to fail. An analysis is presented of the stress field in the neighborhood of the notch tips, based on the integral representation of the biharmonic solution and on numerical methods. Computations were performed for constant loading or constant displacement distributed along one face of the notch. The coefficients in the principal terms of the asymptotic formulae for the circumferential and shear stresses depend on the angle and height of the notch face and on the boundary conditions. The maximal values of these coefficients determine the stress intensity factors for the opening and shear modes. The angles corresponding to the maximal values of the intensity factors indicate the directions of initiation of opening and sliding cracks. Received 30 May 2000; accepted for publication 3 April 2001     

Dynamic response of a crack in a functionally graded interface of two dissimilar piezoelectric half-planes

Summary  In this paper, the dynamic anti-plane crack problem of two dissimilar homogeneous piezoelectric materials bonded through a functionally graded interfacial region is considered. Integral transforms are employed to reduce the problem to Cauchy singular integral equations. Numerical results illustrate the effect of the loading combination parameter λ, material property distribution and crack configuration on the dynamic stress and electric displacement intensity factors. It is found that the presence of the dynamic electric field could impede of enhance the crack propagation depending on the time elapsed and the direction of applied electric impact. Received 4 December 2001; accepted for publication 9 July 2002 This work is supported by the National Natural Science Foundation of China through Grant No. 10132010.     

The analysis of thermal stresses in thermopiezoelastic semi-infinite body with an edge crack

Summary  Thermopiezoelastic materials have recently attracted considerable attention because of their potential use in intelligent or smart structural systems. The governing equations of a thermopiezoelastic medium are more complex due to the intrinsic coupling effects that take place among mechanical, electrical and thermal fields. In this analysis, we deal with the problem of a crack in a semi-infinite, transversely isotropic, thermopiezoelastic material by means of potential functions and Fourier transforms under steady heat-flux loading conditions. The problem is reduced to a singular integral equation that is solved. The thermal stress intensity factor for a crack situated in a cadmium selenide material is calculated. Received 20 March 2001; accepted for publication 18 October 2001     

Finite viscoplasticity of semicrystalline polymers

Summary  A series of uniaxial tensile tests is performed on isotactic polypropylene at room temperature with cross-head speeds ranging from 5 to 100 mm/min. Prior to mechanical testing, injection-molded samples are annealed for 24 h at the temperature 160 ^°C that ensures that the specimens can be drawn up to the Hencky strain of 0.6 without necking. A constitutive model is derived for the viscoplastic behavior of a semicrystalline polymer at finite strains. The stress–strain relations are determined by five adjustable parameters that are found by matching the observations. Fair agreement is demonstrated between the experimental data and the results of numerical simulation. It is shown that the material parameters in the constitutive equations are strongly affected by the strain rate. Received 9 July 2002; accepted for publication 12 November 2002     

On the modeling and design of piezocomposites with prescribed properties

Summary  Piezoelectric transversely isotropic matrix containing spheroidal piezoelectric inclusions with different properties and of, generally, diverse aspect ratios is considered. A full set of ten effective electrostatic constants is derived, using the method of effective field. The case, when the inclusions are circular cylinders (fibers) is analyzed in detail. The results are compared with those of several earlier works. They constitute the theoretical framework for the design of piezocomposites with prescribed overall properties. Received 3 May 2001; accepted for publication 26 June 2001     

A new numerical procedure for free vibrations of pretwisted plates

Summary  A numerical procedure is proposed for the analysis of free vibrations of pretwisted thin plates. An accurate strain–displacement relationship based on the thin-shell theory combined with the finite element method using triangular plate elements with three nodes and nine degrees of freedom for each node is utilized. The vibration characteristics of pretwisted thin plates with different twist rates and aspect ratios are studied. The numerical results are compared with the previous results obtained by various types of finite elements and by the Rayleigh–Ritz method. The effect of the twist rate on the vibration characteristics is studied briefly. Received 28 February 2001; accepted for publication 18 December 2001     

Three-dimensional phenomenological thermodynamic model of pseudoelasticity of shape memory alloys at finite strains

The familiar small strain thermodynamic 3D theory of isotropic pseudoelasticity proposed by Raniecki and Lexcellent is generalized to account for geometrical effects. The Mandel concept of mobile isoclinic, natural reference configurations is used in order to accomplish multiplicative decomposition of total deformation gradient into elastic and phase transformation (p.t.) parts, and resulting from it the additive decomposition of Eulerian strain rate tensor. The hypoelastic rate relations of elasticity involving elastic strain rate are derived consistent with hyperelastic relations resulting from free energy potential. It is shown that use of Jaumann corotational rate of stress tensor in rate constitutive equations formulation proves to be convenient. The formal equation for p.t. strain rate , describing p.t. deformation effects is proposed, based on experimental evidence. Phase transformation kinetics relations are presented in objective form. The field, coupled problem of thermomechanics is specified in rate weak form (rate principle of virtual work, and rate principle of heat transport). It is shown how information on the material behavior and motion inseparably enters the rate virtual work principle through the familiar bridging equation involving Eulerian rate of nominal stress tensor.

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Impact response of an interface crack in a hybrid piezoelectric laminate

Summary  In a hybrid laminate containing an interfacial crack between piezoelectric and orthotropic layers, the dynamic field intensity factors and energy release rates are obtained for electro-mechanical impact loading. The analysis is performed within the framework of linear piezoelectricity. By using integral transform techniques, the problem is reduced to the solution of a Fredholm integral equation of the second kind, which is obtained from one pair of dual integral equations. Numerical results for the dynamic stress intensity factor show the influence of the geometry and electric field. Received 29 June 2001; accepted for publication 3 December 2001