On the longitudinal impact of two phase transforming bars. Elastic versus a rate-type approach. Part II: The rate-type case

We study the propagation of phase transformation fronts induced by the longitudinal impact of two shape memory alloy bars modeled by a general form of a rate-type approach to non-monotone elasticity. We illustrate that such a rate-type law should be seen like a kinetic law for phase transformation. This investigation continues in a comparative way the analysis of the dynamic theory of elastic bar considered in Part I in relation with a viscosity criterion. We focus here on mathematical, thermodynamical and experimental aspects related with the wave structure which accompanies both the forward and reverse transformation. We analyze the propagation of disturbances in a pure phase near and far from their sources, that is the instantaneous waves and the delayed waves as well as the traveling wave solutions and the accompanying dissipation. In the numerical experiments one focuses on the influence of the impact velocity on the way the phase boundary propagates and on the results which can indicate indirectly the existence of a phase transformation like the time of separation, the velocity–time profile at the rear end of the target and the stress history at the impact face.     

On the energy monotony in rate-type viscoelasticity

One proves that the energy function — for the three dimensional rate-type viscoelasticity with linear instantaneous response — has a “monotony property” with respect to the equilibrium hypersurface, in the sense that ordered equilibrium hypersurfaces lead to ordered energy functions.     

(Adiabatic) phase boundaries in a bistable chain with twist and stretch

Mass–spring chains with only extensional degrees of freedom have provided insights into the behavior of crystalline solids, including those capable of phase transitions. Here we add rotational degrees of freedom to the masses in a chain and study the dynamics of phase boundaries across which both the twist and stretch can jump. We solve impact and Riemann problems in the chain by numerical integration of the equations of motion and show that the solutions are analogous to those in a phase transforming rod whose stored energy function depends on both twist and stretch. From the dynamics of phase boundaries in the chain we extract a kinetic relation whose form is familiar from earlier studies involving chains with only extensional degrees of freedom. However, for some combinations of parameters characterizing the energy landscape of our springs we find propagating phase boundaries for which the rate of dissipation, as calculated using isothermal expressions for the driving force, is negative. This suggests that we cannot neglect the energy stored in the oscillations of the masses in the interpretation of the dynamics of mass–spring chains. Keeping this in mind we define a local temperature of our chain and show that it jumps across phase boundaries, but not across sonic waves. Hence, impact problems in our mass–spring chains are analogous to those on continuum thermoelastic bars with Mie–Gruneisen type constitutive laws. At the end of the paper we use our chain to shed some light on experiments involving yarns that couple twist and stretch to perform useful work in response to various stimuli.     

On a variant of the Maxwell and Oldroyd-B models within the context of a thermodynamic basis

In this paper we develop models within a thermodynamic standpoint that are very similar in form to the classical Maxwell and Oldroyd-B models but differ from them in one important aspect, the manner in which they unload instantaneously from the deformed configuration. As long as the response is not instantaneous, the models that are derived cannot be differentiated from the Maxwell and Oldroyd-B models, respectively. The models can be viewed within the context of materials whose natural configuration evolves, the evolution being determined by the maximization of the rate of entropy production of the material. However, the underpinnings to develop the model are quite different from an earlier development by Rajagopal and Srinivasa [8] in that while the total response of the viscoelastic fluid satisfies the constraint of an incompressible material, the energy storage mechanism associated with the elastic response is allowed to be that for a compressible elastic solid and the dissipative mechanism associated with the viscous response allowed to be that for a compressible fluid, the total deformation however being isochoric. The analysis calls for a careful evaluation of firmly held customs in viscoelasticity wherein it is assumed that it is possible to subject a material to a purely instantaneous elastic response without any dissipation whatsoever. Finally, while the model developed by Rajagopal and Srinivasa [8] arises from the linearization of the non-linear elastic response that they chose and leads to a model wherein the instantaneous elastic response is isochoric, here we develop the model within the context of a different non-linear elastic response that need not be linearized but the instantaneous elastic response not necessarily being isochoric.     

On rate-type viscoelasticity and the second law of thermodynamics

We deduce an energy identity which must be satisfied by the smooth solutions of the system of equations governing the dynamics of body with quasilinear rate-type constitutive equation. We give conditions when a unique energy function exists for rate-type viscoelasticity. In the semilinear case we give the conditions when a unique, positive and convex energy function exists and we obtain estimates in energy for the smooth solutions of initial-boundary value problems. A viscoelastic approach to nonlinear elasticity is discussed. Finally, an example shows that the second law of thermodynamics does not imply stability.     

Lonsdaleite Model of Open-Cell Elastic Foams: Theory and Calibration

We formulate a unit-cell model of open-cell elastic foams. In this model, a foam consists of four-bar tetrahedra arranged in the hexagonal diamond structure known as Lonsdaleite. The parameters of the model are the Young??s modulus of the bars and a few geometric parameters, the values of which may be roughly estimated for any given foam. We use the model to simulate a set of experiments in which elastic polyether polyurethane foams in a broad range of densities were tested under five loading conditions, namely tension along the rise direction; compression along the rise direction; compression along a transverse direction; simple shear combined with compression along the rise direction; and hydrostatic pressure combined with compression along the rise direction. With a suitable choice of values of the parameters of the model, the stress?Cstretch curves that we compute using the model compare favorably with the stress?Cstretch curves that were measured in the experiments. In some of the experiments a stress plateau in the stress?Cstretch curve was accompanied by heterogeneous stretch fields, even though the attendant stress fields were homogeneous. For these experiments we show that the model can be used to predict the occurrence of a second-order phase transition, so that the plateau stress can be interpreted as a Maxwell stress and the attendant heterogeneous stretch fields as two-phase fields, consistent with the experimental evidence. In other experiments the stress?Cstretch curve evinced a sudden and pronounced loss of stiffness, but no genuine stress plateau, and the attendant stretch fields remained homogeneous. For these experiments we show that the model can be used to predict the occurrence of a bifurcation of equilibrium in which the stress keeps rising as the deformation continues to increase in the post-buckling stage, so that the stretch fields remain homogeneous throughout, consistent with the experimental evidence. In general, to appraise the goodness of our model we put emphasis on the relation between the stress?Cstretch curve measured in an experiment and the nature of the attendant stretch fields. We submit that this emphasis should remain a guiding methodological trait in the appraisal of constitutive models of open-cell elastic foams.     

Kinetic relations and the propagation of phase boundaries in solids

This paper treats the dynamics of phase transformations in elastic bars. The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition. A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed. For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities.     

The Riemann problem for non-ideal isentropic compressible two phase flows

We consider the Riemann problem for a five-equation, two-pressure (5E2P) model of non-ideal isentropic compressible gas–liquid two-phase flows. This system is more complex due to the extended thermodynamics model for van der Waals gases, that is, typical real gases for gas phase and Tait׳s equation of state for liquid phase. The overall model is strictly hyperbolic and non-conservative form. We investigate the structure of Riemann problem and construct the solution for it. To construct solution of Riemann problem approximately assuming that all waves corresponding to the genuinely non-linear characteristic fields are rarefaction and then we discuss their properties. Lastly, we discuss numerical examples and study the solution influenced by the van der Waals excluded volume.     

Global Continuous Riemann Solver for Nonlinear Elasticity

We consider in this paper an isothermal model of nonlinear elasticity. This model is described by two conservation laws that define a problem of mixed type, both elliptic and hyperbolic. We restrict ourselves to the linearly degenerate case, and consider Riemann data that lies in the hyperbolic regions. The lack of uniqueness of the Riemann problem is solved by the introduction of a so-called kinetic relation, used to narrow the set of admissible subsonic phase transitions. In this situation, we consider the Riemann problem for any data lying in the hyperbolic region, using either explicit computations or geometric arguments. This construction allows us to give sufficient conditions on the kinetic relation in order that the generated Riemann solver possesses properties of uniqueness, globality, and continuous dependence on the initial data in the L ¹ distance. Accepted October 1, 2000?Published online January 22, 2001     

Problem of interaction of thin rigid needle-shaped inclusions located between different elastic materials

We study a piecewise-homogeneous elastic plane composed of two half-planes with different elastic parameters and two thin rigid needle-shaped inclusions located between them. One inclusion is rigidly connected with the environment, and the other inclusion is not, while contacting with it like a smooth rigid punch. We consider the plane deformed state generated by stresses given at infinity. The problem is reduced to a combination of a matrix Riemann boundary-value problem from the theory of analytic functions and a matrix Hilbert problem, which can be solved in terms of integrals through the reduction to two separate scalar Riemann boundary-value problems on a twosheeted Riemann surface.We explicitly obtain the complex potentials of the composite elastic plane, the stress intensity factors near the tips of the inclusion, and the rotation angles of the inclusions. We also present numerical examples illustrating how the stresses near the inclusions depend on the elastic and geometric parameters of the problem.     

Phase mixtures in dynamics of pseudoelasticity

We give a numerical treatment of phase mixtures in pseudoelasticity from a purely mathematical point of view. It is based on a surprising result that the approximate solution may consist of persistent oscillations in strain which resemble the experimentally observed interface patterns. Such a solution is obtained from a sequence of solutions for a rate-type viscoelastic problem with a non-monotone equilibrium stress-strain relation, for which in the limit as the viscosity tends to infinity the viscoelastic problem reduces to the rate-independent elastic problem describing phase transitions. In this manner, it seems to give yet another perspective for the phase mixture from dynamic point of view as the evolution of an unstable state, in contrast to the traditional treatment from stability analysis for phase equilibrium.     

Thermo-elastic aspects of dynamic nucleation

In spite of recent progress in our understanding of the absolute stability of elastic phases under loads, the generic presence of metastable configurations and the possibility of their dynamic breakdown remains a major problem in the mechanical theory of phase transitions in solids. In this paper, by considering the simplest one-dimensional model, we study the interplay between inertial and thermal effects associated with nucleation of a new phase, and address the crucial question concerning the size of a perturbation breaking metastability. We begin by reformulating the nucleation problem as a degenerate Riemann problem. By choosing a specific kinetic relation, originating from thermo-visco-capillary (TVC) regularization, we solve a self-similar problem analytically and demonstrate the existence of two types of solutions: with nucleation and without it. We then show that in the presence of a non-zero latent heat, solution with nucleation may by itself be non-unique. To understand the domain of attraction of different self-similar solutions with and without nucleation, we regularize the model and study numerically the full scale initial value problem with locally perturbed data. Through numerical experiments we present evidence that the TVC regularization is successful in removing deficiencies of the classical thermo-elastic model and is sufficient in specifying the limits of metastability.     

On the response of rubbers at high strain rates—I. Simple waves

In this series of papers, we examine the propagation of waves of finite deformation in rubbers through experiments and analysis; in the present paper, Part I, attention is focused on the propagation of one-dimensional waves in strips of natural, latex and synthetic, nitrile rubber. Tensile wave propagation experiments were conducted at high strain rates by holding one end fixed and displacing the other end at a constant velocity. A high-speed video camera was used to monitor the motion and to determine the evolution of strain and particle velocity in rubber strips. Analysis of the response through the theory of finite waves indicated a need for an appropriate constitutive model for rubber; by quantitative matching between the experimental observations and analytical predictions, an appropriate instantaneous elastic response for the rubbers was obtained. This matching process suggested that a simple power-law constitutive model was capable of representing the high strain-rate response for both rubbers used.     

不同应变率下聚乙烯材料的压缩力学性能

为了研究聚乙烯材料在不同应变率下的压缩力学性能,通过准静态实验和动态实验获得聚乙烯材料不同应变率下的应力应变曲线,分析发现:聚乙烯的弹性模量和屈服强度随应变率增大而增大,具有明显的黏弹塑性;聚乙烯材料进入塑性阶段,其应力应变曲线在不同应变率下具有相近的变化趋势,即塑性切向模量近似相同。根据聚乙烯材料的压缩力学性能,建立了弹性区、屈服点和塑性区的分段本构模型。该模型的屈服点和塑性段与实验结果吻合较好,由于弹性段采用线弹性模型,与实验结果存在一定偏差,可近似描述材料的弹性行为。     

A physically-based,quasilinear viscoelasticity model for the dynamic response of polyurea

Polyurea, a promising material for damage mitigation in impact scenarios, has been investigated through plane-wave, pressure-shear plate impact (PSPI) experiments to obtain its mechanical response at high pressures and high strain rates. Based on these experimental results, a physically-based, quasi-linear, viscoelasticity model is introduced to capture the observed nonlinear pressure-volume behavior, the strong dependence of shearing resistance on pressure, and the strong relaxation of deviatoric stresses. This model has been implemented in finite element software ABAQUS to simulate the response of polyurea P1000 under the impact conditions of a variety of PSPI experiments. Simulation results agree reasonably well with those of the experiments.     

A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage

A micromechanically based non-affine network model for finite rubber elasticity and viscoelasticity was discussed in Parts I and II [Miehe, C., Göktepe, S., Lulei, F., 2004. A micro-macro approach to rubber-like materials. Part I: The non-affine micro-sphere model of rubber elasticity. J. Mech. Phys. Solids 52, 2617-2660; Miehe, C., Göktepe, S., 2005. A micro-macro approach to rubber-like materials. Part II: Viscoelasticity model for polymer networks. J. Mech. Phys. Solids, published on-line, doi:10.1016/j.jmps.2005.04.006.] of this work. In this follow-up contribution, we further extend the micro-sphere network model such that it incorporates a deformation-induced softening commonly referred to as the Mullins effect. To this end, a continuum formulation is constructed by a superimposed modeling of a crosslink-to-crosslink (CC) and a particle-to-particle (PP) network. The former is described by the non-affine elastic network model proposed in Part I. The Mullins-type damage phenomenon is embedded into the PP network and micromechanically motivated by a breakdown of bonds between chains and filler particles. Key idea of the constitutive approach is a two-step procedure that includes (i) the set up of micromechanically based constitutive models for a single chain orientation and (ii) the definition of the macroscopic stress response by a directly evaluated homogenization of state variables defined on a micro-sphere of space orientations. In contrast to previous works on the Mullins effect, our formulation inherently describes a deformation-induced anisotropy of the damage as observed in experiments. We show that the experimentally observed permanent set in stress-strain diagrams is achieved by our model in a natural way as an anisotropy effect. The performance of the model is demonstrated by means of several numerical experiments including the solution of boundary-value problems.     

Nonlinear damping in a micromechanical oscillator

Nonlinear elastic effects play an important role in the dynamics of microelectromechanical systems (MEMS). A Duffing oscillator is widely used as an archetypical model of mechanical resonators with nonlinear elastic behavior. In contrast, nonlinear dissipation effects in micromechanical oscillators are often overlooked. In this work, we consider a doubly clamped micromechanical beam oscillator, which exhibits nonlinearity in both elastic and dissipative properties. The dynamics of the oscillator is measured in both frequency and time domains and compared to theoretical predictions based on a Duffing-like model with nonlinear dissipation. We especially focus on the behavior of the system near bifurcation points. The results show that nonlinear dissipation can have a significant impact on the dynamics of micromechanical systems. To account for the results, we have developed a continuous model of a geometrically nonlinear beam-string with a linear Voigt–Kelvin viscoelastic constitutive law, which shows a relation between linear and nonlinear damping. However, the experimental results suggest that this model alone cannot fully account for all the experimentally observed nonlinear dissipation, and that additional nonlinear dissipative processes exist in our devices.     

Stress wave experiments in plasticity

Plastic wave experiments are reviewed, beginning with the earliest experiments of Bell on the propagation of incremental waves in prestressed bars. Attention is directed to experiments in which the plastic wave profile at different distances of propagation is used to infer information on the dynamic plastic response of the material in which the wave is propagating. Plastic waves in bars, tubes, and plates are considered. Principal results are reviewed on such primary physical features as the velocity of propagation of incremental waves, the dynamic elastic limit, the wave profiles of finite amplitude waves, and the effects of nonproportional loading. Objectives for future research are suggested.     

Method of model analysis for flexible head impacting with elastic plane

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On the Decay Rates of Porous Elastic Systems

In this paper we analyze the porous elastic system. We show that viscoelasticity is not strong enough to make the solutions decay in an exponential way, independently of any relationship between the coefficients of wave propagation speed. However, it decays polynomially with optimal rate. When the porous damping is coupled with microtemperatures, we give an explicit characterization on the decay rate that can be exponential or polynomial type, depending on the relation between the coefficients of wave propagation speed. Numerical experiments using finite differences are given to confirm our analytical results. It is worth noting that the result obtained here is different from all existing in the literature for porous elastic materials, where the sum of the two slow decay processes determine a process that decay exponentially.