Minimum Free Energies for Materials with Finite Memory

Finite memory viscoelastic materials are of interest because (a) they are not necessarily experimentally distinguishable from materials with infinite memory; and (b) the assumption of infinite memory can, in certain contexts, lead to results that run counter to physical intuition. An example of this - the quasi-static viscoelastic membrane in a frictional medium - is discussed. It is shown that, for a finite memory material, the singularity structure of the Fourier transform of the relaxation function derivative is quite different from the infinite memory case in the sense that it is an entire function with all its singularities being essential singularities at infinity. The formula for the minimum free energy [1] is still valid in this case. In contrast to the work function, this quantity, and all other functions of the minimal state, depend only on the values of the history over the period when the relaxation function derivative is nonzero. The factorization required to determine the form of the minimum free energy can be carried out explicitly for simple step-function choices of the relaxation function derivative. The two simplest cases are fully worked through and explicit formulae are given for all relevant quantities.     

Free energies for singleton minimal states

It is assumed that any free energy function exhibits strict periodic behavior for histories that have been periodic for all past times. This is not the case for the work function, which, however, has the usual defining properties of a free energy. Forms given in fairly recent years for the minimum and related free energies of linear materials with memory have this property. Materials for which the minimal states are all singletons are those for which at least some of the singularities of the Fourier transform of the relaxation function are not isolated. For such materials, the maximum free energy is the work function, and free energies intermediate between the minimum free energy and the work function should be given by a linear relation involving these two quantities. All such functionals, except the minimum free energy, therefore do not have strict periodic behavior for periodic histories, which contradicts our assumption. A way out of the difficulty is explored which involves approximating the relaxation function by a form for which the minimal states are no longer singletons. A representation can then be given of an arbitrary free energy as a linear combination of the minimum, maximum and intermediate free energies derived in earlier work. This representation obeys our periodicity assumption. Numerical data are presented, supporting the consistency of this approach.     

On the analytic expression of the free energy in linear viscoelasticity

Two definitions of free energy for a linear viscoelastic material, due to Graffi and to Coleman and Owen, are considered, and the compatibility of these definitions with some expressions of the free energy proposed in the literature is examined. For the expressions of Staverman and Schwarzl and of Breuer and Onat, the two definitions are proved to be equivalent, and the set of all relaxation functions for which the two expressions are indeed free energies is determined. Two more expressions, proposed by Volterra and Graffi and by Morro and Vianello, are taken into consideration. For them, only the classes of relaxation functions for which they are free energies according to the first definition, is completely characterized. All results are established under regularity assumptions weaker than those usually made in the literature.     

多孔硅橡胶有限变形的粘弹性行为

针对孔隙度较大（孔隙度大于50％）的硅橡胶材料在有限变形时的粘弹性行为，从建立描述材料粘弹性特征的松驰函数和变形特征的应变能函数出发，提出了适合多孔隙、可压硅橡胶材料的非线性粘弹性力学行为的本构关系，松驰函数和应变能函数可解耦为等容和体积变形两部分，并引入了拟时间的概念来反映变形对材料特征时间的影响，利用硅橡胶材料的单轴压缩松驰实验与材料模型进行了对比，讨论了多孔硅橡胶的等容变形和体积变形对应力松驰的影响。     

A thermodynamic approach to nonlinear ultrasonics for material state awareness and prognosis

We develop a thermodynamic framework for modeling nonlinear ultrasonic damage sensing and prognosis in materials undergoing progressive damage. The framework is based on the internal variable approach and relies on the construction of a pseudo-elastic strain energy function that captures the energetics associated with the damage progression. The pseudo-elastic strain energy function is composed of two energy functions—one that describes how a material stores energy in an elastic fashion and the other describes how material dissipates energy or stores it in an inelastic fashion. Experimental motivation for the choice of the above two functionals is discussed and some specific choices pertaining to damage progression during fatigue and creep are presented. The thermodynamic framework is employed to model the nonlinear response of material undergoing stress relaxation and creep-like degradation. For each of the above cases, evolution of the nonlinearity parameter with damage as well as with macroscopic measurables like accumulated plastic strain is obtained.     

Dynamic stiffness of chemically and physically ageing rubber vibration isolators in the audible frequency range

The constitutive equations of chemically and physically ageing rubber in the audible frequency range are modelled as a function of ageing temperature, ageing time, actual temperature, time and frequency. The constitutive equations are derived by assuming nearly incompressible material with elastic spherical response and viscoelastic deviatoric response, using Mittag-Leffler relaxation function of fractional derivative type, the main advantage being the minimum material parameters needed to successfully fit experimental data over a broad frequency range. The material is furthermore assumed essentially entropic and thermo-mechanically simple while using a modified William–Landel–Ferry shift function to take into account temperature dependence and physical ageing, with fractional free volume evolution modelled by a nonlinear, fractional differential equation with relaxation time identical to that of the stress response and related to the fractional free volume by Doolittle equation. Physical ageing is a reversible ageing process, including trapping and freeing of polymer chain ends, polymer chain reorganizations and free volume changes. In contrast, chemical ageing is an irreversible process, mainly attributed to oxygen reaction with polymer network either damaging the network by scission or reformation of new polymer links. The chemical ageing is modelled by inner variables that are determined by inner fractional evolution equations. Finally, the model parameters are fitted to measurements results of natural rubber over a broad audible frequency range, and various parameter studies are performed including comparison with results obtained by ordinary, non-fractional ageing evolution differential equations.     

Strain Relaxation in an Alloy Film with a Rough Free Surface

We study strain relief by surface roughness and composition variation in a stressed alloy film. Instead of using common perturbation techniques, we derive a rigorous relaxation formula based on the energy approach in the case of slightly undulating surface and fluctuating composition. We do not require any a priori assumption of elastic isotropy or identical material properties between film and substrate in deriving our result. We show that the change of elastic energy is negative, giving rise to energy relief due to the presence of free surface. We apply our result to the study of compositional and morphological instabilities of a stressed thin layer with a free surface. The critical wave number of instability is determined by the competition between the destabilizing influence of elastic strain energy and the stabilizing influence of chemical and surface energies.     

Viscoelastic solids with unbounded relaxation function

Linear viscoelastic solids are considered where the relaxation function is unbounded in that the initial value of the function and the derivative may be infinite but the function, freed from the equilibrium modulus, is integrable. The second law is given a general form for approximate cycles and then for cycles. Thermodynamic restrictions are derived in connection with cycles and shown to be equivalent to those obtained for bounded relaxation functions. Then the thermodynamic restrictions are shown to be also sufficient for the validity of the second law in the general case of approximate cycles. Finally, a functional is considered which proves to be endowed with the characteristic properties of the free energy.     

A linear dynamic model for a saturated porous medium

A linear isothermal dynamic model for a porous medium saturated by a Newtonian fluid is developed in the paper. In contrast to the mixture theory, the assumption of phase separation is avoided by introducing a single constitutive energy function for the porous medium. An important advantage of the proposed model is it can account for the couplings between the solid skeleton and the pore fluid. The mass and momentum balance equations are obtained according to the generalized mixture theory. Constitutive relations for the stress, the pore pressure are derived from the total free energy accounting for inter-phase interaction. In order to describe the momentum interaction between the fluid and the solid, a frequency independent Biot-type drag force model is introduced. A temporal variable porosity model with relaxation accounting for additional attenuation is introduced for the first time. The details of parameter estimation are discussed in the paper. It is demonstrated that all the material parameters in our model can be estimated from directly measurable phenomenological parameters. In terms of the equations of motion in the frequency domain, the wave velocities and the attenuations for the two P waves and one S wave are calculated. The influences of the porosity relaxation coefficient on the velocities and attenuation coefficients of the three waves of the porous medium are discussed in a numerical example.     

Nonlinear transient thermal stress analysis of thick-walled FGM cylinder with temperature-dependent material properties

In this paper, the problem of generalized thermoelasticity in a thick-walled FGM cylinder with one relaxation time is presented. The material properties are taking as function of temperature and graded in the radial. Due to the nonlinearity of the governing equations, finite element method is adopted to solve such problem. Both the inner and outer curved surfaces of the cylinder are stress free while the inner surface is subjected to thermal shock, while the outer surface is thermally isolated. The effects of temperature-dependent properties, volume fraction parameter and the thermal relaxation time on the physical quantities behavior are evaluated. Results confirm the efficiency of the present algorithm and reveal the significant effects of the temperature-dependent of the material properties.     

On the representation of chemical ageing of rubber in continuum mechanics

In order to represent the chemical ageing behaviour of rubber under finite deformations a three-dimensional theory is proposed. The fundamentals of this approach are different decompositions of the deformation gradient in combination with an additive split of the Helmholtz free energy into three parts. Its first part belongs to the volumetric material behaviour. The second part is a temperature-dependent hyperelasticity model which depends on an additional internal variable to consider the long-term degradation of the primary rubber network. The third contribution is a functional of the deformation history and a further internal variable; it describes the creation of a new network which remains free of stress when the deformation is constant in time. The constitutive relations for the stress tensor and the internal variables are deduced using the Clausius–Duhem inequality. In order to sketch the main properties of the model, expressions in closed form are derived with respect to continuous and intermittent relaxation tests as well as for the compression set test. Under the assumption of near incompressible material behaviour, the theory can also represent ageing-induced changes in volume and their effect on the stress relaxation. The simulations are in accordance with experimental data from literature.     

A Helmholtz free-energy function for a Cu–Al–Ni shape memory alloy

In this paper we develop a free energy function for a Cu–Al–Ni alloy undergoing cubic–orthorhombic phase transitions. We use the irreducible Lagrangian strain polynomial invariants of the cubic austenite parent phase to construct a polynomial expansion for the Helmholtz free energy. Our expansion retains quartic terms of the strain components in order to describe the two-phase material—the cubic austenite phase and six variants of the orthorhombic martensite phase. The coefficients of the free energy polynomial function are temperature dependent and are fitted to appropriate experimental data for austenite and martensite phases from literature. The resulting Helmholtz free energy function is given by (3.20), (4.11). We examine the response predicted by the model for shear in the twinning directions.     

Shape memory behaviour: modelling within continuum thermomechanics

A phenomenological material model to represent the multiaxial material behaviour of shape memory alloys is proposed. The material model is able to represent the main effects of shape memory alloys: the one-way shape memory effect, the two-way shape memory effect due to external loads, the pseudoelastic and pseudoplastic behaviour as well as the transition range between pseudoelasticity and pseudoplasticity.The material model is based on a free energy function and evolution equations for internal variables. By means of the free energy function, the energy storage during thermomechanical processes is described. Evolution equations for internal variables, e.g. the inelastic strain tensor or the fraction of martensite are formulated to represent the dissipative material behaviour. In order to distinguish between different deformation mechanisms, case distinctions are introduced into the evolution equations. Thermomechanical consistency is ensured in the sense that the constitutive model satisfies the Clausius–Duhem inequality.Finally, some numerical solutions of the constitutive equations for isothermal and non-isothermal strain and stress processes demonstrate that the various phenomena of the material behaviour are well represented. This applies for uniaxial processes and for non-proportional loadings as well.     

An Explicit Formula for the Minimum Free Energy in Linear Viscoelasticity

A general explicit formula for the maximum recoverable work from a given state is derived in the frequency domain for full tensorial isothermal linear viscoelastic constitutive equations. A variational approach, developed for the scalar case, is here generalized by virtue of certain factorizability properties of positive-definite matrices. The resultant formula suggests how to characterize the state in the sense of Noll in the frequency domain. The property that the maximum recoverable work represents the minimum free energy according to both Graffi's and Coleman-Owen's definitions is used to obtain an explicit formula for the minimum free energy. Detailed expressions are presented for particular types of relaxation function. This revised version was published online in August 2006 with corrections to the Cover Date.     

Linear dynamic model for porous media saturated by two immiscible fluids

A linear isothermal dynamic model for a porous medium saturated by two immiscible fluids is developed in the paper. In contrast to the mixture theory, phase separation is avoided by introducing one energy for the porous medium. It is an important advantage of the model based on one energy approach that it can account for the couplings between the phases. The volume fraction of each phase is characterized by the porosity of the porous medium and the saturation of the wetting phase. The mass and momentum balance equations are constructed according to the generalized mixture theory. Constitutive relations for the stress, pore pressure are derived from the free energy function. A capillary pressure relaxation model characterizing one attenuation mechanism of the two-fluid saturated porous medium is introduced under the constraint of the entropy inequality. In order to describe the momentum interaction between the fluids and the solid, a frequency independent drag force model is introduced. The details of parameter estimation are discussed in the paper. It is demonstrated that all the material parameters in our model can be calculated by the phenomenological parameters, which are measurable. The equations of motion in the frequency domain are obtained in terms of the Fourier transformation. In terms of the equations of motion in the frequency domain, the wave velocities and the attenuations for three P waves and one S wave are calculated. The influences of the capillary pressure relaxation coefficient and the saturation of the wetting phase on the velocities and attenuation coefficients for the four wave modes are discussed in the numerical examples.     

From dynamic modulus via different relaxation spectra to relaxation and creep functions

The main goal of the paper is to compare predictive power of relaxation spectra found by different methods of calculations. The experimental data were obtained for a new family of propylene random copolymers with 1-pentene as a comonomer. The results of measurements include flow curves, viscoelastic properties, creep curves and rubbery elasticity of copolymer melts. Different relaxation spectra were calculated using independent methods based on different ideas. It lead to various distributions of relaxation times and their “weights”. However, all of them correctly describe the frequency dependencies of dynamic modulus. Besides, calculated spectra were used for finding integral characteristics of viscoelastic behaviour of a material (Newtonian viscosity, the normal stress coefficient, steady-state compliance). In this sense all approaches are equivalent, though it appears impossible to estimate instantaneous modulus. The most crucial arguments in estimating the results of different approaches is calculating the other viscoelastic function and predicting behaviour of a material in various deformation modes. It is the relaxation and creep functions. The results of relaxation curve calculations show that all methods used give rather similar results in the central part of the curves, but the relaxation curves begin to diverge when approaching the high-time (low-frequency) boundary of the relaxation curves. The distributions of retardation times calculated through different approaches also appear very different. Meanwhile, predictions of the creep curves based on these different retardation spectra are rather close to each other and coincide with the experimental points in the wide time range. Relatively slight divergences are observed close to the upper boundary of the experimental window. All these results support the conclusion about a rather free choice of the relaxation time spectrum in fitting experimental data and predicting viscoelastic behaviour of a material in different deformation modes. Received: 15 March 2000 Accepted: 18 September 2000     

Regularity and stability for a viscoelastic material with a singular memory kernel

We consider a linear viscoelastic material whose relaxation function may exhibit an initial singularity. We show that the Laplace transform method is still applicable in order to study existence, uniqueness and asymptotic behaviour of the solution to the dynamic problem. In order to provide these results, we impose on the relaxation function only restrictions deriving from Thermodynamics. Moreover, by using energy estimates, we establish a stability theorem. Finally, for a class of singular kernels, we obtain a regularity result which ensures the asymptotic stability of the solution.This work is supported by G.N.F.M. of C.N.R. and by M.U.R.S.T. 40% and 60% projects.