Thermodynamic Admissibility Research Articles

Thermodynamic admissibility of the pompon model for branched polymers

The pompon model for branched polymers is re-formulated within a general framework of nonequilibrium thermodynamics. While the original model of McLeish and Larson is found to be thermodynamically admissible, nonequilibrium thermodynamics strongly suggests several model modifications. As an immediate consequence, a nonzero second normal-stress coefficient in shear flow is predicted by the modified pompon model.

Note on the Notion of Incompressibility in Thermodynamic Theories of Porous and Granular Materials

We present a simple two-component model of a porous material based on the constraint assumption that the so-called true components are incompressible. In my previous work on this subject [1] I pointed out that many such models are not thermodynamically admissible. Namely the second law of thermodynamics led to the conclusion that an additional field of reaction force on the constraint cannot be introduced, and, consequently, the set of field equations was overdetermined. However I speculated as well that an extension of the set of variables may lead to thermodynamic admissibility. Indeed an example presented in this paper supports this speculation. According to results of this work it seems to be necessary to introduce higher gradients to multicomponent models with constraints in order to satisfy the second law of thermodynamics. The latter is exploited by means of Lagrange multipliers eliminating constraints from the entropy inequality. This procedure is known in the literature as the method of Lagrange-Liu multipliers (see the original work of I-Shih Liu [2]).

A General Time Dependent Constitutive Model: Part I— Theoretical Developments1

Using an internal-variable formalism as a starting point, we describe the viscoelastic complement of a previously-developed viscoplasticity formulation of the complete potential structure type. It is mainly motivated by experimental evidence for the presence of rate/time effects in the so-called quasilinear, reversible, material response range. Several possible generalizations are described, in the general format of hereditary-integral representations for nonequilibrium, stress-type, state variables, both for isotropic as well as anisotropic materials. In particular, thorough discussions are given on the important issues of thermodynamic admissibility requirements for such general descriptions, resulting in a set of explicit mathematical constraints on the associated kernel (relaxation and creep compliance) functions. In addition, a number of explicit, integrated forms are derived, under stress and strain control to facilitate the parametric and qualitative response characteristic studies reported here, as well as to help identify critical factors in the actual experimental characterizations from test data that will be reported in Part II.

On separation of energies in viscoelasticity

Separation of the storaged and dissipated energies in viscoelastic deformation is considered. This is a key problem for the construction of viscoelastic minimum rinciples and for the micromechanics of heterogeneous materials with memory. The notion of the viscoelastic free energy functional is discussed, thermodynamic admissibility conditions are established. An engineering analysis is realized through the method of harmonic strain regimes, influence of the loss and the storage moduli on the dissipation rate is studied. For the Volterra-Frechet integral expansion approach, necessary conditions on the general form of a free energy viscoelastic functional are formulated. The obtained results are used to examine the thermodynamic validity of certain classic viscoelastic models, like that of Staverman-Schwarzl. Through the spectral method, this energy representation is shown to correspond to a generalized Maxwell model.

Modelling the mechanical behavior of shape memory alloys under variant coalescence

In shape memory alloys (SMA) reorientation of crystallographic variants upon stressing takes place in a temperature regime lower than the martensite finish temperature M f , a process which is denoted as variant coalescence (VC). The macroscopic deformation behavior of a polycrystalline Cu-Al-Ni shape memory alloy under VC is modelled by utilizing a combined computational micromechanics and continuum-thermodynamics framework. The kinematical description of the reorientation process is based on the crystallography of the β 1 → γ′ 1 martensitic transformation in Cu-Al-Ni. A thermodynamic field concept for VC is followed by introducing a Gibbs free energy formulation for the polycrystalline mesodomain and the derivation of a criterion for the thermodynamic admissibility of the reorientation process considering the stored elastic energy due to reorientation-induced microstresses and the energy dissipation due to the movement of boundaries between the martensitic variants. The reorientation process on the size scale of self-accommodating plate groups is formulated by the use of simple microscale constitutive assumptions. The reorientation process is the subject of a micromechanical simulation. By checking the thermodynamic admissibility of any small increment of the reorientation process the corresponding required magnitude of externally applied tensile stress can be calculated and the related overall mechanical behavior results with the stress-strain curve. The micro-macro transition is performed by an averaging procedure for each crystallite and a finite element based periodic microfield approach. VC is studied with respect to the mobility of the intervariant boundaries and energy dissipation due to interfacial motion. The effect of the volume change associated with the β 1 → γ′ 1 martensitic transformation in Cu-Al-Ni and the difference in the mechanical behavior in uniaxial tension and compression is quantified.

New class of creep-relaxation functions

The thermodynamic analysis of viscoelastic constitutive laws makes it possible to estimate the class of admissible creep-relaxation functions. This set of functions turns out to be larger than it was traditionally supposed. Except for the well known positive definite, monotone decreasing functions, the new class also includes the so-called non-traditional functions, such as locally non-monotone and even non-positive ones, hence involving the effect of the so-called negative viscosity.It is shown that generalized creep-relaxation functions turn out to be effective in the phenomenologic modelling of the complicated materials behaviour, like that in homogenization of heterogeneous bodies and in modelling of the wave propagation and of the negative Poisson ratio phenomena in composites.New directions for fundamental and applied research, that are considered with the thermodynamic admissibility of viscoelastic models, are discussed. Examples of the application of generalized creep-relaxation functions are given, in particular for the dynamics of composites and for the thermomechanical modelling of wood.

Back to the thermodynamic admissibility of creep-relaxation functions in linear viscoelasticity

The recent discussion on admissibility criteria for the kernels of linear viscoelastic liquids is completed by the results for viscoelastic solids. The necessary and sufficient conditions of thermodynamic admissibility are established. The class of thermodynamically admissible creep-relaxation functions is proved to be wider than it is usually supposed, including as well numerous nonmonotone and even locally negative functions. These nontraditional viscoelasticity kernels find applications in physics of heterogeneous materials, like for instance in mechanics and theology of composites.

A continuum mechanics approach to some problems in subcritical crack propagation

The results of the so-called energetic approach to fracture for the cases of a sharp crack without and with a cohesive zone are briefly reviewed with particular attention to the crack tip singularity analysis and to the issue of energy dissipation due to crack propagation. The case of a crack with a cohesive zone removing all thermomechanical singularities is then further analyzed, focusing the attention on the question of the thermodynamic admissibility of subcritical crack growth, and on some of the hypotheses that lead to the derivation of subcritical crack growth laws. A two-phase cohesive zone model for discontinuous crack growth is presented and its thermodynamics analyzed, followed by an example of its possible application.

Poisson bracket formulation of viscoelastic flow equations of differential type: A unified approach

The Hamiltonian formulation of equations in continuum mechanics through a generalized bracket operation is shown here to reproduce a variety of incompressible viscoelastic fluid models, including the Giesekus model (with particular cases the upper‐convected Maxwell and the Oldroyd‐B models), the FENE–P dumbbell, the Phan‐Thien/Tanner, the Leonov, the Bird/DeAguiar, and the bead–spring chain models. The analysis allows comparison of the differential models on a more fundamental level than previously possible by reformulating the equations in terms of the Hamiltonian (system energy) and the dissipation of the system expressed as functionals involving the velocity vector and structural parameter(s). In fact, all of these models involve only slight variations of the same general Hamiltonian and the dissipation tensor. An advantage of this formulation is the establishment of thermodynamic admissibility criteria which in complex flows can shed light on the range of validity and/or faithfulness of the numerical calculations involving the above models. The usefulness of the generalized bracket formulation lies in the systematic approach that it provides in addressing one of the fundamental problems that the engineer working with complex materials has to deal with: how to transfer information that has been painstakingly provided by the physical chemist, addressing fundamental problems on a molecular level, from the microscopic scale to the macroscopic level where the engineer actually needs the model in dealing with everyday industrial problems. It is hoped that this new formulation can be used in the future to systematically generate continuum constitutive models, which are thermodynamically consistent, and based on microscopic analysis. Thus, it is the purpose here to narrow the gap between detailed (molecular) microscopic descriptions of the motions of polymer chains and (macroscopic phenomenological) continuum approaches. We believe that the generalized bracket formulation, due to its inherent simplicity and symmetry, has the potential to provide an answer to very complex situations, such as multicomponent structured media and coupled transport phenomena.

Thermodynamic admissibility of irrotational flow of two immiscible viscous and heat conducting fluids rotating between two concentric cylinders

A thermodynamically admissible motion of two immiscible fluids rotating irrotationally between two concentric cylinders is presented. The analysis shows that the product of the velocity and the viscosity is continuous at the interface. Thus, slip exists at the interface whenever the fluids have different viscosities. Applying the second law jump condition to the interface, referred to as the Clausius-Duhem inequality, we find that the motion is thermodynamically admissible if the velocity of the outer fluid is smaller or equal to the velocity of the inner fluid. Vulnerability of the interface to Kelvin-Helmoltz instability is tested and the range of wavenumbers for which the fluid-fluid interface of an irrotational admissible motion is stable to small disturbances is determined. It is furthermore shown that if the condition of irrotationality is relaxed there exists a family of rotational admissible solutions satisfying the same boundary conditions at the solid boundaries as the irrotational solution.

On the mechanical inadmissibility and thermodynamic admissibility according to the Clausius-Duhem inequality of the unbounded shear flow of a second-order fluid with negative stress power

Description d'un ecoulement cisaille stationnaire bidimensionnel suggere par l'exemple de Rajagopal, satisfaisant a une inegalite stricte

Relativistic theory of magnetoelastic interactions. II. Constitutive theory

(For pt. I, see abstr. A41616 of 1972). Constitutive equations are obtained for a hyperelastic solid by the use of the thermodynamical admissibility. (Clausius-Duhem inequality), the relevant potential being the magneto-free energy written in an ad hoc invariant form after application of the author's formulation of the principle of material frame indifference. The agreement with equations derived before from a variational formulation is shown.

Mechanical and thermodynamical admissibility of stress-strain functions

Mechanical and thermodynamical admissibility of stress-strain functions