Christina Papenfuss

A sketch of continuum thermodynamics

Abstract Different formulations of non-equilibrium continuum thermodynamics are discussed: Thermodynamics of Irreversible Processes (TIP), Rational Thermodynamics (RAT), Extended Thermodynamics (ET), Mesoscopic Continuum Thermodynamics (MT), and the GENERIC version of thermodynamics. Concepts as constitutive quantity, state space, material frame indifference, exploiting dissipation inequality, mesoscopic variables, and GENERIC balance equations are taken into consideration.

Mesoscopic theory of liquid crystals

The structural roots of the mesoscopic theory of liquid crystals are reviewed and shortly discussed.

Concepts of Mesoscopic Continuum Physics With Application to Biaxial Liquid Crystals

Abstract The mesoscopic concept in continuum mechanics consists of extending the domain of the balance equations by the set of mesoscopic variables and of introducing a local distribution function of these variables as a statistical element. The balance equations defined on this extended domain and an example concerning liquid crystals of biaxial molecules are discussed.

Thermodynamical Frameworks for Higher Grade Material Theories with Internal Variables or Additional Degrees of Freedom

Abstract The objective of the present work is to compare several thermomechanical frameworks, taking into account the influence of strain gradient, internal variables, gradient of internal variables, and temperature gradient on the constitutive behavior of materials. In particular, the restrictions by the second law of thermodynamics are derived. The method of exploitation consists of two steps: an application of the well-known method by Liu and a new method of exploiting the residual inequality. The first example introduces an enlarged set of variables for the constitutive functions including in particular the strain gradient, an internal variable, its gradient, and the temperature gradient. In the second example, the power of internal forces is enriched to incorporate generalized stress measures. In the third example, the classical thermomechanical setting is complemented by a balance-type differential equation for an additional variable. Finally, material theories of grade n are envisaged. It is shown that the free energy density may depend on gradients only in the case that an additional balance equation is introduced. We also demonstrate that for isotropic materials the second law of thermodynamics implies for a large class of state spaces that the entropy flux equals the heat flux divided by temperature.

Exploitation of the Second Law: Coleman–Noll and Liu Procedure in Comparison

Abstract A comparison is made between two classical methods for the exploitation of the second law of thermodynamics: the Coleman–Noll and the Liu procedure. On the example of a rigid heat conductor with general entropy flux, it is shown that the two procedures are equivalent. This equivalence is demonstrated in the case of a state space including the wanted fields, only, as well as in the case of gradients being relevant for constitutive equations, too. Also, the possible importance of an internal variable or an internal degree of freedom is considered.

Thermodynamic approach to generalized continua

Governing equations of dissipative generalized solid mechanics are derived by thermodynamic methods in the Piola–Kirchhoff framework using the Liu procedure. The isotropic small-strain case is investigated in more detail. The connection to the Ginzburg–Landau type evolution, dual internal variables, and a thermodynamic generalization of the standard linear solid model of rheology is demonstrated. Specific examples are chosen to emphasize experimental confirmations and predictions beyond less general approaches.

Constitutive Theory for Two Dimensional Liquid Crystals

Abstract Two dimensional liquid crystal are considered as mathematical surfaces of discontinuity of the bulk fields. The equations of motion for the relevant surface fields in the presence of electromagnetic fields are summarized. These are the mechanical balance equations, Maxwells equations, and an equation of motion for the second order alignment tensor. The restrictions to constitutive functions implied by the Second Law of Thermodynamics are discussed. The stress tensor can be calculated in the frame of this phenomenological theory. For the alignment production a statistical background is needed, what is very shortly outlined.

Griffith cracks in the mesoscopic microcrack theory

The mesoscopic concept is applied to the description of microcracked brittle materials. The mesoscopic equations are solved in a special case when the microcracks are developing according to the Rice–Griffith evolution law. The evolution of the crack distribution function is investigated in the case of simple loading conditions and for two different initial crack distribution functions. The time dependence of the average crack length is also calculated.

Dynamics of the Size and Orientation Distribution of Microcracks and Evolution of Macroscopic Damage Parameters

Abstract We are dealing with damage of brittle materials caused by growth of microcracks. In our model the cracks are penny-shaped. They can only enlarge but not heal. For a single crack a Rice–Griffith growth law is assumed: There is crack growth only if tension is applied normally to the crack surface, exceeding a critical value. Our aim is to investigate the effect of crack growth on macroscopic constitutive quantities. A possible approach taking into account such an internal structure within continuum mechanics is the mesoscopic theory. A distribution of crack lengths and crack orientations within the continuum element is introduced. Macroscopic quantities are calculated as averages with the distribution function. A macroscopic measure of the progressing damage, i.e., a damage parameter, is the average crack length. For this scalar damage parameter we derive an evolution equation. Due to the unilateral growth law for the single crack, it turns out that the form of this differential equation depends explicitly on the initial crack length distribution. In order to treat biaxial loading, it is necessary to introduce a tensorial damage parameter. We define a second-order tensor damage parameter in terms of the crack length and orientation distribution function.

A mesoscopic approach to diffusion phenomena in mixtures

Abstract The mesoscopic concept is applied to the theory of mixtures. The aim is to investigate the diusion phenomenon from a mesoscopic point of view. The domain of the field quantities is extended by the set of mesoscopic variables, here the velocities of the components. Balance equations on this enlarged space are the equations of motion for the mesoscopic fields. Moreover, local distribution functions of the velocities are introduced as a statistical element, and an equation of motion for this distribution function is derived. From this equation of motion, dierential equations for the diusion fluxes and also for higher order fluxes are obtained. These equations are of balance type, as it is postulated in extended thermodynamics. The resulting evolution equation for the diusion flux generalizes Ficks law.