Continuum damage mechanics: combining thermodynamics with a thoughtful characterization of the microstructure

We formulate a macroscopic description of the mechanics of damaged materials. To represent the microstructure, the distribution of crack sizes is captured by way of the Minkowski functionals, or so-called quermass integrals, while a second-rank tensor is used to describe the average orientation of the cracks. A two phase-type approach is adopted to distinguish elastically strained material from unstrained regions in the wake of the cracks. Using nonequilibrium thermodynamic techniques, the driving force for the growth of the microcracks is naturally identified. In particular, Griffith's law is generalized to assemblies of polydisperse crack sizes. Due to the detailed characterization of the microstructure, we are also able to account for the plastic zones at the rims of the cracks that are known to hamper the crack growth, and to discuss possible forms of the damage parameter. The presented approach separates in a transparent fashion the incorporation of fundamental thermodynamic and mechanic principles on one hand, from the specification of the material and details of the crack formation and growth on the other han

Ideal contribution to the macroscopic quasiequilibrium entropy of anisotropic fluids

The Landau-de Gennes free energy plays a central role in the macroscopic theory of anisotropic fluids. Here, the ideal, entropic contribution to this free energy—that is always present in these systems, irrespectively of the detailed form of interactions or applied fields—is derived within the quasiequilibrium ensemble and successfully tested. An explicit and compact form of the macroscopic, ideal entropy is derived. This entropy is nonpolynomial in the order parameter, diverging logarithmically near the fully oriented state and therefore restricting the order parameter to physical admissible values. As an application, it is shown that the isotropic-nematic transition within the Maier-Saupe model is described in a simple and very accurate manner

Microscopically derived free energy of dislocations

The dynamics of large amounts of dislocations is the governing mechanism in metal plasticity. The free energy of a continuous dislocation density profile plays a crucial role in the description of the dynamics of dislocations, as free energy derivatives act as the driving forces of dislocation dynamics. In this contribution, an explicit expression for the free energy of straight and parallel dislocations with different Burgers vectors is derived. The free energy is determined using systematic coarse-graining techniques from statistical mechanics. The starting point of the derivation is the grand-canonical partition function derived in an earlier work, in which we accounted for the finite system size, discrete glide planes and multiple slip systems. In this paper, the explicit free energy functional of the dislocation density is calculated and has, to the best of our knowledge, not been derived before in the present form. The free energy consists of a mean-field elastic contribution and a local defect energy, that can be split into a statistical and a many-body contribution. These depend on the density of positive and negative dislocations on each slip system separately, instead of GND-based quantities only. Consequently, a crystal plasticity model based on the here obtained free energy, should account for both statistically stored and geometrically necessary dislocations

Crystal shapes and crystallization in continuum modeling

A crystallization model appropriate for application in continuum modeling of complex processes is presented. As an extension to the previously developed Schneider equations [ W. Schneider, A. Köppel, and J. Berger, "Non-isothermal crystallization of polymers," Int. Polym. Proc. 2, 151 (1988) ], the model presented here allows one to account for the growth of crystals of various shapes and to distinguish between one-, two-, and three-dimensional growth, e.g., between rod-like, plate-like, and sphere-like growth. It is explained how a priori knowledge of the shape and growth processes is to be built into the model in a compact form and how experimental data can be used in conjunction with the dynamic model to determine its growth parameters. The model is capable of treating transient processing conditions and permits their straightforward implementation. By using thermodynamic methods, the intimate relation between the crystal shape and the driving forces for phase change is highlighted. All these capabilities and the versatility of the method are made possible by the consistent use of four structural variables to describe the crystal shape and number density, irrespective of the growth dimensionality

Backbone of conductivity in two-dimensional metal-insulator composites

In percolation theory, the backbone is defined by chopping off dangling ends from the percolating cluster. For structures with high degree of spatial correlation, as they are typical for porous thin films, trimming of the full structure to reveal the part determining the electrical conductivity is more subtle than the classic definition of the backbone. To expand the applicability of the concept, we present a purely geometric definition for the backbone of a two-dimensional percolating cluster. It is based on a sequence of image analysis operations defining the backbone in terms of an image filter. The change of both area fraction and effective conductivity induced by applying the backbone filter to various binary images and a two-parameter family of sets is assessed by numerical means. It is found that the backbone filter simplifies the geometry of complex microstructures significantly and at the same time preserves their electrical DC behavior. We conclude that the backbone will be useful as a first ingredient for a geometric estimator of the effective conductivity of metal-insulator composites

Dynamic mean-field models from a nonequilibrium thermodynamics perspective

Complicated dynamic models are often approximated by introducing mean-field approximations and closures. The focus here is on examining such mean-field models using nonequilibrium thermodynamics. Two illustrative examples are studied in terms of the double-generator general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework. First, it is shown that a model for the coil-stretch transition of long chains in strong elongation flows as proposed by de Gennes is thermodynamically admissible. In the second example, we study a Gaussian approximation, which is used to simplify the effect of hydrodynamic interactions in polymer solutions. This approximation, which is known to be in conflict with the fluctuation-dissipation theorem, is identified as defective directly when formulated in the thermodynamic formalism