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About this book

This book primarily focuses on rigorous mathematical formulation and treatment of static problems arising in continuum mechanics of solids at large or small strains, as well as their various evolutionary variants, including thermodynamics. As such, the theory of boundary- or initial-boundary-value problems for linear or quasilinear elliptic, parabolic or hyperbolic partial differential equations is the main underlying mathematical tool, along with the calculus of variations. Modern concepts of these disciplines as weak solutions, polyconvexity, quasiconvexity, nonsimple materials, materials with various rheologies or with internal variables are exploited.

This book is accompanied by exercises with solutions, and appendices briefly presenting the basic mathematical concepts and results needed. It serves as an advanced resource and introductory scientific monograph for undergraduate or PhD students in programs such as mathematical modeling, applied mathematics, computational continuum physics and engineering, as well as for professionals working in these fields.

Table of Contents


Static Problems


Chapter 1. Description of Deformable Stressed Bodies

One of our main objectives will be the description of a mechanical response of materials. A key ingredient here is a concept of deformations which identifies a new “shape” of the specimen. Often one can identify an original (reference) configuration of the material. While such configuration is suitable for mathematical considerations because it is fixed, physically relevant is the deformed configuration because there one can measure forces caused by deformations. These forces will manifest themselves in terms of stresses. The Cauchy stress vector reflecting internal material forces compensating for the external ones is introduced axiomatically and its existence is fundamental for continuum mechanics of solids. In order to map stress fields from the deformed to the reference configurations, we introduce the so-called Piola transform which formally allows for the same form of equilibrium equations in both configurations.
Martin Kružík, Tomáš Roubíček

Chapter 2. Elastic Materials

Looking at the force balance (1.2.6), we see that we have only d equations for \(d(d+3)/2\) unknowns (namely d components of y and \(d(d+1)/2\) components of \(T^y\)). Therefore, we complete (1.2.6) by material constitutive relations describing particular materials as far as its elastic properties concerns. Such constitutive relations should involve the stress tensor and the deformation gradient and may be in general implicit, cf. (Rajagopal, Appl Math, 48:279–319 (2003), [409]). Without excluding too many applications, we confine ourselves to the case when the stress is explicitly determined by the strain.
Martin Kružík, Tomáš Roubíček

Chapter 3. Polyconvex Materials: Existence of Energy-Minimizing Deformations

As already mentioned below Theorem , the derivation of Euler-Lagrange equations for the elasticity functional \({\mathcal E}\) is only formal and we cannot rely on the fact that minimizers of \({\mathcal E}\) satisfy these equations.
Martin Kružík, Tomáš Roubíček

Chapter 4. General Hyperelastic Materials: Existence/Nonexistence Results

As we have seen before, polyconvexity ensures existence of a minimizer to the energy functional. Moreover, polyconvex functions are relatively easy to construct and the theory developed allows us to incorporate important physical requirements into the model. On the other hand, polyconvexity is only a sufficient condition for weak lower semicontinuity and one can ask what is a necessary condition. In this chapter, we touch these problems, show a condition called (Morrey’s) quasiconvexity which is the sought necessary and sufficient condition ensuring lower semicontinuity in the weak topology. Moreover, some material models, however, do not allow for polyconvex bulk energy density. Beside St. Venant-Kirchhoff material (2.​3.​33), another prominent example are materials exhibiting clear tendency for creating a microstructure due to purely mechanical reasons as shape memory material, for instance. These materials posses non-quasiconvex stored energy density but if we include an interface energy term for each phase of the material we obtain the existence result as well.
Martin Kružík, Tomáš Roubíček

Chapter 5. Linearized Elasticity

Rather special, but anyhow frequently occurred situations in many engineering applications exhibit deformations whose gradient is relatively very close to identity. In other words, displacement gradient is very small and often even the displacement itself is relatively small. It allows us with a reasonable accuracy to neglect higher order terms and simplified a lot of aspects related to geometrical nonlinearities very substantially. It also facilitates computational algorithms substantially, and there is no wonder that most engineering computations in solid mechanics are just based on such hypotheses. As we will see, linearized elasticity provides us with unique solutions and strongly relies on convexity assumptions.
Martin Kružík, Tomáš Roubíček

Evolution Problems


Chapter 6. Linear Rheological Models at Small Strains

Rheology is a discipline studying relaxation processes in materials and, related to them, the way how materials dissipate energy. Substantial dissipation may typically arise in sudden change of external load (in solids) or, conversely, in a long lasting constant load (like in fluids), or in combination of both. The distinction between solids and fluids is, from the purely mechanical viewpoint, not much lucid. For example, in geophysics, rocks are considered as fluids because they cannot permanently withstand a constant shear load. But they manifest its fluidic character only in observation time scale of millions of years, while in the man’s observation time scale of years, they are well solid, as we all know from our everyday experience. Actually, this paradox is counted in a so-called Deborah number (sometimes denoted by the Hebrew letter ‘daleth’, \(\daleth \)) defined as a ratio between the relaxation time and the observation time. The difference between solids and fluids is thus reflected by this number: large \(\daleth \gg 1\) means that the medium can be well understood as solid while small \(0<\daleth \ll 1\) indicates rather a fluid.
Martin Kružík, Tomáš Roubíček

Chapter 7. Nonlinear Materials with Internal Variables at Small Strains

The concept of internal variables (sometimes called also internal parameters as in [228, 378]) dates back to P. Duhem [160], to the Nobel prize winner P.W. Bridgman [84], and to C. Eckart [164]. Although it may seem a bit artificial, this concept opens extremely fruitful possibilities for modeling in continuum mechanics and often has a direct motivation and mechanical or other physical interpretation.
Martin Kružík, Tomáš Roubíček

Chapter 8. Thermodynamics of Selected Materials and Processes

It is a basic principle (in fact, assumed rather by definition) that the total energy in a closed system conserves. Dissipation of mechanical energy with which we dealt in previous Chaps. 67 then gives rise, beside possible irreversible changes of internal structure of materials contributing to the stored energy, also to a heat (and entropy) production. Except infinitesimally slow processes or very small bodies which are thermally stabilized with the environment, this dissipation of mechanical (or other, e.g. chemical) energy results to varying temperature and the heat transfer throughout the solid-body continuum. In turn, mechanical properties usually depend (sometimes even very substantially) on temperature both as well as stored and dissipation mechanisms concerns.
Martin Kružík, Tomáš Roubíček

Chapter 9. Evolution at Finite Strains

This book has begun with problems at finite or large strains in Chaps. 2–4; recall the Convention 1.1.1 on p. 6. Thus it is expectable to close it with such problems. In dynamical problems at large deformation and finite/large strains, many serious difficulties are pronounced and only very few results are at disposal, in contrast to the static situations. Essentially, the absence of a clearly identifiable reference configuration and of a linear geometry that would allow for defining a time derivative in a unique way is the source of these problems.
Martin Kružík, Tomáš Roubíček


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