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## Über dieses Buch

This collection on „Mechanics of Generalized Continua - from Micromechanical Basics to Engineering Applications“ brings together leading scientists in this field from France, Russian Federation, and Germany. The attention in this publication is be focussed on the most recent research items, i.e., - new models, - application of well-known models to new problems, - micro-macro aspects, - computational effort, - possibilities to identify the constitutive equations, and - old problems with incorrect or non-satisfying solutions based on the classical continua assumptions.

## Inhaltsverzeichnis

### Chapter 1. A Historical Perspective of Generalized Continuum Mechanics

In a period of forty years the author has had the opportunity to work, or to entertain friendly connections, with many actors of the scene of generalized continuum mechanics (GCM). This training and knowledge here is used to the benefit of the readers as an overview of this scene with the aim to delineate further avenues of development within the framework of the trilateral seminar held in Wittenberg (2010). Starting essentially with Pierre Duhem and the Cosserat brothers, this specialized, albeit vast, field of continuum mechanics has developed by successive abandonments of the working hypotheses at the basis of standard continuum mechanics, that mechanics masterly devised by Euler and Cauchy and some of their successors in the 19th century (Piola, Kirchhoff, etc.). In the present survey we briefly analyze successive steps such as the introduction of nonsymmetric stresses, couple stresses, internal degrees of freedom and microstructure, the introduction of strain gradient theories, and material inhomogeneities with a length scale, nonlocality of the weak and strong types, the loss of Euclidean geometry to describe the material manifold, and finally the loss of classical differentiability of basic operations as can occur in a deformable fractal material object.
Gérard A. Maugin

### Chapter 2. Micropolar Shells as Two-dimensional Generalized Continua Models

Using the direct approach the basic relations of the nonlinear micropolar shell theory are considered. Within the framework of this theory the shell can be considered as a deformable surface with attached three unit orthogonal vectors, so-called directors. In other words the micropolar shell is a two-dimensional (2D) Cosserat continuum or micropolar continuum. Each point of the micropolar shell has three translational and three rotational degrees of freedom as in the rigid body dynamics. In this theory the rotations are kinematically independent on translations. The interaction between of any two parts of the shell is described by the forces and moments only. So at the shell boundary six boundary conditions have to be given. In contrast to Kirchhoff-Love or Reissner’s models of shells the drilling moment acting on the shell surface can be taken into account. In the paper we derive the equilibrium equations of the shell theory using the principle of virtual work. The strain measures are introduced on the base of the principle of frame indifference. The boundary-value static and dynamic problems are formulated in Lagrangian and Eulerian coordinates. In addition, some variational principles are presented. For the general constitutive equations we formulate some constitutive restrictions, for example, the Coleman-Noll inequality, the Hadamard inequality, etc. Finally, we discuss the equilibrium of shells made of materials undergoing phase transformations, such as martensitic transformations, and formulate the compatibility conditions on the phase interface.
Holm Altenbach, Victor A. Eremeyev, Leonid P. Lebedev

### Chapter 3. Structural Dynamics and Generalized Continua

This paper deals with the dynamic behavior of reticulated beams made of the periodic repetition of symmetric unbraced frames. Such archetypical cells can present a high contrast between shear and compression deformability, conversely to “massive” media. This opens the possibility of enriched local kinematics involving phenomena of global rotation, inner deformation or inner resonance, according to studied configuration and frequency range. Firstly, the existence of these atypical behaviors is established theoretically through the homogenization method of periodic discrete media. Then, the results are adapted to buildings and confirmed with a numerical example.
Céline Chesnais, Claude Boutin, Stéphane Hans

### Chapter 4. A Bending-gradient Theory for Thick Laminated Plates Homogenization

This work presents a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Love-Kirchhoff theory, to which six components are added representing the gradient of the bending moment. The Bending-gradient theory is an extension to arbitrary multilayered plates of the Reissner-Mindlin theory which appears as a special case when the plate is homogeneous. The new theory is applied to multilayered plates and its predictions are compared to full 3D Pagano’s exact solutions and other approaches. It gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/H goes to infinity.
Arthur Lebée, Karam Sab

### Chapter 5. Internal Length Scale Effects on the Local and Overall Behavior of Polycrystals

A breakthrough in the general hypothesis of spatially homogeneous intragranular fields accepted in mean field approaches based on the classic Eshelby’s inclusion problem (self-consistent schemes, etc.) is proposed. Instead of considering uniform intra-granular plastic strains as usually prescribed in mean field approaches, intragranular slip patterns are modeled in single slip configurations both by distributions of coaxial circular glide loops and by distributions of flat ellipsoids (also called oblate spheroids). Both types of modeling assume slip configurations constrained by spherical grain boundaries, and, mechanical interactions between slip bands are taken into account (for mechanical fields and free energy). It is then found that intra-granular mechanical fields strongly depend on the grain size and the slip band spacing. In addition, in the case of glide loops, the modeling is able to capture different behaviors between near grain boundary regions and grain interiors. In particular, a grain boundary layer with strong gradients of internal stresses (and lattice rotations) is found. These results are confirmed quantitatively by EBSD measurements carried out with orientation imaging mapping (OIM) on deformed Ni polycrystals and on specific grains undergoing quasi single slip. Furthermore, as a result of the computation of the elastic energy, an average back-stress over the grain (in the case of loops) or over slip bands (in the case of oblate spheroids) can be derived so that it is possible to define new interaction laws for polycrystal’s behavior which are naturally dependent on grain size and slip band spacing.
Stéphane Berbenni

### Chapter 6. Formulations of Strain Gradient Plasticity

In the literature, different proposals for a strain gradient plasticity theory exist. So there is still a debate on the formulation of strain gradient plasticity models used for predicting size effects in the plastic deformation of materials. Three such formulations from the literature are discussed in this work. The pros and the cons are pointed out at the light of the original solution of a boundary value problem that considers the shear deformation of a periodic laminate microstructure.
Samuel Forest, Albrecht Bertram

### Chapter 7. On one Model of Generalized Continuum and its Thermodynamical Interpretation

We consider the mechanical model of a two-component medium whose first component is a classical continuum and the other one is a continuum having only rotational degrees of freedom. We show that the proposed model can be used for the description of thermal and dissipative phenomena. It is the presence of additional rotational degrees of freedom and, accordingly, additional inertia and elastic characteristics which can be interpreted as thermodynamical material parameters that distinguish the proposed model among other continuum models. In special cases the mathematical description of the proposed model is proved to reduce to the well-known equations such as the heat conduction, the self-diffusion and the coupled thermoelastic equations. The mathematical description of the proposed mechanical model includes not only the classical formulation of the coupled problem of thermoelasticity but also the formulation of the coupled problem of thermoelasticity with the hyperbolic type heat conduction equation. In the context of the introduced theory we consider the original model of internal damping.
Elena A. Ivanova

### Chapter 8. Micromechanical Bases of Superelastic Behavior of Certain Biopolymers

This work presents a new constitutive theory aiming to describe the truly exceptional, only little known and almost completely uncharacterized thermo-mechanical properties of the whelk egg capsule biopolymer (WECB) which has been recently reported in the literature. The mechanical model is based on the concept of generalized continua. It includes familiar damage-type models and pseudo-elastic models for stress softening (the Mullins effect) in elastomers (natural and synthetic rubbers) and soft tissues. However, the mechanical behavior of WECB is in many aspects very different from the behavior of other elastomeric materials and these differences have been accounted for in the developed constitutive model.
Rasa Kazakevičiūtė-Makovska, Holger Steeb

### Chapter 9. Construction of Micropolar Continua from the Homogenization of Repetitive Planar Lattices

The derivation of the effective mechanical properties of planar lattices made of articulated bars is herewith investigated, relying on the asymptotic homogenization technique to get closed form expressions of the equivalent properties versus the geometrical and mechanical microparameters. Considering lattice microrotations as additional degrees of freedom at both scales, micropolar equivalent continua are constructed from discrete lattices made of a repetitive unit cell, from an extension of the asymptotic homogenization technique. We will show that it is necessary to solve on two different orders a linear system of equations giving the kinematic variables, at both the first and second order. The effective strain and effective curvature appear respectively as the first and second order strain variables. In the case of a centrosymmetric unit cell, there is no coupling between couple stresses and strains nor between stress and curvature. The unknown kinematic variables are determined by solving the translational and rotational equilibrium for the whole lattice. This in turn leads to the expression of the stress vector and couple stress vector, allowing to construct the Cauchy stress and couple stress tensors. The homogenized behavior of the tetragonal and hexagonal lattices is determined in terms of homogenized micropolar moduli.
Francisco Dos Reis, Jean-François Ganghoffer

### Chapter 10. Nonlinear Waves in the Cosserat Continuum with Constrained Rotation

The nonlinear viscoelastic micropolar medium with constrained rotation (the Cosserat pseudo-continuum) is considered. Using the method of bound normal waves, the original nonlinear system describing the dynamics of the medium is transferred to a system of evolutionary equations. It is shown that these evolutionary equations are four nonlinear partial differential equations two of which are the Burgers equations and the other two are the modified Korteweg-de Vries (mKdV) equations. The paper presents the results of the numerical study of nonlinear viscoelastic wave evolution.
Vladimir I. Erofeev, Aleksandr I. Zemlyanukhin, Vladimir M. Catson, Sergey F. Sheshenin

### Chapter 11. Wave Propagation in Quasi-continuous Linear Chains with Self-similar Harmonic Interactions: Towards a Fractal Mechanics

Many systems in nature have arborescent and bifurcated structures such as trees, fern, snails, lungs, the blood vessel system, but also porous materials etc. look self-similar over a wide range of scales. Which are the mechanical and dynamic properties that evolution has optimized by choosing self-similarity so often as an inherent material symmetry? How can we describe the mechanics of self-similar structures in the static and dynamic framework? In order to analyze such material systems we construct self-similar functions and linear operators such as a self-similar variant of the Laplacian and of the D’Alembertian wave operator. The obtained self-similar linear wave equation describes the dynamics of a quasi-continuous linear chain of infinite length with a spatially self-similar distribution of nonlocal inter-particle springs. The dispersion relation of this system is obtained by the negative eigenvalues of the self-similar Laplacian and has the form of a Weierstrass-Mandelbrot function which exhibits self-similar and fractal features. We deduce a continuum approximation that links the self-similar Laplacian to fractional integrals which also yields in the low-frequency regime a power law scaling for the oscillator density with strictly positive exponent leading to a vanishing oscillator density at frequency zero. We suggest that this power law scaling is a characteristic and universal feature of self-similar systems with complexity well beyond of our present model. For more details we refer to our recent paper [7].
Thomas M. Michelitsch, Gérard A. Maugin, Franck C. G. A. Nicolleau, Andrzej F. Nowakowski, Shahram Derogar

### Chapter 12. Nonlinear Dynamic Processes in Media with Internal Structure

The generation of the bell-shaped localized defects in the lattice is studied numerically using essentially nonlinear proper structural model that describes coupling with macro-strains. It is macro-strain localized wave that provides defects generation, the parameters of this wave are very important for localization or delocalization of the variations in the structure of the lattice.
Alexey V. Porubov, Boris R. Andrievsky, Eron L. Aero

### Chapter 13. Buckling of Elastic Composite Rods made of Micropolar Material Subjected to Combined Loads

In the present paper, the stability of a nonlinearly elastic rod with a composite structure is analyzed. It is assumed that the interior (core) of the rod is made of micropolar material, while the behavior of the exterior (coating) is investigated in the framework of a classic non-polar continuum model. The problem is studied for a case of axial compression of a rod under external hydrostatic pressure. Using the linearization method in a vicinity of the basic state, the neutral equilibrium equations have been derived, which describe the perturbed state of a composite rod. By solving these equations numerically for some specific materials, the critical curves and corresponding buckling modes have been found, and the stability regions have been constructed in the planes of loading parameters (relative compression and external pressure). An extensive analysis has been carried out for the influence of a size effect and coating properties on the buckling of elastic composite rod made of micropolar material subject to combined loads.
Denis Sheydakov

### Chapter 14. Theory of Isolated and Continuously Distributed Disclinations and Dislocations in Micropolar Media

The paper deals with nonlinear theory of defects like dislocations and disclinations either isolated or distributed with a certain density in an elastic medium with internal rotational degrees of freedom and couple stresses. The general theory is illustrated by finding the solution of problems of internal stresses induced in an elastic disc with an isolated wedge disclination as well as the distribution of such disclinations. Some results concerning the influence of the microstructure on the possibility of the hole formation along the dislocation line are presented.
Mikhail I. Karyakin, Leonid M. Zubov

### Chapter 15. On the Form-Invariance of Lagrangian Function for Higher Gradient Continuum

In this work, we consider an elastic continuum of third grade. For the sake of simplicity, we do not consider kinetic energy in the Lagrangian function. In this work, we reformulate the problem by considering Lagrangian function depending on the metric tensor $${\mathbf g}$$ and on the affine connection $$\nabla$$ assumed to be compatible with the metric $${\mathbf g}$$, and rewrite the Lagrangian function as $${\fancyscript{L}} ({\mathbf g}, \nabla, \nabla^2).$$ Following the method of Lovelock and Rund, we apply the form-invariance requirement to the Lagrangian $${\fancyscript{L}}.$$ It is shown that the arguments of the function $${\fancyscript{L}}$$ are necessarily the torsion $$\aleph$$ and/or the curvature $$\Re$$ associated with the connection, in addition to the metric $${\mathbf g}.$$ The following results are obtained: (1) $${\fancyscript{L}} ( {\mathbf g}, \nabla )$$ is form-invariant if and only if $${\fancyscript{L}} ( {\mathbf g}, \aleph );$$ (2) $${\fancyscript{L}} ( {\mathbf g}, \nabla^2 )$$ is form-invariant if and only if $${\fancyscript{L}} ( {\mathbf g}, \Re );$$ and (3) $${\fancyscript{L}} ( {\mathbf g}, \nabla, \nabla^2 )$$ is form-invariant if and only if $${\fancyscript{L}} ( {\mathbf g}, \aleph, \Re ).$$
Nirmal Antonio Tamarasselvame, Lalaonirina R. Rakotomanana

### Chapter 16. Cahn-Hilliard Generalized Diffusion Modeling Using the Natural Element Method

In this work, we present an application of two versions of the natural element method (NEM) to the Cahn-Hilliard equation. The Cahn-Hilliard equation is a nonlinear fourth order partial differential equation, describing phase separation of binary mixtures. Numerical solutions requires either a two field formulation with C 0 continuous shape functions or a higher order C 1 continuous approximations to solve the fourth order equation directly. Here, the C 1 NEM, based on Farin’s interpolant is used for the direct treatment of the second order derivatives, occurring in the weak form of the partial differential equation. Additionally, the classical C 0 continuous Sibson interpolant is applied to a reformulation of the equation in terms of two coupled second order equations. It is demonstrated that both methods provide similar results, however the C 1 continuous version needs fewer degrees of freedom to capture the contour of the phase boundaries.
Paul Fischer, Amirtham Rajagopal, Ellen Kuhl, Paul Steinmann

### Chapter 17. Constitutive Models of Mechanical Behavior of Media with Stress State Dependent Material Properties

The behavior of heterogeneous materials is studied. The dependence of the effective elastic properties of micro-heterogeneous materials on the loading conditions are analyzed and corresponding mathematical methods for the description of the observed effects are proposed. The constitutive relations of the theory of elasticity for isotropic solids with stress state dependent deformation properties are considered. The possible approach to the formulation of the constitutive relations for the elastic anisotropic solids that elastic properties depend on the stress state type is considered, and the corresponding constitutive relations are proposed. The method for the determination of material’s functions on the base of experimental data is proposed. The quite satisfactory correspondence between the theoretical results and experimental data is shown.
Evgeny V. Lomakin
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