One of the research direction of Horst Lippmann during his whole scientific career was devoted to the possibilities to explain complex material behavior by generalized continua models. A representative of such models is the Cosserat continuum. The basic idea of this model is the independence of translations and rotations (and by analogy, the independence of forces and moments). With the help of this model some additional effects in solid and fluid mechanics can be explained in a more satisfying manner. They are established in experiments, but not presented by the classical equations. In this paper the Cosserat-type theories of plates and shells are debated as a special application of the Cosserat theory.

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On generalized Cosserat-type theories of plates and shells: a short review and bibliography

Dynamic problems for metamaterials: Review of existing models and ideas for further research

It has been known since the pioneering works by Piola, Cosserat, Mindlin, Toupin, Eringen, Green, Rivlin and Germain that many micro-structural effects in mechanical systems can be still modeled by means of continuum theories. When needed, the displacement field must be complemented by additional kinematical descriptors, called sometimes microstructural fields. In this paper, a technologically important class of fibrous composite reinforcements is considered and their mechanical behavior is described at finite strains by means of a second-gradient, hyperelastic, orthotropic continuum theory which is obtained as the limit case of a micromorphic theory. Following Mindlin and Eringen, we consider a micromorphic continuum theory based on an enriched kinematics constituted by the displacement field u and a second-order tensor field ψ describing microscopic deformations. The governing equations in weak form are used to perform numerical simulations in which a bias extension test is reproduced. We show that second-gradient energy terms allow for an effective prediction of the onset of internal shear boundary layers which are transition zones between two different shear deformation modes. The existence of these boundary layers cannot be described by a simple first-gradient model, and its features are related to second-gradient material coefficients. The obtained numerical results, together with the available experimental evidences, allow us to estimate the order of magnitude of the introduced second-gradient coefficients by inverse approach. This justifies the need of a novel measurement campaign aimed to estimate the value of the introduced second-gradient parameters for a wide class of fibrous materials.

https://hal.archives-ouvertes.fr/hal-00838662/document

Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory

In this paper, the relaxed micromorphic model proposed in Ghiba et al. (Math Mech Solids, 2013), Neff et al. (Contin Mech Thermodyn, 2013) has been used to study wave propagation in unbounded continua with microstructure. By studying dispersion relations for the considered relaxed medium, we are able to disclose precise frequency ranges (band-gaps) for which propagation of waves cannot occur. These dispersion relations are strongly nonlinear so giving rise to a macroscopic dispersive behavior of the considered medium. We prove that the presence of band-gaps is related to a unique elastic coefficient, the so-called Cosserat couple modulus μ
 c
 , which is also responsible for the loss of symmetry of the Cauchy force stress tensor. This parameter can be seen as the trigger of a bifurcation phenomenon since the fact of slightly changing its value around a given threshold drastically changes the observed response of the material with respect to wave propagation. We finally show that band-gaps cannot be accounted for by classical micromorphic models as well as by Cosserat and second gradient ones. The potential fields of application of the proposed relaxed model are manifold, above all for what concerns the conception of new engineering materials to be used for vibration control and stealth technology.

Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps

A 1-dimensional second gradient damage continuum theory is presented within the framework of the variational approach. The action is intended to depend not only with respect to the first gradient of the displacement field and to a scalar damage field, but also to the field of the second gradient of the displacement. Constitutive prescriptions of the stiffness (the constitutive function in front of the squared first gradient term in the action functional) and of the microstructural material length (i.e., the square of the constitutive function in front of the squared second gradient term in the action functional) are prescribed in terms of the scalar damage parameter. On the one hand, as in many other works, the stiffness is prescribed to decrease as far as the damage increases. On the other hand, the microstructural material length is prescribed, in contrast to a certain part of the literature, to increase as far as the damage increases. This last assumption is due to the interpretation that a damage state induces a microstructure in the continuum and that such a microstructure is more important as far as the damage increases. At a given value of the damage parameter, the behavior is referred to second gradient linear elastic material. However, the damage evolution makes the model not only nonlinear but also inelastic. The second principle of thermodynamics can be considered by assuming that the scalar damage parameter does not decrease its value in the process of deformation, and this implies a dissipation for the elastic strain energy. It is finally remarked that damage initiation, in this second gradient continuum damage model, can be induced not only from a prescribed initial lack of stiffness, but also from an external concentrated double force or from suitable boundary conditions, and these last options have advantages that are discussed in the paper. For example, in this last case, it is possible to initiate the damage in the neighborhood of the boundaries, that is, the case in most of the empirical situations. Simple numerical simulations are also presented in order to show the exposed concepts.

A variational approach for a nonlinear 1-dimensional second gradient continuum damage model

Acceleration waves in nonlinear thermoelastic micropolar media are considered. We establish the kinematic and dynamic compatibility relations for a singular surface of order 2 in the media. An analogy to the Fresnel–Hadamard–Duhem theorem and an expression for the acoustic tensor are derived. The condition for acceleration wave’s propagation is formulated as an algebraic spectral problem. It is shown that the condition coincides with the strong ellipticity of equilibrium equations. As an example, a quadratic form for the specific free energy is considered and the solutions of the corresponding spectral problem are presented.

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Acceleration waves and ellipticity in thermoelastic micropolar media

The Cauchy problem for a simplified shallow elastic fluids model, one system of Temple’s type, is studied and a global weak solution is obtained by using the compensated compactness theorem coupled with the total variation estimates on the first and third Riemann invariants, where the second Riemann invariant is singular near the zero layer depth . This work extends in some sense the previous works, (Serre, 1987) and (Leveque and Temple, 1985), which provided the global existence of weak solutions for strictly hyperbolic system and (Heibig, 1994) for strictly hyperbolic system with smooth Riemann invariants.

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Global Solutions for a Simplified Shallow Elastic Fluids Model

The conditions for propagation of accelerating waves in a general nonlinear thermoelastic micropolar media are established. Deformation of micropolar media is described by the time-varying displacement vector r(t) and tensor of microrotation r(t) at each point. We call a surface S(t) an accelerating wave (or a singular surface for a solution of the dynamic problem for the medium) if the points are points of continuity of both r(t) and H(t) and their first spatial and time derivatives while the second spatial and time derivatives (acceleration) of r(t) and H(t) have jumps on S(t) (meaning that their one-sided limits at S(t) differ). So S(t) carries jumps in the acceleration fields as it propagat es through the body. In the thermomechanics of a micropolar continuum, similar propagating surfaces of singularities can exist for the fields of temperature, heat flux, etc. We establish the kinematic and dynamic compatibility relations for the singular surface S(t) in a nonlinear micropolar thermoelastic medium. An analog of Fresnel--Hadamard--Duhem theorem and an expression for the acoustic tensor are derived.

https://hal.archives-ouvertes.fr/docs/00/83/15/58/PDF/v41n2a10.pdf

On the propagation of acceleration waves in thermoelastic micropolar medias

We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. This system is a simplification of a recently proposed system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issue is that the considered system is such that every characteristic field is linearly degenerate. We show an explicit solution for the Cauchy problem with initial data in . We also study the Riemann problem for this system. Under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta-shock type solutions are established.

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Delta Shock Wave for the Suliciu Relaxation System

Existence theorems about bifurcation points of solutions for nonlinear operator equation in Banach spaces are proved. The sucient conditions of bifurcation of solutions of boundary-value problem for Vlasov-Maxwell system are obtained. The analytical method of Lyapunov-Schmidt-Trenogon is employed.

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Leonardo Rendon

Papers

Acceleration waves and ellipticity in thermoelastic micropolar media

Global Solutions for a Simplified Shallow Elastic Fluids Model

On the propagation of acceleration waves in thermoelastic micropolar medias

Delta Shock Wave for the Suliciu Relaxation System

Bifurcation points of nonlinear operators: existence theorems, asymptotics and application to the Vlasov-Maxwell system