Institute of Mathematics of the AV CR

Prague, Czech Republic

Institute of Mathematics of the AV CR

Prague, Czech Republic
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Padovani C.,National Research Council Italy | Silhavy M.,Institute of Mathematics of the AV CR
Mathematics and Mechanics of Solids | Year: 2017

The stress-strain relation of a no-tension material, used to model masonry structures, is determined by the nonlinear projection of the strain tensor onto the image of the convex cone of negative-semidefinite stresses under the fourth-order tensor of elastic compliances. We prove that the stress-strain relation is indefinitely differentiable on an open dense subset O of the set of all strains. The set O consists of four open connected regions determined by the rank k = 0, 1, 2, 3 of the resulting stress. Further, an equation for the derivative of the stress-strain relation is derived. This equation cannot be solved explicitly in the case of a material of general symmetry, but it is shown that for an isotropic material this leads to the derivative established earlier by Lucchesi et al. (Int J Solid Struct 1996; 33: 1961-1994 and Masonry constructions: Mechanical models and numerical applications. Berlin: Springer, 2008) by different means. For a material of general symmetry, when the tensor of elasticities does not have the representation known in the isotropic case, only general steps leading to the evaluation of the derivative are described. © SAGE Publications.

Silhavy M.,Institute of Mathematics of the AV CR
Mathematics and Mechanics of Solids | Year: 2017

Peridynamics is a non-local continuum mechanics that replaces the differential operator embodied by the stress term div S in Cauchy's equation of motion by a non-local force functional L to take into account long-range forces. The resulting equation of motion reads ρü= Ⅎu + b (u = displacement, b = body force, ρ = density). If the characteristic length δ of the interparticle interaction approaches 0, the operator L admits an expansion in δ that, for a linear isotropic material, reads Ⅎu = (λ + μ) ∇ div u + μ Δ u + δ2 Θ2 · ∇4u + δ4Θ3 · ∇6u + ⋯, where λ and μ are the Lamé moduli of the classical elasticity, and the remaining higher-order corrections contain products of the type T s u: = Θ s · ∇ 2 s u of even-order gradients ∇2s u (i.e., the collections of all partial derivatives of u of order 2s) and constant coefficients Θs collectively forming a tensor of order 2s. Symmetry arguments show that the terms T s u have the form δ2s-2 (λs + μs) Δs-1 ∇ div u + δ2s-2 μs Δsu, where λs and μs are scalar constants. This article explicitly determines λ s and μ s in terms of the properties of the material (i.e., of the operator L) in all dimensions n (typically, n = 1, 2 or 3). © SAGE Publications.

Lucchesi M.,University of Florence | Silhavy M.,Institute of Mathematics of the AV CR | Zani N.,University of Florence
Annals of Solid and Structural Mechanics | Year: 2011

In this paper we consider masonry bodies undergoing loads that can be represented by vector valued measures, and prove a result which is an appropriate formulation to this context of the static theorem of the limit analysis. As applications, we study the equilibrium of panels that are subjected both to distributed loads and concentrated forces, and determine equilibrated tensor valued measures. Then, by using an integration procedure for parametric measures, we explicitly calculate equilibrated stress fields that are represented by integrable functions. The obtained solutions are discussed. © 2011 Springer-Verlag.

Lucchesi M.,University of Florence | Silhavy M.,Institute of Mathematics of the AV CR | Zani N.,University of Florence
Continuum Mechanics and Thermodynamics | Year: 2014

The paper deals with membrane reinforced bodies with the membrane treated as a two-dimensional surface with concentrated material properties. The bulk response of the matrix is treated separately in two cases: (a) as a coercive nonlinear material with convex stored energy function expressed in the small strain tensor, and (b) as a no-tension material (where the coercivity assumption is not satisfied). The membrane response is assumed to be nonlinear in the surface strain tensor. For the nonlinear bulk response in Case (a), the existence of states of minimum energy is proved. Under suitable growth conditions, the equilibrium states are proved to be exactly states of minimum energy. Then, under appropriate invertibility condition of the stress function, the principle of minimum complementary energy is proved for equilibrium states. For the no-tension material in Case (b), the principle of minimum complementary energy (in the absence of the invertibility assumption) is proved. Also, a theorem is proved stating that the total energy of the system is bounded from below if and only if the loads can be equilibrated by a stress field that is statically admissible and the bulk stress is negative semidefinite. Two examples are given. In the first, we consider the elastic semi-infinite plate with attached stiffener on the boundary (Melan's problem). In the second example, we present a stress solution for a rectangular panel with membrane occupying the main diagonal plane. © 2013 Springer-Verlag Berlin Heidelberg.

Lucchesi M.,University of Florence | Silhavy M.,Institute of Mathematics of the AV CR | Zani N.,University of Florence
Continuum Mechanics and Thermodynamics | Year: 2013

An approach is outlined to the equilibrium in fiber-reinforced materials in which the fibers are modeled as curves or lines with concentrated material properties. The system of forces representing the interaction of the fibers with the bulk matter is analyzed, and equilibrium of forces is derived from global laws. The displacements of the bulk matter are assumed to have continuous extension to the fibers. This forces the set of admissible deformations superquadratically integrable. This in turn forces the energy of the bulk of superquadratic growth. The material of the bulk matrix therefore cannot be linearly elastic. The energy of fibers can have a slower growth and can be quadratic. A formal set of assumptions is given under which an equilibrium state of minimum energy exists in the given external conditions. A weak form of equilibrium equations is derived for this equilibrium state. An explicitly calculable axisymmetric example is presented with an isotropic and quadratic energy of the matrix (linear elasticity) and linearly stretchable fiber. Since the superquadratic growth assumption is not satisfied, some peculiar features of the solution arise, such as the infinite limit of the radial displacement near the fiber. Nevertheless, from the obtained solution, we can compute the normal force in the fiber and the shear stress at the interface. © 2012 Springer-Verlag.

Silhavy M.,Institute of Mathematics of the AV CR
Journal of Elasticity | Year: 2016

For a given polyconvex function W, among all associated convex functions g of minors there exists the largest one; this function inherits all symmetry properties of W. For a given associated (not necessarily the largest) function g, one can still find an associated (possibly not the largest) function with the symmetry of W. This function is constructed by averaging of symmetry conjugated functions over the symmetry group of W using Haar’s measure. It follows that if a symmetric polyconvex function W has class k=0,…,∞ associated function, then the averaging produces a symmetric associated function that is class k as well. © 2015, Springer Science+Business Media Dordrecht.

Silhavy M.,Institute of Mathematics of the AV CR
Interfaces and Free Boundaries | Year: 2011

The sharp interface limit of a diffuse interface theory of phase transitions is considered in static situations. The diffuse interface model is of the Allen-Cahn type with deformation, with a parameter ε measuring the width of the interface. Equilibrium states of a given elongation and a given interface width are considered and the asymptotics as ε → 0 of the equilibrium energy is determined. The interface energy is defined as the excess energy over the corresponding two-phase state with a sharp interface without the interface energy. It is shown that to within the term of order o(ε) the interface energy is equal to σε where the coefficient σ is given by a new formula that involves the mechanical contribution to the total energy. Also the corresponding equilibrium states are determined and shown to converge to a sharp interface state as ε → 0. © European Mathematical Society 2011.

Silhavy M.,Institute of Mathematics of the AV CR
Journal of Elasticity | Year: 2011

The paper proves the existence of equilibrium two phase states with elastic solid bulk phases and deformation dependent interfacial energy. The states are pairs (y,E) consisting of the deformation y on the body and the region E occupied by one of the phases in the reference configuration. The bulk energies of the two phases are polyconvex functions representing two wells of the substance. The interfacial energy is interface polyconvex. The last notion is introduced and discussed below, together with the interface quasiconvexity and interface null Lagrangians. The constitutive theory and equilibrium theory of the interface are discussed in detail under appropriate smoothness hypotheses. Various forms of the interfacial stress relations for the standard and configurational (Eshelby) interfacial stresses are established. The equilibrium equations are derived by a variational argument emphasizing the roles of outer and inner variations. © The Author(s) 2011.

Silhavy M.,Institute of Mathematics of the AV CR
Technische Mechanik | Year: 2015

A simple proof is given of the characterization of the convexity of the function C ↦ h(det C) on positive definite symmetric matrices due to Lehmich et al. (2014). The proof uses the classical characterization of convex functions depending on a symmetric matrix through its eigenvalues due to Davis (1957). © 2015, Magdeburger Verein fur Technische Mechanik e. V. All rights reserved.

Silhavy M.,Institute of Mathematics of the AV CR
Mathematics and Mechanics of Solids | Year: 2014

The equilibrium equations of no-tension (masonry-like) bodies are analyzed. Unlike the existing proofs of the existence of the solution by Anzellotti or Giaquinta and Giusti, the present proof does not employ the uniform safe load condition. It is based on the assumption of the absence of a suitably defined collapse mechanisms. The collapse mechanism belongs to a generalized space BD(cl Ω) of displacements of bounded deformation on the closure cl Ω of the body Ω. This generalized displacement can have a jump discontinuity on the boundary of the body and the generalized strain is a measure on the closure of the body (instead of the standard interpretation as a measure supported by the interior). The equilibrium solution, however, belongs to the classical space of displacements of bounded deformation BD(Ω). © The Author(s) 2013.

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