Institute of Mathematics of the AV CR

Prague, Czech Republic

Institute of Mathematics of the AV CR

Prague, Czech Republic

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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.


Lucchesi M.,University of Florence | Pintucchi B.,University of Florence | Silhavy M.,Institute of Mathematics of the AV CR | Zani N.,University of Florence
Continuum Mechanics and Thermodynamics | Year: 2015

In this paper, we consider the longitudinal and transversal vibrations of the masonry beams and arches. The basic motivation is the seismic vulnerability analysis of masonry structures that can be modeled as monodimensional elements. The Euler–Bernoulli hypothesis is employed for the system of forces in the beam. The axial force and the bending moment are assumed to consist of the elastic and viscous parts. The elastic part is described by the no-tension material, i.e., the material with no resistance to tension and which accounts for the cases of limitless, as well as bounded compressive strength. The adaptation of this material to beams has been developed in Orlandi (Analisi non lineare di strutture ad arco in muratura. Thesis, 1999) and Zani (Eur J Mech A/Solids 23:467–484, 2004). The viscous part amounts to the Kelvin–Voigt damping depending linearly on the time derivatives of the linearized strain and curvature. The dynamical equations are formulated, and a mathematical analysis of them is presented. Specifically, following Gajewski et al. (Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin, 1974), the theorems of existence, uniqueness and regularity of the solution of the dynamical equations are recapitulated and specialized for our purposes, to support the numerical analysis applied previously in Lucchesi and Pintucchi (Eur J Mech A/Solids 26:88–105, 2007). As usual, for that the Galerkin method has been used. As an illustration, two numerical examples (slender masonry tower and masonry arch) are presented in this paper with the applied forces corresponding to the acceleration in the earthquake in Emilia Romagna in May 29, 2012. © 2014, Springer-Verlag Berlin Heidelberg.


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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