Random homogenization analysis in linear elasticity based on analytical bounds and estimates

In this work, random homogenization analysis of heterogeneous materials is addressed in the context of elasticity, where the randomness and correlation of components’ properties are fully considered and random effective properties together with their correlation for the two-phase heterogeneous material are then sought. Based on the analytical results of homogenization in linear elasticity, when the randomness of bulk and shear moduli, the volume fraction of each constituent material and correlation among random variables are considered simultaneously, formulas of random mean values and mean square deviations of analytical bounds and estimates are derived from Random Factor Method. Results from the Random Factor Method and the Monte-Carlo Method are compared with each other through numerical examples, and impacts of randomness and correlation of random variables on the random homogenization results are inspected by two methods. Moreover, the correlation coefficients of random effective properties are obtained by the Monte-Carlo Method. The Random Factor Method is found to deliver rapid results with comparable accuracy to the Monte-Carlo approach.     

Energetic bounds in finite elasticity

Summary We study the conditions under which the internal work of deformation in an elastic isotropic body in finite deformations may be bounded by results obtained from a suitably defined linear infinitesimal problem. The values of the constants appearing in the principal inequalities are calculated and discussed for a certain class of extensional deformations.     

Homogenization of linear elastic shells

Homogenization techniques were used by Duvaut (1976,1978) in asymptotic analyse of 3-dimensional periodic continuum problems and periodic von Kármán plates.In this paper we homogenize Budiansky-Sanders linear, elastic shells with material parameters rapidly oscillating on the shell surface. We obtain a homogenized shell model which is elliptic and depends on explicitly calculated effective material parameters. We show that the solution of the periodic shell model converges weakly to the solution of the homogenized model when the period tends to zero.     

Internal constraints in linear elasticity

Sufficient conditions are obtained for continuous dependence of solutions of boundary value problems of linear elasticity on internal constraints. Arbitrary hyperelastic materials with arbitrary (linear) internal constraints are included. In particular the results of Bramble and Payne, Kobelkov, Mikhlin for homogeneous, isotropic, incompressible materials are obtained as a special case. In the case of boundary value problem of place, a compatibility condition is obtained between the internal constraints and the boundary data which is necessary for the existence of solutions. With a further coercivity assumption on the compliance tensor, it is shown that the compatibility condition is also sufficient for existence. An orthogonal decomposition theorem for second order tensor fields modeled after Weyl's decomposition of solenoidal and gradient fields leads to the variational formulation of the problem and existence theorems.Almost all the results here apply to materials both with or without internal constraints. For internally constrained materials however, the verification of certain hypothesis is surprisingly non-trivial as indicated by the computation in the appendix.     

Two-plate problems in linear elasticity

Summary Some necessary and/or sufficient conditions for the existence of an equilibrium configuration of a couple of thin, noninterpenetrating plates are given. Various types of boundary data are considered, and some unstable cases discussed in detail.

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Sommario Si danno alcune condizioni necessarie e/o sufficienti per l'esistenza di una configurazione di equilibrio senza interpenetrazione di una coppia di piastre sottili. Si considerano vari tipi di dati al bordo e si discutono in dettaglio alcuni casi instabili.

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This work was supported in party by Istituto di Analisi Numerica del C.N.R., by Gruppo Nazionale di Analisi Funzionale e Applicazioni del C.N.R. and by Ministero della Pubblica Istruzione, Italy.     

Dual extremum principles and error bounds in nonlinear elasticity theory

Using some concepts of convex analysis a mathematical model is considered to derive dual extremum principles and global error bounds for monotone hyperelastic boundary-value problems. Introducing the two vector spaces of displacement gradients and of Piola stresses, which can be put into duality by a bilinear form, conjugate, and complementary energy functions are defined with the aid of Fenchel's transformation. Sufficient conditions for the convexity of the strain energy density are established. If the strain energy density is a convex function in some convex subset of the vector space of displacement gradients, dual extremum principles can be obtained by using Fenchel's inequality. They provide global error bounds for the solution of hyperelastic boundary-value problems.

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Zusammenfassung Mit Hilfe von Ergebnissen der Konvexen Analysis wird ein mathematisches Modell zur Herleitung dualer Extremalprinzipe und zur Berechnung globaler Fehlerschranken für monotone hyperelastische Randwertprobleme betrachtet. Führt man die beiden reellen Vektorräume der Verschiebungsgradiententensoren und der Piolaschen Spannungstensoren ein und setzt beide mit Hilfe einer Bilinear-Form in Dualität, so lassen sich mit der Fenchel Transformation konjugierte und komplementäre Energiedichten definieren. Für die Verformungsenergiedichte werden hinreichende Konvexitätsbedingungen angegeben. Ist die Verformungsenergiedichte konvex in einer konvexen Untermenge des Vektorraumes der Verschiebungsgradiententensoren, so können mit Hilfe der Fenchel-Ungleichung duale Extremalprinzipe erhalten werden, die globale Fehlerschranken für die Lösung des hyperelastischen Randwertproblems liefern.

     

Computing bounds to mixed-mode stress intensity factors in elasticity

A bounding procedure combined with an effective error bound method for linear functionals of the displacements and a simple two points displacement extrapolation method is presented to compute the lower and upper bounds to the stress intensity factors in elastic fracture problems. First, the displacements of two nodes (or node pairs) on the crack edges are used to construct the linear extrapolation to obtain the stress intensity factors at the crack tip, so that stress intensity factors are explicitly expressed as linear functionals of the displacements. Then, a posteriori bounding method is utilized to compute the bounds to the stress intensity factors. Finally, the bounding procedure is verified by a mixed-mode homogenous elastic fracture problem and a bimaterial interface crack problem.     

Chirality in isotropic linear gradient elasticity

Chirality is, generally speaking, the property of an object that can be classified as left- or right-handed. Though it plays an important role in many branches of science, chirality is encountered less often in continuum mechanics, so most classical material models do not account for it. In the context of elasticity, for example, classical elasticity is not chiral, leading different authors to use Cosserat elasticity to allow modelling of chiral behaviour.Gradient elasticity can also model chiral behaviour, however this has received much less attention than its Cosserat counterpart. This paper shows how in the case of isotropic linear gradient elasticity a single additional parameter can be introduced that describes chiral behaviour. This additional parameter, directly linked to three-dimensional deformation, can be either negative or positive, with its sign indicating a discrimination between the two opposite directions of torsion. Two simple examples are presented to show the practical effects of the chiral behaviour.     

Method of integro-differential relations in linear elasticity

Boundary-value problems in linear elasticity can be solved by a method based on introducing integral relations between the components of the stress and strain tensors. The original problem is reduced to the minimization problem for a nonnegative functional of the unknown displacement and stress functions under some differential constraints. We state and justify a variational principle that implies the minimum principles for the potential and additional energy under certain boundary conditions and obtain two-sided energy estimates for the exact solutions. We use the proposed approach to develop a numerical analytic algorithm for determining piecewise polynomial approximations to the functions under study. For the problems on the extension of a free plate made of two different materials and bending of a clamped rectangular plate on an elastic support, we carry out numerical simulation and analyze the results obtained by the method of integro-differential relations.     

Saint-Venant's principle in linear elasticity with microstructure

Toupin's version of the Saint-Venant principle in linear elasticity is generalized to the case of linear elasticity with microstructure. That is, it is shown that, for a straight prismatic bar made of an isotropic linear elastic material with microstructure and loaded by a self-equilibrated force system at one end only, the strain energy stored in the portion of the bar which is beyond a distance s from the loaded end decreases exponentially with the distance s.     

A unilateral contact problem in linear elasticity

In this paper we study the equilibrium of an elastic body which is simply supported, without friction, by a soft elastic plane. We prove that, if the exterior loads satisfy certain conditions of compatibility, the solution exists and is unique. Moreover, we find which regularity properties of the solution hold, provided the data are sufficiently smooth.Istituto di Scienza delle Costruzioni, Università di Pisa     

Amplitude bounds of linear forced vibrations

Summary For reasons of safety and reliability, the maximum amplitudes of vibration responses must be taken into account in engineering. The exact maximum amplitudes are available from solving linear differential equations. However, for large scale systems, it requires too much computation time to be useful. Therefore, simple amplitude bounds are of great interest for engineers. Up to now, only for classically dampled linear systems, some approximate amplitude bounds were presented. In this paper, amplitude bounds of linear forced vibrations are presented for general dampled linear vibrating systems. For transient, harmonic and step excitations, the presented amplitude bounds show simple relations to the system parameters, and are easy to calculate. The advantage is the possibility to judge the level of vibrations, and to choose appropriate parameters at design. Compared with the approximate underestimating amplitude bounds available in literature, the presented amplitude bounds are overestimating the maximum amplitude and, therefore, can be safely applied to general damped vibrating systems.The paper is dedicated to Prof. Dr.-Ing. Dr.-Ing. E.h. Dr.h.c. mult, Erwin Stein, on the occasion of his 65th birthday.     

Eigenvalue bounds for damped linear systems

Lower bounds are obtained on the real and imaginary parts of the eigenvalues of a damped linear system in free vibration. A condition for subcritical damping in all modes is obtained. The bounds have a close relation to the eigenvalue of a one degree-of-freedom system.     

Coaxiality of strain and stress in anisotropic linear elasticity

For a linearly elastic anisotropic body there are at least two rotations of the principal axes of strain such that the stress and strain tensors become coaxial. These rotations correspond to critical points for the stored energy, viewed as a function of the relative orientation between the body and the strain tensor.Supported by Gruppo Nazionale per la Fisica Matematica of C.N.R. (Italy).