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Concept# Linear elasticity

Résumé

Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.
The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations (or strains) and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding.
These assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis.
Mathematical formulation
Equations governing a linear elastic boundary value problem are based on three tensor partial differential equations for the balance of linear momentum and six infinitesimal strain-di

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ME-104: Introduction to structural mechanics

The student will acquire the basis for the analysis of static structures and deformation of simple structural elements. The focus is given to problem-solving skills in the context of engineering design.

ME-331: Solid mechanics

Model the behavior of elastic, viscoelastic, and inelastic solids both in the infinitesimal and finite-deformation regimes.

ME-484: Numerical methods in biomechanics

Students understand and apply numerical methods (FEM) to answer a research question in biomechanics. They know how to develop, verify and validate multi-physics and multi-scale numerical models. They can analyse and comment results in an oral presentation and a written report.

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Déformation élastique

En physique, l'élasticité est la propriété d'un matériau solide à retrouver sa forme d'origine après avoir été déformé. La déformation élastique est une déformation réversible. Un matériau solide se d

Contrainte (mécanique)

vignette|Lignes de tension dans un rapporteur en plastique vu sous une lumière polarisée grâce à la photoélasticité.
En mécanique des milieux continus, et en résistance des matériaux en règle générale

Déformation plastique

La théorie de la plasticité traite des déformations irréversibles indépendantes du temps, elle est basée sur des mécanismes physiques intervenant dans les métaux et alliages mettant en jeu des mouve

Motivated by applications in orthopaedic surgery, new constitutive laws for trabecular (or spongious) bone are developed in the framework of continuum mechanics, implemented in a mechanical analysis computer program, validated by a number of in vitro experiments and illustrated by the simulation of a femoral total hip component. Current knowledge about the morphological and mechanical properties of trabecular bone is reviewed for setting the background and clarifying the contributions of the thesis. Comprehensive 1D and 3D theoretical models based on the approach of standard generalized materials are developed with a specific attention towards irreversible phenomena. The 1D model includes linear elasticity and rate-independent as well as rate-dependent plastic strain flow with damage. Based on a second order fabric tensor, the 3D model includes inhomogeneous, orthotropic linear elasticity and rate-independent plasticity with damage. In order to solve boundary value problems involving complex bone or bone-implant structures, implicit projection algorithms are developed for integrating the plastic flow rules with damage and implemented in the computer program TACT combining the finite element method, the linear iteration method and the finite difference method. The resulting numerical models are illustrated by the means of traction, bending and torsion benchmark tests. A number of pilot in vitro experiments are undertaken on human and bovine trabecular bone specimens in order to validate the theoretical models and identify the material constants. Quasistatic uniaxial and torsion experiments are performed with a method avoiding artefacts due to the inhomogeneous boundary conditions associated with porosity. Anisotropic elasticity, plasticity and damage of trabecular bone prove to be successfully described by the models in terms of structural density and morphology. Finally, the 3D constitutive law is applied to the biomechanical problem of primary stability of a cementless femoral total hip component in order to illustrate its potential.

Fabian Barras, René Carpaij, Jean-François Molinari

Since the early years of the linear elastic theory of fracture [linear elastic fracture mechanics (LEFM)], scientists have sought to understand and predict how fast cracks grow in a material or slip fronts propagate along faults. While shear cracks can travel faster than the shear wave speed, the Rayleigh wave speed is the limiting speed theoretically predicted for tensile failure. This work uncovers the existence of supershear episodes in the tensile (mode I) rupture of linearly elastic materials beyond the maximum allowable (sub-Rayleigh) speed predicted by the classical theory of dynamic fracture. While the admissible rupture speeds predicted by LEFM are verified for smooth crack fronts, we present numerically how a supershear burst can emerge from a discontinuity in crack front curvature. Using a spectral formulation of the three-dimensional elastodynamic equations coupled with a cohesive model of fracture, we study how these short-lived bursts create shock waves persisting far from the discontinuity site. This study provides insight on crack front instabilities present in the rapid tensile failure of brittle materials due to large distortions of the rupture front.

2018Multiscale or multiphysics partial differential equations are used to model a wide range of physical systems with various applications, e.g. from material and natural science to problems in biology or engineering. When the ratio between the smallest scale in the problem and the size of the physical domain (also the size of the solution) is very large, the numerical approximation of the effective behaviour with classical numerical methods, such as the finite element method (FEM), can become computationally prohibitive. Indeed, as the smallest scale in the problem has to be fully resolved, one obtains a discretization of the computational domain with a very large number of degrees of freedom. In the first part of the thesis, we derive a finite element heterogeneous multiscale method (FE-HMM) applied to the wave equation in a linear elastic medium. We state the FE-HMM and give robust a priori error estimates with explicit convergence rates for the macro and micro discretizations. For simplicity, we start with the static highly heterogeneous linear problem and, then, add the time dependency and consider the wave propagation in a highly heteorgeneous linear elastic medium. In the second part of the thesis we are interested in problems in which the scales are well separated only in some regions of the computational domain, with possibly a continuum of scales in the complementary domain. Such problems arise in various situations, for example in heterogeneous composite materials whose effective properties can be well captured by assuming a (locally) periodic microstructure that can however not be valid near defects of the material. In our modeling, the smallest scale is supposed to be still discretized at the continuum level, but for some applications atomistic scale should be considered. Our coupling method is based on a domain decomposition into a family of overlapping domains. Virtual (interface) controls are introduced as boundary conditions, and act as unknown traces or fluxes. Our method is formulated as a minimization problem with states equations as constraints. The optimal boundary controls of two overlapping domains are found by an heterogeneous optimization problem that is based on minimizing the discrepancy between the two models on, at first, the overlapping region, and at second, over the boundary of the overlapping region. The fully discrete optimization based method couples the continuous or discontinuous Galerkin FE-HMM with the FEM. The well-posedness of our method, in continuous and discrete forms, are established and (fully discrete) a priori error estimates are derived.

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