Identification of the mechanical properties of a viscoelastic material depends on characteristics of the observed motion field within the object in question. For certain physical and experimental configurations and certain resolutions and variance within the measurement data, the viscoelastic properties of an object may become non-identifiable. Elastographic imaging methods seek to provide maps of these viscoelastic properties based on displacement data measured by traditional imaging techniques, such as magnetic resonance and ultrasound. Here, 1D analytic solutions of the viscoelastic wave equation are used to generate displacement fields over wave conditions representative of diverse time-harmonic elastography applications. These solutions are tested through the minimization of a least squares objective function suitable for framing the elastography inverse calculation. Analysis shows that the damping ratio and the ratio of the viscoelastic wavelength to the size of the domain play critical roles in the form of this least squares objective function. In addition, it can be shown analytically that this objective function will contain local minima, which hinder discovery of the global minima via gradient descent methods.

Magnetic Resonance Elastography (MRE) is a modality that allows the mapping of the mechanical properties of soft tissues such as brain or liver from Magnetic Resonance Imaging (MRI) data. Specific MRI sequences have been developed in order to estimate the 3D displacement field in biological tissues undergoing harmonic solicitations [1]. The aim of this work consists in comparing the performance of different identification methods proposed for MRE in a situation straightforward enough –while still representative of elastography applications– to permit both analytic and experimental approaches. For the sake of simplicity, we present only the homogeneous case. The mechanical analysis of the problem illustrated in Figure 1 leads, for an adapted set of boundary conditions, to the resolution of the following usual problem. (voir annexe) For a given set of experimental conditions, we determine the complex-valued solution fields. The so-obtained displacement fields can be perturbed to represent experimental noise. These modified displacements are then introduced as input for different identification methods (Finite Element Model Updating [2], Constitutive Equation Gap [3] or Modified Constitutive Equation Gap [4]) in order to characterize their different efficiencies. In parallel, experimental tests are performed on gelatin samples with “controlled” properties in order to characterize these methods using real data.

We use a special duality by perturbation approach in optimization to find two different bi-dual problems of the non-linear Kirchhoff-Love-von Karman plate theory. The first one coincides with that found by means of the intrinsic approach of P.G. Ciarlet, while the second is found by means of a complementary energy introduced by J.J. Telega. (C) 2020 Published by Elsevier Masson SAS.

The periodic homogenisation method is often used to study the behaviour of heterogeneous media, like fibre enforced composites, soils or tissues. In this present work, the method is applied to derive the homogenised behaviour of a one-dimensional elastic heterogeneous medium under non-linear deformations. For the microscopic description of the problem the St. Venant-Kirchhoff model for hyperelastic material is used. An expression corresponding to the homogenised stress at the macroscopic scale is obtained. The model is tested by studying a special case of a heterogeneous medium composed of two materials with constant Young’s moduli. The obtained stress curve represents realistic values. Furthermore, it is shown that different approaches used in the homogenisation process can lead to different results (e.g. linearisation of non-linearities). Therefore, the importance of non-dimensional formulation before homogenisation process is underlined. Finally, an example of numerical computations at the micro- and macroscopic scales is presented. The results are coherent and enabled us to validate the modeling.

We give a presentation of the mathematical and numerical treatment of plate dynamics problems including rotational inertia. The presence of rotational inertia in the equation of motion makes the study of such problems interesting. We employ HCT finite elements for space discretization and the Newmark method for time discretization in FreeFEM++, and test such methods in some significant cases: a circular plate clamped all over its lateral surface, a rectangular plate simply supported all over its lateral surface, and an L-shaped clamped plate.

We present a novel Hybrid High-Order (HHO) discretization of fourth-order elliptic problems arising from the mechanical modeling of the bending behavior of Kirchhoff–Love plates, including the biharmonic equation as a particular case. The proposed HHO method supports arbitrary approximation orders on general polygonal meshes, and reproduces the key mechanical equilibrium relations locally inside each element. When polynomials of degree k ≥ 1 are used as unknowns, we prove convergence in h k +1 (with h denoting, as usual, the meshsize) in an energy-like norm. A key ingredient in the proof are novel approximation results for the energy projector on local polynomial spaces. Under biharmonic regularity assumptions, a sharp estimate in h k +3 is also derived for the L 2 -norm of the error on the deflection. The theoretical results are supported by numerical experiments, which additionally show the robustness of the method with respect to the choice of the stabilization.

The authors use the asymptotic expansion method by P. G. Ciarlet to obtain a Kirchhoff-Love-type plate model for a linear soft ferromagnetic material. They also give a mathematical justification of the obtained model by means of a strong convergence result.

We show that the displacement and strain formulations of the displacement-traction problem of three-dimensional linearized elasticity can be viewed as Legendre-Fenchel dual problems to the stress formulation of the same problem. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new approach to elasticity by means of Legendre-Fenchel duality. Résumé Dualité de Legendre-Fenchel en élasticité. On montre que les formulations en déplacements et en déformations du problème en déplacement-traction de l’élasticité linéarisée tri-dimensionnelle peuvent être vues comme des problèmes duaux de Legendre-Fenchel de la formulation en contraintes de ce même problème. On montre également que chacun des Lagrangiens correspondants a un point-selle, justifiant ainsi complètement cette nouvelle approche de l’élasticité au moyen de la dualité de Legendre-Fenchel. 1. Legendre-Fenchel duality All vector spaces, matrices, etc., considered in this Note are real. The dual space of a normed vector space X is denoted X, and X∗〈·, ·〉X designates the associated duality. The bidual space of X is denoted X; if X is a reflexive Banach space, X will be identified with X by means of the usual canonical isometry. The indicator function IA of a subset A of a set X is the function IA defined by IA(x) := 0 if x ∈ A and IA(x) := +∞ if x / ∈ A. A function g : X → R ∪ {+∞} is proper if {x ∈ X ; g(x) < +∞} 6 = ∅. Let Σ be a normed vector space and let g : Σ → R ∪ {+∞} be a proper function. The Legendre-Fenchel transform of g is the function g : Σ → R ∪ {+∞} defined by g : e ∈ Σ → g(e) := sup σ∈Σ {Σ∗〈e, σ〉Σ − g(σ)}. The next theorem summarizes some basic properties of the Legendre-Fenchel transform when the space Σ is a reflexive Banach space. For proofs, see, e.g., Ekeland & Temam [7] or Brezis [2]. The equality g = g constitutes the Fenchel-Moreau theorem. Email addresses: mapgc@cityu.edu.hk (Philippe G. Ciarlet), giuseppe.geymonat@univ-montp2.fr (Giuseppe Geymonat), krasucki@math.univ-montp2.fr (Francoise Krasucki). Preprint submitted to the Académie des sciences March 4, 2011 Theorem 1.1 Let Σ be a reflexive Banach space, and let g : Σ → R∪{+∞} be a proper, convex, and lower semi-continuous function. Then the Legendre-Fenchel transform g : Σ → R ∪ {+∞} of g is also proper, convex, and lower semi-continuous. Let g : σ ∈ Σ → g(σ) := sup e∈Σ {Σ∗〈e, σ〉Σ − g (e)} denote the Legendre-Fenchel transform of g (recall that X is here identified with X). Then g = g. Given a minimization problem infσ∈Σ G(σ), called (P), with a function G : Σ → R∪{+∞} of the specific form given in Theorem 1.2 below, the following simple result will be the basis for defining two different dual problems of problem (P). The functions L and L̃ defined in the next theorem are the Lagrangians associated with the minimization problem (P). Theorem 1.2 Let Σ and V be two reflexive Banach spaces, let g : Σ → R∪{+∞} and h : V ∗ → R∪{+∞} be two proper, convex, and lower semi-continuous functions, let Λ : Σ → V ∗ be a linear and continuous mapping, let the function G : Σ → R ∪ {+∞} be defined by G : σ ∈ Σ → G(σ) := g(σ) + h(Λσ), and finally, let the two functions L : Σ × Σ → {−∞} ∪ R ∪ {+∞} and L̃ : Σ × V → {−∞} ∪ R ∪ {+∞} be defined by L : (σ, e) ∈ Σ × Σ → L(σ, e) := Σ∗〈e, σ〉Σ − g (e) + h(Λσ), L̃ : (σ, v) ∈ Σ × V → L̃(σ, v) := g(σ) + V ∗〈Λσ, v〉V − h (v). Then inf σ∈Σ G(σ) = inf σ∈Σ sup e∈Σ L(σ, e) = inf σ∈Σ sup v∈V L̃(σ, v). A key issue then consists in deciding whether the infimum found in problem (P) is equal to the supremum found in either one of its dual problems, i.e., for instance in the case of the first dual problem (to fix ideas), whether inf σ∈Σ G(σ) = sup e∈Σ G(e), or equivalently, inf σ∈Σ sup e∈Σ L(σ, e) = sup e∈Σ inf σ∈Σ L(σ, e). If this is the case, the next issue consists in deciding whether the Lagrangian L possesses a saddle-point (σ, e) ∈ Σ × Σ, i.e., that satisfies inf σ∈Σ sup e∈Σ L(σ, e) = inf σ∈Σ L(σ, e) = L(σ, e) = sup e∈Σ L(σ, e) = sup e∈Σ inf σ∈Σ L(σ, e). This is precisely the type of questions addressed in this Note, the point of departure (P) being a classical quadratic minimization problem arising in three-dimensional linearized elasticity. 2. Functional analytic preliminaries Latin indices vary in the set {1, 2, 3}, save when they are used for indexing sequences, and the summation convention with respect to repeated indices is systematically used in conjunction with this rule. A domain in R is a bounded, connected, open subset of R whose boundary, denoted Γ, is Lipschitzcontinuous, the set Ω being locally on a single side of Γ. Spaces of functions, vector fields in R, and 3 × 3 symmetric matrix fields, defined over an open subset of R are respectively denoted by italic capitals, boldface Roman capitals, and special Roman capitals. The inner-product of a ∈ R and b ∈ R is denoted a · b. The notation s : t := sijtij designates the matrix inner-product of two matrices s := (sij) and t := (tij) of order three. The inner product in the space L(Ω) is given by (σ, τ ) ∈ L(Ω) × L(Ω) → ∫ Ω σ : τ dx, and ‖·‖L2(Ω) denotes the corresponding norm. The space L(Ω) will be identified with its dual space. The duality bracket between the space H(Γ) and its dual space will be denoted 〈·, ·〉Γ := H−1/2(Γ)〈·, ·〉H1/2(Γ). 2 For any vector field v = (vi) ∈ D (Ω), the associated linearized strain tensor is the symmetric matrix field ∇sv ∈ D (Ω) defined by ∇sv := 1 2 (∇v T + ∇v). We now recall some functional analytic preliminaries, due to Geymonat & Suquet [10] and Geymonat & Krasucki [8,9]. Given a domain Ω in R, define the space H(div; Ω) := {μ ∈ L(Ω); div μ ∈ L(Ω)}. The set Ω being a domain, the density of the space C(Ω) in the space H(div; Ω) then implies that the mapping μ ∈ C(Ω) → μν|Γ can be extended to a continuous linear mapping from the space H(div; Ω) into H(Γ), which for convenience will be simply denoted μ ∈ H(div; Ω) → μν ∈ H(Γ). Theorem 2.1 The Green formula ∫

Let E3 be the Euclidian space referred to the orthonormal frame (0, e → 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1147.tif"/> , e → 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1148.tif"/> , e → 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1149.tif"/> ) and let Ω− and Ω+ be open, disjoint, connected domains with boundaries ∂Ω− and ∂Ω+ piecewise of class C 2. Ω− and Ω+ are bonded together on a surface S = ∂Ω− ⋂ ∂Ω+. We assume that S is a set of positive 2—measure, is described by a global piecewise C 2 chart, given by x 3 = φ(x 1, x 2). We also set Ω ¯ = Ω ¯ + ∪ Ω ¯ − https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1150.tif"/> . In fact, we describe the bonding of the two solids occupying Ω− and Ω+ by a thin adhesive layer, whose thermomechanical coefficients are small with respect of those of the adherents. More precisely, for a fixed ɛ > 0 we define the domain Ω 0 ε https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1151.tif"/> occupied by the thin layer: Ω ¯ 0 ε = { x → s + ε z e → 3 , 0 ≤ z ≤ 1 , x → s ∈ S } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1152.tif"/>

We present some mathematical convergence results using a two-scale method for a linear elastic isotropic medium containing one layer of parallel periodically distributed heterogeneities located in the interior of the whole domain around a plane surface Σ . The aim of this paper is to study the situation when the rigidity of the linearly isotropic elastic fibres is 1/ ε m the rigidity of the surrounding linearly isotropic elastic material. We use a two-scale convergence method adapted to the geometry of the problem (layer of fibres). In the models obtained Σ behaves for m =1 as a “material surface” without membrane energy in the direction of the plane orthogonal to the direction of the fibres. For m =3 the “material surface” has no bending energy in the direction orthogonal to the fibres.

We present an asymptotic two-dimensional plate model for linear magneto-electro-thermo-elastic sensors and actuators, under the hypotheses of anisotropy and homogeneity. Four different boundary conditions pertaining to electromagnetic quantities are considered, leading to four different models: the sensor–actuator model, the actuator–sensor model, the actuator model and the sensor model. We validate the obtained two-dimensional models by proving weak convergence results. Each of the four plate problems turns out to be decoupled into a flexural problem, involving the transversal displacement of the plate, and a certain partially or totally coupled membrane problem.

We present a mathematical model for linear magneto-electro-thermo-elastic continua, as sensors and actuators can be thought of, and prove the well-posedness of the dynamic and quasi-static problems. The two proofs are accomplished, respectively, by means of the Hille–Yosida theory and of the Faedo–Galerkin method. A validation of the quasi-static hypothesis is provided by a nondimensionalization of the dynamic problem equations. We also hint at the study of the convergence of the solution to the dynamic problem to that to the quasi-static problem as a small parameter — the ratio of the largest propagation speed for an elastic wave in the body to the speed of light — tends to zero.

Linear Donati compatibility conditions guarantee that the components of symmetric tensor fields are those of linearized change of metric or linearized change of curvature tensor fields associated with the displacement vector field arising in a linearly elastic structure when it is subjected to applied forces. These compatibility conditions take the form of variational equations with divergence-free tensor fields as test-functions, by contrast with Saint-Venant compatibility conditions, which take the form of systems of partial differential equations.In this paper, we identify and justify nonlinear Donati compatibility conditions that apply to a nonlinearly elastic plate modeled by the Kirchhoff-von-Karman-Love theory. These conditions, which to the authors' best knowledge constitute a first example of nonlinear Donati compatibility conditions, in turn allow to recast the classical approach to this nonlinear plate theory, where the unknown is the position of the deformed middle surface of the plate, into the intrinsic approach, where the change of metric and change of curvature tensor fields of the deformed middle surface of the plate are the only unknowns. The intrinsic approach thus provides a direct way to compute the stress resultants and the stress couples inside the deformed plate, often the unknowns of major interest in computational mechanics. (C) 2014 Elsevier Masson SAS. All rights reserved.

In this paper we derive a strain gradient plate model from the three-dimensional equations of strain gradient linearized elasticity. The deduction is based on the asymptotic analysis with respect of a small real parameter being the thickness of the elastic body we consider. The body is constituted by a second gradient isotropic linearly elastic material. The obtained model is recognized as a strain gradient Reissner-Mindlin plate model. We also provide a mathematical justification of the obtained plate model by means of a variational weak convergence result.

The aim of this paper is to numerically validate the effectiveness of a matched asymptotic expansion formal method introduced in a pioneering paper by Nguetseng and Sánchez Palencia (1985) [1] and extended in Geymonat et al. (2011) [2,3]. Using this method a simplified model for the influence of small identical heterogeneities periodically distributed on an internal surface to the overall response of a linearly elastic body is derived. In order to validate this formal method a careful numerical study compares the solution obtained by a standard method on a fine mesh to the one obtained by asymptotic expansion. We compute both the zero and the first order terms in the expansion. To efficiently compute the first order term we introduce a suitable domain decomposition method.

Our aim is to demonstrate the effectiveness of the matched asymptotic expansion method in obtaining a simplified model for the influence of small identical heterogeneities periodically distributed on an internal surface on the overall response of a linearly elastic body. The results of several numerical experiments corroborate the precise identification of the different steps, in particular of the outer/inner regions with their normalized coordinate systems and the scale separation, leading to the model.

Let omega be a simply-connected domain in R-2 and let (E-alpha beta) and (F-alpha beta) be two symmetric 2 x 2 matrix fields with components in L-2(omega). In this Note, we identify nonlinear compatibility conditions "of Donati type" that the components E-alpha beta and F-alpha beta must satisfy in order that there exists a vector field (eta(1), eta(2), w) is an element of H-0(1) (omega) x H-0(2) (omega) x H-0(2) (omega) such that:1/2 (partial derivative(alpha)eta(beta) + partial derivative(alpha)w partial derivative(beta)w) = E-alpha beta and partial derivative(alpha beta)w = F-alpha beta in omegaThe left-hand sides of these relations are the components of tensors found in the Kirchhoff-von Karman-Love theory of nonlinearly elastic plates. (C) 2013 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

We give a new derivation, based on the complementary energy formulation, of a simplified model for a multi-structure made up of two anisotropic hyper-elastic bodies connected by a thin strong material layer. The model is obtained by identifying the Mosco-limit of the stored complementary energy functional when the thickness is of order ε and the stiffness of order 1/ε where ε is a positive real adimensional parameter. In order to prove the existence of the displacement associated with the stress we use a suitable weak version of the Saint-Venant compatibility condition also known as Donati’s theorem.

Cette presentation decrit une methode multi-echelle robuste et efficace qui combine developpements asymptotiques raccordes et decomposition de domaines pour resoudre des problemes d'elasticite avec un grand nombre d'heterogeneites.

We show that the displacement and strain formulations of the displacement-traction problem of three-dimensional linearized elasticity can be viewed as Legendre-Fenchel dual problems to the stress formulation of the same problem. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new approach to elasticity by means of Legendre-Fenchel duality. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.