Three-dimensional Nonlinear Elasticity Research Articles

A nonlinear elastic deformable template for soft structure segmentation: application to the heart segmentation in MRI

This paper proposes a nonlinear 3D deformable model for the image segmentation of soft structures. The template is modelled as an elastic body which is deformed by forces derived from the image. It relies on a template, which is a topological, geometrical and material model of the structure to segment. This model is based on the nonlinear three-dimensional elasticity problem with a boundary condition of pure traction. In addition, the applied forces are successive, as they depend on the displacements. For computations, an incremental algorithm is proposed to minimize the global energy of template deformation. Sufficient conditions of the convergence for the incremental algorithm are given. Finally, a discrete algorithm using the finite element method is presented and evaluated on synthetic images and actual MR images of mouse hearts.

Derivation of a plate theory for incompressible materials

We derive a two-dimensional model for elastic plates as a Γ-limit of three-dimensional nonlinear elasticity with the constraint of incompressibility. The energy density of the reduced problem describes plate bending, and is determined from the elastic moduli at the identity of the energy density of the three-dimensional problem. Without the constraint of incompressibility, Γ-convergence to a plate theory was first derived by Friesecke, James and Müller. The main difficulty in the present result is the construction of a recovery sequence which satisfies pointwise the nonlinear constraint of incompressibility. To cite this article: S. Conti, G. Dolzmann, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Nonlinear finite-element analysis of nanoindentation of viral capsids

Recent atomic force microscope (AFM) nanoindentation experiments measuring mechanical response of the protein shells of viruses have provided a quantitative description of their strength and elasticity. To better understand and interpret these measurements, and to elucidate the underlying mechanisms, this paper adopts a course-grained modeling approach within the framework of three-dimensional nonlinear continuum elasticity. Homogeneous, isotropic, elastic, thick-shell models are proposed for two capsids: the spherical cowpea chlorotic mottle virus (CCMV), and the ellipsocylindrical bacteriophage phi29 . As analyzed by the finite-element method, these models enable parametric characterization of the effects of AFM tip geometry, capsid dimensions, and capsid constitutive descriptions. The generally nonlinear force response of capsids to indentation is shown to be insensitive to constitutive particulars, and greatly influenced by geometric and kinematic details. Nonlinear stiffening and softening of the force response is dependent on the AFM tip dimensions and shell thickness. Fits of the models capture the roughly linear behavior observed in experimental measurements and result in estimates of Young's moduli of approximately 280-360 MPa for CCMV and approximately 4.5 GPa for phi29 .

Recursive Derivation of One-Dimensional Models from Three-Dimensional Nonlinear Elasticity

In this paper, we consider a formal asymptotic study of homogeneous slender cylinders made of a Saint-Venant Kirchhoff material submitted to dead loads. Depending on the order of magnitude of the applied loads, we obtain a hierarchy of four one-dimensional models starting from the nonlinear theory of extensible strings to the nonlinear theory of flexible bars. Our approach is based on the resolution of a sequence of recursive minimization problems.

Relaxation theorems in nonlinear elasticity

Relaxation theorems which apply to one, two and three-dimensional nonlinear elasticity are proved. We take into account the fact an infinite amount of energy is required to compress a finite line, surface or volume into zero line, surface or volume. However, we do not prevent orientation reversal.

Another approach to the fundamental theorem of Riemannian geometry in [formula omitted], by way of rotation fields

In 1992, C. Vallée showed that the metric tensor field C = ∇ Θ T ∇ Θ associated with a smooth enough immersion Θ : Ω → R 3 defined over an open set Ω ⊂ R 3 necessarily satisfies the compatibility relation CURL Λ + COF Λ = 0 in Ω , where the matrix field Λ is defined in terms of the field U = C 1 / 2 by Λ = 1 det U { U ( CURL U ) T U − 1 2 ( tr [ U ( CURL U ) T ] ) U } . The main objective of this paper is to establish the following converse: If a smooth enough field C of symmetric and positive-definite matrices of order three satisfies the above compatibility relation over a simply-connected open set Ω ⊂ R 3 , then there exists, typically in spaces such as W loc 2 , ∞ ( Ω ; R 3 ) or C 2 ( Ω ; R 3 ) , an immersion Θ : Ω → R 3 such that C = ∇ Θ T ∇ Θ in Ω. This global existence theorem thus provides an alternative to the fundamental theorem of Riemannian geometry for an open set in R 3 , where the compatibility relation classically expresses that the Riemann curvature tensor associated with the field C vanishes in Ω. The proof consists in first determining an orthogonal matrix field R defined over Ω, then in determining an immersion Θ such that ∇ Θ = R C 1 / 2 in Ω, by successively solving two Pfaff systems. In addition to its novelty, this approach thus also possesses a more “geometrical” flavor than the classical one, as it directly seeks the polar factorization ∇ Θ = RU of the immersion gradient in terms of a rotation R and a pure stretch U = C 1 / 2 . This approach also constitutes a first step towards the analysis of models in nonlinear three-dimensional elasticity where the rotation field is considered as one of the primary unknowns.

Derivation of a rod theory for multiphase materials

A rigorous derivation is given of a rod theory for a multiphase material, starting from three-dimensional nonlinear elasticity. The stored energy density is supposed to be nonnegative and to vanish exactly on a set consisting of two copies of the group of rotations SO(3). The two potential wells correspond to the two crystalline configurations preferred by the material. We find the optimal scaling of the energy in terms of the diameter of the rod and we identify the limit, as the diameter goes to zero, in the sense of Γ-convergence.

A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence

We derive a hierarchy of plate models from three-dimensional nonlinear elasticity by Γ-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume ∼hβ, where h is the thickness of the plate. This is in turn related to the strength of the applied force ∼hα. Membrane theory, derived earlier by Le Dret and Raoult, corresponds to α=β=0, nonlinear bending theory to α=β=2, von Karman theory to α=3, β=4 and linearized vK theory to α>3. Intermediate values of α lead to certain theories with constraints. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [29] which states that for maps v:(0,1)3→ℝ3, the L2 distance of ∇v from a single rotation is bounded by a multiple of the L2 distance from the set SO(3) of all rotations.

Three-dimensional nonlinear analysis of laminated cylindrical shells under cylindrical bending

On the basis of three-dimensional (3D) nonlinear elasticity, asymptotic solutions for the laminated cylindrical shells under cylindrical bending are presented. The basic 3D nonlinear equations such as the relations between finite strains (Green strains) and displacements, the nonlinear stress equilibrium equations in terms of the Kirchhoff stress components and the generalized Hooke's law for a monoclinic elastic material are considered in the present formulation. After introduction of a set of nondimensionalized field variables, asymptotic expansion, consideration of the effects of shear deformations at the leading order problem and then successive integration, we obtain the recursive sets of governing equations for various orders. The von Karman-type first-order shear deformation theory (FSDT) is derived as a first-order approximation to the 3D nonlinear theory. The admissible edge conditions for various orders are derived in the form of generalized force and moment resultants by means of the variational principles for finite deformations. With a set of appropriate edge conditions, the asymptotic solutions of laminated cylindrical shells under cylindrical bending at each order level can be obtained. Since the differential operators for various order problems remain identical, it is shown that the solution procedure can be repeatedly applied to various order problems.

Continuity of a Deformation in H1 as a Function of Its Cauchy-Green Tensor in L1

Let $\Omega$ be a bounded Lipschitz domain in $\BBbR^n$. The Cauchy-Green, or metric, tensor field associated with a deformation of the set $\Omega$, i.e., a smooth-enough orientation-preserving mapping $\bTh\colon\Omega\to\BBbR^n$, is the $n\times n$ symmetric matrix field defined by $\bnabla\bTheta^T(x)\bnabla\bTheta(x)$ at each point $x\in\Omega$. We show that, under appropriate assumptions, the deformations depend continuously on their Cauchy-Green tensors, the topologies being those of the spaces $\bH^1(\Omega)$ for the deformations and $\bL^1(\Omega)$ for the Cauchy-Green tensors. When $n=3$ and $\Omega$ is viewed as a reference configuration of an elastic body, this result has potential applications to nonlinear three-dimensional elasticity, since the stored energy function of a hyperelastic material depends on the deformation gradient field $\bnabla\bTheta$ through the Cauchy-Green tensor.

Derivation of the nonlinear bending-torsion theory for inextensible rods by $\Gamma$ -convergence

We show that the nonlinear bending-torsion theory for inextensible rods arises as \(\Gamma\)-limit of three-dimensional nonlinear elasticity.

A nonlinear model for inextensible rods as a low energy Γ-limit of three-dimensional nonlinear elasticity

Using a variational approach we rigorously deduce a nonlinear model for inextensible rods from three-dimensional nonlinear elasticity, passing to the limit as the diameter of the rod goes to zero. The theory obtained is analogous to the Föppl–von Kármán theory for plates. We also derive an asymptotic expansion of the solution and compare it to a similar expansion which Murat and Sili obtained starting from three-dimensional linear elasticity.

Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence

We show that the nonlinear bending theory of shells arises as a Γ-limit of three-dimensional nonlinear elasticity. To cite this article: G. Friesecke et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Part II : the strongly curved shells

In the first part of this paper we have deduced a classification of asymptotic shallow shell models with respect to the level of applied forces, from the non-linear three-dimensional elasticity. We have used a constructive approach based on a dimensional analysis of the non-linear three-dimensional equilibrium equations, which naturally makes appear dimensionless numbers characterizing the applied forces ( F and G ) and the geometry of the shell (e and C ). To limit our study to one-scale problems, these dimensionless numbers are expressed in terms of the relative thickness e of the shell, considered as the perturbation parameter. In the first part, we have studied the case of shallow shells corresponding to C = e 2 . In the second part of this paper, we will study the case of strongly curved shells for which C = e . The classification that we obtain is then more complex. It depends not only on the force levels, but also on the existence of inextensional displacements which keep invariant the metric of the middle surface of the shell.

Validation of the Variational Asymptotic Beam Sectional Analysis

The computer program VABS (Variational Asymptotic Beam Section Analysis) uses the variational asymptotic method to split a three-dimensional nonlinear elasticity problem into a twodimensional linear cross-sectional analysis and a one-dimensional, nonlinear beam problem. This is accomplished by taking advantage of certain small parameters inherent to beam-like structures. VABS is able to calculate the one-dimensional cross-sectional stiffness constants, with transverse shear and Vlasov refinements, for initially twisted and curved beams with arbitrary geometry and material properties. Several validation cases are presented. First, an elliptic bar is modeled with transverse shear refinement using the variational asymptotic method, and the solution is shown to be identical to that obtained from the theory of elasticity. The shear center locations calculated by VABS for various cross sections agree well with those obtained from common engineering analyses. Comparisons with other composite beam theories prove that it is unnecessary to introduce ad hoc kinematic assumptions to build an accurate beam model. For numerical validation, values of the one-dimensional variables are extracted from an ABAQUS model and compared with results from a one-dimensional beam analysis using cross-sectional constants from VABS. Furthermore, point-wise three-dimensional stress and strain fields are recovered using VABS, and the correlation with the three-dimensional results from ABAQUS is excellent. Finally, classical theory is shown to be insufficient for general-purpose beam modeling. Appropriate refined theories are recommended for some classes of problems.

A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity

AbstractThe energy functional of nonlinear plate theory is a curvature functional for surfaces first proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a Γ‐limit of three‐dimensional nonlinear elasticity theory as the thickness of a plate goes to zero. A key ingredient in the proof is a sharp rigidity estimate for maps v : U → ℝn, U ⊂ ℝn. We show that the L2‐distance of ∇v from a single rotation matrix is bounded by a multiple of the L2‐distance from the group SO(n) of all rotations. © 2002 Wiley Periodicals, Inc.

Energy Scaling of Compressed Elastic Films -Three-Dimensional Elasticity¶and Reduced Theories

We derive an optimal scaling law for the energy of thin elastic films under isotropic compression, starting from three-dimensional nonlinear elasticity. As a consequence we show that any deformation with optimal energy scaling must exhibit fine-scale oscillations along the boundary, which coarsen in the interior. This agrees with experimental observations of folds which refine as they approach the boundary.

Rigorous derivation of nonlinear plate theory and geometric rigidity

We show that nonlinear plate theory arises as a Γ-limit of three-dimensional nonlinear elasticity. A key ingredient in the proof is a sharp rigidity estimate for maps v:(0,1) 3→ R 3 . We show that the L 2 distance of ∇ v from a single rotation is bounded by a multiple of the L 2 distance from the set SO(3) of all rotations. To cite this article: G. Friesecke et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 173–178

The Föppl–von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity

We show that the Föppl–von Kármán theory arises as a low energy Γ-limit of three-dimensional nonlinear elasticity. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [5] that for maps v:(0,1) 3→ R 3 , the L 2 distance of ∇ v from a single rotation is bounded by a multiple of the L 2 distance from the set SO(3) of all rotations. To cite this article: G. Friesecke et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 201–206.

Heterogeneous Thin Films of Martensitic Materials

We study the effective behavior of heterogeneous thin films with three competing length scales: the film thickness and the length scales of heterogeneity and material microstructure. We start with three-dimensional nonhomogeneous nonlinear elasticity enhanced with an interfacial energy of the van der Waals type, and derive the effective energy density as all length scales tend to zero with given limiting ratios. We do not require any a priori selection of asymptotic expansion or ansatz in deriving our results. Depending on the dominating length scale, the effective energy density can be identified by three procedures: averaging, homogenization and thin-film limit. We apply our theory to martensitic materials with multi-well energy density and use a model example to show that the “shape-memory behavior” can crucially depend on the ratios of these length scales. We comment on the effective conductivity of linear composites, and also on multilayers made of shape-memory and elastic materials.