Linearized Strain Tensor Research Articles

Donati compatibility conditions on a surface – Application to shell theory

The Donati compatibility conditions constitute a characterization of symmetric 3×3 matrix fields defined over a domain Ω⊂R3 as linearized strain tensor fields. They take various forms, according to which boundary conditions are to be satisfied by the corresponding displacement vector field. A typical result in this direction is the following one, due to G. Geymonat and F. Krasucki and C. Amrouche, the first author, L. Gratie, and S. Kesavan: Given a symmetric matrix field (eij) with components eij in the space L2(Ω), there exists a displacement vector field (vi) with components vi in the space H01(Ω) such that eij=12(∂jvi+∂ivj) in Ω if, and only if, ∫Ωeijsijdx=0 for all symmetric matrix fields (sij) with sij∈L2(Ω) that satisfy ∂jsij=0 in Ω.The main objective of this paper is to identify and justify various Donati-like compatibility conditions on a surface, guaranteeing that the components of two symmetric matrix fields (cαβ) and (rαβ) with cαβ and rαβ in the space L2(ω), where ω is now a domain in R2, are the covariant components of the linearized change of metric and linearized change of curvature tensors associated with a displacement vector field of a surface θ(ω¯), where θ:ω¯→R3 is a smooth immersion.Such compatibility conditions, the expressions of which again depend on the boundary conditions satisfied by the displacement vector field, in turn allow us to reformulate the quadratic minimization problem for a linearly elastic shell in terms of the fields (cαβ) and (rαβ) as the sole unknowns, thus providing an example of an intrinsic approach to shell theory.

Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

In a previous work, it was shown how the linearized strain tensor field e:=12(∇uT+∇u)∈L2(Ω) can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain Ω⊂R3, instead of the displacement vector field u∈H1(Ω) in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition u=0 on a portion Γ0 of the boundary of Ω can be recast, again as boundary conditions on Γ0, but this time expressed only in terms of the new unknown e∈L2(Ω).

Description of Large Deformation Problem Using Means of Visual Strain

The constitutive equation of large deformation problem is closely related to geometric description. Nowadays, linear strain tensor is no longer unsuitable to describe large deformation. However, the existing non-linear strain tensors have complicated forms as well as no apparent geometric or physical meaning. While, the increment method is used to solve, however, convergence and efficiency are low sometimes. Thus the idea of visual strain tensor is proposed, with distinct meaning and visual image. Beside, it is likely to be used in engineering measurement, and it can be connected with measured constitutive equation directly. Thus this research provides a new idea and method for solving large-deformation problems in practical engineering.

Homogenization of processes in nonlinear visco-elastic composites

The constitutive behaviour of a multiaxial visco-elastic material is here represented by the nonlinear relation e − A(x) : ∫ t 0 σ (x, τ ) dτ ∈ α(σ, x), which generalizes the classical Maxwell model of visco-elasticity of fluid type. Here α(·, x) is a (possibly multivalued) maximal monotone mapping, σ is the stress tensor, e is the linearized strain tensor, and A(x) is a positive-definite fourth-order tensor. The above inclusion is here coupled with the quasi-static force-balance law, − div σ = # f . Existence and uniqueness of the weak solution are proved for a boundary-value problem. Convergence to a two-scale problem is then derived for a composite material, in which the functions α and A periodically oscillate in space on a short length-scale. It is proved that the coarse-scale averages of stress and strain solve a single-scale homogenized problem, and that conversely any solution of this problem can be represented in that way. The homogenized constitutive relation is represented by the minimization of a time-integrated functional, and is rather different from the above constitutive law. These results are also retrieved via De Giorgi’s notion of %-convergence. These conclusions are at variance with the outcome of so-called analogical models, that rest on an (apparently unjustified) mean-field-type hypothesis. Mathematics Subject Classification (2010): 35B27 (primary); 49J40, 73E50, 74Q (secondary).

LAGRANGE MULTIPLIERS IN INTRINSIC ELASTICITY

In an intrinsic approach to three-dimensional linearized elasticity, the unknown is the linearized strain tensor field (or equivalently the stress tensor field by means of the constitutive equation), instead of the displacement vector field in the classical approach. We consider here the pure traction problem and the pure displacement problem and we show that, in each case, the intrinsic approach leads to a quadratic minimization problem constrained by Donati-like relations (the form of which depends on the type of boundary conditions considered). Using the Babuška-Brezzi inf-sup condition, we then show that, in each case, the minimizer of the constrained minimization problem found in an intrinsic approach is the first argument of the saddle-point of an ad hoc Lagrangian, so that the second argument of this saddle-point is the Lagrange multiplier associated with the corresponding constraints. Such results have potential applications to the numerical analysis and simulation of the intrinsic approach to three-dimensional linearized elasticity.

A Cesàro–Volterra formula with little regularity

If a symmetric matrix field e of order three satisfies the Saint-Venant compatibility conditions in a simply-connected domain Ω in R 3 , there then exists a displacement field u of Ω with e as its associated linearized strain tensor, i.e., e = 1 2 ( ∇ u T + ∇ u ) in Ω. A classical result, due to Cesàro and Volterra, asserts that, if the field e is sufficiently smooth, the displacement u ( x ) at any point x ∈ Ω can be explicitly computed as a function of the matrix fields e and CURL e , by means of a path integral inside Ω with endpoint x. We assume here that the components of the field e are only in L 2 ( Ω ) (as in the classical variational formulation of three-dimensional linearized elasticity), in which case the classical path integral formula of Cesàro and Volterra becomes meaningless. We then establish the existence of a “Cesàro–Volterra formula with little regularity”, which again provides an explicit solution u to the equation e = 1 2 ( ∇ u T + ∇ u ) in this case. We also show how the classical Cesàro–Volterra formula can be recovered from the formula with little regularity when the field e is smooth. Interestingly, our analysis also provides as a by-product a variational problem that satisfies all the assumptions of the Lax–Milgram lemma, and whose solution is precisely the unknown displacement field u . It is also shown how such results may be used in the mathematical analysis of “intrinsic” linearized elasticity, where the linearized strain tensor e (instead of the displacement vector u as is customary) is regarded as the primary unknown.

CESÀRO–VOLTERRA PATH INTEGRAL FORMULA ON A SURFACE

If a symmetric matrix field e = (eij) of order three satisfies the Saint-Venant compatibility relations in a simply-connected open subset Ω of ℝ3, then e is the linearized strain tensor field of a displacement field v of Ω, i.e. [Formula: see text] in Ω. A classical result, due to Cesàro and Volterra, asserts that, if the field e is smooth, the unknown displacement field v(x) at any point x ∈ Ω can be explicitly written as a path integral inside Ω with endpoint x, and whose integrand is an explicit function of the functions eij and their derivatives. Now let ω be a simply-connected open subset in ℝ2 and let θ : ω → ℝ3 be a smooth immersion. If two symmetric matrix fields (γαβ) and (ραβ) of order two satisfy appropriate compatibility relations in ω, then (γαβ) and (ραβ) are the linearized change of metric and change of curvature tensor field corresponding to a displacement vector field η of the surface θ(ω). We show here that a "Cesàro–Volterra path integral formula on a surface" likewise holds when the fields (γαβ) and (ραβ) are smooth. This means that the displacement vector η(y) at any point θ(y), y ∈ ω, of the surface θ(ω) can be explicitly computed as a path integral inside ω with endpoint y, and whose integrand is an explicit function of the functions γαβ and ραβ and their covariant derivatives. Such a formula has potential applications to the mathematical analysis and numerical simulation of linear "intrinsic" shell models.

Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl—Reuss model of elastoplasticity

This paper deals with processes in nonlinear inelastic materials whose constitutive behaviour is represented by the inclusionhere we denote by σ the stress tensor, by ε the linearized strain tensor, by B(x) the compliance tensor and by ∂ϕ(·, x) the subdifferential of a convex function ϕ(·, x). This relation accounts for elasto-viscoplasticity, including a nonlinear version of the classical Maxwell model of viscoelasticity and the Prandtl—Reuss model of elastoplasticity.The constitutive law is coupled with the equation of continuum dynamics, and well-posedness is proved for an initial- and boundary-value problem. The function ϕ and the tensor B are then assumed to oscillate periodically with respect to x and, as this period vanishes, a two-scale model of the asymptotic behaviour is derived via Nguetseng's notion of two-scale convergence. A fully homogenized single-scale model is also retrieved, and its equivalence with the two-scale problem is proved. This formulation is non-local in time and is at variance with that based on so-called analogical models that rest on a mean-field-type hypothesis.

Strongly orthotropic continuum mechanics and finite element treatment

AbstractDespite successful application to orthotropic analysis, any Lagrangian strain tensor that is symmetric can be classified as an isotropic metric, while the infinitely orthotropic case can be accurately dealt with using one‐dimensional elements, structural tensors or kinematic constraints. In this paper, we present a strongly orthotropic continuum mechanics basis that models the exact kinematic behavior of the intermediary class of materials and also show its application to multi‐axial media and treatment using the finite element method. By asserting that mechanistic strain metrics must be material property dependent and satisfy equilibrium, we are able to derive a novel orthotropic linear strain tensor that is asymmetric and thus capable of describing all levels of orthotropy, while maintaining generality to the well‐established isotropic approach. Subsequently formulated are a material principal rotation tensor, extended orthotropic compliance tensor and an extended Mohr's plot for strain relying on an additional metric denoted as aspectual strain. Using the developed finite element formulation, it is shown that identical stress results to conventional theory for an orthotropic linear problem are predicted, while offering a more informative analysis. A second numerical example demonstrates the unique capability of this approach to solve the erroneous response of strongly orthotropic materials under trellis shear as compared with a number of conventional and contemporary approaches and thus its ability to produce kinematically exact results. Copyright © 2008 John Wiley & Sons, Ltd.

Homogenization of nonlinear visco-elastic composites

Quasi-static processes in nonlinear visco-elastic materials of solid-type are here represented by the system: (∗) σ − B ( x ) : ∂ ε ∂ t ∈ β ( ε , x ) , − div σ = f → , coupled with initial and boundary conditions. Here σ denotes the stress tensor, ε the linearized strain tensor, B ( x ) the viscosity tensor, β ( ⋅ , x ) a (possibly multi-valued) maximal monotone mapping, and f → an applied load. Existence and uniqueness of the weak solution are proved. A composite material in which the data β and B rapidly oscillate in space is then considered, and a two-scale model is derived via Nguetseng's notion of two-scale convergence. Although neither the stress nor the strain need be mesoscopically uniform, it is proved that their coarse-scale averages solve a global-in-time single-scale homogenized problem ( upscaling). From any solution of the latter a solution of the two-scale problem is then reconstructed ( downscaling). These results are at variance with the outcome of so-called analogical models, that assume a mean-field-type hypothesis. Finally, we represent the system (∗) as a minimum problem, and interpret the above results in terms of two- and single-scale Γ-convergence.

Linear strain tensor and differential geometry

The components of the linear strain tensor in spherical coordinates are calculated using three methods. The first is a change of basis method and does not require elaborate concepts. The second requires the concept of covariant derivative, and the third requires the concept of a Lie derivative. The calculation provides an opportunity to become familiar with both tensorial methods and associated computational tools in the context of an elementary physical situation. The components of the linear strain tensor in an arbitrary curvilinear orthonormal basis are interpreted geometrically as half the sum of two scalar products.

Recovery of a displacement field from its linearized strain tensor field in curvilinear coordinates

We establish that, if a symmetric matrix field defined over a simply-connected open set satisfies the Saint Venant equations in curvilinear coordinates, then its coefficients are the linearized strains associated with a displacement field. Our proof provides an explicit algorithm for recovering such a displacement field, which may be viewed as the linear counterpart of the reconstruction of an immersion from a given flat Riemannian metric. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Characterization of the kernel of the operator CURL CURL

In a simply-connected domain Ω in R 3 , the kernel of the operator CURL CURL acting on symmetric matrix fields from L s 2 ( Ω ) to H s −2 ( Ω ) coincides with the space of linearized strain tensor fields. For not simply-connected domains, Volterra has characterized this kernel for smooth fields. Here we extend this result for domains with a Lipschitz-continuous boundary for fields in L s 2 ( Ω ) . To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Homogenization of the nonlinear Kelvin–Voigt model of viscoelasticity and of the Prager model of plasticity

Denoting by \(\sigma\) the stress tensor, by \(\varepsilon\) the linearized strain tensor, by A the elasticity tensor, and assuming that \(\varphi\) is a convex potential, the inclusion \(\partial \varepsilon / \partial t \in \partial \varphi(\sigma - A : \varepsilon)\) accounts for nonlinear viscoelasticity, and encompasses both the linear Kelvin–Voigt model of solid-type viscoelasticity and the Prager model of rigid plasticity with linear kinematic strain-hardening. This relation is assumed to represent the constitutive behavior of a space-distributed system, and is here coupled with the dynamical equation. An initial- and boundary-value problem is formulated, and the existence and uniqueness of the solution are proved via classical techniques based on compactness and monotonicity. A composite material is then considered, in which the function \(\varphi\) and the tensor A rapidly oscillate in space. A two-scale model is derived via Nguetseng’s notion of two-scale convergence. This provides a detailed account of the mesoscopic state of the system. Any dependence on the fine-scale variable is then eliminated, and the existence of a solution of a new single-scale macroscopic model is proved. The final outcome is at variance with the nonlinear extension of the generalized Kelvin–Voigt model, which is based on an apparently unjustified mean-field-type hypothesis.

On the characterizations of matrix fields as linearized strain tensor fields

Saint Venant's and Donati's theorems constitute two classical characterizations of smooth matrix fields as linearized strain tensor fields. Donati's characterization has been extended to matrix fields with components in L 2 by T.W. Ting in 1974 and by J.J. Moreau in 1979, and Saint Venant's characterization has been extended likewise by the second author and P. Ciarlet, Jr. in 2005. The first objective of this paper is to further extend both characterizations, notably to matrix fields whose components are only in H −1 , by means of different, and to a large extent simpler and more natural, proofs. The second objective is to show that some of our extensions of Donati's theorem allow to reformulate in a novel fashion the pure traction and pure displacement problems of linearized three-dimensional elasticity as quadratic minimization problems with the strains as the primary unknowns. The third objective is to demonstrate that, when properly interpreted, such characterizations are “matrix analogs” of well-known characterizations of vector fields. In particular, it is shown that Saint Venant's theorem is in fact nothing but the matrix analog of Poincaré's lemma.

On Saint Venant's compatibility conditions and Poincaré's lemma

Saint Venant's theorem constitutes a classical characterization of smooth matrix fields as linearized strain tensor fields. This theorem has been extended to matrix fields with components in L 2 by the second author and P. Ciarlet, Jr. in 2005. One objective of this Note is to further extend this characterization to matrix fields whose components are only in H −1 . Another objective is to demonstrate that Saint Venant's theorem is in fact nothing but the matrix analog of Poincaré's lemma. To cite this article: C. Amrouche et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

New formulations of linearized elasticity problems, based on extensions of Donati's theorem

The classical Donati theorem is used for characterizing smooth matrix fields as linearized strain tensor fields. In this Note, we give several generalizations of this theorem, notably to matrix fields whose components are only in H −1 . We then show that our extensions of Donati's theorem allow to reformulate in a novel fashion linearized three-dimensional elasticity problems as quadratic minimization problems with the strains as the primary unknowns. To cite this article: C. Amrouche et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

On free energy-based formulations for kinematic hardening and the decomposition F = fpfe

Within the framework of linear plasticity, based on additive decomposition of the linear strain tensor, kinematical hardening can be described by means of extended potentials. The method is elegant and avoids the need for evolution equations. The extension of small strain formulations to the finite strain case, which is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, proved not straight forward. Specifically, the symmetry of the resulting back stress remained elusive. In this paper, a free energy-based formulation incorporating the effect of kinematic hardening is proposed. The formulation is able to reproduce symmetric expressions for the back stress while incorporating the multiplicative decomposition of the deformation gradient. Kinematic hardening is combined with isotropic hardening where an associative flow rule and von Mises yield criterion are applied. It is shown that the symmetry of the back stress is strongly related to its treatment as a truly spatial tensor, where contraction operations are to be conducted using the current metric. The latter depends naturally on the deformation gradient itself. Various numerical examples are presented.

Generalized Korn's inequality and conformal Killing vectors

Korn's inequality plays an important role in linear elasticity theory. This inequality bounds the norm of the derivatives of the displacement vector by the norm of the linearized strain tensor. The kernel of the linearized strain tensor are the infinitesimal rigid-body translations and rotations (Killing vectors). We generalize this inequality by replacing the linearized strain tensor by its trace free part. That is, we obtain a stronger inequality in which the kernel of the relevant operator are the conformal Killing vectors. The new inequality has applications in General Relativity.

Reissner–Mindlin–Von Kármán type plate model for nonlinear analysis of laminated composite structures

A plate model for analysis and design of laminated composite structures subjected to large deflections is considered. The model is based on a facet approximation of the midsurface and Reissner–Mindlin–Von Kármán type plate equations. The potential energy functional where the linearized strain tensor has been replaced by nonlinear functions, i.e., the Von Kármán model for large deformation is formulated. The nonlinear equations are solved iteratively by Riks’ method with Crisfield’s elliptical constraint for arc length. The linearized equations are discretized by the finite element method. Load–displacement behavior of the structure as well as failure prediction and identification of critical areas of the FE-model are illustrated with a numerical example.