A finite-element discretization of some boundary value problems for nonlinear strain-limiting elastic bodies

It is well known that the linearized theory of elasticity admits the logically inconsistent solution of singular strains when applied to certain naive models of fracture while the theory is a first-order approximation to finite elasticity in the asymptotic limit of infinitesimal displacement gradient. Meanwhile, the strain-limiting models, a special subclass of nonlinear implicit constitutive relations, predict uniformly bounded strain in the whole material body including at the strain-concentrator such as a crack tip or reentrant corner. Such a nonlinear approximation cannot be possible within the standard linearization procedure of either Cauchy or Green elasticity. In this work, we examine a finite-element discretization for several boundary value problems to study the state of stress–strain in the solid body of which response is described by a nonlinear strain-limiting theory of elasticity. The problems of notches, oriented cracks, and an interface crack in anti-plane shear are analyzed. The numerical results indicate that the linearized strain remains below a value that can be fixed a priori, therefore, ensuring the validity of the nonlinear model. In addition, we find high stress values in the neighborhood of the crack tip in every example, thereby suggesting that the crack tip acts as a singular energy sink for a stationary crack. We also calculate the stress intensity factor (SIF) in this study. The computed value of SIF in the nonlinear strain-limiting model is corresponding to that of the classical linear model, and thereby providing a tenet for a possible local criterion for fracture. The framework of strain-limiting theories, within which the linearized strain bears a nonlinear relationship with the stress, can provide a rational basis for developing physically meaningful models to study a crack evolution in elastic solids.

Rigidity of Branching Microstructures in Shape Memory Alloys

We analyze generic sequences for which the geometrically linear energy $$\begin{aligned} E_\eta (u,\chi )\,{:}{=} \,\eta ^{-\frac{2}{3}}\int _{B_{1}\left( 0\right) } \left| e(u)- \sum _{i=1}^3 \chi _ie_i\right| ^2 \, \mathrm {d}x+\eta ^\frac{1}{3} \sum _{i=1}^3 |D\chi _i|({B_{1}\left( 0\right) }) \end{aligned}$$ E η ( u , χ ) : = η - 2 3 ∫ B 1 0 e ( u ) - ∑ i = 1 3 χ i e i 2 d x + η 1 3 ∑ i = 1 3 | D χ i | ( B 1 0 ) remains bounded in the limit $$\eta \rightarrow 0$$ η → 0 . Here $$ e(u) \,{:}{=}\,1/2(Du + Du^T)$$ e ( u ) : = 1 / 2 ( D u + D u T ) is the (linearized) strain of the displacement u, the strains $$e_i$$ e i correspond to the martensite strains of a shape memory alloy undergoing cubic-to-tetragonal transformations and the partition into phases is given by $$\chi _i:{B_{1}\left( 0\right) } \rightarrow \{0,1\}$$ χ i : B 1 0 → { 0 , 1 } . In this regime it is known that in addition to simple laminates, branched structures are also possible, which if austenite was present would enable the alloy to form habit planes. In an ansatz-free manner we prove that the alignment of macroscopic interfaces between martensite twins is as predicted by well-known rank-one conditions. Our proof proceeds via the non-convex, non-discrete-valued differential inclusion $$\begin{aligned} e(u) \in \bigcup _{1\le i\ne j\le 3} {\text {conv}} \{e_i,e_j\}, \end{aligned}$$ e ( u ) ∈ ⋃ 1 ≤ i ≠ j ≤ 3 conv { e i , e j } , satisfied by the weak limits of bounded energy sequences and of which we classify all solutions. In particular, there exist no convex integration solutions of the inclusion with complicated geometric structures.

On an Implicit Model Linear in Both Stress and Strain to Describe the Response of Porous Solids

We study some mathematical properties of a novel implicit constitutive relation wherein the stress and the linearized strain appear linearly that has been recently put into place to describe elastic response of porous metals as well as materials such as rocks and concrete. In the corresponding mixed variational formulation the displacement, the deviatoric and spherical stress are three independent fields. To treat well-posedness of the quasi-linear elliptic problem, we rely on the one-parameter dependence, regularization of the linear-fractional singularity by thresholding, and applying the Browder–Minty existence theorem for the regularized problem. An analytical solution to the nonlinear problem under constant compression/extension is presented.

Efficient Finite Element Formulation for Geometrically Exact Beams

This paper proposes a new approach to the modeling of geometrically exact beams based on motion interpolation schemes. Motion interpolation schemes yield simple expressions for the sectional strains and linearized strain-motion relationships at the mesh nodes. The classical formulation of the finite element method starts from the weak form of the continuous governing equations obtained from a variational principle. Approximations, typically of a polynomial nature, are introduced to express the continuous displacement field in term of its nodal values. Introducing these approximations into the weak form of the governing equations then yields nonlinear discrete that can be solved with the help of a linearization process. In the proposed approach, the order of the first two steps of the procedure is reversed: approximations are introduced in the variational principle directly and the discrete equations of the problem then follow. This paper has shown that for geometrically exact beams, the discrete equations obtained from the two procedure differ significantly: far simpler discrete equations are obtained from the proposed approach.

On nonlinear strain theory for a viscoelastic material model and its implications for calving of ice shelves

ABSTRACTIn the current ice-sheet models calving of ice shelves is based on phenomenological approaches. To obtain physics-based calving criteria, a viscoelastic Maxwell model is required accounting for short-term elastic and long-term viscous deformation. On timescales of months to years between calving events, as well as on long timescales with several subsequent iceberg break-offs, deformations are no longer small and linearized strain measures cannot be used. We present a finite deformation framework of viscoelasticity and extend this model by a nonlinear Glen-type viscosity. A finite element implementation is used to compute stress and strain states in the vicinity of the ice-shelf calving front. Stress and strain maxima of small (linearized strain measure) and finite strain formulations differ by ~ 5% after 1 and by ~ 30% after 10 years, respectively. A finite deformation formulation reaches a critical stress or strain faster, thus calving rates will be higher, despite the fact that the exact critical values are not known. Nonlinear viscosity of Glen-type leads to higher stress values. The Maxwell material model formulation for finite deformations presented here can also be applied to other glaciological problems, for example, tidal forcing at grounding lines or closure of englacial and subglacial melt channels.

On the behaviour of spherical inclusions in a cylinder under tension loads

In the present paper the behaviour of a hyperelastic body is studied, considering the presence of one, two and more spherical inclusions, under the effect of an external tension load. The inclusions are modeled as nonlinear elastic bodies that undergo small strains. For the material constitutive relation, a relatively new type of model is used, wherein the strains (linearized strain) are assumed to be nonlinear functions of the stresses. In particular, it is used a function such that the strains are always small, independently of the magnitude of the external loads. In order to simplify the problem, the hyperelastic medium and the inclusions are modelled as axial-symmetric bodies. The finite element method is used to obtain results for these boundary value problems. The objective of using these new models for elastic bodies in the case of the inclusions is to study the behaviour of such bodies in the case of concentration of stresses, which happens near the interface with the surrounding matrix. From the results presented in this paper, it is possible to observe that despite the relatively large magnitude for the stresses, the strains for the inclusions remain small, which would be closer to the actual behaviour of real inclusions made of brittle materials, which cannot show large strains.

Intrinsic formulation of the displacement-traction problem in linearized elasticity

The displacement-traction problem of linearized elasticity is a system of partial differential equations and boundary conditions whose unknown is the displacement field inside a linearly elastic body. We explicitly identify here the corresponding boundary conditions satisfied by the linearized strain tensor field associated with such a displacement field. Using this identification, we are then able to provide an intrinsic formulation of the displacement-traction problem of linearized elasticity, by showing how it can be recast into a boundary value problem whose unknown is the linearized strain tensor field.

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.

WEAKENED CONDITIONS OF ADMISSIBILITY OF SURFACE FORCES APPLIED TO LINEARLY ELASTIC MEMBRANE SHELLS

We consider a family of linearly elastic shells of the first kind (as defined in [2]), also known as non inhibited pure bending shells [7]. This family is indexed by the half-thickness ε. When ε approaches zero, the averages across the thickness of the shell of the covariant components of the displacement of the points of the shell converge strongly towards the solution of a "2D generalized membrane shell problem" provided the applied forces satisfy admissibility conditions [1,3]. The identification of the admissible applied forces usually requires delicate analysis. In the first part of this paper, we simplify the general admissibility conditions when applied forces h are surface forces only, and obtain conditions that no longer depend on ε [5]: find hαβ = hαβ in L2(ω) such that for all η = (ηi) in V(ω), ∫ω hi ηi dω = ∫ω hαβγαβ(η)dω where ω is a domain of ℝ2, θ is in [Formula: see text] and [Formula: see text] is the middle surface of the shells, where (γαβ (η)) is the linearized strain tensor of S and V(ω) = {η ∈ H1(ω), η = 0 on γ0}, the shells being clamped along Γ0 = θ(γ0). In the second part, since the simplified admissibility formulation does not allow to conclude directly to the existence of hαβ, we seek sufficient conditions on h for hαβ to exist in L2(ω). In order to get them, we impose more regularity to hαβ and boundary conditions. Under these assumptions, we can obtain from the weak formulation a system of PDE's with hαβ as unknowns. The existence of solutions depends both on the geometry of the shell and on the choice of h. We carry through the study of four representative geometries of shells and identify in each case a special admissibility functional space for h.