Venant Strain Tensor Research Articles

Residual stresses for a new class of transversely isotropic nonlinear elastic solid

We study the response of a class of transversely elastic bodies, wherein the Green–Saint Venant strain tensor is a function of the second Piola–Kirchhoff stress tensor, when the body is residually stressed. The notion of such non-Cauchy elastic bodies being transversely isotropic is defined in Rajagopal (Mech. Res. Commun. 64, 2015, 38–41), and by a body being residually stressed, we mean the interior of the body is not in a stress-free state although the boundary is free of traction as considered by Coleman and Noll (Arch. Ration. Mech. Anal. 15, 1964, 87–111) and by Hoger (Arch. Ration. Mech. Anal. 88, 1985, 271–289).

Dynamics of pull and release of graphene nanoribbons

The pull and release molecular dynamics simulations of graphene nanoribbons (GNR) lead to their length dependent several folded conformations in the ground states at finite temperatures. The folding occurs when the potential energy of the stretched GNR is sufficient to overcome the energy barriers between the flat GNR and its folded conformations. We explore the dynamics of GNRs during their pull and release as the flat GNR develops intrinsic ripples as soon as thermal fluctuations are introduced and rich mechanical phenomena are found even before the folding process starts. Nudged elastic band calculations were performed to understand the nature of energy barriers and the minimum energy paths between the flat and the rippled nanoribbons. The potential and the kinetic energies of nanoribbons show strong oscillatory characters with attenuation in amplitudes at the start and the end of the pulling process, but as soon as the release process starts the oscillatory nature becomes weak giving rise to small fluctuations. Both thermostatted and unthermostatted nanoribbons were studied with some important observations. Different equivalent continuum models were analyzed which can mimic these experiments. It has been found that the pull and release of the GNRs without any intrinsic rippling can be successfully simulated with a thin plate continuum model using Hooke’s law which does not account for geometric nonlinearities and therefore is aptly able to capture small deformations of the flat GNR. This relation fails when the GNR develops ripples and then undergoes through the pull and release experiments. The rippled GNR was modeled by introducing ripples in the flat continuum GNR by nonlinear curve fitting of the atomistic z-coordinates with a sinusoidal function. Then three different constitutive relations were tried and it was found that modeling GNRs as St. Venant–Kirchhoff materials, which assume a linear relation between the second Piola–Kirchhoff stress tensor and the Green-St. Venant strain tensor correctly reproduce the atomistic behavior before the folding starts. The temperature results were also reproduced by solving the energy balance equation where a thermoelastic constitutive relation between the second Piola–Kirchhoff stress tensor and the Green-St. Venant strain tensor subtracted by the thermal strain tensor was assumed.

Variational differential quadrature: A technique to simplify numerical analysis of structures

• Proposing a novel solution methodology called VDQ. • Presenting a highly accurate integral operator using Taylor series and DQ rule. • Presenting a matrix representation for the nonlinear Green-St Venant strain tensor. • Presenting symmetric formulation for equations arising from the fully nonlinear elasticity theory. • Unified and standard formulating the nonlinear discretized equations independent of material type, structure type and even selected theory. A new solution method in the area of computational mechanics is developed in this article, which is called variational differential quadrature (VDQ). The main idea of this method is based on the accurate and direct discretization of the energy functional in the structural mechanics. In the VDQ method, through developing an efficient matrix formulation and using an accurate integral operator, the discretized governing equations are derived directly from the weak form of the equations with no need for the analytical derivation of the strong form. This technique provides an alternative way to discretize the energy functional, which avoids the local interpolation and the assembly process of the methods of this kind. We first implement the VDQ method for the nonlinear elasticity theory considering the Green-St. Venant strain tensor; then we simplify the formulation further for the first-order shear deformable beam and plate theories. The final formulation of these cases demonstrates the simplicity of the implementation for the VDQ method in the numerical analysis of the structures, which is a major goal for this article. Using these examples, one can easily learn and apply this technique to other structures. To assess the performance of the VDQ method, we compare it with the generalized differential quadrature (GDQ) method and finite element method (FEM) in the case of bending analysis of Mindlin plates. It is indicated that computational cost of VDQ is less than that of GDQ, and the convergence rate of VDQ is faster than that of FEM.

A Nonlinear Korn Inequality Based on the Green-Saint Venant Strain Tensor

A nonlinear Korn inequality based on the Green-Saint Venant strain tensor is proved, whenever the displacement is in the Sobolev space $W^{1,p}$, $p\geq 2$, under Dirichlet conditions on a part of the boundary. The inequality can be useful in proving the coercivity of a nonlinear elastic energy.

Nonlinear finite element analysis of functionally graded structures by enhanced assumed strain shell elements

In this work, large elastic deformation of structures made of functionally graded materials (FGMs) is investigated. Due to highly nonlinear nature of the problem, a total Lagrangian finite element formulation for a seven-parameter shell model is developed. The model accounts explicitly for thickness change, as well as normal stress and strain components through the thickness of the structure. The Green–Saint Venant strain tensor in conjunction with the Kirchhoff–Saint Venant constitutive equation is used to capture the geometric nonlinearity. To circumvent locking phenomenon at large distortions, an enhanced assumed strain formulation is adopted. Five different examples have been analyzed and the effect of enhanced strains and that of FGM exponent is investigated.

On Monotonicity for Strain-Limiting Theories of Elasticity

In a previous paper, we studied strong ellipticity for explicit strain-limiting theories of elasticity where the Green-St. Venant strain tensor is defined as a nonlinear response function of the second Piola-Kirchhoff stress tensor. The approach to strong ellipticity studied in that paper requires that the Frechet derivative of the response function be invertible as a fourth-order tensor. In this paper, a weaker convexity notion is introduced in the case that the Frechet derivative of the response function either fails to exist or is not invertible. We generalize the classical notion of monotonicity to a class of nonlinear strain-limiting models. It is shown that the generalized monotonicity holds for sufficiently small Green-St. Venant strains and fails (through demonstration by counterexample) when the small strain constraint is relaxed.

Delamination in sandwich panels due to local water slamming loads

We study delamination in a sandwich panel due to transient finite plane strain elastic deformations caused by local water slamming loads and use the boundary element method to analyze motion of water and the finite element method to determine deformations of the panel. The cohesive zone model is used to study delamination initiation and propagation. The fluid is assumed to be incompressible and inviscid, and undergo irrotational motion. A layer-wise third order shear and normal deformable plate/shell theory is employed to simulate deformations of the panel by considering all geometric nonlinearities (i.e., all non-linear terms in strain–displacement relations) and taking the panel material to be St. Venant–Kirchhoff (i.e., the second Piola–Kirchhoff stress tensor is a linear function of the Green–St. Venant strain tensor). The Rayleigh damping is introduced to account for structural damping that reduces oscillations in the pressure acting on the panel/water interface. Results have been computed for water entry of (i) straight and circular sandwich panels made of Hookean materials with and without consideration of delamination failure, and (ii) flat sandwich panels made of the St. Venant–Kirchhoff materials. The face sheets and the core of sandwich panels are made, respectively, of fiber reinforced composites and soft materials. It is found that for the same entry speed (i) the peak pressure for a curved panel is less than that for a straight panel, (ii) the consideration of geometric nonlinearities significantly increases the peak hydrodynamic pressure, (iii) delamination occurs in mode-II, and (iv) the delamination reduces the hydroelastic pressure acting on the panel surface and hence alters deformations of the panel.

Analysis of post-buckling and delamination in laminated composite St. Venant–Kirchhoff beams using CZM and layer-wise TSNDT

A layer-wise third order shear and normal deformable plate/shell theory (TSNDT) incorporating a cohesive zone model (CZM) is used to study the initiation and growth of delamination in straight and curved laminated beams. Upon satisfaction of the delamination criteria at a point on the interface between two layers, displacements there of two abutting points on the interface between the two layers are made discontinuous. Delaminations under mode-I, mode-II and mixed-mode static and transient loadings have been studied. All geometric nonlinearities, including the von Karman nonlinearity, are considered. The material of each layer is assumed to be St. Venant–Kirchhoff for which the second Piola–Kirchhoff stress tensor is a linear function of the Green-St. Venant strain tensor. Example problems studied also include delamination growth during axial buckling of a three-layer beam. It is found that the consideration of inertia forces noticeably delays the buckling load and significantly affects the deformed shape of an axially compressed laminated beam.

Finite deformations of curved laminated St. Venant–Kirchhoff beam using layer-wise third order shear and normal deformable beam theory (TSNDT)

A layer-wise third order shear and normal deformable plate/shell theory (TSNDT) incorporating all geometric nonlinearities is used to study finite transient deformations of a curved laminated beam composed of a St. Venant–Kirchhoff material. In the TSNDT all displacement components of a point are expressed as 3rd order polynomials in the thickness coordinate in each layer while maintaining the displacement continuity across adjoining layers. No shear correction factor is used. Transverse shear and transverse normal stresses are found from the computed displacement fields and the constitutive relation (i.e., no stress recovery technique is employed). For the St. Venant–Kirchhoff material the strain energy density is a quadratic function of the Green-St. Venant strain tensor appropriate for finite deformations. The software based on the finite element method (FEM) capable of solving static and transient nonlinear problems has been verified by using the method of manufactured solutions. Furthermore, results computed with the TSNDT have been found to agree well with those obtained using the commercial software ABAQUS, and C3D20 elements. Significant contributions of the work include developing a TSNDT considering all geometric nonlinearities and a materially objective constitutive relation, using the method of manufactured solutions to verify the numerical solution of transient nonlinear problems, and showing that results from the plate theory agree well with those from the analysis of plane strain nonlinear problems using the finite elasticity theory. Plate problems using the TSNDT can be analyzed with piecewise linear basis functions in the FEM.

A nonlinear Korn inequality with boundary conditions and its relation to the existence of minimizers in nonlinear elasticity

We establish a nonlinear Korn inequality with boundary conditions showing that the H 1 -distance between two mappings from Ω ⊂ R n into R n preserving orientation is bounded, up to a multiplicative constant, by the L 2 -distance between their metrics. This inequality is then used to show the existence of a unique minimizer to the total energy of a hyperelastic body, under the assumptions that the L p -norm of the density of the applied forces is small enough and the stored energy function is bounded from below by a positive definite quadratic function of the Green–Saint Venant strain tensor.

Decomposition of the Deformations of a Thin Shell. Asymptotic Behavior of the Green-St Venant’s Strain Tensor

We investigate the behavior of the deformations of a thin shell, whose thickness $\delta$ tends to zero, through a decomposition technique of these deformations. The terms of the decomposition of a deformation $v$ are estimated in terms of the $L^2$-norm of the distance from $\nabla v$ to $SO(3)$. This permits in particular to derive accurate nonlinear Korn's inequalities for shells (or plates). Then we use this decomposition technique and estimates to give the asymptotic behavior of the Green-St Venant's strain tensor when the {\it ''strain energy''} is of order less than $\delta^{3/2}$.

Decomposition of shell deformations – Asymptotic behavior of the Green–St Venant strain tensor

This Note deals with a new method, based on a decomposition of the deformations, to study thin shells. In particular, we give the asymptotic behavior of the Green–St Venant's strain tensor. To cite this article: D. Blanchard, G. Griso, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Torsion of a Viscoelastic Cylinder

Finite torsional deformations of an incompressible viscoelastic circular cylinder are studied with its material modeled by two constitutive relations. One of these is a linear relation between the determinate part of the second Piola-Kirchhoff stress tensor and the time history of the Green-St. Venant strain tensor, and the other a linear relation between the deviatoric Cauchy stress tensor and the left Cauchy-Green tensor, its inverse, and the time history of the relative Green-St. Venant strain tensor. It is shown that the response predicted by the latter constitutive relation is in better agreement with the test data, and this constitutive relation is used to compute energy dissipated during torsional oscillations of the cylinder. [S0021-8936(00)00502-X]

Static equilibrium of hyperelastic thin shell: symbolic and numerical computation

We here examine the natural shapes of an hyperelastic thin shell called a Carpentier's joint, when the terminal position is known. More specifically we study a rectangular strip that is a flexible thin shell with a constant curvature in its width and a null curvature in its length, at its unconstrained state. We use the theory of large displacement and small strain for hyperelastic material. We first consider an appropriate parameterization of the joint. Then we compute the Green-St Venant strain tensor with a symbolic computation system and we generate the numerical code to compute the elastic energy. In particular, we make strong use of symbolic elements to resolve some problems with zero division. Numerical minimization of this energy is used to find the shape and a couple of simulation are presented.

Linear constitutive relations in isotropic finite elasticity

For simple shearing and simple extension deformations of a homogeneous and isotropic elastic body, it is shown that a linear relation between the second Piola-Kirchhoff stress tensor and the Green-St. Venant strain tensor does not predict a physically reasonable response of the body. This constitutive relation implies that the slope of the curve between an appropriate component of the first Piola-Kirchhoff stress tensor and a deformation measure is an increasing functions of the deformation measure.

静的引張り‐ねじり組合せ荷重下におけるＦＲＰ材の強度および応力‐ひずみ特性

Static tests under tension/compression-torsion biaxial loading were conducted using tubular FRP specimens reinforced by plain woven glass cloth. Stress-strain curves were observed with a biaxial extensometer developed by the authors. In order to take account of the large deformation which is one of the unique charactaristics of FRP, the strength and S-S curve were evaluated with the Piola-Kirchhoff stress tensor and the Green-Saint Venant strain tensor, and these results were also compared with the S-S curves based on the nominal stress and nominal strain. The effects of normal stress to shear stress ratio (i.e, combined stress ratio) on both strength and steess-strain relations were also investigated. The following results were obtained;(1) For orthotropic materials having large deformation capability, their properties (strength and stress-strain relations) must be evaluated with Piola-Kirchhoff stress tensor and Green-Saint Venant strain tensor.(2) The combined stress ratio has no effect on the stress-strain relation within an elastic range, but it affects not only on the stress level where the yield point appears but also on modulus decay after yielding.

Mathematical characterization of the nonlinear thermorheological behavior of the vascular tissue.

A characterization of the passive nonlinear thermorheological response of incompressible, curvilinearly orthotropic arterial tissue is presented in the framework of modern continuum thermodynamics. The stress tensor, the specific entropy, the specific internal energy and the heat flux vector and expressed as functionals of the histories of local deformation, temperature and the temperature gradient. These functionals are systematically reduced by subjecting them to the requirements of Clausius-Duhem inequality and material frame indifference. The reduced functionals are then specialized to reflect the material frame indifference. The reduced functionals are then specialized to reflect the material symmetry characterizing the tissue by using the histories of the joint invariants of the Green-St. Venant strain tensor and temperature as the independent argument functions. The functionals are expressed in terms of series of multiple integrals and terms upto and including second order integrals are retained. An approach toward experimental determination of the 14 constitutive functions to describe two stress differences is outlined. It is believed that the characterization presented here will provide a rational basis for simpler thermorheological descriptions and experimental programs to include important thermorheologic considerations.