Linearized Strain Tensor Research Articles

An iteration method to study nonlinear wave propagation for a non-Green elastic 1D bar

Abstract The problem of propagation of nonlinear waves in a 1D bar is studied, wherein the linearized strain tensor is considered as a function of the Cauchy stress tensor. Specifically, two constitutive equations for non-Green elastic solids are investigated, introducing a novel numerical iterative method capable of obtaining approximate solutions of one nonlinear constitutive equation for rock, and one constitutive equation that shows a strain-limiting behaviour. The numerical results are compared with exact solutions for the case of a linearized elastic solid.

Heat Conduction of Micropolar Solids Sensitive to Mirror Reflections of Three-Dimensional Space

This article considers a variant of the heat conduction theory of thermal conductivity, in which the heat flux pseudovector has a weight of 1. The pseudoinvariants associated to the heat flux pseudovector are sensitive to mirror reflections and inversions of threedimensional space. The primary purpose of the study was to find a heat flux vector that is algebraically equivalent to the microrotation pseudovector and to measure elementary volumes and areas using pseudoinvariants that are sensitive to mirror reflections. To represent spinor displacements, a contravariant microrotation pseudovector with a weight of +1 was selected. Thus, the heat flux and mass density were expressed as odd-weight pseudotensors. The Helmholtz free energy per unit doublet pseudoinvariant volume was employed as the thermodynamic state potential of the following functional arguments: absolute temperature, symmetric parts, and accompanying vectors for the linear asymmetric strain tensor and the wryness pseudotensor. The results obtained show that the thermal conductivity coefficient and heat capacity of elastic micropolar solids are pseudoscalars of odd weight, indicating their sensitivity to mirror reflections.

Approximate determination of shear stresses for a 2D beam made of a non-Green elastic solid

Shear stresses for a two-dimensional (2D) beam are calculated modifying the classical method developed by Jouravski, for the case the linearized strain tensor is assumed to be a nonlinear function of the Cauchy stresses. Two problems are studied, namely, the case of a cantilever beam with a point load on its free edge (considering a rectangular and a circular cross-section) and the three-point flexural test for a beam of rectangular cross-section. Numerical results are obtained for the particular case of a bimodular constitutive model for rock, and the results for the shear stresses are compared with the predictions of the classical theory of strength of materials for such problems, assuming a linearized elastic body.

Analysis of the propagation of small-amplitude waves in nonlinear elastic solids for problems with infinitesimal strains

A relatively new type of constitutive equation for elastic bodies, wherein the linearized strain tensor is a function of the Cauchy stress tensor, is studied for the particular case the stresses are divided into two parts. One is an initial time-independent stress to which is added an incremental (small in comparison with that initial stress) time-dependent stress. Incremental equations are derived that can be used to study the propagation of small-amplitude waves. The behaviour of such waves is studied for three boundary value problems, where the distributions of the initial stresses are inhomogeneous. The problems are: the inflation of a hollow sphere, the inflation of an infinitely long cylindrical annulus, and the circumferential shear and inflation of a cylindrical annulus. The behaviour of the small-amplitude waves is compared for different external loads, investigating how the initial stress affects such waves. A nonlinear model for a class of isotropic dry elastic rock is considered.

A novel nonlinear constitutive model for rock: Numerical assessment and benchmarking

In this article, we assess and benchmark a novel nonlinear constitutive relation for modeling the behavior of rock, in which the linearized strain tensor is a function of the Cauchy stress tensor. In stark contrast with the linearized theory of elasticity, the main feature of this novel nonlinear constitutive model is that a different behavior is obtained in compression than in tension, which is consisting with the experimental evidence. Four problems are solved using the finite element method: the compression of a cylinder, the biaxial compression of a slab with a circular hole and with an elliptic hole, and the shear of a slab with an elliptic hole. The results are compared with the predictions of the linearized theory of elasticity. In this comparison, it is found that the maximum stresses and their locations are significantly affected by the choice of the constitutive equation.

A spectral approach for nonlinear transversely isotropic elastic bodies, for a new class of constitutive equation: Applications to rock mechanics

A constitutive equation is provided for nonlinear transversely isotropic elastic solids, wherein the linearized strain tensor is assumed to be a function of the Cauchy stress tensor; this elastic constitutive equation belongs to a subclass of a more general set of implicit constitutive relations proposed in the recent years. The proposed constitutive equation is valid for both compressible and incompressible bodies and can be simply modified, to exclude the mechanical influence of compressed fibres and to model inextensible fibres. A crude specific constitutive model is proposed to compare with a uniaxial experimental data on Marcellus shale. Some simple boundary value problems are analyzed.

Origin of stabilization of macrotwin boundaries in martensites

The origin of stabilization of complex microstructures along macrotwin boundaries in martensites is explained by comparing two models based on Ginzburg-Landau theory. The first model incorporates a geometrically nonlinear strain tensor to ensure that the Landau energy is invariant under rigid body rotations, while the second model uses a linearized strain tensor under the assumption that deformations and rotations are small. We show that the approximation in the second model does not always hold for martensites and that the experimental observations along macrotwin boundaries can only be reproduced by the geometrically nonlinear (exact) theory.

An efficient general curvilinear coordinates finite element method for the linear dynamic study of thickness-independent shells

Abstract yyTo date, a large number of finite element methods have been developed to study the dynamics of shell structures. Most of them are generally based on the degenerated solid approach and other less in shell theories, but introducing, in this last case, some assumptions to analyze this problem: some of them refer to shallow shells (slightly curved shells), or consider thin shells neglecting shear deformation, or dispense some terms in their stress-strain developments like the off-diagonal components of the curvature or metric tensors (orthogonal coordinates). In the present work, we present an improved finite element method for the linear dynamic analysis of shells, from thin to moderately thick and thick shells, developed in general curvilinear coordinates, based on a refined shear deformation shell theory and free of the well-known shear locking effect. Exact constitutive equations, including higher order moments-strains relations, are also deduced for the adequate analysis of thick shells. To circumvent the shear locking problems, the mixed interpolation of the tensorial components (MITC) of the linear strain tensor is used. An exhaustive study of different surfaces is performed, especially in doubly curved shells, and interesting conclusions of the higher order modes of vibration and the strain energy of the element are derived. Other desirable features like a low computational effort, a straightforward extension to nonlinear formulation and applications for composite shells are found in this novel and general formulation. Very good results in the proposed practical cases have been found.

The intrinsic theory of linearly elastic plates

In an intrinsic approach to a problem in elasticity, the only unknown is a tensor field representing an appropriate ‘measure of strain’, instead of the displacement vector field in the classical approach. The objective of this paper is to study the displacement traction problem in the special case where the elastic body is a linearly elastic plate of constant thickness, clamped over a portion of its lateral face. In this respect, we first explicitly compute the intrinsic three-dimensional boundary condition of place in terms of the Cartesian components of the linearized strain tensor field, thus avoiding the recourse to covariant components in curvilinear coordinates and providing an interesting example of actual computation of an intrinsic boundary condition of place in three-dimensional elasticity. Second, we perform a rigorous asymptotic analysis of the three-dimensional equations as the thickness of the plate, considered as a parameter, approaches zero. As a result, we identify the intrinsic two-dimensional equations of a linearly elastic plate modelled by the Kirchhoff–Love theory, with the linearized change of metric and change of curvature tensor fields of the middle surface of the plate as the new unknowns, instead of the displacement field of the middle surface in the classical approach.

An Implicit SPH Formulation for Incompressible Linearly Elastic Solids

AbstractWe propose a novel smoothed particle hydrodynamics (SPH) formulation for deformable solids. Key aspects of our method are implicit elastic forces and an adapted SPH formulation for the deformation gradient that—in contrast to previous work—allows a rotation extraction directly from the SPH deformation gradient. The proposed implicit concept is entirely based on linear formulations. As a linear strain tensor is used, a rotation‐aware computation of the deformation gradient is required. In contrast to existing work, the respective rotation estimation is entirely realized within the SPH concept using a novel formulation with incorporated kernel gradient correction for first‐order consistency. The proposed implicit formulation and the adapted rotation estimation allow for significantly larger time steps and higher stiffness compared to explicit forms. Performance gain factors of up to one hundred are presented. Incompressibility of deformable solids is accounted for with an ISPH pressure solver. This further allows for a pressure‐based boundary handling and a unified processing of deformables interacting with SPH fluids and rigids. Self‐collisions are implicitly handled by the pressure solver.

Correction to ‘On a new class of electroelastic bodies. I’

Proc. R. Soc. A 469 , 20120521. (Published online November 14, 2012) ([doi:10.1098/rspa.2012.0521][1]) In §3c of [1] (Case 2), the linearized strain tensor e is given as (see eqn (3.21) therein) e = f ^ ( τ , E , a ) , where τ is the total stress tensor, E is the electric field and a is a direction field such that ∥ a ∥ = 1 (the direction where the body is transversely isotropic). There are two missing terms in eqn (3.22) of that paper for the explicit representation for e (see [2]). The … [1]: http://dx.doi.org/10.1098/rspa.2012.0521

A note on the wave equation for a class of constitutive relations for nonlinear elastic bodies that are not Green elastic

A class of constitutive relations for elastic bodies has been proposed recently, where the linearized strain tensor is expressed as a nonlinear function of the stress tensor. Considering this new type of constitutive equation, the initial boundary value problem for such elastic bodies has been expressed only in terms of the stress tensor. In this communication, this new type of nonlinear wave equation is studied for the case of a one-dimensional straight bar. Conditions for the existence of the travelling wave solutions are given and some self-similar solutions are obtained.

A note on incremental equations for a new class of constitutive relations for elastic bodies

Some new classes of constitutive relations for elastic bodies have been proposed in the literature, wherein the stresses and strains are obtained from implicit constitutive relations. A special case of the above relations corresponds to a class of constitutive equations where the linearized strain tensor is given as a nonlinear function of the stresses. For such constitutive equations we consider the problem of decomposing the stresses into two parts: one corresponds to a time-independent solution of the boundary value problem, plus a small (in comparison with the above) time-dependent stress tensor. The effect of this initial time-independent stress in the propagation of a small wave motion is studied for an infinite medium.

A finite element analysis of some boundary value problems for a new type of constitutive relation for elastic bodies

Recently, there has been interest in the study of a new class of constitutive relation, wherein the linearized strain tensor is assumed to be a function of the stresses. In this communication, some boundary value problems are solved using the finite element method and the solid material being described by such a constitutive relation, where the stresses can be arbitrarily ‘large’, but strains remain small. Three problems are analyzed, namely the traction of a plate with hyperbolic boundaries, a plate with a point load, and the traction of a plate with an elliptic hole. The results for the stresses and strains are compared with the predictions that are obtained by using the constitutive equation of the classical linearized theory of elasticity.

Growing avascular tumours as elasto-plastic bodies by the theory of evolving natural configurations

The aim of this article is to propose a simple way of describing a tumour as a linear elastic material from a reference configuration that is continuously evolving in time due to growth and remodelling. The main assumption allowing this simplification is that the tumour mass is a very ductile material, so that it can only sustain moderate stresses while the deformation induced by growth, that can actually be quite big, mainly induces a plastic reorganisation of malignant cells. In mathematical terms this means that the deformation gradient can be split into a volumetric growth term, a term describing the reorganisation of cells, and a term that can be approximated by means of the linear strain tensor. A dimensional analysis of the importance of the different terms also allows to introduce a second simplification consisting of decoupling the equations describing the growth of the tumour mass from those describing the flow of the interstitial fluid.

Revisiting the Link between Lipid Membrane Elasticity and Microscopic Continuum Models

Fluid lipid membranes can be described by a continuum-elastic Hamiltonian that features two central parameters mean and Gaussian curvature modulus. Of those two, the Gaussian modulus is much less understood, since it affects the membrane energetics only through topology or boundaries and is thus difficult to measure experimentally. Moreover, recent work by Hu et al. [Biophys. J. 102, 1403 (2012)] to determine this modulus from computational studies revealed discrepancies with more microscopic expectations, according to which this modulus should be given by the second moment of the lateral trans-membrane stress profile. Our goal in the present study is to revisit the arguments linking the Gaussian modulus to properties of thin sheets, using thin plate theory as a starting point, but allowing for a number of generalizations. For instance, membranes are more generally described as anisotropic continua with in-plane fluidity and a nontrivial distribution of pre-stresses. It is easy to see that in this case linear elasticity, combined with the additional but common approximation of a linear strain tensor, will predict that the Gaussian curvature modulus vanishes. We therefore need to include the possibility of nonlinear strains (while still using linear constitutive equations), but this in turn leads to some subtle inconsistencies within Monge gauge. Another difficulty is that virtually all existing theories treat Young's modulus and the Poisson ratio as constant throughout the width of the membrane. Starting from fundamental linear elasticity theory, amended by a nonlinear strain tensor, and using consistent geometrical constructions, we then investigate the elastic properties of membranes, especially the Gaussian curvature modulus.

A note on some new classes of constitutive relations for elastic bodies

The class of elastic bodies, that is bodies incapable of dissipation in whatever motion that they undergo, has been significantly enlarged recently (see Rajagopal 2003, On implicit constitutive theories. Appl. Math., 48, 279–319; Rajagopal 2007, The elasticity of elasticity. Z. Angew. Math. Phys. 58, 309–317; Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids. Proc. R. Soc. Lond. A, 463, 357–367). The new classes include fully implicit constitutive relations for the stress and the deformation gradient, and the interesting sub-class wherein the Cauchy–Green tensor or the linearized strain tensor bears a non-linear relationship to the stress. While a fully thermodynamic treatment of such elastic bodies, when defined through implicit constitutive relations between the Piola stress and the Green–St. Venant strain, within a 3D framework has been carried out (see Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids, Proc. R. Soc. Lond. A, 463, 357–367), other possible implicit relationships between other stress and kinematic measures have not been investigated. This paper is devoted to the determination of the consequences of thermodynamics on the new class of elastic bodies, when they are expressed through implicit relations for different stress and stretch/strain measures.

Direct determination of stresses from the stress equations of motion and wave propagation for a new class of elastic bodies

For a new class of elastic bodies, where the linearized strain tensor is given as a function of the Cauchy stress tensor, the problem of considering unsteady motions is studied. A system of partial differential equations that only depends on the stress tensor is found from the equation of motion, which is a system of six partial differential equations for the six components of the stress tensor. A simple boundary value problem is solved for a 1D bar using exact and numerical methods.

Modeling fracture in the context of a strain-limiting theory of elasticity: A single plane-strain crack

Implicit constitutive relations afford a logically consistent framework for formulating strain-limiting, nonlinear elastic constitutive models utilizing the classical linearized strain tensor. This is in marked contrast to traditional Cauchy or Green elastic formulations which give rise to customary linear elasticity in the infinitesimal strain limit. Within this strain-limiting elastic constitutive setting, the present paper investigates the asymptotic behavior of the stress field at the tip of a straight plane-strain fracture. It is shown that within the general class of crack-tip asymptotic expansions considered, the only cases satisfying the required boundary conditions correspond to bounded stresses. This is in agreement with previous results for the corresponding anti-plane shear fracture problem.

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.