Showing papers on "Linear elasticity published in 2015"

Phase-field modeling of a fluid-driven fracture in a poroelastic medium

Abstract: In this paper, we present a phase field model for a fluid-driven fracture in a poroelastic medium. In our previous work, the pressure was assumed given. Here, we consider a fully coupled system where the pressure field is determined simultaneously with the displacement and the phase field. To the best of our knowledge, such a model is new in the literature. The mathematical model consists of a linear elasticity system with fading elastic moduli as the crack grows, which is coupled with an elliptic variational inequality for the phase field variable and with the pressure equation containing the phase field variable in its coefficients. The convex constraint of the variational inequality assures the irreversibility and entropy compatibility of the crack formation. The phase field variational inequality contains quadratic pressure and strain terms, with coefficients depending on the phase field unknown. We establish existence of a solution to the incremental problem through convergence of a finite dimensional approximation. Furthermore, we construct the corresponding Lyapunov functional that is linked to the free energy. Computational results are provided that demonstrate the effectiveness of this approach in treating fluid-driven fracture propagation.

Scale{dependent homogenization of random composites as micropolar continua

Abstract: A multitude of composite materials ranging from polycrystals to rocks, concrete, and masonry overwhelmingly display random morphologies. While it is known that a Cosserat (micropolar) medium model of such materials is superior to a Cauchy model, the size of the Representative Volume Element (RVE) of the effective homogeneous Cosserat continuum has so far been unknown. Moreover, the determination of RVE properties has always been based on the periodic cell concept. This study presents a homogenization procedure for disordered Cosserat-type materials without assuming any spatial periodicity of the microstructures. The setting is one of linear elasticity of statistically homogeneous and ergodic two-phase (matrix-inclusion) random microstructures. The homogenization is carried out according to a generalized Hill–Mandel type condition applied on mesoscales, accounting for non-symmetric strain and stress as well as couple-stress and curvature tensors. In the setting of a two-dimensional elastic medium made of a base matrix and a random distribution of disk-shaped inclusions of given density, using Dirichlet-type and Neumann-type loadings, two hierarchies of scale-dependent bounds on classical and micropolar elastic moduli are obtained. The characteristic length scales of approximating micropolar continua are then determined. Two material cases of inclusions, either stiffer or softer than the matrix, are studied and it is found that, independent of the contrast in moduli, the RVE size for the bending micropolar moduli is smaller than that obtained for the classical moduli. The results point to the need of accounting for: spatial randomness of the medium, the presence of inclusions intersecting the edges of test windows, and the importance of additional degrees of freedom of the Cosserat continuum.

A variational approach for a nonlinear 1-dimensional second gradient continuum damage model

Abstract: A 1-dimensional second gradient damage continuum theory is presented within the framework of the variational approach. The action is intended to depend not only with respect to the first gradient of the displacement field and to a scalar damage field, but also to the field of the second gradient of the displacement. Constitutive prescriptions of the stiffness (the constitutive function in front of the squared first gradient term in the action functional) and of the microstructural material length (i.e., the square of the constitutive function in front of the squared second gradient term in the action functional) are prescribed in terms of the scalar damage parameter. On the one hand, as in many other works, the stiffness is prescribed to decrease as far as the damage increases. On the other hand, the microstructural material length is prescribed, in contrast to a certain part of the literature, to increase as far as the damage increases. This last assumption is due to the interpretation that a damage state induces a microstructure in the continuum and that such a microstructure is more important as far as the damage increases. At a given value of the damage parameter, the behavior is referred to second gradient linear elastic material. However, the damage evolution makes the model not only nonlinear but also inelastic. The second principle of thermodynamics can be considered by assuming that the scalar damage parameter does not decrease its value in the process of deformation, and this implies a dissipation for the elastic strain energy. It is finally remarked that damage initiation, in this second gradient continuum damage model, can be induced not only from a prescribed initial lack of stiffness, but also from an external concentrated double force or from suitable boundary conditions, and these last options have advantages that are discussed in the paper. For example, in this last case, it is possible to initiate the damage in the neighborhood of the boundaries, that is, the case in most of the empirical situations. Simple numerical simulations are also presented in order to show the exposed concepts.

Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients

Abstract: In the present paper, a two-dimensional solid consisting of a linear elastic isotropic material, for which the deformation energy depends on the second gradient of the displacement, is considered. The strain energy is demonstrated to depend on 6 constitutive parameters: the 2 Lame constants ( $${\lambda}$$ and $${\mu}$$ ) and 4 more parameters (instead of 5 as it is in the 3D-case). Analytical solutions for classical problems such as heavy sheet, bending and flexure are provided. The idea is very simple: The solutions of the corresponding problem of first gradient classical case are imposed, and the corresponding forces, double forces and wedge forces are found. On the basis of such solutions, a method is outlined, which is able to identify the six constitutive parameters. Ideal (or Gedanken) experiments are designed in order to write equations having as unknowns the six constants and as known terms the values of suitable experimental measurements.

A quasi-static phase-field approach to pressurized fractures

Abstract: In this paper we present a quasi-static formulation of a phase-field model for a pressurized crack in a poroelastic medium. The mathematical model represents a linear elasticity system with a fading Gassman tensor as the crack grows, that is coupled with a variational inequality for the phase-field variable containing an entropy inequality. We introduce a novel incremental approximation that decouples displacement and phase-field problems. We establish convergence to a solution of the quasi-static problem, including Rice's condition, when the time discretization step goes to zero. Numerical experiments confirm the robustness and efficiency of this approach for multidimensional test cases.

Mathematical Methods in Elasticity Imaging

Abstract: This book is the first to comprehensively explore elasticity imaging and examines recent, important developments in asymptotic imaging, modeling, and analysis of deterministic and stochastic elastic wave propagation phenomena. It derives the best possible functional images for small inclusions and cracks within the context of stability and resolution, and introduces a topological derivative-based imaging framework for detecting elastic inclusions in the time-harmonic regime. For imaging extended elastic inclusions, accurate optimal control methodologies are designed and the effects of uncertainties of the geometric or physical parameters on stability and resolution properties are evaluated. In particular, the book shows how localized damage to a mechanical structure affects its dynamic characteristics, and how measured eigenparameters are linked to elastic inclusion or crack location, orientation, and size. Demonstrating a novel method for identifying, locating, and estimating inclusions and cracks in elastic structures, the book opens possibilities for a mathematical and numerical framework for elasticity imaging of nanoparticles and cellular structures.

Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells

Abstract: A geometrically non-linear theory is developed for shells of generic shape allowing for third-order thickness and shear deformation and rotary inertia by using eight parameters; geometric imperfections are also taken into account. The geometrically non-linear strain–displacement relationships are derived retaining full non-linear terms in all the 8 parameters, i.e. in-plane and transverse displacements, rotations of the normal and thickness deformation parameters; these relationships are presented in curvilinear coordinates, ready to be implemented in computer codes. Higher order terms in the transverse coordinate are retained in the derivation so that the theory is suitable also for thick laminated shells. Three-dimensional constitutive equations are used for linear elasticity. The theory is applied to circular cylindrical shells complete around the circumference and simply supported at both ends to study initially static finite deformation. Both radially distributed forces and displacement-dependent pressure are used as load and results for different shell theories are compared. Results show that a 6 parameter non-linear shell theory is quite accurate for isotropic shells. Finally, large-amplitude forced vibrations under harmonic excitation are investigated by using the new theory and results are compared to other available theories. The new theory with non-linearity in all the 8 parameters is the only one to predict correctly the thickness deformation; it works accurately for both static and dynamics loads.

Isotropic incompressible hyperelastic models for modelling the mechanical behaviour of biological tissues: a review.

Abstract: Modelling the mechanical behaviour of biological tissues is of vital importance for clinical applications. It is necessary for surgery simulation, tissue engineering, finite element modelling of soft tissues, etc. The theory of linear elasticity is frequently used to characterise biological tissues; however, the theory of nonlinear elasticity using hyperelastic models, describes accurately the nonlinear tissue response under large strains. The aim of this study is to provide a review of constitutive equations based on the continuum mechanics approach for modelling the rate-independent mechanical behaviour of homogeneous, isotropic and incompressible biological materials. The hyperelastic approach postulates an existence of the strain energy function--a scalar function per unit reference volume, which relates the displacement of the tissue to their corresponding stress values. The most popular form of the strain energy functions as Neo-Hookean, Mooney-Rivlin, Ogden, Yeoh, Fung-Demiray, Veronda-Westmann, Arruda-Boyce, Gent and their modifications are described and discussed considering their ability to analytically characterise the mechanical behaviour of biological tissues. The review provides a complete and detailed analysis of the strain energy functions used for modelling the rate-independent mechanical behaviour of soft biological tissues such as liver, kidney, spleen, brain, breast, etc.

Elastic buckling of a sandwich beam with variable mechanical properties of the core

Abstract: This paper deals with the analytical and numerical studies of elastic buckling of a three-layered beam with metal foam core. Mechanical properties of the core are variable along the z -axis. There are two schemes of displacement of the faces and core of the beam: a broken line hypothesis and a non-linear hypothesis. The mathematical models for both types of displacements are presented. The governing differential equations of the sandwich beam are derived. Numerical analysis of sandwich beams is conducted in ANSYS environment. The finite elements analysis has been performed using a linear elastic buckling model. The analysis with constant and variable Young׳s modulus of the core of the beam is carried out. The values of the critical load obtained by the analytical and numerical (FEM) methods are compared.

Strain gradient elasticity with geometric nonlinearities and its computational evaluation

Abstract: The theory of linear elasticity is insufficient at small length scales, e.g., when dealing with micro-devices. In particular, it cannot predict the “size effect” observed at the micro- and nanometer scales. In order to design at such small scales an improvement of the theory of elasticity is necessary, which is referred to as strain gradient elasticity. There are various approaches in literature, especially for small deformations. In order to include geometric nonlinearities we start by discussing the necessary balance equations. Then we present a generic approach for obtaining adequate constitutive equations. By combining balance equations and constitutive relations nonlinear field equations result. We apply a variational formulation to the nonlinear field equations in order to find a weak form, which can be solved numerically by using open-source codes. By using balances of linear and angular momentum we obtain the so-called stress and couple stress as tensors of rank two and three, respectively. Since dealing with tensors an adequate representation theorem can be applied. We propose for an isotropic material a stress with two and a couple stress with three material parameters. For understanding their impact during deformation the numerical solution procedure is performed. By successfully simulating the size effect known from experiments, we verify the proposed theory and its numerical implementation. Based on representation theorems a self consistent strain gradient theory is presented, discussed, and implemented into a computational reality.

A model for compression-weakening materials and the elastic fields due to contractile cells

Abstract: We construct a homogeneous, nonlinear elastic constitutive law that models aspects of the mechanical behavior of inhomogeneous fibrin networks. Fibers in such networks buckle when in compression. We model this as a loss of stiffness in compression in the stress–strain relations of the homogeneous constitutive model. Problems that model a contracting biological cell in a finite matrix are solved. It is found that matrix displacements and stresses induced by cell contraction decay slower (with distance from the cell) in a compression weakening material than linear elasticity would predict. This points toward a mechanism for long-range cell mechanosensing. In contrast, an expanding cell would induce displacements that decay faster than in a linear elastic matrix.

Analysis of an HDG method for linear elasticity

Abstract: Summary We present the first a priori error analysis for the first hybridizable discontinuous Galerkin method for linear elasticity proposed in Internat. J. Numer. Methods Engrg. 80 (2009), no. 8, 1058–1092. We consider meshes made of polyhedral, shape-regular elements of arbitrary shape and show that, whenever piecewise-polynomial approximations of degree k≥0 are used and the exact solution is smooth enough, the antisymmetric part of the gradient of the displacement converges with order k, the stress and the symmetric part of the gradient of the displacement converge with order k + 1/2, and the displacement converges with order k + 1. We also provide numerical results showing that the orders of convergence are actually sharp. Copyright © 2014 John Wiley & Sons, Ltd.

A unifying variational framework for stress gradient and strain gradient elasticity theories

Abstract: Stress gradient elasticity and strain gradient elasticity do constitute distinct continuum theories exhibiting mutual complementary features This is probed by a few variational principles herein presented and discussed, which include: i) For stress gradient elasticity, a (novel) principle of minimum complementary energy and an (improved-form) principle of stationarity of the Hellinger–Reissner type; ii) For strain gradient elasticity, a (known) principle of minimum total potential energy and a (novel) principle of stationarity of the Hu–Washizu type Additionally, the higher order boundary conditions for stress gradient elasticity, previously derived by the author (Polizzotto, Int J Solids Struct 51, 1809–1818, (2014)) in the form of higher order boundary compatibility equations, are here revisited and reinterpreted with the aid of a discrete model of the body's boundary layer The reasons why the latter conditions need to be relaxed for beam and plate structural models are explained

A three-dimensional finite element solution of frictional wheel–rail rolling contact in elasto-plasticity:

Abstract: Wear, rolling contact fatigue, and plastic deformation are the major failure modes of railway wheels and rails. Proper analyses of the failure mechanisms as well as improvement in design and maintenance require an accurate evaluation of the stress and strain states. Solution of frictional rolling contact between wheel and rail in elasto-plasticity seems, however, still to lack in the literature. This paper presents a model for such a solution. A 3D finite element model is built up to simulate the rolling contact. The focus is on the tangential problem, namely the distributions of surface shear stress and micro-slip as well as the distinction of areas of adhesion and slip in the contact patch. With the presented model the assumptions of half space, linear elasticity and quasi-static or steady state, which are often employed in existing solution of rolling contact, are dropped. A bilinear elasto-plastic material with isotropic hardening is employed to examine the effects of plasticity on frictional rolling....

Convergence of a Cell-Centered Finite Volume Discretization for Linear Elasticity

Abstract: We show convergence of a cell-centered finite volume discretization for linear elasticity. The discretization, termed the MPSA method, was recently proposed in the context of geological applications, where cell-centered variables are often preferred. Our analysis utilizes a hybrid variational formulation, which has previously been used to analyze finite volume discretizations for the scalar diffusion equation. The current analysis deviates significantly from the previous in three respects. First, additional stabilization leads to a more complex saddle-point problem. Second, a discrete Korn's inequality has to be established for the global discretization. Finally, robustness with respect to the Poisson ratio is analyzed. The stability and convergence results presented herein provide the first rigorous justification of the applicability of cell-centered finite volume methods to problems in linear elasticity.

A linear elastic-brittle interface model: application for the onset and propagation of a fibre-matrix interface crack under biaxial transverse loads

Abstract: A new linear elastic and perfectly brittle interface model for mixed mode is presented and analysed. In this model, the interface is represented by a continuous distribution of springs which simulates the presence of a thin elastic layer. The constitutive law for the continuous distribution of normal and tangential initially-linear-elastic springs takes into account possible frictionless elastic contact between adherents once a portion of the interface is broken. A perfectly brittle failure criterion is employed for the springs, which enables the study of crack onset and propagation. This interface failure criterion takes into account the variation of the interface fracture toughness with the fracture mode mixity. A unified way to represent several phenomenological both energy and stress based failure criteria is introduced. A proof relating the energy release rate and tractions at an interface point (not necessarily a crack tip point) is introduced for this interface model by adapting Irwin’s crack closure technique for the first time. The main advantages of the present interface model are its simplicity, robustness and computational efficiency, even in the presence of snap-back and snap-through instabilities, when the so-called sequentially linear (elastic) analysis is applied. This model is applied here in order to study crack onset and propagation at the fibre-matrix interface in a composite under tensile/compressive remote biaxial transverse loads. Firstly, this model is used to obtain analytical predictions about interface crack onset, while investigating a single fibre embedded in a matrix which is subjected to uniform remote transverse loads. Then, numerical results provided by a 2D boundary element analysis show that a fibre-matrix interface failure is initiated by the onset of a finite debond in the neighbourhood of the interface point where the failure criterion is first reached (under increasing proportional load); this debond further propagates along the interface in mixed mode or even, in some configurations, with the crack tip under compression. The analytical predictions of the debond onset position and associated critical load are used for several parametric studies of the influence of load biaxiality, fracture-mode sensitivity and brittleness number, and for checking the computational procedure implemented.

Peridynamics and Material Interfaces

Abstract: The convergence of a peridynamic model for solid mechanics inside heterogeneous media in the limit of vanishing nonlocality is analyzed. It is shown that the operator of linear peridynamics for an isotropic heterogeneous medium converges to the corresponding operator of linear elasticity when the material properties are sufficiently regular. On the other hand, when the material properties are discontinuous, i.e., when material interfaces are present, it is shown that the operator of linear peridynamics diverges, in the limit of vanishing nonlocality, at material interfaces. Nonlocal interface conditions, whose local limit implies the classical interface conditions of elasticity, are then developed and discussed. A peridynamics material interface model is introduced which generalizes the classical interface model of elasticity. The model consists of a new peridynamics operator along with nonlocal interface conditions. The new peridynamics interface model converges to the classical interface model of linear elasticity.