Showing papers on "Linear elasticity published in 2011"

Deformation of an elastic substrate by a three-phase contact line.

Abstract: Young's classic analysis of the equilibrium of a three-phase contact line ignores the out-of-plane component of the liquid-vapor surface tension. While it is expected that this unresolved force is balanced by the elastic response of the solid, a definitive analysis has remained elusive because of an apparent divergence of stress at the contact line. While a number of theories have been presented to cut off the divergence, none of them have provided reasonable agreement with experimental data. We measure surface and bulk deformation of a thin elastic film near a three-phase contact line using fluorescence confocal microscopy. The out-of-plane deformation is well fit by a linear elastic theory incorporating an out-of-plane restoring force due to the surface tension of the solid substrate. This theory predicts that the deformation profile near the contact line is scale-free and independent of the substrate elastic modulus.

Linear elastic trusses leading to continua with exotic mechanical interactions

Abstract: We study the statics of some trusses, i.e. networks of nodes linked by linear springs. The trusses are designed in such a way that a few number of floppy modes are present and remain when considering the homogenized limit of the truss. We then obtain linear elastic materials with exotic mechanical interactions which cannot be described in the classical framework of Cauchy stress theory. For aim of simplicity, the structures described here are two-dimensional. The extension to the 3D case does not present any difficulty.

A Comparison of All Hexagonal and All Tetrahedral Finite Element Meshes for Elastic and Elasto-plastic Analysis

Abstract: This paper compares the accuracy of elastic and elastio-plastic solid continuum finite element analyses modeled with either all hexagonal or all tetrahedral meshes. Eigenvalues of element stiffness matrices, linear static displacements and stresses, dynamic modal frequencies, and plastic flow values in are computed and compared. Elements with both linear and quadratic displacement functions are evaluated. Linear incompressibility conditions are also investigated. A simple bar with a rectangular cross-section, fixed at one end, is modeled and results are compared to known analytical solutions wherever possible. The evaluation substantiates a strong preference for linear displacement hexagonal finite elements when compared solely to linear tetrahedral finite elements. The use of quadratic displacement formulated finite elements significantly improve the performance of the tetrahedral as well as the hexahedral elements. The nonlinear elastoplastic comparison indicates that linear hexagonal elements may be superior to even quadratic tetrahedrons when shear stress in dominant. Results of this work may serve as a guide in selecting appropriate finite element types to be used in three dimensional elastic and elastic plastic analysis. INTRODUCTION Consideration of the convergence characteristics of two dimensional solutions of elastic continuum problems, using both quadrilateral and triangular elements, has been covered in previous studies and some finite element textbooks[1,2]. Such studies conclude that the significant factors that effect convergence characteristics of finite element solutions include the element's basic shape, element distortion, polynomial order of the element, completeness of polynomial functions, integration techniques, and material incompressibility. It is generally accepted that simplex triangular elements are inferior when compared to bilinear quadrilaterals. For example, statements such as “... for reasons of better accuracy and efficiency, quadrilateral elements are preferred for two-dimensional meshes and hexahedral elements for three-dimensional meshes. This preference is clear in structural analysis and seems to also hold for other engineering disciplines.”[2] However, it is also generally accepted that triangular elements, with higher order displacement assumptions, provide acceptable accuracy and convergence characteristics. However, mesh locking due to material incompressibility as reported by Hughes[3], is a serious shortcoming of triangular elements. The current focus for developing rapidly converging finite element procedures is to incorporate h-p adaptive techniques.[4] Of particular note for this study is an article by Lo and Lee[5] which investigates the convergence of mixed element in h-p adaptive finite element analysis. A significant conclusion from this paper is, that by carefully controlling quality and grading, quadrilateral elements provide an increase in efficiency in h-p adaptivity over pure triangular elements. A few studies have been published comparing the convergence characteristics of hexahedral verse tetrahedral meshes. Cifuentes and Kalbag [6] conclude that the results obtained with quadratic tetrahedral elements, compared to bilinear hexahedral elements, were equivalent in terms of both accuracy and CPU time. Bussler and Ramesh [7] report more accuracy using the same order hexahedral elements over tetrahedrons. Weingarten [8] indicates that both quadratic tetrahedrons and hexahedrons were equivalent in accuracy and efficiency and recommends using p method tetrahedrons to achieve desired accuracy. No studies were found incorporating incompressibility or plasticity aspects relating to the convergence of hexagonal and tetrahedral elements. In this paper, stiffness matrix eigenvalues of a square geometrical volume, meshed with a single hexahedron is compared to the same geometrical volume meshed with five tetrahedrons. Next, results of a linear elastic, fixed end bar, meshed with either all hexahedrons or all tetrahedrons are compared. Both bending and torsional results are considered. The computed vibration modes of the fixed end bar problem are then evaluated. Finally, elasto-plastic calculations of the fixed end bar again meshed with both types of elements are evaluated. STIFFNESS MATRIX EIGENVALUES The evaluation of the eigenvalues and eigenvectors of a stiffness matrix is important when studying the convergence characteristics of any finite element.[9] Properly formulated elements have a zero valued eigenvalue associated with each rigid body motion. In addition, since the displacement based finite element technique overestimates the stiffness of a body, the smaller the eigenvalues for a particular deformation mode, the more effective is the element. Therefore, to provide an initial assessment of the effectiveness of simplex tetrahedrons compared with bilinear hexahedrons, the eigenvalues of equivalent models were computed. A regular unit cube volume, with a Young’s Modulus of 30,000,000 and Poisson’s Ratio of .3 was modeled with a single hexahedron and five tetrahedrons as shown in Figure 1. Note the configuration shown at the bottom of the figure 1 shows how the five tetrahedrons are positioned to fill the unit cube. The internal tetrahedron’s position results in some directional properties of the stiffness matrix. The eigenvalues of the hexahedron were computed from (1) the stiffness matrix generated

Phase field modeling of crack propagation

Abstract: Fracture is a fundamental mechanism of materials failure. Propagating cracks can exhibit a rich dynamical behavior controlled by a subtle interplay between microscopic failure processes in the crack tip region and macroscopic elasticity. We review recent approaches to understand crack dynamics using the phase field method. This method, developed originally for phase transformations, has the well-known advantage of avoiding explicit front tracking by making material interfaces spatially diffuse. In a fracture context, this method is able to capture both the short-scale physics of failure and macroscopic linear elasticity within a self-consistent set of equations that can be simulated on experimentally relevant length and time scales. We discuss the relevance of different models, which stem from continuum field descriptions of brittle materials and crystals, to address questions concerning crack path selection and branching instabilities, as well as models that are based on mesoscale concepts for crack tip ...

Probing the mechanical properties of graphene using a corrugated elastic substrate

Abstract: We examine the mechanical properties of graphene samples of thicknesses ranging from 1 to 17 atomic layers, placed on a microscale-corrugated elastic substrate. Using atomic force microscopy, we show that the graphene adheres to the substrate surface and can substantially deform the substrate, with larger graphene thicknesses creating greater deformations. We use linear elasticity theory to model the deformations of the composite graphene-substrate system. We compare experiment and theory, and thereby extract information about graphene’s bending rigidity, adhesion, critical stress for interlayer sliding, and sample-dependent tension.

Introduction to Finite Element Analysis: Formulation, Verification and Validation

Abstract: About the Authors. Series Preface. Preface. 1 Introduction. 1.1 Numerical simulation. 1.2 Why is numerical accuracy important? 1.3 Chapter summary. 2 An outline of the finite element method. 2.1 Mathematical models in one dimension. 2.2 Approximate solution. 2.3 Generalized formulation in one dimension. 2.4 Finite element approximations. 2.5 FEM in one dimension. 2.6 Properties of the generalized formulation. 2.7 Error estimation based on extrapolation. 2.8 Extraction methods. 2.9 Laboratory exercises. 2.10 Chapter summary. 3 Formulation of mathematical models. 3.1 Notation. 3.2 Heat conduction. 3.3 The scalar elliptic boundary value problem. 3.4 Linear elasticity. 3.5 Incompressible elastic materials. 3.6 Stokes' flow. 3.7 The hierarchic view of mathematical models. 3.8 Chapter summary. 4 Generalized formulations. 4.1 The scalar elliptic problem. 4.2 The principle of virtual work. 4.3 Elastostatic problems. 4.4 Elastodynamic models. 4.5 Incompressible materials. 4.6 Chapter summary. 5 Finite element spaces. 5.1 Standard elements in two dimensions. 5.2 Standard polynomial spaces. 5.3 Shape functions. 5.4 Mapping functions in two dimensions. 5.5 Elements in three dimensions. 5.6 Integration and differentiation. 5.7 Stiffness matrices and load vectors. 5.8 Chapter summary. 6 Regularity and rates of convergence. 6.1 Regularity. 6.2 Classification. 6.3 The neighborhood of singular points. 6.4 Rates of convergence. 6.5 Chapter summary. 7 Computation and verification of data. 7.1 Computation of the solution and its first derivatives. 7.2 Nodal forces. 7.3 Verification of computed data. 7.4 Flux and stress intensity factors. 7.5 Chapter summary. 8 What should be computed and why? 8.1 Basic assumptions. 8.2 Conceptualization: drivers of damage accumulation. 8.3 Classical models of metal fatigue. 8.4 Linear elastic fracture mechanics. 8.5 On the existence of a critical distance. 8.6 Driving forces for damage accumulation. 8.7 Cycle counting. 8.8 Validation. 8.9 Chapter summary. 9 Beams, plates and shells. 9.1 Beams. 9.2 Plates. 9.3 Shells. 9.4 The Oak Ridge experiments. 9.5 Chapter summary. 10 Nonlinear models. 10.1 Heat conduction. 10.2 Solid mechanics. 10.3 Chapter summary. A Definitions. A.1 Norms and seminorms. A.2 Normed linear spaces. A.3 Linear functionals. A.4 Bilinear forms. A.5 Convergence. A.6 Legendre polynomials. A.7 Analytic functions. A.8 The Schwarz inequality for integrals. B Numerical quadrature. B.1 Gaussian quadrature. B.2 Gauss Lobatto quadrature. C Properties of the stress tensor. C.1 The traction vector. C.2 Principal stresses. C.3 Transformation of vectors. C.4 Transformation of stresses. D Computation of stress intensity factors. D.1 The contour integral method. D.2 The energy release rate. E Saint-Venant's principle. E.1 Green's function for the Laplace equation. E.2 Model problem. F Solutions for selected exercises. Bibliography. Index.

Interaction of highly nonlinear solitary waves with linear elastic media.

Abstract: We study the interaction of highly nonlinear solitary waves propagating in granular crystals with an adjacent linear elastic medium. We investigate the effects of interface dynamics on the reflection of incident waves and on the formation of primary and secondary reflected waves. Experimental tests are performed to correlate the linear medium geometry, materials, and mass with the formation and propagation of reflected waves. We compare the experimental results with theoretical analysis based on the long-wavelength approximation and with numerical predictions obtained from discrete particle models. Experimental results are found to be in agreement with theoretical analysis and numerical simulations. This preliminary study establishes the foundation for utilizing reflected solitary waves as novel information carriers in nondestructive evaluation of elastic material systems.

The influence of the cohesive process zone in hydraulic fracturing modelling

Abstract: This paper studies the importance of the cohesive zone in the modelling of a fluid driven fracture under plain strain conditions. The fracture is driven by pumping of an incompressible viscous fluid at the fracture inlet. Rock deformation is modeled for linear elastic and poroelastic solids. Fluid flow in the fracture is modeled by lubrication theory. The cohesive zone approach is used as the fracture propagation criterion. Finite element analysis was used to compute the solution for the crack length, the fracture opening and propagation pressure as a function of the time and distance from the wellbore. It is demonstrated that the crack profiles and the propagation pressures are larger in the case of elastic-softening cohesive model compared to the results of the rigid-softening cohesive model for both elastic and poroelastic cohesive solids. It is found that the results are affected by the slope of the loading branch of the cohesive model and they are nearly unaffected from the exact form of the softening branch. Furthermore, the size of the process zone, the fracture geometry and the propagation pressure increase with increasing confining stresses. These results may explain partially the discrepancies in net-pressures between field measurements and conventional model predictions.

Supported Axisymmetric Tunnels Within Linear Viscoelastic Burgers Rocks

Abstract: An exact closed form solution is derived for the mechanical behaviour of a linear viscoelastic Burgers rock around an axisymmetric tunnel, supported by a linear elastic ring. Analytical formulae are provided for the displacement of the rock/lining interface and for the pressure exerted by the rock on the lining, taking into account the stiffness and its installation time. Results calculated from these formulae do validate the corresponding numerical results of a 2D finite differences code. Further, comparison to previous existing solutions for the same viscoelastic model indicates similarities and differences. A parametric study is performed to investigate the effect of the viscoelastic constants, the stiffness and installation time of the support. The derived closed form solution is used to construct the time-dependent Supported Ground Reaction Curves of the viscoelastic rock, i.e. the time contour plots on the convergence confinement diagram. The importance of the effect of the support on the restrained rock creep and the exerted pressure on the lining, during the design life of a structure, is examined.

BEM analysis of crack onset and propagation along fiber–matrix interface under transverse tension using a linear elastic–brittle interface model

Abstract: The behavior of the fiber–matrix interface under transverse tension is studied by means of a new linear elastic–brittle interface model. Similar models, also called weak or imperfect interface models, are frequently applied to describe the behavior of adhesively bonded joints. The interface is modeled by a continuous distribution of linear-elastic springs which simulates the presence of a thin adhesive layer (interphase). In the present work a new linear elastic–brittle constitutive law for the continuous distribution of springs is introduced. In this law the normal and tangential stresses across the undamaged interface are, respectively, proportional to the relative normal and tangential displacements. This model not only allows for the study of crack growth but also for the study of crack onset. An important feature of this law is that it takes into account the variation of the fracture toughness with the fracture mode mixity of a crack growing along the interface between bonded solids, in agreement with previous experimental results. The present linear elastic–brittle interface model is implemented in a 2D boundary element method (BEM) code to carry out micromechanical analysis of the fiber–matrix interface failure in fiber-reinforced composite materials. It is considered that the behavior of the fiber–matrix interphase can be modeled by the present model although, strictly speaking, there is usually no intermediate material between fiber and matrix. A linear-elastic isotropic behavior of both fiber and matrix is assumed, the fiber being stiffer than the matrix. The failure mechanism of an isolated fiber under transverse tension, i.e., the onset and growth of the fiber–matrix interface crack, is studied. The present model shows that failure along the interface initiates with an abrupt onset of a partial debonding between the fiber and the matrix, caused by presence of the maximum radial stress at the interface, and this debonding further develops as a crack growing along the interface.

Nonaffine Displacements in Flexible Polymer Networks

Abstract: The validity of the affine assumption in model flexible polymer networks is explored. To this end, the displacements of fluorescent tracer beads embedded in polyacrylamide gels are quantified by confocal microscopy under shear deformation, and the deviations of these displacements from affine responses are recorded. Nonaffinity within the gels is quantified as a function of polymer chain density and cross-link concentration. Observations are compared with current theories of nonaffinity in random elastic media. We note that the mean-squared nonaffine deviation is proportional to the square of the applied strain in the linear elasticity regime, as per theoretical predictions. The measured degree of nonaffinity in the polyacrylamide gels suggests the presence of structural inhomogeneities which likely result from heterogeneous reaction kinetics during gel preparation. In addition, the macroscopic elasticity of the polyacrylamide gels is confirmed to behave in accordance with standard models of flexible poly...