Showing papers on "Linear elasticity published in 2012"
TL;DR: In this article, a comprehensive review of the current state of the art of the study of elastic properties, the establishments of correlations between elastic moduli and properties/features, and the elastic models and elastic perspectives of metallic glasses is presented.
1,070 citations
Book•
17 Jan 2012
TL;DR: In this paper, the authors present an extension of the Dirichlet problem for the case of Perforated Domains with a Non-Periodic Structure, where the boundary value problem is solved with Neumann conditions on the outer part of the boundary and on the surface of the Cavities.
Abstract: Some Mathematical Problems of the Theory of Elasticity. Some Functional Spaces and Their Properties. Auxiliary Propositions. Korn's Inequalities. Boundary Value Problems of Linear Elasticity. Perforated Domains with a Periodic Structure. Extension Theorems. Estimates for Solutions of Boundary Value Problems of Elasticity in Perforated Domains. Periodic Solutions of Boundary Value Problems for the System of Elasticity. Saint-Venant's Principle for Periodic Solutions of the Elasticity System. Estimates and Existence Theorems for Solutions of the Elasticity System in Unbounded Domains. Strong G -Convergence of Elasticity Operators. Homogenization of the System of Linear Elasticity. Composites and Perforated Materials. The Mixed Problem in a Perforated Domain with the Dirichlet Boundary Conditions on the Outer Part of the Boundary and the Neumann Conditions on the Surface of the Cavities. The Boundary Value Problem with Neumann Conditions in a Perforated Domain. Asymptotic Expansions for Solutions of Boundary Value Problems of Elasticity in a Perforated Layer. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Elasticity System in a Perforated Domain. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Biharmonic Equation. Some Generalizations for the Case of Perforated Domains with a Non-Periodic Structure. Homogenization of the System of Elasticity with Almost-Periodic Coefficients. Homogenization of Stratified Structures. Estimates for the Rate of G -Convergence of Higher-Order Elliptic Operators. Spectral Problems . Some Theorems from Functional Analysis. Spectral Problems for Abstract Operators. Homogenization of Eigenvalues and Eigenfunctions of Boundary Value Problems for Strongly Non-Homogeneous Elastic Bodies. On the Behaviour of Eigenvalues and Eigenfunctions of the Dirichlet Problem for Second Order Elliptic Equations in Perforated Domains. Third Boundary Value Problem for Second Order Elliptic Equations in Domains with Rapidly Oscillating Boundary. Free Vibrations of Bodies with Concentrated Masses. On the Behaviour of Eigenvalues of the Dirchlet Problem in Domains with Cavities Whose Concentration is Small. Homogenization of Eigenvalues of Ordinary Differential Operators. Asymptotic Expansion of Eigenvalues and Eigenfunctions of the Sturm-Liouville Problem for Equations with Rapidly Oscillating Coefficients. On the Behaviour of the Eigenvalues and Eigenfunctions of a G -Convergent Sequence of Non-Self-Adjoint Operators. References.
838 citations
TL;DR: In this article, a strain smoothing procedure for the extended finite element method (XFEM) is presented, which is tailored to linear elastic fracture mechanics and, in this context, to outperform the standard XFEM.
210 citations
Book•
04 Feb 2012TL;DR: In this paper, the conservation laws of linear Elastostatics of Inhomogeneous Bernoulli-Euler Beams have been studied and compared to the classical theory of linear elasticity.
Abstract: 1 Mathematical Preliminaries.- 1.1 General Remarks.- 1.2 What is a Conservation Law?.- 1.3 Trivial Conservation Laws.- 1.4 System with a Lagrangian Noether's Method.- 1.5 System without a Lagrangian Neutral-Action Method.- 1.6 Discussion.- 2 Linear Theory of Elasticity.- 2.1 General Remarks.- 2.2 Elements of Linear Elasticity.- 2.3 Conservation Laws of Linear Elastostatics.- 2.4 Alternative Derivations of Conservation Laws.- 3 Properties of the Eshelby Tensor.- 3.1 General Remarks 81.- 3.2 Physical Interpretation of the Components of the Eshelby Tensor.- 3.3 Invariants, Principal Values, Principal Directions and Extremal Values of the Eshelby Tensor.- 4 Linear Elasticity with Defects.- 4.1 General Remarks.- 4.2 Path-Independent Integrals and Energy-Release Rates.- 4.3 Example: Hole-Dislocation Interaction.- 4.4 Path-Independent Integrals of Fracture Mechanics.- 5 Inhomogeneous Elastostatics.- 5.1 General Remarks.- 5.2 Symmetry Transformations.- 5.3 The Homogeneous Case.- 5.4 The Inhomogeneous Case.- 5.5 Relation to Stress-Intensity Factors.- 5.6 Examples.- 6 Elastodynamics.- 6.1 General Remarks.- 6.2 Time t as an Additional Independent Variable.- 6.3 Convolution in Time.- 6.4 Domain-Independent Integrals.- 6.5 Energy-Release Rates.- 6.6 Wave Motion.- 7 Dissipative Systems.- 7.1 General Remarks.- 7.2 Diffusion Equation.- 7.3 Non-Linear Wave Equation.- 7.4 Viscoelasticity.- 8 Coupled Fields.- 8.1 General Remarks.- 8.2 Piezoelectricity.- 8.3 Thermoelasticity.- 8.4 Mechanics of a Porous Medium.- 9 Bars, Shafts and Beams.- 9.1 General Remarks.- 9.2 Elements of Strength-of-Materials.- 9.3 Balance and Conservation Laws for Bars and Shafts.- 9.4 Balance and Conservation Laws for Beams.- 9.5 Energy-Release Rates and Stress-Intensity Factors.- 9.6 Examples.- 10 Plates and Shells.- 10.1 General Remarks.- 10.2 Plate Theories.- 10.3 Conservation Laws for Elastostatics of Mindlin Plates.- 10.4 Reduction to the Classical Theory.- 10.5 Conservation Laws for Shells.- Appendix A.- Conservation Laws for Inhomogeneous Bars under Arbitrary Axial Loading.- Appendix B.- B.1 Elastodynamics of Inhomogeneous Bernoulli-Euler Beams.- B.2 Reduction to Statics.- Appendix C.- C.1 Elastodynamics of Inhomogeneous Mindlin Plates.- C.2 Reduction to Statics.- References.- Symbol Index.- Author Index.
136 citations
TL;DR: In this paper, the fundamental dynamic properties of the flexible rocking system are compared with those of similar linear elastic oscillators and rigid rocking structures, revealing the distinct characteristics of flexible rocking structures.
Abstract: Numerous structures uplift under the influence of strong ground motion. Although many researchers have investigated the effects of base uplift on very stiff (ideally rigid) structures, the rocking response of flexible structures has received less attention. Related practical analysis methods treat these structures with simplified 'equivalent' oscillators without directly addressing the interaction between elasticity and rocking. This paper addresses the fundamental dynamics of flexible rocking structures. The nonlinear equations of motion, derived using a Lagrangian formulation for large rotations, are presented for an idealized structural model. Particular attention is devoted to the transition between successive phases; a physically consistent classical impact framework is utilized alongside an energy approach. The fundamental dynamic properties of the flexible rocking system are compared with those of similar linear elastic oscillators and rigid rocking structures, revealing the distinct characteristics of flexible rocking structures. In particular, parametric analysis is performed to quantify the effect of elasticity on uplift, overturning instability, and harmonic response, from which an uplifted resonance emerges. The contribution of stability and strength to the collapse of flexible rocking structures is discussed. © 2012 John Wiley & Sons, Ltd.
126 citations
TL;DR: In this article, the authors analyzed the time-dependent problem of a simply-supported laminated beam, composed of two elastic layers connected by a viscoelastic interlayer, whose response is modeled by a Prony's series of Maxwell elements.
111 citations
TL;DR: In this paper, two simple one dimensional models that conclude to first and second strain gradient elastic theories being identical to the corresponding ones proposed by Mindlin are presented, where in both models the derived internal length scale parameters are correlated to the size of the considered unit cell.
109 citations
TL;DR: In this article, an incrementally linear elastic, orthotropic constitutive model is suggested to represent the equivalent continuum pre-failure mechanical behavior of the jointed rock mass by incorporating the effect of joint geometry network by the fracture tensor components.
104 citations
TL;DR: In this article, a comparison of different discrete element models of a rock-type material is presented, which employs spherical particles with the cohesive interaction model combining linear elastic behaviour with brittle failure.
88 citations
TL;DR: In this article, the 3D Finite Element method is applied to mixed fracture under anti-plane loading of a straight through-the-thickness crack in a linear elastic plate.
83 citations
TL;DR: In this article, a micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented.
01 Jan 2012
TL;DR: The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach and the resulting flows are given by Allen-Cahn and Cahn-Hilliard type dynamics coupled to a linear elasticity system.
Abstract: The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach. Two steepest descent approaches based on L2- and H-1-gradient flow dynamics are discussed. The resulting flows are given by Allen-Cahn and Cahn-Hilliard type dynamics coupled to a linear elasticity system. We finally compare numerical results obtained from the two different approaches.
TL;DR: In this article, the authors provided a method to determine the load causing delamination along an interface in a composite structure based on elastic interface model, according to which the interface is equivalent to a bed of linear elastic springs, and on Finite Fracture Mechanics, a crack propagation criterion recently proposed for homogeneous structures.
TL;DR: In this article, the structural response of a composite laminate T-piece specimen subjected to a mechanical pull-off load case is investigated. And the authors show that the failure is controlled by crack propagation, and that using realistic cohesive maximum strength values requires a very fine mesh.
Abstract: This paper presents a rigorous numerical investigation into the structural response of a composite laminate T-piece specimen subjected to a mechanical “pull-off” load case. Initially, a linear elastic stress analysis is conducted, showing very high stresses at the free-edge. In a further analysis, special-purpose interface elements are then inserted where appropriate and used to predict both the crack pattern and the load to failure. It is demonstrated that that using realistic cohesive maximum strength values requires a very fine mesh. Reducing the values to ensure initiation occurs leads to conservative and mesh independent predictions and that a suitable choice leads to good correlation with the experimental results. This study also shows that the T-piece failure is controlled by crack propagation.
TL;DR: In this article, the Young modulus of hydrogenated graphene was measured by a computer experiment and it was shown that hydrogenation generally leads to a much smaller longitudinal extension upon loading than in pristine graphene.
Abstract: We blend together continuum elasticity and first-principles calculations to measure by a computer experiment the Young modulus of hydrogenated graphene. We provide evidence that hydrogenation generally leads to a much smaller longitudinal extension upon loading than in pristine graphene. Furthermore, the Young modulus is found to depend upon the loading direction for some specific conformers, characterized by an anisotropic linear elastic behavior.
TL;DR: In this article, a connection between bond forces and the stress in the classical (local) theory is established for the calculation of peridynamic stress, which is calculated by summing up bond forces passing through or ending at the cross section of a node.
Abstract: Peridynamics is a theory of continuum mechanics employing a nonlocal model that can simulate fractures and discontinuities (Askari et al. J Phys 125:012---078, 2008; Silling J Mech Phys Solids 48(1):175---209, 2000). It reformulates continuum mechanics in forms of integral equations rather than partial differential equations to calculate the force on a material point. A connection between bond forces and the stress in the classical (local) theory is established for the calculation of peridynamic stress, which is calculated by summing up bond forces passing through or ending at the cross section of a node. The peridynamic stress and the constitutive law in elasticity are used for the derivation of one- and three-dimensional numerical micromoduli. For three-dimensional discretized peridynamics, the numerical micromodulus is larger than the analytical micromodulus, and converges to the analytical value as the horizon to grid spacing ratio increases. A comparison of material responses in a three-dimensional discretized peridynamic model using numerical and analytical micromoduli, respectively, is performed for different horizons. As the horizon increases, the boundary effect is more conspicuous, and the errors increase in the back-calculated Young's modulus and strains. For the simulation of materials of Poisson's ratios other than 1/4, a pairwise compensation scheme for discretized peridynamics is proposed. Compared with classical (local) elasticity solutions, the computational results by applying the proposed scheme show good agreement in the strain, the resultant Young's modulus and Poisson's ratio.
TL;DR: In this paper, a singular edge-based smoothed finite element method (sES-FEM) is further developed for dynamic crack analysis in two-dimensional elastic solids.
TL;DR: In this paper, the displacement-traction problem of three-dimensional linearized elasticity can be posed as three different minimization problems, depending on whether the displacement vector field or the stress tensor field is the unknown.
Abstract: The displacement-traction problem of three-dimensional linearized elasticity can be posed as three different minimization problems, depending on whether the displacement vector field, or the stress tensor field, or the strain tensor field, is the unknown. The objective of this paper is to put these three different formulations of the same problem in a new perspective, by means of Legendre-Fenchel duality theory. More specifically, we show that both the displacement and strain formulations can be viewed as Legendre-Fenchel dual problems to the stress formulation. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new duality approach to elasticity.
Book•
01 Apr 2012TL;DR: In finite homogeneous regions -image stresses 9. Inhomogeneities 10. Dislocations in infinite homogenous regions 11. Interactions between point defects and stresses 12. Interaction between dislocations and stresses 14. Interfaces 15.Interactions between interfaces and stresses 16.
Abstract: 1. Introduction 2. Basic linear elasticity 3. Methods 4. Green's functions for unit point force 5. Interactions between defects and stress 6. Inclusions in infinite homogeneous regions 7. Interactions between inclusions and imposed stresses 8. Inclusions in finite homogeneous regions - image stresses 9. Inhomogeneities 10. Point defects in infinite homogeneous regions 11. Interactions between point defects and stresses 12. Dislocations in infinite homogeneous regions 13. Interactions between dislocations and stresses 14. Interfaces 15. Interactions between interfaces and stresses 16. Interactions between defects Appendices Index.
TL;DR: In this article, the authors analyzed temperature effects on the static and dynamic response of suspended inclined cables through a continuous monodimensional model including geometric nonlinearities, and the significance of the temperature-dependent variation of tension and sag are parametrically investigated.
TL;DR: In this article, an experimental method is presented to obtain the effective in-plane compliance matrices of cellular structures using Nomex® honeycomb cores without a priori assumptions such as orthotropy.
TL;DR: In this paper, the capabilities of an interface model to predict failure behavior of steel fiber reinforced cementitious composites (SFRCCs) are evaluated at both macro and mesoscale levels of observation.
Abstract: In this work the capabilities of an interface model to predict failure behavior of steel fiber reinforced cementitious composites (SFRCCs) are evaluated at both macro and mesoscale levels of observation. The interface model is based on a hyperbolic maximum strength criterion defined in terms of the normal and shear stress components acting on the joint plane. Pre-peak regime is considered linear elastic, while the post-peak behavior is formulated in terms of the fracture energy release under failure mode I and/or II. The well-known “Mixture Theory” is adopted for modeling the interactions between fibers and the surrounding cementitious composite. The effects of both the axial forces on the fibers induced by normal relative displacements, as well as the dowel action due to tangential relative displacements in the interfaces are considered in the formulation of the interaction mechanisms between fibers and cementitious composites. After describing the interface model, this work focuses on numerical analyses of SFRCC failure behavior. Firstly, the validation analysis of the interface model is performed at the constitutive level by comparing its numerical predictions against experimental results available in scientific literature. Then, the sensitivity of the interface theory for SFRCC regarding the variation of main parameters of the composite constituents is evaluated. Finally, the attention is focused on Finite Element (FE) analysis of SFRCC failure behavior at meso and macroscopic levels of observation. The results demonstrate the capabilities of the interface theory based on the Mixture Theory to reproduce the main features of failure behavior of SRFCC in terms of fiber content and involved fracture modes.
TL;DR: Two new methods for linear elasticity that simultaneously yield stress and displacement approximations of optimal accuracy in both the mesh size and polynomial degree are presented.
Abstract: We present two new methods for linear elasticity that simultaneously yield stress and displacement approximations of optimal accuracy in both the mesh size $$h$$ and polynomial degree $$p$$ . This is achieved within the recently developed discontinuous Petrov---Galerkin (DPG) framework. In this framework, both the stress and the displacement approximations are discontinuous across element interfaces. We study locking-free convergence properties and the interrelationships between the two DPG methods.
TL;DR: In this paper, a general approach based on the superposition principle is proposed to study the problem of a finite elastic body with an arbitrarily shaped and located inclusion, which consists in decomposing the initial inclusion problem into the issue of the inclusion embedded in the corresponding infinite body and the auxiliary problem of the finite body subjected to the appropriate boundary loading provided by solving the former problem.
TL;DR: In this article, the energy functional of linear elasticity is obtained as Γ-limit of suitable rescalings of the energies of finite elasticity, and the quadratic control from below of the energy density W ( ∇ v ) for large values of the deformation gradient ∇v is replaced by the weaker condition W( ∆ v ) ⩾ | ∆v | p, for some p > 1.
TL;DR: In this paper, a new enriched weight function for meshless methods is proposed for the numerical treatment of multiple arbitrary cracks in two dimensions, which allows a more straightforward implementation and simulation of the presence of multiple cracks, crack branching and crack propagation in a meshless framework without using any of the existing algorithms such as visibility, transparency, and diffraction and without using additional unknowns and additional equations for the evolution of the level-sets.
Abstract: SUMMARY
A new enriched weight function for meshless methods is proposed for the numerical treatment of multiple arbitrary cracks in two dimensions. The main novelty consists in modifying the weight function with an intrinsic enrichment which is discontinuous over the finite length of the crack, represented by a segment, but continuous all around the crack tips. An analytical function is used to introduce discontinuities that are incorporated in the kernel in a simple, multiplicative manner.
The resulting method allows a more straightforward implementation and simulation of the presence of multiple cracks, crack branching and crack propagation in a meshless framework without using any of the existing algorithms such as visibility, transparency, and diffraction and without using additional unknowns and additional equations for the evolution of the level-sets, as in extrinsic partition of unity-based methods.
Stress intensity factors calculated using the J-integral demonstrate excellent agreement with analytical solutions for classical fracture mechanics benchmarks. Copyright © 2011 John Wiley & Sons, Ltd.
TL;DR: In this article, a gradient-enhanced homogenization procedure is proposed for fiber reinforced materials, where the fibers are assumed to remain linear elastic while the matrix material is modeled as elasto-plastic coupled with a damage law described by a non-local constitutive model.
TL;DR: In this paper, a high order theory for functionally graded (FG) axisymmetric cylindrical shells based on the expansion of the linear elasticity for functional graded materials (FGMs) into Fourier series in terms of Legendre's polynomials is presented.
TL;DR: In this paper, the authors show how to determine the fourth-order elasticity constant of soft (incompressible, isotropic) solids by coupling a small-amplitude wave with a small but finite pre-deformation.
TL;DR: In this article, a decomposition of the stochastic elasticity tensor on a deterministic tensor basis was introduced, and a generalized model for random elasticity matrices was proposed to take into account constraints on both the level of anisotropy and statistical fluctuations.
Abstract: This work is concerned with the construction of stochastic models for random elasticity matrices, allowing either for the generation of elasticity tensors exhibiting some material symmetry properties almost surely (integrating the statistical dependence between the random stiffness components) or for the modeling of random media that requires the mean of a stochastic anisotropy measure to be controlled apart from the level of statistical fluctuations. To this aim, we first introduced a decomposition of the stochastic elasticity tensor on a deterministic tensor basis and considered the probabilistic modeling of the random components, having recourse to the MaxEnt principle. Strategies for random generation and estimation were further reviewed, and the approach was exemplified in the case of a material that was transversely isotropic almost surely. In a second stage, we made use of such derivations to propose a generalized model for random elasticity matrices that took into accountconstraints on both the level of stochastic anisotropy and the level of statistical fluctuations.