Showing papers on "Linear elasticity published in 2007"

Microstructural Randomness and Scaling in Mechanics of Materials

Abstract: PREFACE BASIC RANDOM MEDIA MODELS Probability Measure of Geometric Objects Basic Point Fields Directional Data Random Fibers, Random Line Fields, Tessellations Basic Concepts and Definitions of Random Microstructures RANDOM PROCESSES AND FIELDS Elements of One-Dimensional Random Fields Mechanics Problems on One-Dimensional Random Fields Elements of Two- and Three-Dimensional Random Fields Mechanics Problems on Two- and Three-Dimensional Random Fields Ergodicity The Maximum Entropy Method PLANAR LATTICE MODELS: PERIODIC TOPOLOGIES AND ELASTOSTATICS One-Dimensional Lattices Planar Lattices: Classical Continua Applications in Mechanics of Composites Planar Lattices: Nonclassical Continua Extension-Twist Coupling in a Helix LATTICE MODELS: RIGIDITY, RANDOMNESS, DYNAMICS, AND OPTIMALITY Rigidity of Networks Spring Network Models for Disordered Topologies Particle Models Michell Trusses: Optimal Use of Material TWO- VERSUS THREE-DIMENSIONAL CLASSICAL ELASTICITY Basic Relations The CLM Result and Stress Invariance Poroelasticity TWO- VERSUS THREE-DIMENSIONAL MICROPOLAR ELASTICITY Micropolar Elastic Continua Classical vis-a-vis Nonclassical (Elasticity) Models Planar Cosserat Elasticity The CLM Result and Stress Invariance Effective Micropolar Moduli and Characteristic Lengths of Composites MESOSCALE BOUNDS FOR LINEAR ELASTIC MICROSTRUCTURES Micro-, Meso-, and Macroscales Volume Averaging Spatial Randomness Hierarchies of Mesoscale Bounds Examples of Hierarchies of Mesoscale Bounds Moduli of Trabecular Bone RANDOM FIELD MODELS AND STOCHASTIC FINITE ELEMENTS Mesoscale Random Fields Second-Order Properties of Mesoscale Random Fields Does There Exist a Locally Isotropic, Smooth Elastic Material? Stochastic Finite Elements for Elastic Media Method of Slip-Lines for Inhomogeneous Plastic Media Michell Trusses in the Presence of Random Microstructure HIERARCHIES OF MESOSCALE BOUNDS FOR NONLINEAR AND INELASTIC MICROSTRUCTURES Physically Nonlinear Elastic Microstructures Finite Elasticity of Random Composites Elastic-Plastic Microstructures Rigid-Perfectly Plastic Microstructures Viscoelastic Microstructures Stokes Flow in Porous Media Thermoelastic Microstructures Scaling and Stochastic Evolution in Damage Phenomena Comparison of Scaling Trends MESOSCALE RESPONSE IN THERMOMECHANICS OF RANDOM MEDIA From Statistical Mechanics to Continuum Thermodynamics Extensions of the Hill Condition Legendre Transformations in (Thermo)Elasticity Thermodynamic Orthogonality on the Mesoscale Complex versus Compound Processes: The Scaling Viewpoint Toward Continuum Mechanics of Fractal Media WAVES AND WAVEFRONTS IN RANDOM MEDIA Basic Methods in Stochastic Wave Propagation Toward Spectral Finite Elements for Random Media Waves in Random 1D Composites Transient Waves in Heterogeneous Nonlinear Media Acceleration Wavefronts in Nonlinear Media BIBLIOGRAPHY INDEX

Robust strategies for automated AFM force curve analysis--I. Non-adhesive indentation of soft, inhomogeneous materials.

Abstract: The atomic force microscope (AFM) has found wide applicability as a nanoindentation tool to measure local elastic properties of soft materials. An automated approach to the processing of AFM indentation data, namely, the extraction of Young's modulus, is essential to realizing the high-throughput potential of the instrument as an elasticity probe for typical soft materials that exhibit inhomogeneity at microscopic scales. This paper focuses on Hertzian analysis techniques, which are applicable to linear elastic indentation. We compiled a series of synergistic strategies into an algorithm that overcomes many of the complications that have previously impeded efforts to automate the fitting of contact mechanics models to indentation data. AFM raster data sets containing up to 1024 individual force-displacement curves and macroscopic compression data were obtained from testing polyvinyl alcohol gels of known composition. Local elastic properties of tissue-engineered cartilage were also measured by the AFM. All AFM data sets were processed using customized software based on the algorithm, and the extracted values of Young's modulus were compared to those obtained by macroscopic testing. Accuracy of the technique was verified by the good agreement between values of Young's modulus obtained by AFM and by direct compression of the synthetic gels. Validation of robustness was achieved by successfully fitting the vastly different types of force curves generated from the indentation of tissue-engineered cartilage. For AFM indentation data that are amenable to Hertzian analysis, the method presented here minimizes subjectivity in preprocessing and allows for improved consistency and minimized user intervention. Automated, large-scale analysis of indentation data holds tremendous potential in bioengineering applications, such as high-resolution elasticity mapping of natural and artificial tissues.

Mixed finite element methods for linear elasticity with weakly imposed symmetry

Abstract: In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approxima- tions to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.

The elasticity of elasticity

Abstract: In this note we assert that the usual interpretation of what one means by “elasticity” is much too insular and illustrates our thesis by introducing implicit constitutive theories that can describe the non-dissipative response of solids. There is another important aspect to the introduction of such an implicit approach to the non-dissipative response of solids, the development of a hierarchy of approximations wherein, while the strains are infinitesimal the relationship between the stress and the linearized strain is non-linear. Such approximations would not be logically consistent within the context of explicit theories of Cauchy elasticity or Green elasticity that are currently popular.

Topology optimization of acoustic-structure interaction problems using a mixed finite element formulation

Abstract: The paper presents a gradient-based topology optimization formulation that allows to solve acoustic–structure (vibro-acoustic) interaction problems without explicit boundary interface representation. In acoustic–structure interaction problems, the pressure and displacement fields are governed by Helmholtz equation and the elasticity equation, respectively. Normally, the two separate fields are coupled by surface-coupling integrals, however, such a formulation does not allow for free material re-distribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process. In this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables (a u/p-formulation). The Helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating to zero. Hence, by spatial variation of the mass density, shear and bulk moduli we are able to solve the coupled problem by the mixed formulation. Using this modelling approach, the topology optimization procedure is simply implemented as a standard density approach. Several two-dimensional acoustic–structure problems are optimized in order to verify the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.

On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity

Abstract: The non-local peridynamic theory describes the displacement field of a continuous body by the initial-value problem for an integro-differential equation that does not include any spatial derivative. The non-locality is determined by the so-called peridynamic horizon $\delta$ which is the radius of interaction between material points taken into account. Well-posedness and structural properties of the peridynamic equation of motion are established for the linear case corresponding to small relative displacements. Moreover the limit behavior as $\delta \rightarrow 0$ is studied.

Modulus-density scaling behaviour and framework architecture of nanoporous self-assembled silicas.

Abstract: Natural porous materials such as bone, wood and pith evolved to maximize modulus for a given density. For these three-dimensional cellular solids, modulus scales quadratically with relative density. But can nanostructuring improve on Nature's designs? Here, we report modulus-density scaling relationships for cubic (C), hexagonal (H) and worm-like disordered (D) nanoporous silicas prepared by surfactant-directed self-assembly. Over the relative density range, 0.5 to 0.65, Young's modulus scales as (density)n where n(C)

Identification of Spring Parameters for Deformable Object Simulation

Abstract: Mass spring models are frequently used to simulate deformable objects because of their conceptual simplicity and computational speed. Unfortunately, the model parameters are not related to elastic material constitutive laws in an obvious way. Several methods to set optimal parameters have been proposed but, so far, only with limited success. We analyze the parameter identification problem and show the difficulties, which have prevented previous work from reaching wide usage. Our main contribution is a new method to derive analytical expressions for the spring parameters from an isotropic linear elastic reference model. The method is described and expressions for several mesh topologies are derived. These include triangle, rectangle, and tetrahedron meshes. The formulas are validated by comparing the static deformation of the MSM with reference deformations simulated with the finite element method.

Computation of Cavity Expansion Pressure and Penetration Resistance in Sands

Abstract: A cavity expansion-based theory for calculation of cone penetration resistance qc in sand is presented. The theory includes a completely new analysis to obtain cone resistance from cavity limit pressure. In order to more clearly link the proposed theory with the classical cavity expansion theories, which were based on linear elastic, perfectly plastic soil response, linear equivalent values of Young's modulus, Poisson’s ratio and friction and dilatancy angles are given in charts as a function of relative density, stress state, and critical-state friction angle. These linear-equivalent values may be used in the classical theories to obtain very good estimates of cavity pressure. A much simpler way to estimate qc —based on direct reading from charts in terms of relative density, stress state, and critical-state friction angle—is also proposed. Finally, a single equation obtained by regression of qc on relative density and stress state for a range of values of critical-state friction angle is also proposed. ...

Vibration analysis of spherical structural elements using the GDQ method

Abstract: This paper deals with the dynamical behaviour of hemispherical domes and spherical shell panels. The First-order Shear Deformation Theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is conducted within the theory of linear elasticity, when the material behaviour is assumed to be homogeneous and isotropic. The governing equations of motion, written in terms of internal resultants, are expressed as functions of five kinematic parameters, by using the constitutive and the congruence relationships. The boundary conditions considered are clamped (C), simply supported (S) and free (F) edge. Numerical solutions have been computed by means of the technique known as the Generalized Differential Quadrature (GDQ) Method. These results, which are based upon the FSDT, are compared with the ones obtained using commercial programs such as Abaqus, Ansys, Femap/Nastran, Straus, Pro/Engineer, which also elaborate a three-dimensional analysis. The effect of different grid point distributions on the convergence, the stability and the accuracy of the GDQ procedure is investigated. The convergence rate of the natural frequencies is shown to be fast and the stability of the numerical methodology is very good. The accuracy of the method is sensitive to the number of sampling points used, to their distribution and to the boundary conditions.

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

Abstract: In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length. To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple-stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.

Analysis and Numerical Approximation of an Integro-differential Equation Modeling Non-local Effects in Linear Elasticity

Abstract: Long-range interactions for linearly elastic media resulting in nonlinear dispersion relations are modeled by an initial-value problem for an integro-differential equation (IDE) that incorporates non-local effects. Interpreting this IDE as an evolutionary equation of second order, well-posedness in L ∞(ℝ) as well as jump relations are proved. Moreover, the construction of the micromodulus function from the dispersion relation is studied. A numerical approximation based upon quadrature is suggested and carried out for two examples, one involving jump discontinuities in the initial data corresponding to a Riemann-like problem.

A variational model for fracture mechanics - Numerical experiments

Abstract: In the variational model for brittle fracture proposed in Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342], the minimum problem is formulated as a free discontinuity problem for the energy functional of a linear elastic body. A family of approximating regularized problems is then defined, each of which can be solved numerically by a finite element procedure. Here we re-formulate the minimum problem within the context of finite elasticity. The main change is the introduction of the dependence of the strain energy density on the determinant of the deformation gradient. This change requires new, more general existence and Γ -convergence results. The results of some two-dimensional numerical simulations are presented, and compared with corresponding simulations made in Bourdin et al. [2000. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48, 797–826] for the linear elastic model.

Three-Dimensional Vibration Analyses of Functionally Graded Plates under Various Boundary Conditions

Abstract: Three-dimensional vibration solutions are presented for rectangular functionally graded plates with different boundary conditions, based on the small strain linear elasticity theory. The material properties are assumed to vary as a power form of the thickness coordinate. The Ritz method with Chebyshev displacement functions is used to solve the vibration problem of functionally graded plates. The convergence and comparison studies demonstrate the accuracy and correctness of the present method. The effects of aspect and thickness ratios, and gradient index, on the free vibration frequencies are investigated. Some results are given in the form of tables and figures, which can serve as the benchmark for further research.

Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem

Abstract: We consider several inverse problems motivated by elastography. Given the (possibly transient) displacement field measured everywhere in an isotropic, compressible, linear elastic solid, and given density ρ, determine the Lame parameters λ and μ. We consider several special cases of this problem: (a) for μ known a priori, λ is determined by a single deformation field up to a constant. (b) Conversely, for λ known a priori, μ is determined by a single deformation field up to a constant. This includes as a special case that for which the term λ∇ ⋅ u ≡ 0. (c) Finally, if neither λ nor μ is known a priori, but Poisson's ratio ν is known, then μ and λ are determined by a single deformation field up to a constant. This includes as a special case plane stress deformations of an incompressible material. Exact analytical solutions valid for 2D, 3D and transient deformations are given for all cases in terms of quadratures. These are used to show that the inverse problem for μ based on the compressible elasticity equations is unstable in the limit λ → ∞. Finally, we use the exact solutions as a basis to compute non-trivial modulus distributions in a simulated example.

The embedded discontinuous Galerkin method : Application to linear shell problems

Abstract: A new discontinuous Galerkin method for elliptic problems which is capable of rendering the same set of unknowns in the final system of equations as for the continuous displacement-based Galerkin method is presented. Those equations are obtained by the assembly of element matrices whose structure in particular cases is also identical to that of the continuous displacement approach. This makes the present formulation easily implementable within the existing commercial computer codes. The proposed approach is named the embedded discontinuous Galerkin method. It is applicable to any system of linear partial differential equations but it is presented here in the context of linear elasticity. An application of the method to linear shell problems is then outlined and numerical results are presented.

A bridging domain and strain computation method for coupled atomistic–continuum modelling of solids

Abstract: We present a multiscale method that couples atomistic models with continuum mechanics. The method is based on an overlapping domain-decomposition scheme. Constraints are imposed by a Lagrange multiplier method to enforce displacement compatibility in the overlapping subdomain in which atomistic and continuum representations overlap. An efficient version of the method is developed for cases where the continuum can be modelled as a linear elastic material. An iterative scheme is utilized to optimize the coupled configuration. Conditions for the regularity of the constrained matrices are determined. A method for computing strain in atomistic models and handshake domains is formulated based on a moving least-square approximation which includes both extensional and angle-bending terms. It is shown that this method exactly computes the linear strain field. Applications to the fracture of defected single-layer atomic sheets and nanotubes are given. Copyright © 2006 John Wiley & Sons, Ltd.

Numerical inversion of the Laplace–Carson transform applied to homogenization of randomly reinforced linear viscoelastic media

Abstract: Homogenization of linear viscoelastic materials is possible using the viscoelastic correspondence principle (VCP) and homogenization solutions obtained for linear elastic materials. The VCP involves a Laplace–Carson Transform (LCT) of the material phases constitutive theories and in most cases, the time domain solution must be obtained through numerical inversion of the LCT. The objective of this paper is to develop and test numerical algorithms to invert LCT which are encountered in the context of homogenization of linear viscoelastic materials. The homogenized properties, as well as the stress concentration and strain localization tensors, are considered. The algorithms suggested have the following two key features: (1) an acceptance criterion which allows to reject solutions of unacceptable accuracy and (2) some algorithms lead to solutions for the homogenized properties where the thermodynamics restrictions imposed on linear viscoelastic materials are encountered. These two features are an improvement over the previous algorithms. The algorithms are tested on many examples and the accuracy of the inversion is excellent in most cases.

J-integral evaluation for U- and V-blunt notches under Mode I loading and materials obeying a power hardening law

Abstract: The paper deals with calculations of the J-integral for a plate weakened by U- and V-blunt notches under mode I loading in the case of a linear and nonlinear elastic material. The main aim of the study is to suggest simple equations suitable for rapid calculations of the J-integral. The semicircular arc of the notch, which is traction free, is assumed as integration path and the J-integral is given as a function of the strain energy over the notch edge. For a numerical investigation of the strain energy density distribution on the notch edge the equation W(θ)=Wmax cosδ(θ) has been assumed, where δ has been determined from finite element analyses. In particular, the following values of the notch acuity a/ρ and the opening angle 2α have been analyzed: 4 ≤ a/ρ ≤ 400 and 0 ≤ 2α ≤ 3π /4. Considering plates weakened by lateral and central notches under symmetric mode I loading, the approximate relationships for the strain energy density, which require the presence of a non zero notch radius for their application, and the J-integral are discussed firstly considering a linear elastic material and then a material obeying a power hardening law during the loading phase. The predicted results of the J-integral are consistent with those directly obtained from finite element analyses.

Structural interfaces in linear elasticity. Part I: Nonlocality and gradient approximations

Abstract: Many biological and optimal materials, at multiple scales, consist of what can be idealized as continuous bodies joined by structural interfaces. Mechanical characterization of the microstructure defining the interface can nowadays be accurately done; however, such interfaces are usually analyzed employing models where those properties are overly simplified. To introduce into the analysis the microstructure properties, a new model of structural interfaces is proposed and developed: a true structure is introduced in the transition zone, joining continuous bodies, with geometrical and material properties directly obtained from those of the interfacial microstructure. First, the case of an elliptical inclusion connected by a structural interface to an infinite matrix is solved analytically, showing that nonlocal effects follow directly from the introduction of the structure, related to the inclination of the connecting elements. Second, starting from a discrete structure, a continuous model of a structural interface is derived. The usual zero-thickness linear interface model is shown to be a special case of this more general continuous structural interface model. Then, a gradient approximation of the interface constitutive law is rigorously derived: it is the first example of the analytical derivation of a nonlocal interface model from the microstructure properties. The effects introduced in the mechanical behavior by both the continuous model and its gradient approximation are illustrated by solving, for the first time, the problem of a circular inclusion connected to an infinite matrix by a structural interface and subject to remote uniform stress.

Lower Order Rectangular Nonconforming Mixed Finite Elements for Plane Elasticity

Abstract: In this paper, we present two stable rectangular nonconforming mixed finite element methods for the equations of linear elasticity in two space dimensions which produce direct approximations for the stress and displacement. In the first method, the normal stress space of the matrix-valued stress space is taken as the second order rotated Brezzi-Douglas-Fortin-Marini element space [F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991], the enriched nonconforming rotated $Q_1$ element [Q. Lin, L. Tobiska, and A. H. Zhou, IMA J. Numer. Anal., 25 (2005), pp. 160-181] is taken for the shear stress, and the lowest order Raviart-Thomas element space [P. A. Raviart and J. M. Thomas, in Mathematical Aspects of the Finite Element Method, Lecture Notes in Math. 606, Springer-Verlag, New York, 1977, pp. 292-315] is employed to approximate the vector displacement field. The second method is obtained from the first one through dropping the interior degrees of the normal stress on each element. A first order convergence rate is obtained for both the stress and the displacement for these methods based on the superconvergence of the enriched nonconforming rotated $Q_1$ element.

A linear curved-beam model for the analysis of galloping in suspended cables

Abstract: A linear model of curved, prestressed, no-shear, elastic beam, loaded by wind forces, is formulated. The beam is assumed to be planar in its reference configuration, under its own weight and static wind forces. The incremental equilibrium equations around the prestressed state are derived, in which shear forces are condensed. By using a linear elastic constitutive law and accounting for damping and inertial effects, the complete equations of motion are obtained. They are then greatly simplified by estimating the order of magnitude of all their terms, under the hypotheses of small sag-to-span ratio, order-1 aspect ratio of the (compact) section, characteristic section radius much smaller than length (slender cable), small transversal-to-longitudinal and transversal-to-torsional wave velocity ratios. A system of two integrodifferential equations is drawn in the two transversal displacements only. A simplified model of aerodynamic forces is then developed according to a quasisteady formulation. The nonlinear, nontrivial equilibrium path of the cable subjected to increasing static wind forces is successively evaluated, and the influence of the angle of twist on the equilibrium is discussed. Then stability is studied by discretizing the equations of motion via a Galerkin approach and analyzing the small oscillations around the nontrivial equilibrium. Finally, the role of the angle of twist on the dynamic stability of the cable is discussed for some sample cables.