Showing papers on "Linear elasticity published in 2010"

A simple geometric model for elastic deformations

Abstract: We advocate a simple geometric model for elasticity: distance between the differential of a deformation and the rotation group. It comes with rigorous differential geometric underpinnings, both smooth and discrete, and is computationally almost as simple and efficient as linear elasticity. Owing to its geometric non-linearity, though, it does not suffer from the usual linearization artifacts. A material model with standard elastic moduli (Lame parameters) falls out naturally, and a minimizer for static problems is easily augmented to construct a fully variational 2nd order time integrator. It has excellent conservation properties even for very coarse simulations, making it very robust.Our analysis was motivated by a number of heuristic, physics-like algorithms from geometry processing (editing, morphing, parameterization, and simulation). Starting with a continuous energy formulation and taking the underlying geometry into account, we simplify and accelerate these algorithms while avoiding common pitfalls. Through the connection with the Biot strain of mechanics, the intuition of previous work that these ideas are "like" elasticity is shown to be spot on.

Nonlinear Mechanics of Crystals

Abstract: Introduction.- Mathematical foundations.- Kinematics of Crystalline Solids.- Thermomechanics of Crystalline Solids.- Thermoelasticity.- Elastoplasticity.- Residual Deformation from Lattice Defects.- Mechanical Twinning in Crystal Plasticity.- Generalized Inelasticity.- Dielectrics and piezoelectricity.- Chrystal Symmetries and Elastic Constants.- Lattice Statics and Dynamics.- Discrete Defects in Linear Elasticity.- SI Units and Fundamental Constants.- Kinematic Derivations.- References.- Index.

A new elasticity element made for enforcing weak stress symmetry

Abstract: We introduce a new mixed method for linear elasticity. The novelty is a simplicial element for the approximate stress. For every positive integer k, the row-wise divergence of the element space spans the set of polynomials of total degree k. The degrees of freedom are suited to achieve continuity of the normal stresses. What makes the element distinctive is that its dimension is the smallest required for enforcing a weak symmetry condition on the approximate stress. This is achieved using certain "bubble matrices", which are special divergence-free matrix-valued polynomials. We prove that the approximation error is of order k + 1 in both the displacement and the stress, and that a postprocessed displacement approximation converging at order k + 2 can be computed element by element. We also show that the globally coupled degrees of freedom can be reduced by hybridization to those of a displacement approximation on the element boundaries.

Topological optimization of structures subject to Von Mises stress constraints

Abstract: The topological asymptotic analysis provides the sensitivity of a given shape functional with respect to an infinitesimal domain perturbation. Therefore, this sensitivity can be naturally used as a descent direction in a structural topology design problem. However, according to the literature concerning the topological derivative, only the classical approach based on flexibility minimization for a given amount of material, without control on the stress level supported by the structural device, has been considered. In this paper, therefore, we introduce a class of penalty functionals that mimic a pointwise constraint on the Von Mises stress field. The associated topological derivative is obtained for plane stress linear elasticity. Only the formal asymptotic expansion procedure is presented, but full justifications can be deduced from existing works. Then, a topology optimization algorithm based on these concepts is proposed, that allows for treating local stress criteria. Finally, this feature is shown through some numerical examples.

Elastic Properties and Frequencies of Free Vibrations of Single-Layer Graphene Sheets

Abstract: We determine the basal plane stiffness and Poisson’s ratio of single layer graphene sheets (SLGSs) in armchair and zigzag directions by using molecular mechanics simulations of their uniaxial tensile deformations with the MM3 potential, and of their axial and bending vibrations. Both approaches give the basal plane stiffness equal to ∼340 N/m which agrees well with that reported in the literature and derived from results of indentation experiments on SLGSs and from the first principle calculations. The computed value of Poisson’s ratio equals 0.21 in both armchair and zigzag directions. Assuming that the response of a SLGS is the same as that of a plate made of a linear elastic, homogeneous, and isotropic material having Poisson’s ratio = 0.21, the in-plane stiffness of ∼340 N/m and the total mass equal to that of the SLGS, the thickness of the SLGS is found to be ∼1 A. Thus Young’s modulus and the shear modulus of a SLGS equal ∼3.4 TPa and ∼1.4 TPa, respectively. It is shown that mode shapes corresponding to the several lowest frequencies of the SLGS differ noticeably from those of an equivalent thin layer made of a linear elastic isotropic material with Young’s modulus = 3.4 TPa and the shear modulus = 1.4 TPa. Furthermore, a free– free SLGS vibrates about a plane bisecting its width rather than its thickness as predicted by the Euler Bernoulli beam theory. We also investigate the effect of pretension on the natural frequencies of SLGSs using MM simulations and correlate it to that of 1 A thick linear elastic plate found by analyzing its three-dimensional deformations. These results will help design SLGS nanomechanical resonators having frequencies in the THz range.

The bending of single layer graphene sheets: the lattice versus continuum approach

Abstract: The out-of-plane bending behaviour of single layer graphene sheets (SLGSs) is investigated using a special equivalent atomistic-continuum model, where the C-C bonds are represented by deep shear bending and axial stretching beams and the graphene properties by a homogenization approach. SLGS models represented by circular and rectangular plates are subjected to linear and nonlinear geometric point loading, similar to what is induced by an atomic force microscope (AFM) tip. The graphene models are developed using both a lattice and a continuum finite element discretization of the partial differential equations describing the mechanics of the graphene. The minimization of the potential energy allows us to identify the thickness, elastic parameters and force/displacement histories of the plates, in good agreement with other molecular dynamic (MD) and experimental results. We note a substantial equivalence of the linear elastic mechanical properties exhibited by circular and rectangular sheets, while some differences in the nonlinear geometric elastic regime for the two geometrical configurations are observed. Enhanced flexibility of SLGSs is observed by comparing the nondimensional force versus displacement relations derived in this work and the analogous ones related to equivalent plates with conventional isotropic materials.

An efficient multigrid method for the simulation of high-resolution elastic solids

Abstract: We present a multigrid framework for the simulation of high-resolution elastic deformable models, designed to facilitate scalability on shared memory multiprocessors. We incorporate several state-of-the-art techniques from multigrid theory, while adapting them to the specific requirements of graphics and animation applications, such as the ability to handle elaborate geometry and complex boundary conditions. Our method supports simulation of linear elasticity and corotational linear elasticity. The efficiency of our solver is practically independent of material parameters, even for near-incompressible materials. We achieve simulation rates as high as 6 frames per second for test models with 256K vertices on an 8-core SMP, and 1.6 frames per second for a 2M vertex object on a 16-core SMP.

Topological derivative for multi‐scale linear elasticity models applied to the synthesis of microstructures

Abstract: This paper proposes an algorithm for the synthesis/optimization of microstructures based on an exact formula for the topological derivative of the macroscopic elasticity tensor and a level set domain representation. The macroscopic elasticity tensor is estimated by a standard multi-scale constitutive theory where the strain and stress tensors are volume averages of their microscopic counterparts over a representative volume element. The algorithm is of simple computational implementation. In particular, it does not require artificial algorithmic parameters or strategies. This is in sharp contrast with existing microstructural optimization procedures and follows as a natural consequence of the use of the topological derivative concept. This concept provides the correct mathematical framework to treat topology changes such as those characterizing microstuctural optimization problems. The effectiveness of the proposed methodology is illustrated in a set of finite element-based numerical examples.Copyright © 2010 John Wiley & Sons, Ltd.

Electroactive Elastomeric Actuator for All-Polymer Linear Peristaltic Pumps

Abstract: This paper describes the concept of a lightweight and deformable structure with intrinsic distributed electromechanical actuation, potentially suitable to develop soft linear peristaltic pumps for incompressible fluids The proposed system is represented by a series of radially expanding flexible tubular modules, made of dielectric elastomers (DEs), as one of the most promising classes of electroactive polymers for actuation Each module consists of a cylindrical hollow DE actuator, working in purely radial mode with specific boundary constraints The static electromechanical transduction performance of such a module was investigated analytically, numerically, and experimentally For this purpose, predictions obtained from an analytical model, derived in the hypothesis of linear elasticity, were compared with those provided by a finite-element method Both models were validated by means of a comparison with experimental data, obtained from a silicone-made prototype module Results permitted to obtain a simple tool of simulation, suitable to predict the performance of the system in terms of both displaced volume and driving pressure, as a function of the material elastic modulus and the applied voltage

Numerically determined enrichment functions for the extended finite element method and applications to bi‐material anisotropic fracture and polycrystals

Abstract: Strain singularities appear in many linear elasticity problems. A very fine mesh has to be used in the vicinity of the singularity in order to obtain acceptable numerical solutions with the finite element method (FEM). Special enrichment functions describing this singular behavior can be used in the extended finite element method (X-FEM) to circumvent this problem. These functions have to be known in advance, but their analytical form is unknown in many cases. Li et al. described a method to calculate singular strain fields at the tip of a notch numerically. A slight modification of this approach makes it possible to calculate singular fields also in the interior of the structural domain. We will show in numerical experiments that convergence rates can be significantly enhanced by using these approximations in the X-FEM. The convergence rates have been compared with the ones obtained by the FEM. This was done for a series of problems including a polycrystalline structure. Copyright © 2010 John Wiley & Sons, Ltd.

Cosserat model for periodic masonry deduced by nonlinear homogenization

Abstract: The paper deals with the problem of the determination of the in-plane behavior of periodic masonry material. The macromechanical equivalent Cosserat medium, which naturally accounts for the absolute size of the constituents, is derived by a rational homogenization procedure based on the Transformation Field Analysis. The micromechanical analysis is developed considering a Cauchy model for masonry components. In particular, a linear elastic constitutive relationship is considered for the blocks, while a nonlinear constitutive law is adopted for the mortar joints, accounting for the damage and friction phenomena occurring during the loading history. Some numerical applications are performed on a Representative Volume Element characterized by a selected commonly used texture, without performing at this stage structural analyses. A comparison between the results obtained adopting the proposed procedure and a nonlinear micromechanical Finite Element Analysis is presented. Moreover, the substantial differences in the nonlinear behavior of the homogenized Cosserat material model with respect to the classical Cauchy one, are illustrated.

Computationally efficient pseudo-viscoelastic models for evaluation of residual stresses in thermoset polymer composites during cure

Abstract: In this study of the constitutive modelling of thermoset polymers during cure, we compare what we call the “cure hardening instantaneously linear elastic (CHILE)” approach with the more computationally intensive viscoelastic approach. The CHILE approach is popular compared to the viscoelastic approach as the cost of material characterization, data reduction, finite element model development and implementation, and computer run time is significantly lower. However, CHILE models suffer from the fact that the justification for their validity is essentially anecdotal, rather than based on a clear linkage to viscoelastic theory; and in related manner, materials characterization is done at an intuitively low but essentially arbitrary frequency. In this work we show that there are approximations that allow the full viscoelastic approach to be simplified progressively, and that these approximations are appropriate for the typical cure cycle undergone by a thermoset polymer. We present the functions of time at which the elastic modulus of the polymer should be calibrated for these simplified ‘pseudo-viscoelastic’ models, and show that for the uniaxial loading of a fully constrained block of polymer undergoing a given cure cycle, the predicted residual stresses compare very well with those computed using the full viscoelastic model. For further simplification, at the price of slightly lower accuracy and generality, a constant time or frequency can be chosen to evaluate the modulus. In general, we show that the CHILE approach, when properly calibrated, is a valid and efficient pseudo-viscoelastic (PVE) model, and that there is a continuum of trade-off of investment versus accuracy as we go from a full viscoelastic approach to the simplest CHILE approach.

On the existence of solution in the linear elasticity with surface stresses

Abstract: The mathematical investigation of the initial-boundary and boundary value problems in the linear elasticity considering surface stresses is presented. Weak setup of the problems based on mechanical variational principles is studied. Theorems of uniqueness and existence of the weak solution in energy spaces of static and dynamic problems are formulated and proved. Some properties of the spectrum of the problems under consideration are established. The studies are performed applying the functional analysis techniques. Finally, the Rayleigh principle for eigenfrequencies is constructed.

Aortic valve leaflet mechanical properties facilitate diastolic valve function

Abstract: This work was concerned with the numerical simulation of the behaviour of aortic valves whose material can be modelled as non-linear elastic anisotropic Linear elastic models for the valve leaflets with parameters used in previous studies were compared with hyperelastic models, incorporating leaflet anisotropy with pronounced stiffness in the circumferential direction through a transverse isotropic model The parameters for the hyperelastic models were obtained from fits to results of orthogonal uniaxial tensile tests on porcine aortic valve leaflets The computational results indicated the significant impact of transverse isotropy and hyperelastic effects on leaflet mechanics; in particular, increased coaptation with peak values of stress and strain in the elastic limit The alignment of maximum principal stresses in all models follows approximately the coarse collagen fibre distribution found in aortic valve leaflets The non-linear elastic leaflets also demonstrated more evenly distributed stress and strain which appears relevant to long-term scaffold stability and mechanotransduction

Plane strain bifurcations of elastic layered structures subject to finite bending: theory versus experiments

Abstract: Finite plane strain bending is solved for a multilayered elastic–incompressible thick plate. This multilayered solution, previously considered only in the case of homogeneity, is in itself interesting and reveals complex stress states such as the existence of more than one neutral axis for certain geometries. The bending solution is employed to investigate possible incremental bifurcations. The analysis reveals that a multilayered structure can behave in a completely different way from the corresponding homogeneous plate. For a thick plate of neo-Hookean material, for instance, the presence of a stiff coating strongly affects the bifurcation critical angle. Experiments designed and performed to substantiate our theoretical findings demonstrate that the theory can be effectively used as a design tool for predicting the capability of an elastic multilayered structure to be subject to a finite bending without suffering localized crazing.

Nanoscale tensile, shear, and failure properties of layered silicates as a function of cation density and stress

Abstract: Mechanical properties of layered silicates on the nanometer scale have been associated with large uncertainty. We attempt to clarify the linear elastic properties including tensile moduli, shear mo...