Showing papers on "Linear elasticity published in 2014"

Efficient fixed point and Newton---Krylov solvers for FFT-based homogenization of elasticity at large deformations

Abstract: In recent years the FFT-based homogenization method of Moulinec and Suquet has been established as a fast, accurate and robust tool for obtaining effective properties in linear elasticity and conductivity problems. In this work we discuss FFT-based homogenization for elastic problems at large deformations, with a focus on the following improvements. Firstly, we exhibit the fixed point method introduced by Moulinec and Suquet for small deformations as a gradient descent method. Secondly, we propose a Newton---Krylov method for large deformations. We give an example for which this methods needs approximately 20 times less iterations than Newton's method using linear fixed point solvers and roughly $$100$$100 times less iterations than the nonlinear fixed point method. However, the Newton---Krylov method requires 4 times more storage than the nonlinear fixed point scheme. Exploiting the special structure we introduce a memory-efficient version with 40 % memory saving. Thirdly, we give an analytical solution for the micromechanical solution field of a two-phase isotropic St.Venant---Kirchhoff laminate. We use this solution for comparison and validation, but it is of independent interest. As an example for a microstructure relevant in engineering we discuss finally the application of the FFT-based method to glass fiber reinforced polymer structures.

Atomistic Studies of Mechanical Properties of Graphene

Abstract: Recent progress of simulations/modeling at the atomic level has led to a better understanding of the mechanical behaviors of graphene, which include the linear elastic modulus E, the nonlinear elastic modulus D, the Poisson’s ratio ν, the intrinsic strength σint and the corresponding strain eint as well as the ultimate strain emax (the fracture strain beyond which the graphene lattice will be unstable). Due to the two-dimensional geometric characteristic, the in-plane tensile response and the free-standing indentation response of graphene are the focal points in this review. The studies are based on multiscale levels: including quantum mechanical and classical molecular dynamics simulations, and parallel continuum models. The numerical studies offer useful links between scientific research with engineering application, which may help to fulfill graphene potential applications such as nano sensors, nanotransistors, and other nanodevices.

Thermomechanics of monolayer graphene: Rippling, thermal expansion and elasticity

Abstract: Thermomechanical properties of monolayer graphene with thermal fluctuation are studied by both statistical mechanics analysis and molecular dynamics (MD) simulations. While the statistical mechanics analysis in the present study is limited by a harmonic approximation, significant anharmonic effects are revealed by MD simulations. The amplitude of out-ofplane thermal fluctuation is calculated for graphene membranes under both zero stress and zero strain conditions. It is found that the fluctuation amplitude follows a power-law scaling with respect to the linear dimension of the membrane, but the roughness exponents are different for the two conditions due to anharmonic interactions between bending and stretching modes. Such thermal fluctuation or rippling is found to be responsible for the effectively negative in-plane thermal expansion of graphene at relatively low temperatures, while a transition to positive thermal expansion is predicted as the anharmonic interactions suppress the rippling effect at high temperatures. Subject to equi-biaxial tension, the amplitude of thermal rippling decreases nonlinearly, and the in-plane stress-strain relation of graphene becomes nonlinear even at infinitesimal strain, in contrast with classical theory of linear elasticity. It is found that the tangent biaxial modulus of graphene depends on strain non-monotonically, decreases with increasing temperature, and depends on membrane size. Both statistical mechanics and MD simulations suggest considerable entropic contribution to the thermomechanical properties of graphene, and as a result thermal rippling is intricately coupled with thermal expansion and thermoelasticity for monolayer graphene membranes.

Dynamics of periodic mechanical structures containing bistable elastic elements: From elastic to solitary wave propagation

Abstract: We investigate the nonlinear dynamics of a periodic chain of bistable elements consisting of masses connected by elastic springs whose constraint arrangement gives rise to a large-deformation snap-through instability. We show that the resulting negative-stiffness effect produces three different regimes of (linear and nonlinear) wave propagation in the periodic medium, depending on the wave amplitude. At small amplitudes, linear elastic waves experience dispersion that is controllable by the geometry and by the level of precompression. At moderate to large amplitudes, solitary waves arise in the weakly and strongly nonlinear regime. For each case, we present closed-form analytical solutions and we confirm our theoretical findings by specific numerical examples. The precompression reveals a class of wave propagation for a partially positive and negative potential. The presented results highlight opportunities in the design of mechanical metamaterials based on negative-stiffness elements, which go beyond current concepts primarily based on linear elastic wave propagation. Our findings shed light on the rich effective dynamics achievable by nonlinear small-scale instabilities in solids and structures.

Flattening of a patterned compliant solid by surface stress.

Abstract: We measured the shape change of periodic ridge surface profiles in gelatin organogels resulting from deformation driven by their solid–vapor surface stress. A gelatin organogel was molded onto poly-dimethylsiloxane (PDMS) masters having ridge heights of 1.7 and 2.7 μm and several periodicities. Gel replicas were found to have a shape deformed significantly compared to their PDMS master. Systematically larger deformations in gels were measured for lower elastic moduli. Measuring the elastic modulus independently, we estimate a surface stress of 107 ± 7 mN m−1 for the organogels in solvent composed of 70 wt% glycerol and 30 wt% water. Shape changes are in agreement with a small strain linear elastic theory. We also measured the deformation of deeper ridges (with height 13 μm), and analysed the resulting large surface strains using finite element analysis.

A homogenization approach for nonwoven materials based on fiber undulations and reorientation

Abstract: This paper presents a new micromechanical based approach for the modeling of the highly anisotropic and non-linear stiffening response of fibrous materials with random network microstructure at finite strains. The first key aspect of the proposed approach arises from the experimentally justified need to model the elastic microscopic response of the constituent fibers, which are one-dimensional elements, as linear elastic. This linear elastic behavior is modified in the lower strain regime to account for the inherent fiber undulations and the associated fiber unfolding phenomena. Another key aspect is the reorientation of these fibers which is identified as one primary mechanism for the overall macroscopic stiffening. The one-dimensional elements are statistically distributed as unit vectors in a non-uniform manner over an affine referential network space of orientations represented by a unit circle in the two-dimensional case of interest here. A physically motivated reorientation of these unit vectors is achieved by a bijective map which asymptotically aligns them with the maximum loading direction in the referential orientation space. A rate-independent evolution law for this map is sought by a physically motivated assumption to maintain the overall elastic framework of the proposed formulation. A closed form solution to the new evolution law is also presented which allows faster computation of updating orientations without resorting to numerical integration or storing history variables. The unit vectors upon reorientation in the referential orientation space are then mapped to the spatial orientation space by the macrodeformation gradient to compute the macroscopic Kirchhoff stress and the associated spatial elasticity modulus. Reorientation of these unit vectors results in the evolution of the associated probability density function which is also computed in closed form depending on the initial probability density. However, it is shown that for a bijective reorientation map, the homogenization of micro-variables over the referential orientation space is independent of the evolving probability density function. Homogeneous deformation tests are performed to highlight the elastic framework of the proposed formulation. A direct comparison of the numerical results with the experimental results from the literature is made which demonstrates the predictive capabilities of the proposed formulation.

A family of conforming mixed finite elements for linear elasticity on triangular grids

Abstract: This paper presents a family of mixed finite elements on triangular grids for solving the classical Hellinger-Reissner mixed problem of the elasticity equations. In these elements, the matrix-valued stress field is approximated by the full $C^0$-$P_k$ space enriched by $(k-1)$ $H(\d)$ edge bubble functions on each internal edge, while the displacement field by the full discontinuous $P_{k-1}$ vector-valued space, for the polynomial degree $k\ge 3$. The main challenge is to find the correct stress finite element space matching the full $C^{-1}$-$P_{k-1}$ displacement space. The discrete stability analysis for the inf-sup condition does not rely on the usual Fortin operator, which is difficult to construct. It is done by characterizing the divergence of local stress space which covers the $P_{k-1}$ space of displacement orthogonal to the local rigid-motion. The well-posedness condition and the optimal a priori error estimate are proved for this family of finite elements. Numerical tests are presented to confirm the theoretical results.

Molecular Approaches for Multifield Continua: origins and current developments

Abstract: The mechanical behaviour of complex materials, characterised at finer scales by the presence of heterogeneities of significant size and texture, strongly depends on their microstructural features. Attention is centred on multiscale approaches which aim to deduce properties and relations at a given macroscale by bridging information at proper underlying microlevel via energy equivalence criteria. Focus is on physically–based corpuscular–continuous models originated by the molecular models developed in the 19th century to give an explanation per causas of elasticity. In particular, the ‘mechanistic–energetistic’ approach by Voigt and Poincare who, when dealing with the paradoxes deriving from the search of the exact number of elastic constants in linear elasticity, respectively introduced molecular models with moment and multi–body interactions is examined. Thus overcoming the experimental discrepancies related to the so–called central–force scheme, originally adopted by Navier, Cauchy and Poisson.

Fractional continua for linear elasticity

Abstract: Fractional continua is a generalisation of the classical continuum body. This new concept shows the application of fractional calculus in continuum mechanics. The advantage is that the obtained description is non-local. This natural non-locality is inherently a consequence of fractional derivative definition which is based on the interval, thus variates from the classical approach where the definition is given in a point. In the paper, the application of fractional continua to one-dimensional problem of linear elasticity under small deformation assumption is presented.

Viscous interfaces as source for material creep: A continuum micromechanics approach

Abstract: It is generally agreed upon that fluids may play a major role in the creep behavior of materials comprising heterogeneous microstructures and fluid-filled porosities at small length scales. In more detail, nanoconfined fluid-filled interfaces are typically considered to act as a lubricants, once electrically charged solid surfaces start to glide along fluid sheets, with the fluid being typically in a liquid crystal state, which refers to an “adsorbed”, “ice-like”, or “glassy” structure of fluid molecules. Here, we aim at translating this interface behavior into apparent creep laws at the continuum scale of materials consisting of one non-creeping solid matrix with embedded fluid-filled interfaces. To this end, we consider a linear relationship between (i) average interface dislocations and (ii) corresponding interface tractions, with an interface viscosity as the proportionality constant. Homogenization schemes for eigenstressed heterogeneous materials are used to upscale this interface behavior to the much larger observation scale of a matrix-inclusion composite comprising an isotropic and linear elastic solid matrix, as well as interacting parallel interfaces of circular shape, which are embedded in the aforementioned matrix. This results in exponentially decaying macroscopic viscoelastic phenomena, with both creep and relaxation times increasing with increasing interface size and viscosity, as well as with decreasing elastic stiffness of the solid matrix; while only the relaxation time decreases with increasing interface density. Accordingly, non-asymptotic creep of hydrated (quasi-) crystalline materials at higher load intensities may be readily explained through non-stationarity, i.e. spreading, of liquid crystal interfaces throughout solid elastic matrices.

Generalized multiscale finite element method for elasticity equations

Abstract: In this paper, we discuss the application of generalized multiscale finite element method (GMsFEM) to elasticity equation in heterogeneous media. We consider steady state elasticity equations though some of our applications are motivated by elastic wave propagation in subsurface where the subsurface properties can be highly heterogeneous and have high contrast. We present the construction of main ingredients for GMsFEM such as the snapshot space and offline spaces. The latter is constructed using local spectral decomposition in the snapshot space. The spectral decomposition is based on the analysis which is provided in the paper. We consider both continuous Galerkin and discontinuous Galerkin coupling of basis functions. Both approaches have their cons and pros. Continuous Galerkin methods allow avoiding penalty parameters though they involve partition of unity functions which can alter the properties of multiscale basis functions. On the other hand, discontinuous Galerkin techniques allow gluing multiscale basis functions without any modifications. Because basis functions are constructed independently from each other, this approach provides an advantage. We discuss the use of oversampling techniques that use snapshots in larger regions to construct the offline space. We provide numerical results to show that one can accurately approximate the solution using reduced number of degrees of freedom.

Numerical Study of Shear Stress Distribution for Discrete Columns in Liquefiable Soils

Abstract: Discrete columns, such as stone and soil-cement columns, are often used to improve the liquefaction resistance of loose sandy ground potentially subjected to strong shaking. The shear stress reduction in the loose ground resulting from the reinforcing effect of these stiffer discrete columns is often considered as a contributing mechanism for liquefaction mitigation. Current design practice often assumes that discrete columns and soil deform equally in pure shear (i.e., shear strain–compatible deformation). In addition, because the discrete column is stiffer than the soil, it is assumed to attract higher shear stress, thereby reducing the shear stress in the surrounding soil. In this paper, shear stress reduction in liquefiable soils and shear strain distribution between liquefiable soil and discrete columns along with the potential of development of tensile cracks is investigated using three-dimensional linear elastic, finite-element analysis. Parametric analyses are performed for a range of geom...

Phase-field-crystal models and mechanical equilibrium.

Abstract: Phase-field-crystal (PFC) models constitute a field theoretical approach to solidification, melting, and related phenomena at atomic length and diffusive time scales. One of the advantages of these models is that they naturally contain elastic excitations associated with strain in crystalline bodies. However, instabilities that are diffusively driven towards equilibrium are often orders of magnitude slower than the dynamics of the elastic excitations, and are thus not included in the standard PFC model dynamics. We derive a method to isolate the time evolution of the elastic excitations from the diffusive dynamics in the PFC approach and set up a two-stage process, in which elastic excitations are equilibrated separately. This ensures mechanical equilibrium at all times. We show concrete examples demonstrating the necessity of the separation of the elastic and diffusive time scales. In the small-deformation limit this approach is shown to agree with the theory of linear elasticity.

A path-independent integral for fracture of solids under combined electrochemical and mechanical loadings

Abstract: In this study, we first demonstrate that the J-integral in classical linear elasticity becomes path-dependent when the solid is subjected to combined electrical, chemical and mechanical loadings. We then construct an electro-chemo-mechanical J-integral that is path-independent under such combined multiple driving forces. Further, we show that this electro-chemo-mechanical J-integral represents the rate at which the grand potential releases per unit crack growth. As an example, the path-independent nature of the electro-chemo-mechanical J-integral is demonstrated by solving the problem of a thin elastic film delaminated from a thick elastic substrate.

Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases

Abstract: We review the theoretical bounds on the effective properties of linear elastic inhomogeneous solids (including composite materials) in the presence of constituents having non-positive-definite elastic moduli (so-called negative-stiffness phases). Using arguments of Hill and Koiter, we show that for statically stable bodies the classical displacement-based variational principles for Dirichlet and Neumann boundary problems hold but that the dual variational principle for traction boundary problems does not apply. We illustrate our findings by the example of a coated spherical inclusion whose stability conditions are obtained from the variational principles. We further show that the classical Voigt upper bound on the linear elastic moduli in multi-phase inhomogeneous bodies and composites applies and that it imposes a stability condition: overall stability requires that the effective moduli do not surpass the Voigt upper bound. This particularly implies that, while the geometric constraints among constituents in a composite can stabilize negative-stiffness phases, the stabilization is insufficient to allow for extreme overall static elastic moduli (exceeding those of the constituents). Stronger bounds on the effective elastic moduli of isotropic composites can be obtained from the Hashin–Shtrikman variational inequalities, which are also shown to hold in the presence of negative stiffness.