Framework Of Linear Elasticity Research Articles

Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites

We report on the mathematical analysis of two different, FFT-based, numerical schemes for the homogenization of composite media within the framework of linear elasticity: the basic scheme of Moulinec and Suquet (1994, 1998) [9,10], and the energy-based scheme of Brisard and Dormieux (2010) [13]. Casting these two schemes as Galerkin approximations of the same variational problem allows us to assert their well-posedness and convergence. More importantly, we extend in this work their domains of application, by relieving some stringent conditions on the reference material which were previously thought necessary. The origins of the flaws of each scheme are identified, and a third scheme is proposed, which seems to combine the strengths of the basic and energy-based schemes, while leaving out their weaknesses. Finally, a rule is proposed for handling heterogeneous pixels/voxels, a situation frequently met when images of real materials are used as input to these schemes.

Bovine Cortical Bone Stiffness and Local Strain are Affected by Mineralization and Morphology

A multiscale analysis of the mechanical behavior of bovine Haversian cortical bone is presented in the frame-work of linear elasticity. Cortical bone displays a complex microstructure that includes four phases: Haversian canals, osteons, cement lines, and interstitial bone. Based on close experimental observations, a Monte Carlo algorithm is implemented to build the natural bone composite microstructure. To represent the hierarchical nature of bone, the algorithm incorporates macroscopic morphological components, such as its porosity and osteonal volume fraction, as well as microscopic parameters, such as the characterized distributions of the osteons diameters. Bone local mechanical properties are measured by nanoindentation and microextensometry. The three-dimensional microstructures are discretized by a finite element method in order to evaluate the representative volume element of bovine cortical bone. The numerical model calculates the macroscopic bulk and material Young’s moduli and describes the local stress and strain. How geometrical or mechanical factors affect bone failure is investigated through a comparison of the macroscopic anisotropy and local strain to experimental data.

Mechanical characterization and numerical simulation of polyether–ether–ketone (PEEK) cranial implants

Cranial implants have experienced a significant evolution in the last decade in different aspects such as materials, method of fixation, and the structure. In addition, patient-specific cranial implants have recently been started to be developed. To achieve this objective, efficient mechanical characterization and numerical modeling of the implant are required to guarantee its functionality on each patient as well as to facilitate further developments. In this work, mechanical characterization and numerical models have been performed for patient-specific Polyaryletherketone (PEEK) scaffold cranial implants. Mechanical characterization has been performed at the scaffold and the whole implant levels under displacement control tests. Two different implant designs for the same patient but with different scaffold structure were experimentally characterized, and finite element models of the implants were developed within the framework of linear elasticity. Two types of finite element models were developed: a detailed finite element model with the actual scaffold geometry, and a solid shell-like model with effective material properties. These effective material properties were obtained by means of the Asymptotic Expansion Homogenization (AEH) theory which accounts for the periodicity of the underlying structure of the material. Experimental results showed a linear response of the material and the implant up to failure, therefore supporting the use of linear elastic models for simulation. Numerical models showed excellent agreement with experiments regarding load–displacement response. Models also showed a very consistent behavior with regard to the location and the value of the maximum principal stress in the implant when subjected to the maximum load of the experiments. The two numerical models were compared. The homogenized model gave results that were very close to those obtained with the detailed model, while reducing the number of degrees of freedom by 90%, and therefore the overall computational burden. The results showed that the models are able to reproduce experimental results conducted on actual implants, offering a valid alternative to be used in the design of customized cranial implants with a scaffold structure.

Coefficient identification and fault detection in linear elastic systems; one dimensional problems

The determination of parameter distributions, including fault detection, in elastic structures is a subject of great importance in structural engineering and related areas of applied mathematics. In this article we explore, in both continuous and discrete settings, some methods for approximate solution of such identification problems in a one dimensional linear elasticity framework. Methods for related optimization problems based on the matrix trace norm are described. The main objective of the paper is to introduce a method, believed new with this article, for which we suggest the names adjoint null space method or complementary projection method. Computational results for sample problems based on this technique are presented.

Effective properties of fiber and platelet systems and related phase arrangements in n-phase heterogenous media

For the structures which are neither aggregates nor inclusion-reinforced matrices, relevant property estimates are missing. We address composites with several, all or none co-continuous phases. Property estimates from various fibre systems and platelet systems that are likely to well approach the effective properties of such microstructures are examined. We provide arguments in favour of the relevancy of some of the investigated systems to represent phase co-continuity (or co-discontinuity), the reported results being supported by theoretical, topological and experimental considerations. The discussion is held in the linear elasticity framework with open extensions to other properties and behaviours.

The 3-D tectonic stress fields in and around Japan inverted from centroid moment tensor data of seismic events

[1] We can regard the occurrence of earthquakes as the partial release of tectonic stress by sudden brittle rupture. In the framework of linear elasticity, any indigenous source including earthquake rupture is represented by a moment tensor. The moment tensor is mathematically equivalent to the volume integral of stress release over the whole elastic region surrounding the source, and so we can quantitatively relate the centroid moment tensor (CMT) of seismic events with an unknown tectonic stress field. On the basis of such an idea and Bayesian statistical inference theory, we developed an inversion method to estimate the 3-D pattern of tectonic stress from CMT data. Applying the CMT data inversion method to 12,500 seismic events in and around Japan, we obtained precise 3-D tectonic stress patterns that illuminate the present-day (Quaternary) complex tectonic motion of Japanese islands. The stress pattern of the Kuril-Japan-Nankai arc is basically E–W compression, but the direction of intermediate principal stress changes from N–S (reverse faulting type) in northeast Japan to vertical (strike-slip faulting type) in southwest Japan. On the other hand, the stress pattern of the Ryukyu and Izu-Bonin back-arc regions is basically trench perpendicular tension (normal faulting type). In addition to these basic stress patterns governed by mechanical interaction between the Eurasian, North American, Pacific, and Philippine Sea plates, we can recognize several characteristic local stress patterns corresponding to the horizontal motion of the Kuril fore-arc sliver, the collision of the Izu Peninsula with the mainland of Japan, and the opening of the Beppu-Shimabara rift zone.

Iterative method for solving fully-coupled fluid solid systems

This work is concerned with the modelling of the interaction of a fluid with a rigid or a flexible elastic cylinder in the presence of axial or cross-flow. A partitioned procedure is involved to perform the computation of the fully-coupled fluid solid system. The fluid flow is governed by the incompressible Navier-Stokes equations and modeled by using a fractional step scheme combined with a co-located finite volume method for space discretisation. The motion of the fluid domain is accounted for by a moving mesh strategy through an Arbitrary Lagrangian-Eulerian (ALE) formulation. Solid dyncamics is modeled by a finite element method in the linear elasticity framework and a fixed point method is used for the fluid solid system computation. In the present work two examples are presented to show the method robustness and efficiency.

Step interactions on Pt(111) vicinal surfaces determined by grazing incidence X-ray diffraction: Influence of the step orientation

We have studied the step–step interactions on the Pt(997) vicinal surface. Grazing incidence X-ray diffraction (GIXD) allowed us to measure the elastic atomic relaxations near the surface due to the steps. By means of the model of buried elastic dipoles, within the framework of anisotropic linear elasticity (ALE) calculations, the surface stress of Pt(111), and the elastic interaction between steps are deduced. The values so-obtained are compared to the values previously measured on the Pt(779) surface with the same technique. The comparison shows the strong influence of step geometry on step interactions.

Crystal mismatched layers in pentagonal nanorods and nanoparticles

Pentagonal nanorods (PNRs) and nanoparticles (PNPs) covered by mismatched shell layers are theoretically investigated. Mechanical stresses and elastic energies of such objects are calculated analytically and analyzed in the framework of linear isotropic elasticity. Difference between elastic modules of core and shell is taken into account. The threshold radiuses as the minimal radiuses of PNR and PNP for which the formation of the shell layer is energetically favorable are found. The threshold radius is approximately 10 nm for PNPs and 100 nm for PNRs of typical face-centered cubic (FCC) metals. The optimal magnitudes of mismatch parameter giving the maximal energy release for shelled PNRs and PNPs are determined.

Effect of surface stress on the stiffness of cantilever plates: Influence of cantilever geometry

Numerous measurements have indicated that surface stress can significantly modify the stiffness of cantilever sensors. In contrast, theoretical calculations using classical beam theory predict that stiffness is independent of surface stress. Using a three-dimensional analysis, we recently showed that surface stress does indeed have an effect within the framework of linear elasticity. However, only cantilevers of rectangular geometry were explored. Here, we vary cantilever geometry and find that it plays a critical role, with V-shaped cantilevers displaying greatly enhanced sensitivity in comparison to rectangular cantilevers. Tuning cantilever geometry therefore provides a sensitive route to controlling the effects of surface stress.

Dissecting the Impact of Matrix Anchorage and Elasticity in Cell Adhesion

Extracellular matrices determine cellular fate decisions through the regulation of intracellular force and stress. Previous studies suggest that matrix stiffness and ligand anchorage cause distinct signaling effects. We show herein how defined noncovalent anchorage of adhesion ligands to elastic substrates allows for dissection of intracellular adhesion signaling pathways related to matrix stiffness and receptor forces. Quantitative analysis of the mechanical balance in cell adhesion using traction force microscopy revealed distinct scalings of the strain energy imparted by the cells on the substrates dependent either on matrix stiffness or on receptor force. Those scalings suggested the applicability of a linear elastic theoretical framework for the description of cell adhesion in a certain parameter range, which is cell-type-dependent. Besides the deconvolution of biophysical adhesion signaling, site-specific phosphorylation of focal adhesion kinase, dependent either on matrix stiffness or on receptor force, also demonstrated the dissection of biochemical signaling events in our approach. Moreover, the net contractile moment of the adherent cells and their strain energy exerted on the elastic substrate was found to be a robust measure of cell adhesion with a unifying power-law scaling exponent of 1.5 independent of matrix stiffness.

Comprehensive Study of the Fabrication of SGOI Substrates by the Ge Condensation Technique: Oxidation Kinetics and Relaxation Mechanism

We report on the fabrication and characterization of thin SiGe on insulator (SGOI) with high Ge content by the condensation technique. While one of the objectives of this report is to describe the process and give a complete characterization of the obtained substrates, the main intention is to focus on the fundamental understanding of key phenomena involved in the Ge condensation method. In particular, we will describe the oxidation kinetic using the conventional Deal and Grove model and show that for such process, since the thickness of oxide grown is large and since the oxide formed is pure SiO2, the oxidation kinetic can be modeled with the same parameters as those used for pure Si. Next, we will discuss the relaxation of the formed SGOI layer using linear elastic theory framework and evidence that even though the condensation method makes use of long high temperature treatments, the obtained layer are not in their equilibrium state and their relaxation is frustrated. However, we will show that it is still possible to predict their relaxation rate and that it is observed to increase with increasing Ge content. This poses some limitations in terms of defectivity of such substrates as they are inevitably partially relaxed. Expanding the analysis of relaxation and defect nucleation, we will finally discuss an approach to decrease the defect density and consider its drawbacks.

Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers

In the framework of linear elasticity, we study the limit of a class of discrete free energies modeling strain-alignment-coupled systems by a rigorous coarse-graining procedure, as the number of molecules diverges. We focus on three paradigmatic examples: magnetostrictive solids, ferroelectric crystals and nematic elastomers, obtaining in the limit three continuum models consistent with those commonly employed in the current literature. We also derive the correspondent macroscopic energies in the presence of displacement boundary conditions and of various kinds of applied external fields.

Mixed finite element formulation for the general anti-plane shear problem, including mode III crack computations, in the framework of dipolar linear gradient elasticity

A mixed formulation is developed and numerically validated for the general 2D anti-plane shear problem in micro-structured solids governed by dipolar strain gradient elasticity. The current mixed formulation employs the form II statement of the gradient elasticity theory and uses the double stress components and the displacement field as main variables. High order, C0-continuous, conforming basis functions are employed in the finite element approximations (p-version). The results for the mode III crack problem reveal that, with proper mesh refinement at the areas of high solution gradients, the current approximation method captures the exact solution behaviour at different length scales, which depend on the size of material micro-structure. The latter is of vital importance because, near the crack tip, the nature of the exact solution, changes radically as we proceed from the macro- to micro-scale.

On the conforming contact problem in a reinforced pin-loaded structure with a non-zero second Dundurs’ constant

Within the framework of two-dimensional linear elasticity, the unilateral frictionless contact between two conformal cylindrical surfaces is governed by an integral equation in which the first and second Dundurs’ constants are involved. In the case of elastic similarity characterized by the zero second Dundurs’ constant, the integral equation is considerably simplified so as to lend itself to a closed-form solution. However, in the case of elastic dissimilarity defined by the non-zero second Dundurs’ constant, the question of obtaining a closed-form solution to the integral equation is a much tougher one. Starting from the integral equation established by To et al. [To, Q.D., He Q.-C., Cossavella, M., Morcant, K., Panait, A., 2007. Closed-form solution for the contact problem of reinforced pin-loaded joints used in glass structures. Int. J. Solids Struct. 44, 3887–3903] for the conformal contact problem originating from a reinforced pin-loaded joint used in tempered glass structures, the present work proposes a new approximate analytical method to solve it in the case of elastic dissimilarity by minimizing an error function. The derived closed-form solution, valid not only for the conformal contact between a pin and an infinite holed plate but also for the one between a pin and a finite holed plate, is shown to be in very good agreement with available numerical results.

Energy-Consistent CoRotational Schemes for Frictional Contact Problems

In this paper, we consider the unilateral frictional contact problem of a hyperelastic body in the case of large displacements and small strains. In order to retain the linear elasticity framework, we decompose the deformation into a large global rotation and a small elastic displacement. This corotational approach is combined with a primal-dual active set strategy to tackle the contact problem. The resulting algorithm preserves both energy and angular momentum.

Explicit models for elastic and piezoelastic surface waves

Explicit models for the Rayleigh and Bleustein-Gulyaev surface waves are extracted from the original 2D formulations within the general framework of linear elasticity and electroelasticity. The derivations are based on perturbing in slow time the self-similar solutions for homogeneous surface wave. Both the proposed models involve hyperbolic equations on the surface along with elliptic equations over the interior, emphasizing the dual nature of a surface wave. Comparisons with exact solutions are presented, including that for the plane Lamb problem.

Discussion: “A Paradox in Sliding Contact Problems With Friction” (Adams, G. G., Barber, J. R., Ciavarella, M., and Rice, J. R., 2005, ASME J. Appl. Mech., 72, pp. 450–452)

In this interesting paper, the authors address an anomaly which arises when a rigid, square-ended block is pressed against a linear elastic half plane and slid along. The authors note that, within the framework of linear elasticity, the singularity in the contact pressure, and hence shearing traction, produces, adjacent to the edges, regimes in which the implied local relative slip direction dominates the rigid-body sliding velocity, and hence produces a violation of the Coulomb friction law. They seek to resolve the paradox by appealing to a more sophisticated strain definition. All of this is within the context of a quasistatic formulation. The authors recognize, of course, that in any real problem the paradox is unlikely to arise because of (a) the finite strength of the contact giving rise to a yield zone, and (b) the absence of an atomically sharp corner at the contact edge where there is, in all probability, a finite edge radius. Here, we wish to address these issues quantitatively, and so demonstrate that it is unlikely that the paradox described, though interesting, will have any bearing in a real contact.

The Reissner–Mindlin plate theory via Γ-convergence

We obtain the energy functional of Reissner–Mindlin plates as the Γ-limit of a family of three-dimensional energy functionals within the framework of second-order linear elasticity. The choice of the family of functionals, as well as of the candidate limiting functional, is guided by a formal scaling argument. To cite this article: R. Paroni et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Configurational force on a lattice dislocation and the Peierls stress

The solution for a crystalline edge dislocation is presented within a framework of continuum linear elasticity, and is compared with the Peierls–Nabarro solution based on a semi-discrete method. The atomic disregistry and the shear stress across the glide plane are discussed. The Peach–Koehler configurational force is introduced as the gradient of the strain energy with respect to the dislocation position between its two consecutive equilibrium positions. The core radius is assumed to vary periodically between equilibrium positions of the dislocation. The critical force is expressed in terms of the core radii or the energies of the stable and unstable equilibrium configurations. This is used to estimate the Peierls stress for both wide and narrow dislocations.