Variational Formulation and Upper Bounds for Degenerate Scales in Plane Elasticity

International audienceDegenerate scales appear when certain plane boundary value problems solved using Boundary Integral Equations do not have a unique solution. The main contribution of this paper is to prove four inequalities that constrain the degenerate scales for plane elasticity. These results are based on a new variational formulation. It is shown that the degenerate scales depend only on Poisson’s ratio. The bounds on the degenerate scales for plane elasticity in a given boundary are obtained mainly from the degenerate scales obtained from the Laplace equation for the same boundary, which are well documented

Fast, Scalable, and Interactive Software for Landau-de Gennes Numerical Modeling of Nematic Topological Defects

Numerical modeling of nematic liquid crystals using the tensorial Landau-de Gennes (LdG) theory provides detailed insights into the structure and energetics of the enormous variety of possible topological defect configurations that may arise when the liquid crystal is in contact with colloidal inclusions or structured boundaries. However, these methods can be computationally expensive, making it challenging to predict (meta)stable configurations involving several colloidal particles, and they are often restricted to system sizes well below the experimental scale. Here we present an open-source software package that exploits the embarrassingly parallel structure of the lattice discretization of the LdG approach. Our implementation, combining CUDA/C++ and OpenMPI, allows users to accelerate simulations using both CPU and GPU resources in either single- or multiple-core configurations. We make use of an efficient minimization algorithm, the Fast Inertial Relaxation Engine (FIRE) method, that is well-suited to large-scale parallelization, requiring little additional memory or computational cost while offering performance competitive with other commonly used methods. In multi-core operation we are able to scale simulations up to supra-micron length scales of experimental relevance, and in single-core operation the simulation package includes a user-friendly GUI environment for rapid prototyping of interfacial features and the multifarious defect states they can promote. To demonstrate this software package, we examine in detail the competition between curvilinear disclinations and point-like hedgehog defects as size scale, material properties, and geometric features are varied. We also study the effects of an interface patterned with an array of topological point-defects.Comment: 16 pages, 6 figures, 1 youtube link. The full catastroph

A recursive-faulting model of distributed damage in confined brittle materials

We develop a model of distributed damage in brittle materials deforming in triaxial compression based on the explicit construction of special microstructures obtained by recursive faulting. The model aims to predict the effective or macroscopic behavior of the material from its elastic and fracture properties; and to predict the microstructures underlying the microscopic behavior. The model accounts for the elasticity of the matrix, fault nucleation and the cohesive and frictional behavior of the faults. We analyze the resulting quasistatic boundary value problem and determine the relaxation of the potential energy, which describes the macroscopic material behavior averaged over all possible fine-scale structures. Finally, we present numerical calculations of the dynamic multi-axial compression experiments on sintered aluminum nitride of Chen and Ravichandran [1994. Dynamic compressive behavior of ceramics under lateral confinement. J. Phys. IV 4, 177–182; 1996a. Static and dynamic compressive behavior of aluminum nitride under moderate confinement. J. Am. Soc. Ceramics 79(3), 579–584; 1996b. An experimental technique for imposing dynamic multiaxial compression with mechanical confinement. Exp. Mech. 36(2), 155–158; 2000. Failure mode transition in ceramics under dynamic multiaxial compression. Int. J. Fracture 101, 141–159]. The model correctly predicts the general trends regarding the observed damage patterns; and the brittle-to-ductile transition resulting under increasing confinement

Topology design of 2D and 3D elastic material microarchitectures with crystal symmetries displaying isotropic properties close to their theoretical limits

This paper evaluates the effect that different imposed crystal symmetries have on the topology design of two-phase isotropic elastic composites ruled by the target of attaining extreme theoretical properties. Extreme properties are defined by the Cherkaev–Gibiansky bounds, for 2D cases, or the Hashin–Shtrikman bounds, for 3D cases. The topology design methodology used in this study is an inverse homogenization technique which is mathematically formulated as a topology optimization problem. The crystal symmetry is imposed on the material configuration within a predefined design domain, which is taken as the primitive cell of the underlying Bravais lattice of the crystal system studied in each case. The influence of imposing crystal symmetries to the microstructure topologies is evaluated by testing five plane groups of the hexagonal crystal system for 2D problems and four space groups of the cubic crystal systems for 3D problems. A discussion about the adequacy of the tested plane or space groups to attain elastic properties close to the theoretical bounds is presented. The extracted conclusions could be meaningful for more general classes of topology design problems in the thermal, phononic or photonic fields.Peer ReviewedPostprint (published version

Non degenerate anisotropic green's function for 3D magneto-electro-elasticity and bem shape sensitivity framework for 3D contact in anisotropic elasticity

The first part of the thesis presents a new expression for the magneto electro elastic (MEE) fundamental solution which is explicit in terms of the Stroh’s eigenvalues, remains welldefined for repeated Stroh’s eigenvalues and is exact. We then define a fast and robust numerical scheme to evaluate the function and its derivatives based on a double Fourier series representation. These newly developed expressions allow to compute the Fourier coefficients for any material symmetry or anisotropy, and is done only once for a given material. One evaluates the Green’s function and its derivatives through simple trigonometric formulas. Several results are presented for elastic, piezoelectric/piezomagnetic and magneto-eletro-elastic materials. The second part of the thesis provides a BEM-based formulation for shape sensitivity analysis of anisotropic elastic media, also including contact conditions, and based on the newly presented Green’s functions. The parameter sensitivity is evaluated using the complex step (CS) method: An approach similar to finite differentiation (FD), with the advantage of being step-size independent, therefore an extremely robust method. A convergence study on shape sensitivity is provided, proving the efficiency of the CS-BEM approach. We solve Hertz and non-Hertzian type contact problems as well as an application example of a dovetail joint found in gas turbines. We analyzed several parmeter sensitivities to shape variation, such as contact pressure, shear stress, as well as Von Mises stress, for both isotropic and anisotropic materials. The results showed good agreement with analytical solutions, as well as other works from the literature. In comparison with FD, which did not converged for an example case, the CS method showed excellent stability and precision for a broad range of step sizes.A primeira parte da tese apresenta uma nova expressão para a solução fundamental Magneto-Eletro-Elástica explícita em termos de autovalores de Stroh, bem definida para autovalores repetidos, e exata. Em seguida, uma série de Fourier dupla é utilizada como uma forma rápida e robusta para avaliar a solução fundamental e as suas derivadas. As expressões recém-desenvolvidas permitem calcular os coeficientes de Fourier para qualquer simetria ou anisotropia de material, o que é feito apenas uma vez para um dado material. Diversos resultados são apresentados para materiais elásticos, piezoelétricos e magneto-eletro-elásticos. A segunda parte desta tese apresenta uma formulação completa para análise de sensibilidade em estruturas elasticas anisotrópicas baseada nestas funções de Green recém apresentadas, incluindo condições de contato. A sensibilidade à parâmetros é avaliada utilizando o método do incremento complexo, método extremamente robusto, similar a diferenciação finita (FD), mas independente do tamanho do incremento. Problemas de contato de Hertz e não Hertzianos foram resolvidos, assim como um estudo de aplicação de uma palheta de turbinas a gás. Foi avaliada a sensibilidade à variação de forma das tensões de contato, tensões cisalhantes máximas e também nas tensões equivalentes de Von Mises, em diferentes materiais anisotrópicos. Os resultados mostraram boa correlação com soluções analíticas assim como em outros trabalhos da literatura. Quando comparado com FD, que não obteve convergência em um dos exemplos, o método CS demonstrou excelente estabilidade e precisão para uma larga faixa de tamanhos de incremento

Anti-plane Shear of Cylinders and Layered Systems: Cohesive Fracture and Instability

This research examines the mechanics of mode-III cohesive fracture by defect initiation and quasi-static growth in both cylinder and layered systems. The analysis, which is exact, is based on the solution of two fundamental elasticity problems: i) a cylinder subject to an arbitrary shear on one end cap and an equilibrating torque on the other and, ii) a layer subject to arbitrary anti-plane shear traction on one surface and an equilibrating uniform traction on the other. For a particular geometry and defect configuration, these solutions are shown to lead to a pair of interfacial integral equations whose derived cohesive surface fields capture the entire defect evolution process from incipient growth through complete failure. The anti-plane shear separation/slip process is assumed to be modeled by Needleman-type traction-separation relations (e.g., bilinear, Xu-Needleman, frictional) characterized by a shear cohesive strength, a characteristic force length and, in the case of the bilinear law, a finite decohesion cutoff length and possibly other parameters as well. Symmetrically arrayed cohesive surface defects are modeled by a cohesive surface strength function which varies with surface coordinate. Infinitesimal strain equilibrium solutions, which allow for rigid body movement, are found by eigenfunction approximation of the solution of the governing interfacial integral equations. General features of the solutions to anti-plane shear cohesive fracture in both cylindrical and layered geometries indicate that quasi-static defect initiation and propagation occur under monotonically increasing load. For small values of characteristic force length, brittle behavior occurs that is readily identifiable with the growth of a sharp crack, i.e., the existence of a strong local stress concentration. At larger values of characteristic force length, ductile response occurs which is more typical of a linear “spring” cohesive surface, i.e., more distributed stress and slip distribution. Both behaviors ultimately give rise to abrupt failure of the cohesive surface. Results for the stiff, strong cohesive surface under a small applied load show consistency with static linear elastic fracture mechanics solutions in the literature. By superimposing a frictional part onto the cohesive law, the solution can be used to predict frictional response. Both decohesion and friction dominated cases show similar quasi-static defect propagation process, stable defect growth till a maximum load is reached, then defect growth becomes dynamic and unstable. However, the difference lies in that the friction dominated cohesive surface can still sustain certain load even after response becomes dynamic, but the decohesion dominated case will not. For friction dominated cohesive surfaces, the cylinder cases have smooth deformation processes whereas the layered systems experience a noticeable displacement jump. Both cylinder and layered systems predict the principal plane (perpendicular to principal stress direction) to be close to 45 degrees which helps to explain the orientation of mode-I microcracks in layered systems and the initiation of a spiral crack plane in cylinder geometries. The cohesive fracture solution to layered geometries can be extended to obtaining traction fields for more complicated defect geometries (array of cracks and subsurface cracks in nonuniform bilayer) which can be used to predict the sequence of defect propagation. The bifurcation analysis of the uniform two-sublayer system shows the phenomenon of non-unique slip for the same loading. The bifurcation analysis for the multi-sublayer system with such non-uniqueness gives an explanation of the asymmetric configuration. For the analysis of nonuniform multi-sublayer systems, no additional difficulty occurs in the problem-solving process. By studying different geometries and crack patterns, the current study discussed the combined effects of interlaminar and intralaminar crack interaction which are important in predicting the most vulnerable place in the system while multiple defects exist