Elastostatic Fields Research Articles

A fast algorithm for fracture simulations representing fibre breakage and matrix failure in three-dimensional fibre composites

A linear, periodic, three-dimensional shear-lag model of unidirectionally-reinforced composites that allows for fibre breakage, and matrix failure is proposed. Matrix failure can take the form of matrix splitting or interfacial debonding. A computationally efficient scheme for its solution is developed. This scheme exploits the translation invariance of the elastostatic fields due to failed elements in the periodic cell, and is asymptotically faster than the classical eigensolution-based approach. The new computational scheme is used to illustrate the influence of matrix failure on the elastostatic fields induced by small clusters of fibre breaks in several test problems. Monte Carlo simulations of fracture in model three-dimensional composite specimen with Weibull-distributed fibre segment strengths are also performed. Matrix failure is found to considerably alter fracture development, to weaken the median specimen, and to reduce the variability in composite strength.

An asymptotic finite plane deformation analysis of the elastostatic fields at a crack tip in the framework of hyperelastic, isotropic, and nearly incompressible neo-Hookean materials under mode-I loading

In this work, stress and displacement fields were computed around a crack tip in the case of nearly incompressible and isotropic neo-Hookean material. The constitutive equation was linearized, so that the Cauchy stress tensor could be written as a sum of two components: the linear response in term of elastic Hooke’s law and the nonlinear one. Based on this decomposition, an asymptotic analysis has been developed, the fields of linear elastic fracture mechanics (LEFM-theory) are the zero-order terms of the asymptotic expansion. The validity of the proposed theory has been checked in the case of a mode-I crack problem. A numerical model was constructed using a finite element method. It has shown that the computed fields arising from this theory are qualitatively in agreement with those of the finite element simulations.

A plane strain analysis of the elastostatic fields near the notch-tip of a Blatz-Ko material

An asymptotic plane strain analysis is carried out in order to study the notch problem in a homogeneous isotropic compressible hyperelastic material whose behavior is governed by the Blatz-Ko constitutive law. The work is realized within the framework of the fully nonlinear fracture mechanics at large strain. Close to the notch tip, a stress-free state is assumed in the notch edges, while at infinity (sufficiently far to the notch tip) fracture modes I and II are considered. The results emphasizes many discrepancies with linear theory predictions like the emergence of a logarithmic singularity and show that solution depends on the notch initial opening angle.

A Damped Dynamic Finite Difference Approach for Modeling Static Stress–Strain Fields

Modeling dynamic and static responses of an elastic medium often employs different numerical schemes. By introducing damping into the system, we show how the widely used time-marching staggered finite difference (FD) approach in solving elastodynamic wave equation can be used to model time-independent elastostatic problems. The damped FD method can compute elastostatic stress and strain fields of a model subject to the influence of an external field via prescribed boundary conditions. We also show how to obtain an optimized value for the damping factor. We verified the damped FD approach by comparing results against the analytical solutions for a borehole model and a laminated model. We also validated our approach numerically for an inclusion model by comparing the results computed by a finite element method. The damped FD showed excellent agreement with both the analytical results and the finite element results.

The anti-plane shear elasto-static fields near a crack terminating at an isotropic hyperelastic bi-material interface

In the framework of hyperelasticity, we treat the configuration of a terminated crack at the interface of an incompressible full plane composite. Considering the traction-free boundary conditions, three particular cases are discussed when a cylinder is subjected to an anti-plane shear transformation. Taking all these conditions into account, an asymptotic analysis is performed to identify the sufficient orders contributing to the singular form of the Cauchy stress static fields. Adding to that, an inquiry about the presence of logarithmic singularities was achieved using the approach of Dempsey and Sinclair.

Three-dimensional singular elastostatic fields in a cracked Neo-Hookean hyperelastic solid

The boundary-value problem of Neo-Hookean incompressible hyperelastic cracked solid under a superposition of a plane deformation to an anti-plane one is formulated. An asymptotic analysis is then employed to compute the elastostatic fields near the crack front and their principal properties are illustrated. In a particular basis, the crack is bound to open independently of the magnitude and the mode of the boundary conditions at infinity. The stress field components have different singularities and each one can posses more than one singular term. Some disagreements with the linear theory are evidenced.

Singular elastostatic fields near the notch vertex of a Mooney–Rivlin hyperelastic body

Within the framework of fully nonlinear fracture mechanics under plane deformation conditions, boundary value problem equations for the V-notch vertex problem in an incompressible Mooney–Rivlin material are deduced. Using an asymptotic procedure, the deformation, the Lagrange multiplier and stress fields near the notch vertex are computed and their principal properties are illustrated. First, it is shown that the singularity order depend on the V-notch angle. Second, a change of the singular term for a critical angle was revealed and a logarithmic singular term appears. Based on the results obtained, the deformed V-notch vertex lips near the tip was drawn. The V-notch vertex is shown to open, but not necessary in a symmetric manner. Analysis of the Cauchy stress tensor shows that the component σ 22 dominates the stress field. Some discrepancy with the linear theory have emerged.

The anti-plane shear elastostatic fields near the wedge vertex of an incompressible hyperelastic bimaterial

The anti-plane shear transformation corresponding to a bimaterial wedge problem is studied within the framework of finite elasticity. Two wedges of arbitrary angles are assumed to be incompressible hyperelastic Neo–Hookean materials, and have a common perfect interface and a traction free surface. The resolution of the boundary value problem near the vertex of a bimaterial wedge by means of asymptotic procedure leads to an eigenequation connecting wedges angles and materials properties. Results proved that, contrary to linear elasticity predictions, it is needed to extend development to higher orders because of their contribution in stress singularity.

Elastostatic Fields of an Embedded Circular Rigid Nano/Micro-Fiber with Interfacial Damage in Anti-Plane Couple Stress Elasticity

It is well-known that classical continuum theory has certain deficiencies in predicting the nanoscopic behavior of materials in the vicinity of defects. Couple stress theory is one of the higher order continuum theories which can overcome such difficulties by introducing new characteristic length. An accurate analytical determination of the elastic fields of an embedded circular nano/micro-rigid fiber with damaged interface under remote anti-plane loading in the context of couple stress elasticity is of particular interest.

A fast multipole method for the evaluation of elastostatic fields in a half-space with zero normal stress

In this paper, we present a fast multipole method (FMM) for the half-space Green's function in a homogeneous elastic half-space subject to zero normal stress, for which an explicit solution was given by Mindlin (Physics 7, 195---202 1936). The image structure of this Green's function is unbounded, so that standard outgoing representations are not easily available. We introduce two such representations here, one involving an expansion in plane waves and one involving a modified multipole expansion. Both play a role in the FMM implementation.

A criterion based on the local strain energy density for the fracture assessment of cracked and V-notched components made of incompressible hyperelastic materials

The present contribution is devoted to the theoretical and numerical study of the elasto-static fields at a vertex notch under mode I loading. The analysis is based on the plane deformation hyperelasticity theory for an incompressible Mooney–Rivlin material. While for cracked components some contributions can be found in the recent and past literature, studies on V-notched components are instead very limited. The aim of this paper is to partially fill this lack, providing a fracture criterion for the assessment of components weakened by sharp V-notches. In the first part of the paper, a brief description of the analytical frame available for V-notches in hyperelastic material is reported. A Williams’ type diagram reporting the degree of singularity for a material obeying a Mooney–Rivlin behavior is present. The asymptotic stress field and the local strain energy density are investigated by means of non-linear finite element analyses. In the second part of the paper, a criterion based on the local energy is proposed and successfully applied to a set of experimental data taken from the literature. Future works are surely necessary to validate the criterion considering more sets of data from sharp and blunt V-notches.

Cavity/Inclusion Detection in Plane Linear Elastic Bodies Using Linear Sampling Method

In this paper, solution of inverse problems in elastostatic fields is investigated. For this purpose, we propose a qualitative inverse approach based on linear sampling method (LSM) for cavity/inclusion detection in a two-dimensional (2D) isotropic linear elastic body using measurement of data on the boundary. The LSM is an effective approach to image the geometrical features of unknown targets. Although the LSM has been used in the context of inverse scattering problems such as acoustics, and electromagnetism, there is no specific attempt to apply this method for identification of cavities/inclusions in inverse elastostatic problems. This study emphasizes the implementation of the LSM coupled with the finite element method (FEM). A set of numerical simulations on 2D elastostatic problems is presented to highlight many effective features of the proposed LSM fast qualitative identification method.

Elastostatic computations on aggregates of grains with sharp interfaces, corners, and triple-junctions

We present a fast algorithm for the calculation of elastostatic fields in two-dimensional assemblies of elastic grains, separated by sharp grain boundaries. The algorithm uses an integral equation approach, combined with the fast multipole method and recursive compression to resolve stress concentrations also very close to grain boundary junctions. Singular basis functions on analytic form are not required. Accurate results can be obtained at a cost roughly proportional to the number of grains in the assembly. Large-scale problems, with thousands of grains, are solved using modest computational resources.

An asymptotic finite plane deformation analysis of the elastostatic fields at a notch vertex of an incompressible hyperelastic material

This Note is devoted to the theoretical study of the elastostatic fields at a vertex notch under general far-field loading conditions. The analysis is based on the finite plane deformation hyperelasticity theory for an incompressible Mooney–Rivlin material. We approach the solution, near the singularity, by a mixed asymptotic development. We show that the shape of the solution depends on the opening angle of the notch and that there is singularity if the notch is concave. Furthermore, we show that a pure loading mode II gives rise to the opening of the notch vertex in contrast to the linear elasticity. To cite this article: M. Arfaoui et al., C. R. Mecanique 336 (2008).

The mechanics of pyramids

The elastostatic fields and free-vibration characteristics are studied for a class of continuous solids in the shape of homogeneous or layered pentahedral pyramids. The pyramids possess an arbitrary number of linear elastic layers containing dissimilar elastic constants. The layers are assumed to be perfectly bonded in the out-of-plane direction of the pyramid. The change in elastic stiffness across each interface requires a model that allows for a jump in displacement gradient at these location. A discrete-layer representation is used that combines the Ritz method with polynomial in-plane approximations with one-dimensional Lagrangian interpolation polynomials in the thickness direction. The free-vibration characteristics are examined for a variety of isotropic and anisotropic materials and representative bulk stiffness estimates are given for homogeneous pyramids under static deformation.

Effects of elastic anisotropy on the self-organized ordering of quantum dotsuperlattices

The vertical and lateral ordering of quantum dot superlattices is dependent on theelastic stress/strain fields induced by the buried quantum dots within theprevious superlattice layers. It has been found that quantum dots may bevertically correlated or may exhibit anti-correlation for different materialsystems. These differences are caused by the elastic anisotropy as well as thecrystallographic orientation of the materials. The finite element method isused in this work to investigate the effects of anisotropy of the elasticproperty and crystal orientation on the elastostatic fields of quantum dotsuperlattices. The strain energy density distribution obtained from thecalculations can be used to deduce the locations whereby subsequentquantum dot islands will nucleate by locating the minima positions. Atetragonal body centred arrangement is observed for the CdSe/ZnSe(001)system with a sufficiently thick spacer layer. Through comparison with aless anisotropic system (InAs/GaAs), it is shown that an increase in thedegree of anisotropy favours an anti-correlated island growth. Spatialcorrelation in different crystal orientations is also investigated and discussed.

Elastostatic fields in an anisotropic substrate due to a buried quantum dot

We present an efficient and accurate continuum-mechanics approach for the numerical prediction of displacement, stress, strain, and strain energy density fields in an anisotropic substrate (modeled as a half-space) due to a buried quantum dot (QD). Our approach is based on Green’s function solution in anisotropic and linearly elastic half-space combined with the Betti’s reciprocal theorem. Numerical examples clearly show that the crystalline anisotropy of the III–V semiconductor group has great influence on the elastic fields, as compared to the isotropic solution. In particular, it is found that the hydrostatic strain and strain energy density on the surface of anisotropic half-space made of different crystalline materials due to a cubic QD can be substantially different, and therefore, the isotropy approximation neglecting their differences should not be used in general. Furthermore, the hydrostatic strains on the surface of an anisotropic half-space due to a finite-size (cubic) QD and an equal-intensity point QD at relatively large depth (about twice the side length of the cubic QD) can still be quite different, in contrast to the corresponding isotropic result. These observations indicate that in modeling and analyzing the mechanical and electronic behaviors of QD semiconductor structures, the effect of crystalline anisotropy should be considered in general.

Analysis of Concentrated Colloidal Dispersions

A long-standing area of research in colloid science concerns the determination of electrostatic, magnetic, and elastostatic fields in dispersions comprising densely packed particles whose material properties differ markedly from that of the background. As new materials with complex microstructures are being developed, there is a corresponding need for analytic and numerical methods to predict and understand their properties. Dense systems with large numbers of close-to-touching particles require very fine discretization which causes the linear system of equations that arise to be highly ill-conditioned. A framework for solving the many-body interface problems is provided by a mathematical transformation of the boundary value integral equation. The transformation avoids the use of asymptotics and involves no uncontrolled approximations which distinguishes this work from previous methods. Pointwise analysis of local charge densities in the bulk and at interfaces is now possible.

On the numerical evaluation of elastostatic fields in locally isotropic two-dimensional composites

We present a fast algorithm for the calculation of elastostatic fields in locally isotropic composites. The method uses an integral equation approach due to Sherman, combined with the fast multipole method and an adaptive quadrature technique. Accurate solutions can be obtained with inclusions of arbitrary shape at a cost roughly proportional to the number of points needed to resolve the interface. Large-scale problems, with hundreds of thousands of interface points can be solved using modest computational resources.

An Efficient Boundary Element Method for Solution of 2-D Elastic Fields with Small Circular Inclusions.

In the conventional boundary element technique, all the boundaries of the body including those of inclusions must be discretized even when the inclusions are comparatively small. Therefore the size of the coefficient matrix of the resulting system of linear algebraic equations becomes quite large for a problem with many small inclusions. The authors have previously proposed an approach to circumvent the discretization of the boundaries of small inclusions for two-dimensional potential problems. In that approach the distribution of the potential over circular inclusions is approximated with a simple harmonic function with a small number of freedoms. In the present paper, this approach is extended to the small circular inclusions in two-dimensional elastostatic fields. The effectiveness of the present formulation is illustrated in several numerical examples.