Elastostatic Problems Research Articles

Elastostatic problems in multicomponent, multilayered periodic composites

The object of the analysis is a two-dimensional elastostatic problem for multicomponent, multilayered periodic composites. The equations of equilibrium for this composite are obtained within the framework of tolerance modelling procedure. The paper presents two examples of solutions of boundary value problems.

Stress-driven two-phase integral elasticity for torsion of nano-beams

Size-dependent structural behavior of nano-beams under torsion is investigated by two-phase integral elasticity. An effective torsional model is proposed by convexly combining the purely nonlocal integral stress-driven relation with a local phase. Unlike Eringen's strain-driven mixture, the projected model does not exhibit singular behaviors and leads to well-posed elastostatic problems in all cases of technical interest. The new theory is illustrated by studying torsional responses of cantilever and doubly-clamped nano-beams under simple loading conditions. Specifically, the integral convolution of the two-phase mixture is done by considering the special bi-exponential kernel. With this choice, the stress-driven two-phase model is shown to be equivalent to a differential problem equipped with higher-order constitutive boundary conditions. Exact solutions are established and comparisons with pertinent results obtained by the Eringen strain-driven two-phase mixture and by the strain gradient theory of elasticity are carried out. The outcomes could be useful for the design and optimization of nano-devices and provide new benchmarks for numerical analyses.

Longitudinal vibrations of nano-rods by stress-driven integral elasticity

ABSTRACTIn elastic continuous structures defined on bounded domains, Eringen strain-driven integral model leads to ill-posed elastostatic problems since the constitutive stress field, got by convoluting the elastic strain field with an averaging kernel, conflicts with equilibrium requirements. The innovative stress-driven integral model, recently proposed in literature, is instead able to overcome inconsistencies of Eringen theory. In the present paper, size-dependent free vibrations of nano-rods are investigated by adopting the stress-driven nonlocal model equipped with Helmholtz kernel. Natural frequencies and mode shapes of nano-rods are established by an effective analytical solution strategy and are compared with those obtained by strain gradient theory.

Accelerating isogeometric boundary element analysis for 3‐dimensional elastostatics problems through black‐box fast multipole method with proper generalized decomposition

SummaryThe isogeometric approach to computational engineering analysis makes use of nonuniform rational B‐splines to discretise both the geometry and the analysis field variables, giving a higher‐fidelity geometric description and leading to improved convergence properties of the solution over conventional piecewise polynomial descriptions. Because of its boundary‐only modelling, with no requirement for a volumetric nonuniform rational B‐spline geometric definition, the boundary element method is an ideal choice for isogeometric analysis of solids in 3D. An isogeometric boundary element analysis (IGABEM) algorithm is presented for the solution of such problems in elasticity and is accelerated using the black‐box fast multipole method (bbFMM). The bbFMM scheme is of O(n) complexity, giving a general kernel‐independent separation that can be easily integrated into existing conventional IGABEM codes with little modification. In the bbFMM scheme, an important process of obtaining a low‐rank approximation of moment‐to‐local operators has been hitherto based on singular value decomposition, which can be very time consuming for large 3D problems, and this motivates the present work. We introduce the proper generalized decomposition method as an alternative approach, and this is demonstrated to enhance efficiency in comparison with schemes that rely on the singular value decomposition. In the worst case, a factor of approximately 2 performance gain is achieved. Numerical examples show the performance gains that are achievable in comparison to standard IGABEM solutions and demonstrate that solution accuracy is not affected. The results illustrate the potential of this numerical technique for solving arbitrary large scale elastostatics problems directly from computer‐aided design models.

A Regularized Singular Boundary Method for Inverse Cauchy Problem in Three-Dimensional Elastostatics

A Regularized Singular Boundary Method for Inverse Cauchy Problem in Three-Dimensional Elastostatics

Fast Solution of Boundary Integral Equations for Elasticity Around a Crack Network: A Comparative Study

Because of the non-local nature of the integral kernels at play, the discretization of boundary integral equations leads to dense matrices, which would imply high computational complexity. Acceleration techniques, such as hierarchical matrix strategies combined with Adaptive Cross Approximation (ACA), are available in literature. Here we apply such a technique to the solution of an elastostatic problem, arising from industrial applications, posed at the surface of highly irregular cracks networks.

On topology optimization for elastostatic problem in 3D dominated by body force

On topology optimization for elastostatic problem in 3D dominated by body force

Exact solutions of inflected functionally graded nano-beams in integral elasticity

Abstract The elastostatic problem of a Bernoulli-Euler functionally graded nanobeam is formulated by adopting stress-driven nonlocal elasticity theory, recently proposed by G. Romano and R. Barretta. According to this model, elastic bending curvature is got by convoluting bending moment interaction with an attenuation function. The stress-driven integral relation is equivalent to a differential problem with higher-order homogeneous constitutive boundary conditions, when the special bi-exponential kernel introduced by Helmholtz is considered. Simple solution procedures, based on integral and differential formulations, are illustrated in detail to establish the exact expressions of nonlocal transverse displacements of inflected nano-beams of technical interest. It is also shown that all the considered nano-beams have no solution if Eringen's strain-driven integral model is adopted. The solutions of the stress-driven integral method indicate that the stiffness of nanobeams increases at smaller scales due to size effects. Local solutions are obtained as limit of the nonlocal ones when the characteristic length tends to zero.

Identification of multiple cracks in 2D elasticity by means of the reciprocity principle and cluster analysis

An inverse 2D elastostatic problem is considered. It is assumed that an isotropic, linear elastic body can contain a finite number of rectilinear, well-separated cracks. The surfaces of the cracks are assumed to be free of the loads. A method is developed for reconstruction the cracks by means of the applied loads and displacements on the boundary of the body, obtained in a single static test. The method is based on the reciprocity principle, elements of the theory of distributions, and cluster analysis. Numerical examples are considered.

Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithm

This paper deals with the optimal design of a titanium mesoscale implant in a cortical bone for which the apparent elasticity tensor is modeled by a non-Gaussian random field at mesoscale, which has been experimentally identified. The external applied forces are also random. The design parameters are geometrical dimensions related to the geometry of the implant. The stochastic elastostatic boundary value problem is discretized by the finite element method. The objective function and the constraints are related to normal , shear, and von Mises stresses inside the cortical bone. The constrained nonconvex optimization problem in presence of uncertainties is solved by using a probabilistic learning algorithm that allows for considerably reducing the numerical cost with respect to the classical approaches.

On Boussinesq’s problem for a cracked halfspace

This paper examines the axisymmetric elastostatic problem that deals with the action of a concentrated normal force on the surface of an isotropic elastic halfspace containing a penny-shaped crack. The mathematical formulation of the elasticity problem should take into consideration the sense of action of the concentrated force. The paper presents the development of Fredholm integral equations of the second-kind that are associated with this category of problem and indicates the numerical technique that is adopted for their solution. The numerical results are presented for the stress intensity factors generated at the penny-shaped crack experiencing either opening or closure.

Nano-beams under torsion: a stress-driven nonlocal approach

Purpose This study aims to model scale effects in nano-beams under torsion. Design/methodology/approach The elastostatic problem of a nano-beam is formulated by a novel stress-driven nonlocal approach. Findings Unlike the standard strain-driven nonlocal methodology, the proposed stress-driven nonlocal model is mathematically and mechanically consistent. The contributed results are useful for the design of modern devices at nanoscale. Originality/value The innovative stress-driven integral nonlocal model, recently proposed in literature for inflected nano-beams, is formulated in the present submission to study size-dependent torsional behavior of nano-beams.

Analysis of stress field in a single anisotropic diamond/square shaped inclusion using the volume integral equation method and the numerical equivalent inclusion method

A volume integral equation method (VIEM) is used to study elastostatic problems in an unbounded elastic solid containing a single diamond/square shaped inclusion subject to uniform tensile stress at infinity. The inclusion is assumed to be a long parallel diamond/square cylinder composed of isotropic or anisotropic elastic materials and perfectly bonded to the isotropic matrix. The solid is assumed to be under plane strain on the plane normal to the cylinder. A detailed analysis of the stress field at the interface between the isotropic matrix and the single isotropic/orthotropic diamond/square shaped inclusion is carried out. The effects of a single isotropic/orthotropic diamond/square shaped inclusion on the stress field at the interface between the matrix and the inclusion are investigated in detail. The accuracy of the volume integral equation method for the interfacial stress field is validated and compared by the numerical equivalent inclusion method (NEIM) and the finite element method (FEM) using ADINA. Through detailed analysis of plane elastostatic problems using the parallel volume integral equation method (PVIEM) in an unbounded isotropic matrix with multiple isotropic diamond shaped inclusions under uniform remote tensile loading, it is demonstrated that the volume integral equation method can also be applied to solve general two- and three-dimensional elastostatic problems involving multiple isotropic/anisotropic inclusions whose shape and number are arbitrary.

A scaled boundary finite element formulation with bubble functions for elasto-static analyses of functionally graded materials

This manuscript presents an extension of the recently-developed high order complete scaled boundary shape functions to model elasto-static problems in functionally graded materials. Both isotropic and orthotropic functionally graded materials are modelled. The high order complete properties of the shape functions are realized through the introduction of bubble-like functions derived from the equilibrium condition of a polygon subjected to body loads. The bubble functions preserve the displacement compatibility between the elements in the mesh. The heterogeneity resulting from the material gradient introduces additional terms in the polygon stiffness matrix that are integrated analytically. Few numerical benchmarks were used to validate the developed formulation. The high order completeness property of the bubble functions result in superior accuracy and convergence rates for generic elasto-static and fracture problems involving functionally graded materials.

Injectivity and self-contact in second-gradient nonlinear elasticity

We prove the existence of globally injective weak solutions in mixed boundary-value problems of second-gradient nonlinear elastostatics via energy minimization. This entails the treatment of self-contact. In accordance with the classical (first-gradient) theory, the model incorporates the unbounded growth of the potential energy density as the local volume ratio approaches zero. We work in a class of admissible vector-valued deformations that are injective on the interior of the domain. We first establish a rigorous Euler–Lagrange variational inequality at a minimizer. We then define a self-contact coincidence set for an admissible deformation in a natural way, which we demonstrate to be confined to a closed subset of the boundary of the domain. We then prove the existence of a non-negative (Radon) measure, vanishing outside of the coincidence set, which represents the normal contact-reaction force distribution. With this in hand, we obtain the weak form of the equilibrium equations at a minimizer.

Stochastic spline fictitious boundary element method for random vibration analysis of plane elastic problems with structural uncertainties

A stochastic spline fictitious boundary element method (SFBEM) is proposed for random vibration analysis of plane elastic problems with both structural and loading uncertainties modeled as random fields and random processes, respectively. Two sets of governing differential equations with respect to the means and deviations of dynamic responses are derived by including the first order terms of deviations. As these equations are in similar forms to those of deterministic plane elastostatic problems, they can be solved by the traditional procedure of SFBEM with deterministic fundamental solutions, leading to the mean and covariance solutions to dynamic responses. To deal with the domain integrals of equivalent loads involved in the covariance solution, the random fields and processes considered are effectively represented by Karhunen–Loeve (KL) expansion in conjunction with the Galerkin projection. Numerical examples indicate that the results of the proposed method agree well with those of Monte-Carlo simulation (MCS) with small variations of structural parameters, and the present approach has higher efficiency than the perturbation stochastic finite element method (FEM) with KL expansion techniques.

The solution of elastostatic and dynamic problems using the boundary element method based on spherical Hankel element framework

SummaryIn this paper, new spherical Hankel shape functions are used to reformulate boundary element method for 2‐dimensional elastostatic and elastodynamic problems. To this end, the dual reciprocity boundary element method is reconsidered by using new spherical Hankel shape functions to approximate the state variables (displacements and tractions) of Navier's differential equation. Using enrichment of a class of radial basis functions (RBFs), called spherical Hankel RBFs hereafter, the interpolation functions of a Hankel boundary element framework has been derived. For this purpose, polynomial terms are added to the functional expansion that only uses spherical Hankel RBF in the approximation. In addition to polynomial function fields, the participation of spherical Bessel function fields has also increased robustness and efficiency in the interpolation. It is very interesting that there is no Runge phenomenon in equispaced Hankel macroelements, unlike equispaced classic Lagrange ones. Several numerical examples are provided to demonstrate the effectiveness, robustness and accuracy of the proposed Hankel shape functions and in comparison with the classic Lagrange ones, they show much more accurate and stable results.

Fluid–structure interaction solver for compressible flows with applications to blast loading on thin elastic structures

In this study, we report the development and application of a fluid–structure interaction (FSI) solver for compressible flows with large-scale flow-induced deformation of the structure. The FSI solver utilizes a partitioned approach to strongly couple a sharp interface immersed boundary method-based flow solver with an open-source finite-element structure dynamics solver. The flow solver is based on a higher-order finite-difference method using a Cartesian grid, where it employs the ghost-cell methodology to impose boundary conditions on the immersed boundary. Higher-order accuracy near the immersed boundary is achieved by combining the ghost-cell approach with a weighted least squares error method based on a higher-order approximate polynomial. We present validations for two-dimensional canonical acoustic wave scattering on a rigid cylinder at a low Mach number and for flow past a circular cylinder at a moderate Mach number. The second order spatial accuracy of the flow solver was established in a grid refinement study. The structural solver was validated according to a canonical elastostatics problem. The FSI solver was validated based on comparisons with published measurements and simulations of the large-scale deformation of a thin elastic steel panel subjected to blast loading in a shock tube. The solver correctly predicted the oscillating behavior of the tip of the panel with reasonable fidelity and the computed shock wave propagation was qualitatively consistent with the published results. In order to demonstrate the fidelity of the solver and to investigate the coupled physics of the shock–structure interaction for a thin elastic plate, we employed the solver to simulate a 6.4 kg TNT blast loading on the thin elastic plate. The initial conditions for the blast were taken from previously reported field tests. Using numerical schlieren, the shock front propagation, Mach reflection, and vortex shedding at the tip of the plate were visualized during the impact of the shock wave on the plate. We discuss the coupling between the nonlinear dynamics of the plate and blast loading. The plate oscillates under the influence of blast loading and the restoration of elastic forces. The time-varying displacement of the tip of the plate is the superimposition of two dominant frequencies, which correspond to the first and second modes of the natural frequency of a vibrating plate. The effects of the material properties and length of the plate on the flow-induced deformation are briefly discussed. The proposed FSI solver is a versatile computational tool for simulating the impact of a blast wave on thin elastic structures and the results presented in this study may facilitate the design of thin structures subjected to realistic blast loadings.

Isogeometric symmetric Galerkin boundary element method for three-dimensional elasticity problems

The isogeometric analysis (IGA) is applied for the weakly singular symmetric Galerkin boundary element method (SGBEM) to analyzelinear elastostatics problems in three-dimensional domains. The background of the proposed method is to use non-uniform rational B-splines (NURBS) as the basis functions for the approximation of both geometry and field variables (i.e. displacement and traction) of the governing integral equations. Same as weakly singular SGBEM, the basic ingredient of the method is a pair of weakly singular weak-form integral equations for the displacement and traction on the boundary of the domain. These integral equations are solved approximately using standard Galerkin approximation. In addition to the advantages that IGA owned, the proposed method exploits the common boundary representation of CAD model and boundary element method. Various numerical examples of both simple and complex geometries are examined to validate the accuracy and efficiency of the proposed method. Through the numerical examples, it is observed that the IGA–SGBEM produces highly accurate results.

On the coupling of local multilevel mesh refinement and ZZ methods for unilateral frictional contact problems in elastostatics

This paper addresses an efficient and robust automatic adaptive local multilevel mesh refinement strategy for unilateral frictional contact problems in elastostatics. The proposed strategy couples the Local Defect Correction multigrid method LDC (Hackbusch, 1984) with the ZZ (Zienkiewicz and Zhu, 1987) a posteriori error estimator. An extension of LDC method to frictional contact problems is introduced. An interesting feature of this extended LDC algorithm is that it still only lies on interpolations of displacement fields. Neither forces conservation nor exchange of contact status is required between the refinement levels. The ZZ a posteriori error estimator is exploited to automatically build the sub-grids of the LDC method. A criterion linked to the relative error in stress is used. The efficiency of the proposed strategy is analyzed on examples derived from nuclear engineering. Practical numerical choices are proposed and justified. The refinement process automatically varies and stops with respect to a given tolerance. Post-treatments show that the sub-grids focus around the contact areas and that the converged LDC solution always respects the prescribed tolerance.