Elastostatic Problems Research Articles

Hygro-thermal bending behavior of porous FG nano-beams via local/nonlocal strain and stress gradient theories of elasticity

This work studies the bending response of porous functionally graded Bernoulli-Euler nano-beams under hygro-thermo-mechanical loadings. The governing equations of the elastostatic problems associated with both local/nonlocal stress- and strain-driven gradient models of elasticity were derived by using the virtual work principle. A Wolfram language code in Mathematica was then written to carry out a numerical investigation for different boundary conditions including cantilever and simply-simply conditions. The effects of the different parameters, such as porosity volume fraction, gradient index, nonlocal parameter, gradient length parameter and mixture parameter are presented. It is shown how the proposed approach is able to capture the structural behavior of porous functionally graded Bernoulli–Euler nano-beams under a hygro-thermal environment.

A Coupled RBF Method for the Solution of Elastostatic Problems

Radial basis function (RBF) has been widely used in many scientific computing and engineering applications, for instance, multidimensional scattered data interpolation and solving partial differential equations. However, the accuracy and stability of the RBF methods often strongly depend on the shape parameter. A coupled RBF (CRBF) method was proposed recently and successfully applied to solve the Poisson equation and the heat transfer equation (Appl. Math. Lett., 2019, 97: 93–98). Numerical results show that the CRBF method completely overcomes the troublesome issue of the optimal shape parameter that is a formidable obstacle to global schemes. In this paper, we further extend the CRBF method to solve the elastostatic problems. Discretization schemes are present in detail. With two elastostatic numerical examples, it is found that both numerical solutions of the CRBF method and the condition numbers of the discretized matrices are almost independent of the shape parameter. In addition, even if the traditional RBF methods take the optimal shape parameter, the CRBF method achieves better accuracy.

A computational approach for solving elasto-statics problems

Meshless methods are the new computational techniques to solve elasto-statics problem. This technique has an advantage as it does not require any surface discretization unlike finite element method (FEM). These techniques also provide best results and save computational timing. The elasto-statics problem pertaining to external loading often defined using high order differential equation, which become a challenging task for analytical modelling and simulation. Thus, in present work new computational meshless methods viz. Element Free Galerkin Method (EFGM) and Meshless Local Petro-Galerkin (MLPG) have been used to solve elasto-statics problems for obtaining the higher approximation value compare to FEM. This technique takes the information of local nodes in order to develop the equilibrium equation, as a consequence it converge faster and accurate.

A stable RBF partition of unity local method for elliptic interface problems in two dimensions

The interface problems are faced with multiple connected domains and consequently, their solutions or derivatives might be discontinuous. This paper proposes the use of collocation based radial basis function partition of unity method (RBF-PUM) for solving two-dimensional elliptic interface problems. The RBF-PUM is a local method that allows overcoming the high computational cost associated with the global RBF methods. In the RBF-PUM, the domain is split into overlapping patches forming a covering of it. However, this method suffers from instability when the RBF shape parameter ε tends to zero. To overcome this issue, we use the RBF-QR algorithm which offers stable computations for all values of ε and provides higher accuracy. To obtain the appropriate solution in the vicinity of the interface, the domain decomposition technique is used. In this technique, the approximation in each subdomain is built separately, and proper jump conditions are then imposed across the interface. We illustrate how to apply the proposed method to Sturm-Liouville, Sturm-Liouville eigenvalue and elastostatic interface problems. The proposed method in dealing with arbitrary interfaces within different domain sizes is validated. We present some numerical examples in which the results are compared with exact solutions and those provided by other numerical methods.

A modification of the method of fundamental solutions for solving 2D problems with concave and complicated domains

To solve various problems in complicated and concave domains by the method of fundamental solutions (MFS), it is required to consider a large number of source and collocation points that increases the computational time of the analysis. This paper suggests a modification to the MFS, which can make it more efficient and reliable for solving applied problems in complicated domains. In the proposed method, each pseudo source is converted to two or more number of sub-sources with equal intensities. Therefore, the total number of sources is increased while the number of unknowns is not increased. By this approach we will be able to reduce the distance between the main and the pseudo boundaries and therefore modeling the problems with complicated boundaries can be performed effectively. The proposed method is investigated for a scalar field problem, i.e. the Laplace equation, and a vector field problem, i.e. elastostatic problem. It is shown that by using the proposed method, in addition to reducing the calculation time, the condition number of the coefficient matrix also significantly decreases. By solving several numerical example problems, it is observed that the presented method can obtain accurate solutions by using a small number of collocation points.

An improved response function based stochastic meshless method for problems in elasto-statics

The current study proposes an improved response function (IRF) based stochastic element free Galerkin method (SEFGM) for the analysis of problems in elasto-statics, wherein Young’s modulus is modelled as a homogeneous random field with symmetric distribution characteristics. The proposed SEFGM approximates displacement as the sum of a deterministic part and a stochastic part. The stochastic part is modelled with the help of an IRF, which is a function of discretized set of random variables. Moving least square shape functions are employed to discretize the random field. Utilizing Taylor series expansions of stiffness matrix and force vector and IRF approximation of displacement, explicit expressions for system responses in terms of random variables are derived. Stochastic informations of system responses are evaluated by employing Monte Carlo Simulation (MCS) on the response function, which eliminates the need of construction and simulation of system matrices at each set of sample generation. 1D and 2D numerical examples in elasto-statics are solved using proposed method. Results are validated with those obtained by direct simulation of system of equations using MCS and also compared with other methods like second order perturbation and ad-hoc response function based SEFGM. Normalized computational times required for all the methods are also compared. It is found that the proposed method is computationally efficient and can produce accurate results even for higher coefficient of variation of input random fields.

Geometrically exact three-dimensional modeling of cable-driven parallel manipulators for end-effector positioning

A geometrically exact analytical model of cable-driven parallel manipulators operating in a three-dimensional space is formulated and discussed in this work. The model aims at investigating the exact positioning of a rigid body in a Cartesian space by a m-cables-driven parallel manipulator taking into account the cables mass, their elasticity and, consequently, the sag effect. The direct kinematic problem is formulated by means of 3(m+1) nonlinear equations expressing compatibility and equilibrium in the same number of unknowns, namely, 3m components of the constraint reactions and 3 finite rotations. A novel approach to the solution of the inverse kinematic problem is formulated by adding a set of m target equations to obtain 4m+3 nonlinear algebraic equations solved in terms of an equal number of unknowns. The latter are defined by the 3(m+1) increments of the variables of the direct problem and m additional unknowns, i.e., the increments of the cable unstretched lengths. The elastostatic problem of point mass end-effector is then derived. Consequently, the direct and the inverse kinematic problems are formulated by means of 3m and 4m nonlinear equations, respectively, in the equal number of unknowns. Examples with an increasing number of cables are considered, showing that, although solving the inverse kinematic problem may imply very slack cables configurations, the proposed approach allows to find a solution.

Approximating the influence coefficients of non-planar elastic solids for conformal contact analysis

The exact contact theory is an efficient alternative to the more general yet computationally expensive Finite Element Method for the detailed study of elastostatic contact problems. For its application in conformal contact problems, the exact contact theory needs to be fed with influence coefficients (ICs) appropriate for non-planar solids. An analytical approximation of the ICs for non-planar solids was proposed in a previous work, avoiding the involved process generally necessary to obtain ICs accurately. This work presents further developments of this approximation, further comparison with numerically obtained ICs, and evaluates the errors incurred when using approximated ICs in conformal contact.

The Stokes Paradox in Inhomogeneous Elastostatics

We prove that the displacement problem of inhomogeneous elastostatics in a two–dimensional exterior Lipschitz domain has a unique solution with finite Dirichlet integral \(\boldsymbol{u}\), vanishing uniformly at infinity if and only if the boundary datum satisfies a suitable compatibility condition (Stokes paradox). Moreover, we prove that it is unique under the sharp condition \(\boldsymbol{u}=o(\log r)\) and decays uniformly at infinity with a rate depending on the elasticities. In particular, if these last ones tend to a homogeneous state at large distance, then \(\boldsymbol{u}=O(r^{-\alpha })\), for every \(\alpha <1\).

An inverse elastic method of crack identification based on sparse strain sensing sheet

Reliable damage detection over large areas of structures can be achieved by spatially quasi-continuous structural health monitoring enabled by two-dimensional sensing sheets. They contain dense arrays of short-gauge sensors, which increases the probability to have sensors in direct contact with damage (e.g. crack opening) and thus identify (i.e. detect, localize, and quantify) it at an early stage. This approach in damage identification is called direct sensing. Although the sensing sheet is a reliable and low-cost technology, the overall structural health monitoring system that is using it might become complex due to large number of sensors. Hence, intentional reduction in number of sensors might be desirable. In addition, malfunction of sensors can occur in real-life settings, which results in unintentional reduction in the number of functioning sensors. In both cases, reduction in the number of (functioning) sensors may lead to lack of performance of sensing sheet. Therefore, it is important to explore the performance of sparse arrays of sensors, in the cases where sensors are not necessarily in direct contact with damage (indirect sensing). The aim of this research is to create a method for optimizing the design of arrays of sensors, that is, to find the smallest number of sensors while maintaining a satisfactory reliability of crack detection and accuracy of damage localization and quantification. To achieve that goal, we first built a phase field finite element model of cracked structure verified by the analytical model to determine the crack existence (detection), and then we used the algorithm of inverse elastostatic problem combined with phase field finite element model to determine the crack length (quantification) and location (localization) by minimizing the difference between the sensor measurements and the phase field finite element model results. In addition, we experimentally validated the method by means of a reduced-scale laboratory test and assessed the accuracy and reliability of indirect sensing.

Boundary value problems in elastostatics with singular data

We consider themain boundary value problems of linear elastostatics with nonregular data. We prove existence and uniqueness results for bounded and exterior domains of ℝ3 of class Ck (k ⩾ 2). In the case of isotropic body, we prove the results for domains of class C1,α (α ∈ (0, 1]) and of class C1 in the case of the displacement problem.

First Principles Derivation of Displacement and Stress Function for Three-Dimensional Elastostatic Problems, and Application to the Flexural Analysis of Thick Circular Plates

In this study, stress and displacement functions of the three-dimensional theory of elasticity for homogeneous isotropic bodies are derived from first principles from the differential equations of equilibrium, the generalized stress – strain laws and the geometric relations of strain and displacement. It is found that the stress and displacement functions must be biharmonic functions. The derived functions are used to solve the elasticity problem of finding stresses and displacement fields in a thick circular plate with clamped edges for the case of uniformly distributed transverse load over the plate domain. Superposition of second to sixth order Legendre polynomials which are biharmonic functions are used in the thick circular plate problem as the stress function with the unknown constants as the parameters to be determined. Use of the stresses and displacement fields derived in terms of the stress and displacement function yielded the stress fields and displacement fields in terms of the unknown constants of the biharmonic stress function. Enforcement of the boundary conditions yielded the unknown constants, leading to a complete determination of the stress and displacement function for the stress fields and the displacement fields. The solutions obtained are comparable to solutions in the technical literature.

Two-dimensional elastodynamic and free vibration analysis by the method of fundamental solutions

In this paper we will present a formulation for solving elastodynamic problems in 2D using the method of fundamental solutions (MFS). The governing equation for the displacement in the elastodynamic problem is expressed by the Navier equation with an additional non-homogeneous inertial term, which involves the second derivative of displacement with respect to time. The inertial term is approximated by the Houbolt finite difference formula. In each time step, the solution for the displacement is separated into a homogeneous solution and a particular solution. The homogeneous solution is constructed in terms of the fundamental solutions of the elastostatic problem. The particular solution corresponds to the non-homogeneous inertial term, which is obtained by approximating the inertial term with radial basis functions (RBFs). In addition, a formulation based on the MFS for free vibration analysis of 2D bodies is presented in this paper. By solving several numerical examples, we have demonstrated that the proposed methods are robust and accurate. The convergence of the results with respect to the number of collocation points, the number of internal points, the value of the RBF parameter, and the time step size is also studied.

The BEM based on conformal Duffy-distance transformation for three-dimensional elasticity problems

Here, we describe the robust and efficient application of the conventional 3D BEM in solving elasticity problems. We have focused on the precise computation of weakly singular integrals. The conformal Duffy-distance transformation was employed to alleviate near singularities caused from two aspects: (1) the large aspect ratio of elements, i.e., element shape distortions; and (2) the closeness of element boundaries to field points, i.e., ill-shaped patches. Then, the rigid body motion method was employed to evaluate strongly singular integrals. Numerical solutions of 3D elastostatic problems demonstrated the high accuracy of the proposed method with coarse meshes and high convergence rates with mesh refinement. Compared with the Duffy transformation and original polar coordinate transformations, the proposed method is insensitive to element shapes.

Enhanced numerical integration scheme based on image-compression techniques: application to fictitious domain methods

In the present work, we propose a new approach, the so-called compressed adaptive integration scheme (C-AIS), for the computation of the stiffness and mass matrices in fictitious domain methods requiring the integration of discontinuous functions. The novel approach extends the conventional quadtree-decomposition-based adaptive integration scheme (AIS) by an additional step, in which established image-compression techniques are exploited to decrease the number of integration sub-cells. The benefits of the C-AIS are manifold: First, the compression of the sub-cells inevitably leads to significant savings in terms of computational time required by the numerical integration. Second, the compression procedure, which is executed directly after the quadtree-decomposition algorithm, can be easily included in existing codes. Third, if applied to polynomial integrands, the C-AIS yields exactly the same accuracy as the conventional AIS. Finally, the fourth advantage is seen in the fact that the C-AIS can readily be combined with other approaches seeking a reduction of the number of integration points such as the Boolean-FCM. The efficiency of the C-AIS approach is presented in the context of the FCM based on Cartesian meshes applied to problems of linear elastostatics and modal analysis, while it is also suitable for the quadrature in other fictitious domain approaches, e.g., CutFEM and cgFEM.

Integral equations in direct and inverse problems of elastostatics for crack detection in layered structures

This study deals with both direct and inverse problems for interfacial crack identification in laminates. The main focus is given to the case of an elastic substrate coated by a film made of a different elastic material. It is assumed that delamination can be developed on the interface between these materials. It is modelled as a combined open-sliding interface crack or by a pure sliding crack (slip). Its position may not be specified. The boundary conditions on the interface assume continuity of the stress vector across the whole interface and continuity of the displacements outside the crack. In the case of the slip the normal displacements are assumed to be continuous. The inverse boundary value considered is of the Cauchy type; it assumes that both stress and displacement vectors are known on the external boundary of the structure. In the case of the slip the problem is overdetermined on a part of the boundary, where three conditions are imposed, and undetermined on its remainder where just one condition is imposed. It is further referred to as a semi-inverse formulation. Therefore, these problems are ill-posed with the specified boundary conditions. Inverse, semi-inverse and direct problems are reduced to integral equations derived from basic properties of holomorphic functions followed by applications of Fourier transforms. Analytical solutions are found for the formulations considered and difficulties of numerical implementation are discussed in brief.

Singular integrals in axisymmetric problems of elastostatics

Singular integral equations arisen in axisymmetric problems of elastostatics are under consideration in this paper. These equations are received after applying the integral transformation and Gauss–Ostrogradsky’s theorem to the Green tensor for equilibrium equations of the infinite isotropic medium. Initially, three-dimensional problems expressed in Cartesian coordinates are transformed to cylindrical ones and integrated with respect to the circumference coordinate. So, the three-dimensional axisymmetric problems are reduced to systems of one-dimensional singular integral equations requiring the evaluation of linear integrals only. The thorough analysis of both displacement and traction kernels is accomplished, and similarity in behavior of both kernels is established. The kernels are expressed in terms of complete elliptic integrals of first and second kinds. The second kind elliptic integrals are nonsingular, and standard Gaussian quadratures are applied for their numerical evaluation. Analysis of external integrals proved the existence of logarithmic and Cauchy’s singularities. The numerical treatment of these integrals takes into account the presence of this integrable singularity. The numerical examples are provided to testify accuracy and efficiency of the proposed method including integrals with logarithmic singularity, Catalan’s constant, the Gaussian surface integral. The comparison between analytical and numerical data has proved high precision and availability of the proposed method.

On the Asymptotic Behavior of D-Solutions to the Displacement Problem of Linear Elastostatics in Exterior Domains

We study the asymptotic behavior of solutions with finite energy to the displacement problem of linear elastostatics in a three-dimensional exterior Lipschitz domain.

Analysis of High-order Approximations by Spectral Interpolation Applied to One- and Two-dimensional Finite Element Method

The implementation of high-order (spectral) approximations associated with FEM is an approach to overcome the difficulties encountered in the numerical analysis of complex problems. This paper proposes the use of the spectral finite element method, originally developed for computational fluid dynamics problems, to achieve improved solutions for these types of problems. Here, the interpolation nodes are positioned in the zeros of orthogonal polynomials (Legendre, Lobatto, or Chebychev) or equally spaced nodal bases. A comparative study between the bases in the recovery of solutions to 1D and 2D elastostatic problems are performed. Examples are evaluated, and a significant improvement is observed when the SFEM, particularly the Lobatto approach, is used in comparison to the equidistant base interpolation.

Method of manufactured solutions code verification of elastostatic solid mechanics problems in a commercial finite element solver

Much progress has been made in advancing and standardizing verification, validation, and uncertainty quantification practices for computational modeling in recent decades. However, examples of rigorous code verification for solid mechanics problems in literature remain scarce, particularly for commercial software and for the non-trivial large-deformation analyses and nonlinear materials typically needed to simulate medical devices. Here, we apply the method of manufactured solutions (MMS) to verify a commercial finite element code for elastostatic solid mechanics analyses using linear-elastic, hyperelastic (neo-Hookean), and quasi-hyperelastic (Hencky) constitutive models. Analytical source terms are generated using Python/SymPy and are implemented in ABAQUS/Standard without modification to solver source code. Source terms for the three constitutive models are found to vary nearly six orders of magnitude in the number of mathematical operations they contain. Refinement studies reveal second-order displacement and first-order (or higher) stress and strain convergence in response to mesh refinement for all constitutive models and first-order displacement convergence in response to pseudo-time increment refinement for the Hencky-elastic case. We also investigate the sensitivity of convergence order to quantitatively minor changes to the underlying mathematical model using an exploratory case. Code used to generate the MMS source terms and simulation input files are provided as Supplemental Material.