Linear Elastic Problems Research Articles

Impedance operator of a curved thin layer in linear elasticity with voids

We consider a two‐dimensional transmission problem of linear elasticity with voids in a fixed domain coated by a curved thin layer . Our aim is to model the effect of the thin layer on the fixed domain by an impedance boundary condition. For that, we use the techniques of asymptotic expansion to approximate the transmission problem by an impedance problem set in the fixed domain , and we prove an error estimate for the approximate impedance problem.

Closed-form solutions for axially non-uniform Timoshenko beams and frames under static loading

This paper presents the Green’s Functions Stiffness Method (GFSM) for solving linear elastic static problems in arbitrary axially non-uniform Timoshenko beams and frames subjected to general external loads and bending moments. The GFSM is a mesh reduction method that seamlessly integrates elements from the Stiffness Method (SM), Finite Element Method (FEM), and Green’s Functions (GFs), resulting in a highly versatile methodology for structural analysis. It incorporates fundamental concepts such as stiffness matrices, shape functions, and fixed-end forces, in line with SM and FEM frameworks. Leveraging the capabilities of GFs, the method facilitates the derivation of closed-form solutions, addressing a gap in existing methods for analyzing non-uniform reticular structures which are typically limited to simple cases like single-span beams with specific axial variations and loading scenarios. The effectiveness of the GFSM is demonstrated through three practical examples, showcasing its applicability in analyzing non-uniform beams and plane frames, thereby broadening the scope of closed-form solutions for axially non-uniform Timoshenko structures.

Linel2D-Net: A deep learning approach to solving 2D linear elastic boundary value problems on image domains

Linel2D-Net: A deep learning approach to solving 2D linear elastic boundary value problems on image domains

Discrete Helmholtz Decompositions of Piecewise Constant and Piecewise Affine Vector and Tensor Fields

AbstractDiscrete Helmholtz decompositions dissect piecewise polynomial vector fields on simplicial meshes into piecewise gradients and rotations of finite element functions. This paper concisely reviews established results from the literature which all restrict to the lowest-order case of piecewise constants. Its main contribution consists of the generalization of these decompositions to 3D and of novel decompositions for piecewise affine vector fields in terms of Fortin–Soulie functions. While the classical lowest-order decompositions include one conforming and one nonconforming part, the decompositions of piecewise affine vector fields require a nonconforming enrichment in both parts. The presentation covers two and three spatial dimensions as well as generalizations to deviatoric tensor fields in the context of the Stokes equations and symmetric tensor fields for the linear elasticity and fourth-order problems. While the proofs focus on contractible domains, generalizations to multiply connected domains and domains with non-connected boundary are discussed as well.

The lowest-order weak Galerkin finite element method for linear elasticity problems on convex polygonal grids

This paper presents the lowest-order weak Galerkin finite element method for linear elasticity problems on the convex polygonal meshes. This method uses piecewise constant vector-valued spaces on element interiors and edges. The discrete weak gradient space introduced by this paper is the matrix version of CW0 space. The discrete weak divergence space is piecewise constant space on each element. This method is simple, efficient, stabilizer-free and symmetric positive-definite. The optimal error estimates in discrete H1 and L2 norms are presented. Numerical results are given to demonstrate the efficiency of algorithm and the locking-free property.

Dimension reduction and homogenization of composite plate with matrix pre-strain

This paper focuses on the simultaneous homogenization and dimension reduction of periodic composite plates within the framework of non-linear elasticity. The composite plate in its reference (undeformed) configuration consists of a periodic perforated plate made of stiff material with holes filled by a soft matrix material. The structure is clamped on a cylindrical part. Two cases of asymptotic analysis are considered: one without pre-strain and the other with matrix pre-strain. In both cases, the total elastic energy is in the von-Kármán (vK) regime ( ε 5 ). A new splitting of the displacements is introduced to analyze the asymptotic behavior. The displacements are decomposed using the Kirchhoff–Love (KL) plate displacement decomposition. The use of a re-scaling unfolding operator allows for deriving the asymptotic behavior of the Green St. Venant’s strain tensor in terms of displacements. The limit homogenized energy is shown to be of vK type with linear elastic cell problems, established using the Γ-convergence. Additionally, it is shown that for isotropic homogenized material, our limit vK plate is orthotropic. The derived results have practical applications in the design and analysis of composite structures.

M‐PINN: A mesh‐based physics‐informed neural network for linear elastic problems in solid mechanics

AbstractPhysics‐informed neural networks (PINNs) have emerged as a promising approach for solving a wide range of numerical problems. Nevertheless, conventional PINNs frequently face challenges in model convergence and stability when optimizing complex loss functions containing complex gradients. In this study, a new mesh‐based PINN method, called M‐PINN, is proposed drawing the ideas of the finite element method (FEM). By partitioning the solution domain into several subdomains and incorporating finite element data distribution constraints to the prior estimates of the predicted data distribution of PINN on the solution domain, the M‐PINN approach effectively reduces the optimization difficulty of conventional PINNs. Moreover, it is sometimes difficult to directly obtain precise boundary conditions in some practical applications. This method can be used to solve PINN problems with unknown boundary conditions, thus having wider applicability. In this study, the efficiency of M‐PINN was demonstrated through a standard 2D linear elastic solid mechanics simulation experiment, and its applicability was investigated in depth. The results indicate that the M‐PINN method outperforms traditional PINN and exhibits superior applicability and convergence, especially in cases involving unknown boundary conditions.

A parameter-free mixed formulation for the Stokes equations and linear elasticity with strongly symmetric stress

In this paper, we design and analyze a parameter-free mixed method of arbitrary polynomial orders for the Stokes equations and linear elasticity problem, where the symmetry of stress is strongly imposed. Equal-order polynomials are exploited for the stress and velocity spaces in which the stress is approximated by discontinuous polynomials. In contrast, the velocity is approximated by an H(div;Ω) conforming space. As a consequence, the proposed scheme yields divergence-free velocity approximations and is pressure-robust for the Stokes equations. The stable Stokes pair are integrated to facilitate the proof of the inf-sup condition. The comprehensive convergence error estimates for all the variables are explored for the Stokes equations and linear elasticity problem. In particular, we explicitly track the dependence of the error estimates on the physical parameters and illustrate the locking-free property of the proposed scheme, which is non-trivial under the proposed formulation. The judicious balancing of the mixed formulation and the equivalent primal formulation enable us to achieve the robust error estimates for the linear elasticity problem. Several numerical experiments for the Stokes equations and linear elasticity problem are carried out to verify the proposed theories.

A family of three-dimensional virtual elements for Hellinger-Reissner elasticity problems

We present a family of Virtual Element Methods for three-dimensional linear elasticity problems based on the Hellinger-Reissner variational principle. A convergence and stability analysis is developed. Moreover, using the hybridization technique and exploiting the information derived from this procedure, we show how to compute a better approximation for the displacement field. The numerical experiments confirm the theoretical predictions.

Analysis of axisymmetric hollow cylinder under surface loading using variational principle

In this work, a variational principle–based approach has been adopted to analyze one of the classical linear elasticity problem of the axisymmetric cylinder under surface loading. The use of variational principle results in a set of governing partial differential equations with associated boundary conditions. The equations have been solved using the separation of variable approach and the Frobenius method. A general solution has been derived and used to solve two test cases. The proposed solution is capable of meeting all the boundary conditions. The solution has been validated by comparing it with a finite element–based numerical solution and considering a special limiting condition of a solid cylinder, for which results are available in the literature. Further various studies have been carried out to understand the robustness and limitation of the presented solution.

Quantitative stability estimate for the inverse coefficients problem in linear elasticity

In this article we consider the inverse problem of reconstructing piece-wise Lamé coefficients from boundary measurements. We reformulate the inverse problem into a minimization one using a Kohn-Vogelius type functional. We study the stability of the parameters when the jump of the discontinuity is perturbed. Using tools of shape calculus, we give a quantitative stability result for local optimal solution. Dans cet article, nous considérons le problème inverse de reconstruction des coefficients de Lamé constants par morceaux à partir de mesures au bord. Nous reformulons le problème inverse en un problème de minimisation utilisant une fonctionnelle de type Kohn-Vogelius. Nous étudions la stabilité des paramètres lorsque le saut de la discontinuité est perturbé. En utilisant les outils du calcul de forme, nous donnons un résultat de stabilité quantitative pour une la solution optimale locale.

POD method for solving elastodynamics problems of functionally gradient materials based on radial integration boundary element method

Functionally graded materials (FGMs) are widely utilized due to their excellent mechanical properties, prompting a need to explore the dynamic response of FGMs. To accurately and efficiently analyze elastodynamics problems, the radial integration boundary element method (RIBEM) is coupled with the proper orthogonal decomposition (POD) method in this paper. The POD method can transform a high-dimensional model into a low-dimensional system with high precision. Using the basic solution (Kelvin solution), the governing partial differential equation for the linear elastic dynamics problem of FGMs is transformed into a boundary-domain integral equation. The radial integration method is employed to convert the domain integral arising from material inhomogeneity and inertia terms into an equivalent boundary integral. This establishes a solution for the elastic dynamics problem of FGMs without the need for internal grid RIBEM. The Newmark time integration scheme is applied to solve the second-order ordinary differential equations. The displacement field obtained by RIBEM is utilized to construct the snapshots matrix, from which the POD reduced-order model is established. Numerical examples validate that the POD method significantly reduces the model's order, enhancing computational efficiency while also demonstrating the high accuracy of the POD method.

Signorini problem as a variational limit of obstacle problems in nonlinear elasticity

<abstract><p>An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint.</p></abstract>

A co‐rotational virtual element method for 2D elasticity and plasticity

AbstractThe virtual element method (VEM) allows discretization of the problem domain with polygons in 2D. The polygons can have an arbitrary number of sides and can be concave or convex. These features, among others, are attractive for meshing complex geometries. VEM applied to linear elasticity problems is now well established. Nonlinear problems involving plasticity and hyperelasticity have also been explored by researchers using VEM. Clearly, techniques for extending the method to nonlinear problems are attractive. In this work, a novel first‐order consistent VEM is applied within a static co‐rotational framework. To the author's knowledge, this has not appeared before in the literature with virtual elements. The formulation allows for large displacements and large rotations in a small strain setting. For some problems avoiding the complexity of finite strains, and alternative stress measures, is warranted. Furthermore, small strain plasticity is easily incorporated. The basic method, VEM specific implementation details for co‐rotation, and representative benchmark problems are illustrated. Consequently, this research demonstrates that the co‐rotational VEM formulation successfully solves certain classes of nonlinear solid mechanics problems. The work concludes with a discussion of results for the current formulation and future research directions.

Method of solution composition in contact problems with friction of deformable bodies

A new iterative method for solving static contact problems of two deformable bodies is proposed. The method is based on alternately solving the unilateral contact problem for the first body and the linear elasticity problem with natural boundary conditions for the second body. Fulfillment of Coulomb's friction law involves correction of tangential nodal forces in the sliding area and setting kinematic boundary conditions in the sticking area for the contact boundary of the first body. The goal of solving the linear elasticity problem for the second body is to gradually equalize contact loads on the interacting surfaces. The advantages of the method are demonstrated by solving a number of model examples, including unilateral contact of a linear-elastic plate with a solid foundation, bilateral contact of pressing a deformable block into the foundation, the Hertz problem of contact of two deformable cylinders etc. The method can solve problems on flat and curvilinear contact boundaries.

A hierarchy of asymptotic models for a fluid-loaded elastic layer

A hierarchy of asymptotic models governing long-wave low-frequency in-plane motion of a fluid-loaded elastic layer is established. In contrast to a layer with traction-free faces, modelled by Neumann boundary conditions, a fluid-loaded one assumes more involved conditions along the interfaces, dictating a special asymptotic scaling. The latter corresponds to a fluid-borne bending wave, controlled by elastic stiffness of the layer and fluid inertia. In this case, the transverse inertia of the layer and fluid compressibility do not appear at zero-order approximation. The first-order approximation is associated with a Kirchhoff plate, immersed into incompressible fluid. In the studied free vibration setup, the fluid compressibility has to be taken into account only at third order, along with elastic rotary inertia. Transverse shear deformation enters the second-order approximation along with a few other corrections. The conventional impenetrability condition has to be also refined at second order. Dispersion relations corresponding to the developed asymptotic models are compared with the polynomial expansions of the full dispersion relation, obtained from the plane-strain problem of linear elasticity.

FFT-Based Numerical Method for Nonlinear Elastic Contact

In theoretical research pertaining to sealing, a contact model must be used to obtain the leakage channel. However, for elastoplastic contact, current numerical methods require a long calculation time. Hyperelastic contact is typically simplified to a linear elastic contact problem, which must be improved in terms of calculation accuracy. Based on the fast Fourier transform, a numerical method suitable for elastoplastic and hyperelastic frictionless contact that can be used for solving two-dimensional and three-dimensional (3D) contact problems is proposed herein. The nonlinear elastic contact problem is converted into a linear elastic contact problem considering residual deformation (or the equivalent residual deformation). Results from numerical simulations for elastic, elastoplastic, and hyperelastic contact between a hemisphere and a rigid plane are compared with those obtained using the finite element method to verify the accuracy of the numerical method. Compared with the existing elastoplastic contact numerical methods, the proposed method achieves a higher calculation efficiency while ensuring a certain calculation accuracy (i.e., the pressure error does not exceed 15%, whereas the calculation time does not exceed 10 min in a 64 × 64 grid). For hyperelastic contact, the proposed method reduces the dependence of the approximation result on the load, as in a linear elastic approximation. Finally, using the sealing application as an example, the contact and leakage rates between complicated 3D rough surfaces are calculated. Despite a certain error, the simplified numerical method yields a better approximation result than the linear elastic contact approximation. Additionally, the result can be used as fast solutions in engineering applications.

Fracture analysis in 2D plane strain problems for composite materials containing hard inclusions and voids using an extended consecutive-interpolation quadrilateral element

This paper investigates fracture mechanics in particle-reinforced composites by using the extended finite element method enhanced by the consecutive-interpolation quadrilateral element. These composite materials have discontinuous boundaries such as cracks, voids, holes, and soft inclusions. And the extended consecutive-interpolation quadrilateral element (XCQ4) is employed to model these boundaries in two-dimensional linear elastic deformation problems. XCQ4 combines the enrichment functions in the traditional extended finite element method with the consecutive interpolation on a 4-node quadrilateral element. This element uses both nodal values and averaged nodal gradients as interpolated conditions. In fracture analysis, the stress intensity factors (SIFs) are important parameters that must be defined.  In this study, the values of SIFs at the crack tips are evaluated with the help of the interaction integrals approach. The obtained numerical results are compared with other reliable results showing high accuracy and convergence rate of the XCQ4 element.

An FFT-Based MAC scheme for Stokes equations with periodic boundary conditions and its application to elasticity problems

This paper presents a fast Fourier transform (FFT) based finite difference MAC scheme to compute the Stokes equations with periodic boundary conditions. The numerical implementation of this solver is very convenient and only requires three simple steps. The remarkable superiorities of such a solver compared to classical iterative approaches are computational time-saving and memory storage saving. The application of the proposed solver to nearly incompressible linear elasticity problems is also introduced. In addition, following the similar spirit used in Dong et al. (2020) for Stokes equations with Dirichlet boundary conditions, a rigorous error analysis is provided for Stokes equations with periodic boundary conditions. The results include: a second-order convergence in the ℓ2-norm for velocity, pressure as well as the gradient of velocity; a second-order accuracy in maximum norm for both velocity and its gradient. Moreover, the properties of the discrete Green functions play an important part in maximum error estimation. Most notably, the major difficulties of the extension from Dirichlet boundary conditions to periodic boundary conditions are the discrete version of Green’s formulae, Poincaré inequality and LBB condition. A variety of two-dimensional and three-dimensional numerical examples are given to illustrate the efficiency of the proposed FFT solver and validate the theoretical results.

A low-rank isogeometric solver based on Tucker tensors

We propose an isogeometric solver for Poisson problems that combines (i) low-rank tensor techniques to approximate the unknown solution and the system matrix, as a sum of a few terms having Kronecker product structure, (ii) a Truncated Preconditioned Conjugate Gradient solver to keep the rank of the iterates low, and (iii) a novel low-rank preconditioner, based on the Fast Diagonalization method where the eigenvector multiplication is approximated by the Fast Fourier Transform. Although the proposed strategy is written in arbitrary dimension, we focus on the three-dimensional case and adopt the Tucker format for low-rank tensor representation, which is well suited in low dimension. We show by numerical tests that this choice guarantees significant memory saving compared to the full tensor representation. We also extend and test the proposed strategy to linear elasticity problems.