Nonlinear Elastic Problems Research Articles

Resolution of a nonlinear elasticity problem subject to friction laws

Resolution of a nonlinear elasticity problem subject to friction laws

Mixed displacement–pressure formulations and suitable finite elements for multimaterial problems with compressible and incompressible models

Multimaterial problems in linear and nonlinear elasticity are some of the least explored using mixed finite element formulations with higher-order elements. The fundamental issue in adapting the mixed displacement–pressure formulations with linear and higher-order continuous elements for the pressure field is their inability to capture pressure and stress jumps across material interfaces. In this paper, for the first time in literature, we perform comprehensive studies of multimaterial problems in elasticity consisting of compressible and incompressible material models using the mixed displacement–pressure formulation to assess the performance of different element types in accurately resolving pressure fields within the domains and pressure jumps across material interfaces. In particular, inf–sup stable displacement–pressure combinations with element-wise discontinuous pressure for triangular and tetrahedral elements are considered and their performance is assessed along with the Q1/P0 element and Taylor–Hood elements using several numerical examples. The results show that Taylor–Hood elements fail to capture the stress jumps due to the continuity of DOFs across elements, the Crouzeix–Raviart (P2b/P1dc) element yields substantially poor pressure fields despite a significant increase in pressure degrees of freedom and that the P3/P1dc element produces superior quality results fields when compared with the P2b/P1dc element.

Revisiting the nonlinear elastic problem of internally constrained beams in a perturbation perspective

Unshearable and inextensible planar beams, in a static regime of finite displacements, are studied in this paper. A nonlinear mixed model is derived via a direct approach, in which displacements and reactive internal forces are taken as unknowns. The elasto-static problem is then addressed, and the role of the boundary conditions is systematically discussed. The relevant solutions for selected classes of problems are pursued via a perturbation method. It is shown that each considered class calls for a specific algorithm, accounting for a proper scaling and expansion of the variables. Finally, the asymptotic solutions are compared with benchmark numerical computations, grounded on finite-element analyses. The paper is focused on the case of longitudinal force significantly smaller than the buckling load, leaving the case of large force to future developments, where a different perturbation scheme is required.

Effect of compressibility on the mechanics of hyperelastic membranes

Elastic membranes are usually studied assuming material incompressibility. However, in several applications they are made of compressible materials such as polymeric foams, hydrogels, and certain kinds of elastomers. Only a few works attempted to incorporate volume changes into membrane problems, but with significant limitations. The models proposed were designed for nearly incompressible materials and lacked a foundation in experimental data, leading to results of limited value. In this work, we investigate the effect of compressibility in membrane problems adopting a consistent model based on the real response of materials to large volume changes. We consider three benchmark problems of nonlinear elasticity: (i) inflation of a circular flat membrane; (ii) inflation of a thin-walled cylindrical tube; (iii) inflation of a thin-walled spherical balloon. Four types of materials divided by increasing degree of compressibility are studied. The results indicate that volumetric deformations have a significant impact on both the limit pressure and the deformed shape. The proposed solutions represent benchmarks for developing new applications of compressible membranes made of polymeric foams and hydrogels, playing an increasingly important role in engineering technologies.

Parametric control of quasi-zero stiffness mechanisms for vibration isolation at near-zero frequencies

Infra-frequency vibration, including near-zero frequencies, has become a critical limiting factor for developers of next-generation ultra-high precision measurement systems. Mechanisms involving quasi-zero stiffness show potential to solve this new vibration problem. An approach for designing a novel type of the mechanisms for this class is presented. An algorithm is developed and tested for numerical solution of nonlinear elasticity problem in designing the mechanism structural and parametric elements based on FEM models. The elements are spatially prestressed multi-layer designs of polymer composites for modeling elastic postbuckling in large and the effect of a “floating” binder. An algorithm and models following the Nelder-Mead method are used for optimizing the designs. Examples of Euler-beam-shaped composite elements under postbuckling demonstrate a qualitative leap in mechanisms design and efficiency, specifically, in terms of compactness, lightness, and strength. A dramatically expanded control range for the linear section of operating travel with quasi-zero stiffness is obtained. A damping threshold is evaluated, and an optimal type diagram of the mechanism is designed and enumerated as necessary conditions for vibration isolation at near-zero frequencies. The validity and effectiveness of the approach are demonstrated with the results of theoretical design and testing of the prototype for vibration isolation of the ultra-high precision optical system. The learning capability of the mechanisms is estimated by an adapted version of the Levenberg-Marquardt algorithm with stepwise refinement in the test mode using Rosenbrock’s function.

Adaptive neural network surrogate model for solving the nonlinear elastic inverse problem via Bayesian inference

Abstract In this paper, we consider a Bayesian method for nonlinear elastic inverse problems. As a working model, we are interested in the inverse problem of restoring elastic properties from measured tissue displacement. In order to reduce the computational cost, we will use the following multi-fidelity model approach. First, we construct a surrogate low-fidelity DNNs-based model in the prior distribution, then use a certain number of simulations of high fidelity model associated with an adaptive strategy online to update the low-fidelity model locally. Numerical examples show that the proposed method can solve nonlinear elastic inverse problems efficiently and accurately.

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>

On Solving Nonlinear Elasticity Problems Using a Boundary-Elements-Based Solution Method

The attractiveness of the boundary element method—the reduction in the problem dimension by one—is lost when solving nonlinear solid mechanics problems. The point collocation method applied to strong-form differential equations is appealing because it is easy to implement. The method becomes inaccurate in the presence of traction boundary conditions, which are inevitable in solid mechanics. A judicious combination of the point collocation and the boundary integral formulation of Navier’s equation allows a pure boundary element method to be obtained for the solution of nonlinear elasticity problems. The potential of the approach is investigated in some simple examples considering isotropic and anisotropic material models in the total Lagrangian framework.

Asymptotically correct 3D displacement of the Mooney–Rivlin model using VAM

The current work focuses on the nonlinear analytical analysis (dimensional reduction and recovery relations) of a hyperelastic plate governed by the compressible Mooney–Rivlin model using the Variational Asymptotic Method (VAM). The geometric nonlinearity is accommodated through finite deformations, and generalized 3D warping functions, while material nonlinearity through the hyperelastic material model. VAM mathematically splits the 3D nonlinear elastic problem into the 1D through the thickness analysis and 2D nonlinear plate analysis, using the inherent small parameters. These are the geometric small parameter (ratio of thickness to the characteristic dimension), and the physical small parameter (moderate strains). This work results include the derivation of closed-form analytical expressions of 3D warping functions, 2D nonlinear constitutive relation, and recovery relation to express the 3D displacement field for a plate. The 2D nonlinear constitutive relation is given as an input to the in-house developed 2D nonlinear finite element analysis of the reference surface to determine the 2D displacements and 2D strains. In order to validate the current theory, standard test cases are solved and compared with 3D nonlinear Finite Element Analysis (FEA).

Geometrically Non-Linear Plane Elasticity Problem in the Area of an Angular Boundary Cut-Out

A relevant problem in the development and improvement of numeric analytical methods for the research of structures, buildings and construction is studying the stress–strain state of structures and construction with boundaries that have complex shapes. Deformations and stresses arise in a domain with a geometrically non-linear shape of the boundary (cut-outs and cuts). These stresses and deformations have great values and gradients. Experiments carried out using the photoelasticity method show a change in the deformation order ratios for different subareas of the boundary cut-out area depending on proximity to the apex of the angular cut-out. Areas with minor deformations are observed, and areas where linear deformations and shears are more significant than rotations are also observed. In addition, areas where section rotations are more significant than linear and shear deformations are observed. According to the experimental data, the mathematical model of the SSS in the area of the apex of the cut-out of the domain boundary should take into account non-linear deformations. Hence, it is necessary to formulate the boundary value problem of the theory of elasticity, taking into account the geometrical non-linearity. The research aim of this paper is to formulate the problem of the elasticity theory taking into account the geometrical non-linearity in furtherance of the proposed mathematical model justified by the experimental data obtained using the photoelasticity method. The obtained formulation of the elasticity theory problem allows analyzing the form of the system of equations of the boundary value problem depending on the proximity of the considered area to the irregular point of the boundary, i.e., taking into account the difference in the effect of linear and shear deformations, rotations and forced deformations on the solution to the geometrically non-linear elastic problem dealing with forced deformations in the area of an angular cut-out of the boundary of the plane domain.

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.

An immersed peridynamics model of fluid-structure interaction accounting for material damage and failure

This paper develops and benchmarks an immersed peridynamics method to simulate the deformation, damage, and failure of hyperelastic materials within a fluid-structure interaction framework. The immersed peridynamics method describes an incompressible structure immersed in a viscous incompressible fluid. It expresses the momentum equation and incompressibility constraint in Eulerian form, and it describes the structural motion and resultant forces in Lagrangian form. Coupling between Eulerian and Lagrangian variables is achieved by integral transforms with Dirac delta function kernels, as in standard immersed boundary methods. The major difference between our approach and conventional immersed boundary methods is that we use peridynamics, instead of classical continuum mechanics, to determine the structural forces. We focus on non-ordinary state-based peridynamic material descriptions that allow us to use a constitutive correspondence framework that can leverage well-characterized nonlinear constitutive models of soft materials. The convergence and accuracy of our approach are compared to both conventional and immersed finite element methods using widely used benchmark problems of nonlinear incompressible elasticity. We demonstrate that the immersed peridynamics method yields comparable accuracy with similar numbers of structural degrees of freedom for several choices of the size of the peridynamic horizon. We also demonstrate that the method can generate grid-converged simulations of fluid-driven material damage growth, crack formation and propagation, and rupture under large deformations.

A Class of Nonlinear Elasticity Problems with No Local but Many Global Minimizers

A Class of Nonlinear Elasticity Problems with No Local but Many Global Minimizers

Existence of the positive solutions for a class of nonlinear elasticity problems involving Orlicz spaces via sub-supersolution method

In this paper, we study nonlinear elastic problem of the type{−ΔΦu+K(x)u−α|u|LΛγ=λup|u|LΨq,inΩ,u=0,on∂Ω, where q,γ are positive constants and p,α∈(0,1). |⋅|LΨ, |⋅|LΛ and Φ are defined later. Based on the pseudomonotone operators theory, we establish some new sub-supersolution theorems, that generalize the results in [12]. In addition, based on the obtained theorems, we prove the existence of positive solutions to the above mentioned under certain assumptions about the parameters. Our work improves the results of some past researches.

Coupling of Finite and Boundary Elements for Singularly Nonlinear Transmission and Contact Problems

Abstract This article discusses the well-posedness and error analysis of the coupling of finite and boundary elements for interface problems in nonlinear elasticity. It concerns 𝑝-Laplacian-type Hencky materials with an unbounded stress-strain relation, as they arise in the modelling of ice sheets, non-Newtonian fluids or porous media. We propose a functional analytic framework for the numerical analysis and obtain a priori and a posteriori error estimates for Galerkin approximations to the resulting boundary/domain variational inequality.

Discontinuous Mappings and the Limit Load in Boundary Value Problems of Nonlinear Elasticity

Discontinuous Mappings and the Limit Load in Boundary Value Problems of Nonlinear Elasticity

A HDG formulation for nonlinear elasticity problems featuring finite deformations and frictionless contact constraints

A HDG formulation for nonlinear elasticity problems featuring finite deformations and frictionless contact constraints

A second-order reduced homogenization for nonlinear structures with periodic configurations in cylindrical coordinates

A high-efficiency second-order reduced homogenization (SORH) method is established for the inelastic problem of periodic heterogeneous structures in cylindrical coordinate. The composite structures investigated in this paper are periodical in radial, axial and circumferential directions. The new second-order linear/nonlinear local solutions at microscale are given by introducing various multiscale auxiliary functions. And, the nonlinear high-order homogenization solutions are obtained at macroscale using a modified asymptotic expansion method. The significant novelties of the work are (i) an effective reduced model on the basis of transformation field analysis given to solve the nonlinear multiscale problems in cylindrical coordinates with less computational amount and (ii) a new second-order nonlinear multiscale algorithm established using asymptotic expansion technique for simulating the heterogeneous cylindrical structure. Finally, by three typical numerical examples including nonlinear elastic and elasto-plastic periodic problems, the validity and accuracy of the proposed algorithms are verified.

Discontinuous mappings and the limit load in boundary value problems of nonlinear elasticity

The article discusses a boundary value problem of nonlinear elasticity for mapping (deformation) in two weak formulations: in the form of a variational equilibrium equation and in the form of minimization of a multidimensional integral functional. Some questions of a mathematical correctness of the nonlinear elasticity BVP are discussed. Using methods of calculus of variations, by the example of two simple problems, it is proved that for some nonlinear elastic models in the corresponding boundary value problems there can be mappings that have discontinuities of the slip type, as well as there is the limit load - such a finite value of external forces, above which the boundary value problem has no solution at all.

Differential variational–hemivariational inequalities with application to contact mechanics

In this paper we study a dynamical system which consists of the Cauchy problem for a nonlinear evolution equation of first order coupled with a nonlinear time-dependent variational–hemivariational inequality with constraint in Banach spaces. The evolution equation is considered in the framework of evolution triple of spaces, and the inequality which involves both the convex and nonconvex potentials. We prove existence of solution by the Kakutani–Ky Fan fixed point theorem combined with the Minty formulation and the theory of hemivariational inequalities. We illustrate our findings by examining a nonlinear quasistatic elastic frictional contact problem for which we provide a result on existence of weak solution.