Problems In Finite Elasticity Research Articles

A kinematically stabilized linear tetrahedral finite element for compressible and nearly incompressible finite elasticity

We propose a stabilized linear tetrahedral finite element method for static, finite elasticity problems involving compressible and nearly incompressible materials. Our approach relies on a mixed formulation, in which the nodal displacement unknown filed is complemented by a nodal Jacobian determinant unknown field. This approach is simple to implement in practical applications (e.g., in commercial software), since it only requires information already available when computing the Newton–Raphson tangent matrix associated with irreducible (i.e., displacement-based) finite element formulations. By nature, the proposed method is easily extensible to nonlinear models involving visco-plastic flow. An extensive suite of numerical tests in two and three dimensions is presented, to demonstrate the performance of the method.

Mixed peridynamic formulations for compressible and incompressible finite deformations

The large flexibility of meshfree solution schemes makes them attractive for many kinds of engineering applications, like Additive Manufacturing or cutting processes. While numerous meshfree methods were developed over the years, the accuracy and robustness are still challenging and critical issues. Stabilization techniques of various kinds are typically used to overcome these problems, but often require the tuning of unphysical parameters. The Peridynamic Petrov–Galerkin method is a generalization of the peridynamic theory of correspondence materials and offers a stable and robust alternative. In this work, the stabilization free approach is extended to three dimensional problems of finite elasticity. Locking-free mixed formulations for nearly incompressible and incompressible materials are developed and investigated in convergence studies. In general, an efficient implicit quasi-static framework based on Automatic Differentiation is presented. The numerical examples highlight the convergence properties and robustness of the proposed formulations.

3D printing of carbon nanotubes reinforced thermoplastic polyimide composites with controllable mechanical and electrical performance

Existing aerospace metal parts or structural parts have the problems of finite elasticity, large mass, difficult manufacture of complex parts, expensive equipment, narrow range of shielding and electrostatic regulation. This study analyzed the tensile and bending properties of pure thermoplastic polyimide (TPI) 3D printing specimens printed by the nozzles with different diameters. The effects of different carbon nanotube (CNT) contents on the thermal and electrical properties of carbon nanotube composites were investigated. The relationship between the electrical conductivity, tensile properties, bending properties, resistivity and cyclic bending deformation of 3D printed specimens with different carbon nanotube contents was studied. Compared with the 0.4 mm nozzle, the tensile and bending strength of pure TPI specimen printed by a 0.8 mm nozzle were increased by 40% and 20%, respectively. In order to ensure the excellent mechanical and electrical properties of 3D printed composites, 3%wt carbon nanotubes-thermoplastic polyimide (CNTs-TPI) composites with appropriate tensile and bending strength were used to print the characteristic structure, and then the relationship between the deformation and conductive resistance was studied. By optimizing the 3D printing parameters and the filling contents of CNTs, it is possible to print complex parts which can be used in aerospace and industrial fields with special properties, such as electrical conductivity, mechanical properties, temperature resistance and anti-electromagnetic shielding.

On improving the numerical convergence of highly nonlinear elasticity problems

Finite elasticity problems commonly include material and geometric nonlinearities and are solved using various numerical methods. However, for highly nonlinear problems, achieving convergence is relatively difficult and requires small load step sizes. In this work, we present a new method to transform the discretized governing equations so that the transformed problem has significantly reduced nonlinearity and, therefore, Newton solvers exhibit improved convergence properties. We study exponential-type nonlinearity in soft tissues and geometric nonlinearity in compression, and propose novel formulations for the two problems. We test the new formulations in several numerical examples and show significant reduction in iterations required for convergence, especially at large load steps. Notably, the proposed formulation is capable of yielding convergent solution even when 10–100 times larger load steps are applied. The proposed framework is generic and can be applied to other types of nonlinearities as well.

On variational formulations with rigid-body constraints for finite elasticity: Applications to 2D and 3D finite element simulations

We investigate a recently proposed variational principle with rigid-body constraints and present an extension of its implementation in three dimensional finite elasticity problems. Through numerical examples, we illustrate that the proposed variational principle with rigid-body constraints is applicable to both single field and mixed finite elements of arbitrary order and geometry, e.g. triangular/tetrahedral and quadrilateral/hexagonal elements, in two and three dimensions. Moreover, we demonstrate that, as compared to the commonly adopted approach of discretizing the rigid domains with standard finite elements, the proposed formulation requires neither discretization nor numerical integration in the interior of each rigid domain. As a comparative result, the variational formulation may reduce the total number of degrees of freedom of the resulting finite element system and provide improved accuracy.

A paradigm for higher-order polygonal elements in finite elasticity using a gradient correction scheme

Recent studies have demonstrated that polygonal elements possess great potential in the study of nonlinear elastic materials under finite deformations. On the one hand, these elements are well suited to model complex microstructures (e.g. particulate microstructures and microstructures involving different length scales) and incorporating periodic boundary conditions. On the other hand, polygonal elements are found to be more tolerant to large localized deformations than the standard finite elements, and to produce more accurate results in bending and shear. With mixed formulations, lower order mixed polygonal elements are also shown to be numerically stable on Voronoi-type meshes without any additional stabilization treatment. However, polygonal elements generally suffer from persistent consistency errors under mesh refinement with the commonly used numerical integration schemes. As a result, non-convergent finite element results typically occur, which severely limit their applications. In this work, a general gradient correction scheme is adopted that restores the polynomial consistency by adding a minimal perturbation to the gradient of the displacement field. With the correction scheme, the recovery of optimal convergence for solutions of displacement-based and mixed formulations with both linear and quadratic displacement interpolants is confirmed by numerical studies of several boundary value problems in finite elasticity. In addition, for mixed polygonal elements, the various choices of the pressure field approximations are discussed, and their performance on stability and accuracy are numerically investigated. We present applications of those elements in physically-based examples including a study of filled elastomers with interphasial effect and a qualitative comparison with cavitation experiments for fiber reinforced elastomers.

Analytical solutions to general anti-plane shear problems in finite elasticity

This paper presents a pure complementary energy variational method for solving a general anti-plane shear problem in finite elasticity. Based on the canonical duality–triality theory developed by the author, the nonlinear/nonconvex partial differential equations for the large deformation problem are converted into an algebraic equation in dual space, which can, in principle, be solved to obtain a complete set of stress solutions. Therefore, a general analytical solution form of the deformation is obtained subjected to a compatibility condition. Applications are illustrated by examples with both convex and nonconvex stored strain energies governed by quadratic-exponential and power-law material models, respectively. Results show that the nonconvex variational problem could have multiple solutions at each material point, the complementary gap function and the triality theory can be used to identify both global and local extremal solutions, while the popular convexity conditions (including rank-one condition) provide mainly local minimal criteria and the Legendre–Hadamard condition (i.e., the so-called strong ellipticity condition) does not guarantee uniqueness of solutions. This paper demonstrates again that the pure complementary energy principle and the triality theory play important roles in finite deformation theory and nonconvex analysis.

Polygonal finite elements for finite elasticity

SUMMARYNonlinear elastic materials are of great engineering interest, but challenging to model with standard finite elements. The challenges arise because nonlinear elastic materials are characterized by non‐convex stored‐energy functions as a result of their ability to undergo large reversible deformations, are incompressible or nearly incompressible, and often times possess complex microstructures. In this work, we propose and explore an alternative approach to model finite elasticity problems in two dimensions by using polygonal discretizations. We present both lower order displacement‐based and mixed polygonal finite element approximations, the latter of which consist of a piecewise constant pressure field and a linearly‐complete displacement field at the element level. Through numerical studies, the mixed polygonal finite elements are shown to be stable and convergent. For demonstration purposes, we deploy the proposed polygonal discretization to study the nonlinear elastic response of rubber filled with random and periodic distributions of rigid particles, as well as the development of cavitation instabilities in elastomers containing vacuous defects. These physically‐based examples illustrate the potential of polygonal finite elements in studying and modeling nonlinear elastic materials with complex microstructures under finite deformations. Copyright © 2014 John Wiley & Sons, Ltd.

A plane strain quadrilateral Cosserat point element (CPE) for nonlinear orthotropic elastic materials — An extension to initially distorted geometry and general orthotropic directions

The objective of the present paper is to develop a four-noded quadrilateral Cosserat point element (CPE) for the numerical solution of plane strain problems in finite elasticity of orthotropic materials with general orientation and initially distorted geometry. Generally speaking, the Cosserat point approach connects the kinetic quantities to derivatives of a strain energy function and once the strain energy of the CPE has been specified, the procedure needs no integration over the element region and it ensures that the response of the CPE is hyperelastic. In the present paper a functional form for the strain energy that controls the inhomogeneous deformations and allowing for accurate modeling of distorted meshes is proposed. A number of example problems, which compare the performance of the developed quadrilateral CPE with that of other elements based on the mixed formulation, reduced integration with enhanced hourglass control, and enhanced strains/incompatible modes methods, are considered. These examples demonstrate that CPE is free of hourglass instabilities, and it is a robust user friendly element that can be used for modeling finite deformations of orthotropic elastic materials.

DRM—MFS for two-dimensional finite elasticity

Abstract This paper presents a meshless method for the solution of problems in finite elasticity. The method is based on coupling the method of fundamental solutions (MFS) with dual reciprocity method (DRM). The solution is obtained by adding the homogeneous solution generated by MFS to the particular solution obtained by the radial basis function employed in DRM. The procedure has the advantage of eliminating domain integration. The proposed method is tested through two numerical examples that confirm its efficiency and accuracy.

The Equations of Equilibrium in Orthogonal Curvilinear Reference Coordinates

Analytical solutions to problems in finite elasticity are most often derived using the semi-inverse approach along with the spatial form of the equations of motion involving the Cauchy stress tensor. This procedure is somewhat indirect since the spatial equations involve derivatives with respect to spatial coordinates while the unknown functions are in terms of material coordinates, thus necessitating the use of the chain rule. In this classroom note, we derive compact expressions for the components of the divergence, with respect to orthogonal material coordinates, of the first Piola-Kirchhoff stress tensor. The spatial coordinate system is also assumed to be an orthogonal curvilinear one, although, not necessarily of the same type as the material coordinate system. We show by means of some example applications how analytical solutions can be derived more directly using the derived results.

Substrate stretching and reorganization of stress fibers as a finite elasticity problem

It is well known that substrate stretching reorganizes the actin cytoskeleton of an adherent cell. Experiments have proved that the stress fibers are reoriented into one or two distinct directions. It is further pointed out that reorientation of the stress fibers phenomena are observed with quite high strains, where linear elasticity theory is not valid. Therefore, the existing linear theories for mechanics interpretation of those phenomena are not satisfactory. The experimental evidence of the existence of two distinct directions of the reorientation indicates a co-existence of phases non-linear elastic phenomenon. Further that observation is quite strong evidence that the strain energy density function should be non-convex. The reorganization of the stress fibers under uniaxial substrate stretching is studied as a non-linear elastic stability problem. Adapting the global (Maxwell’s) criterion for stability, various phenomena, concerning the reorientation of the stress fibers, are interpreted into the context of finite elasticity. The experiments demonstrate co-existence of phases phenomena that are justified in the present study. The influence of various factors like contractility and extracellular stretching is demonstrated. Furthermore, the non-convexity factor is attributed to the influence of the small GTPase Rho regulating the formation of the actin stress fibers. The predominant final stress fibers placement that is transverse to the extracellular stress direction and appears in long time after the co-existence of phases placement is also justified.

A polyconvex model for materials with cubic symmetry

A framework for developing polyconvex strain energy functions for hyperelastic materials with cubic anisotropy is proposed. Polyconvexity of the strain energy density guarantees existence of solutions of boundary value problems in finite elasticity. The anisotropy is described by a single fourth order structural tensor for which the minimal polynomial basis is identified and used for the formulation of the strain energy functions. After proving the polyconvexity of some polynomial terms, a model based on a simple strain energy function is proposed. The model response is investigated analytically under simple deformations and its versatility and robustness are demonstrated via numerical simulations of problems in two and three dimensions involving large anisotropic deformations. In particular, the model is verified in the infinitesimal limit against the exact solution for an anisotropic plate with a hole subjected to remote unidirectional loading.

Diamond elements: a finite element/discrete‐mechanics approximation scheme with guaranteed optimal convergence in incompressible elasticity

AbstractWe present a finite element discretization scheme for the compressible and incompressible elasticity problems that possess the following properties: (i) the discretization scheme is defined on a triangulation of the domain; (ii) the discretization scheme is defined—and is identical—in all spatial dimensions; (iii) the displacement field converges optimally with mesh refinement; and (iv) the inf–sup condition is automatically satisfied. The discretization scheme is motivated both by considerations of topology and analysis, and it consists of the combination of a certain mesh pattern and a choice of interpolation that guarantees optimal convergence of displacements and pressures. Rigorous proofs of the satisfaction of the inf–sup condition are presented for the problem of linearized incompressible elasticity. We additionally show that the discretization schemes can be given a compelling interpretation in terms of discrete differential operators. In particular, we develop a discrete analogue of the classical tensor differential complex in terms of which the discrete and continuous boundary‐value problems are formally identical. We also present numerical tests that demonstrate the dimension‐independent scope of the scheme and its good performance in problems of finite elasticity. Copyright © 2007 John Wiley & Sons, Ltd.

Method and Apparatus for Soft Tissue Material Parameter Estimation Using Tissue Tagged Magnetic Resonance Imaging

We describe an experimental method and apparatus for the estimation of constitutive parameters of soft tissue using Magnetic Resonance Imaging (MRI), in particular for the estimation of passive myocardial material properties. MRI tissue tagged images were acquired with simultaneous pressure recordings, while the tissue was cyclically deformed using a custom built reciprocating pump actuator A continuous three-dimensional (3D) displacement field was reconstructed from the imaged tag motion. Cavity volume changes and local tissue microstructure were determined from phase contrast velocity and diffusion tensor MR images, respectively. The Finite Element Method (FEM) was used to solve the finite elasticity problem and obtain the displacement field that satisfied the applied boundary conditions and a given set of material parameters. The material parameters which best fit the FEM predicted displacements to the displacements reconstructed from the tagged images were found by nonlinear optimization. The equipment and method were validated using inflation of a deformable silicon gel phantom in the shape of a cylindrical annulus. The silicon gel was well described by a neo-Hookian material law with a single material parameter C1=8.71+/-0.06kPa, estimated independently using a rotational shear apparatus. The MRI derived parameter was allowed to vary regionally and was estimated as C1 =8.80+/-0.86kPa across the model. Preliminary results from the passive inflation of an isolated arrested pig heart are also presented, demonstrating the feasibility of the apparatus and method for isolated heart preparations. FEM based models can therefore estimate constitutive parameters accurately and reliably from MRI tagging data.

A stability study of some mixed finite elements for large deformation elasticity problems

We consider the finite elasticity problem for incompressible materials, proposing a simple bidimensional problem for which we provide an indication on the solution stability. Furthermore, we study the stability of discrete solutions, obtained by means of some well-known mixed finite elements, and we present several numerical experiments.

Complementarity, polarity and triality in non‐smooth, non–convex and non–conservative Hamilton systems

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Yang Gao David 2001Complementarity, polarity and triality in non‐smooth, non–convex and non–conservative Hamilton systemsPhil. Trans. R. Soc. A.3592347–2367http://doi.org/10.1098/rsta.2001.0855SectionRestricted accessComplementarity, polarity and triality in non‐smooth, non–convex and non–conservative Hamilton systems David Yang Gao David Yang Gao Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA () Google Scholar Find this author on PubMed Search for more papers by this author David Yang Gao David Yang Gao Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA () Google Scholar Find this author on PubMed Search for more papers by this author Published:15 December 2001https://doi.org/10.1098/rsta.2001.0855AbstractThis paper presents a unified critical–point theory in non–smooth, non–convex and dissipative Hamilton systems. The canonical dual/polar transformation methods and the associated bi–duality and triality theories proposed recently in non–convex variational problems are generalized into fully nonlinear dissipative dynamical systems governed by non–smooth constitutive laws and boundary conditions. It is shown that, by this method, non–smooth and non–convex Hamilton systems can be reformulated into certain smooth dual, complementary and polar variational problems. Based on a newly proposed polar Hamiltonian, a nice bipolarity variational principle is established for three–dimensional non–smooth elastodynamical systems, and a potentially powerful complementary variational principle can be used for solving unilateral variational inequality problems governed by non–smooth boundary conditions. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Qiu Z and Xia H (2021) Symplectic perturbation series methodology for non-conservative linear Hamiltonian system with damping, Acta Mechanica Sinica, 10.1007/s10409-021-01076-0, 37:6, (983-996), Online publication date: 1-Jun-2021. Gao D and Ali E (2019) A Novel Canonical Duality Theory for Solving 3-D Topology Optimization Problems Advances in Mathematical Methods and High Performance Computing, 10.1007/978-3-030-02487-1_13, (209-246), . Gao D (2019) Canonical Duality-Triality Theory: Unified Understanding for Modeling, Problems, and NP-Hardness in Global Optimization of Multi-Scale Systems Advances in Mathematical Methods and High Performance Computing, 10.1007/978-3-030-02487-1_1, (3-50), . Gao D (2018) On topology optimization and canonical duality method, Computer Methods in Applied Mechanics and Engineering, 10.1016/j.cma.2018.06.027, 341, (249-277), Online publication date: 1-Nov-2018. Gao D, Ruan N and Latorre V (2017) Canonical Duality-Triality Theory: Bridge Between Nonconvex Analysis/Mechanics and Global Optimization in Complex System Canonical Duality Theory, 10.1007/978-3-319-58017-3_1, (1-47), . Gao D (2015) Analytical solutions to general anti-plane shear problems in finite elasticity, Continuum Mechanics and Thermodynamics, 10.1007/s00161-015-0412-y, 28:1-2, (175-194), Online publication date: 1-Mar-2016. Gao D (2009) Canonical duality theory: Unified understanding and generalized solution for global optimization problems, Computers & Chemical Engineering, 10.1016/j.compchemeng.2009.06.009, 33:12, (1964-1972), Online publication date: 1-Dec-2009. Gao D and Sherali H (2009) Canonical Duality Theory: Connections between Nonconvex Mechanics and Global Optimization Advances in Applied Mathematics and Global Optimization, 10.1007/978-0-387-75714-8_8, (257-326), . Penot J (2009) Ekeland Duality as a Paradigm Advances in Applied Mathematics and Global Optimization, 10.1007/978-0-387-75714-8_10, (349-376), . Yuan Y (2008) Optimal solutions to a class of nonconvex minimization problems with linear inequality constraints, Applied Mathematics and Computation, 10.1016/j.amc.2008.04.016, 203:1, (142-152), Online publication date: 1-Sep-2008. Gao D and Yu H (2008) Multi-scale modelling and canonical dual finite element method in phase transitions of solids, International Journal of Solids and Structures, 10.1016/j.ijsolstr.2007.08.027, 45:13, (3660-3673), Online publication date: 1-Jun-2008. Gao D (2007) Duality-Mathematics Wiley Encyclopedia of Electrical and Electronics Engineering, 10.1002/047134608X.W2412.pub2 Gao D (2016) Complementary Principle, Algorithm, and Complete Solutions to Phase Transitions in Solids Governed by Landau-Ginzburg Equation, Mathematics and Mechanics of Solids, 10.1177/1081286504038455, 9:3, (285-305), Online publication date: 1-Jun-2004. Gao D (2003) Perfect duality theory and complete solutions to a class of global optimization problems*, Optimization, 10.1080/02331930310001611501, 52:4-5, (467-493), Online publication date: 1-Aug-2003. Gao D (2003) Nonconvex Semi-Linear Problems and Canonical Duality Solutions Advances in Mechanics and Mathematics, 10.1007/978-1-4613-0247-6_5, (261-312), . This Issue15 December 2001Volume 359Issue 1789Theme Issue ‘Non-smooth mechanics’ compiled by F. G. Pfeiffer Article InformationDOI:https://doi.org/10.1098/rsta.2001.0855Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Published online15/12/2001Published in print15/12/2001 License: Citations and impact Keywordsnon–convex variational problemsnon–smooth elastodynamicstrialitypolaritynon–conservative Hamilton systemscomplementarity

A symmetric Galerkin boundary/domain element method for finite elastic deformations

The Symmetric Galerkin Boundary Element Method (SGBEM) is reformulated for problems of finite elasticity with hyperelastic material and incompressibility, using fundamental solutions related to a (fictitious) homogeneous isotropic and compressible linear elastic material. The proposed formulation contains, besides the standard boundary integrals, domain integrals which account for the problem's nonlinearities through some (fictitious) initial strain and stress fields required to satisfy appropriate “consistency” equations. The boundary/domain integral equation problem so obtained is shown to admit a stationarity principle (a consequence of the Hu-Washizu one), which covers a number of particular cases, including the compressible case, as well as the fully linear case, either incompressible and compressible (so recovering known results of the literature). The continuum rate problem is also addressed and suitably discretized by boundary and domain elements. The related algebraic solving equation system is shown to exhibit symmetry and some sign definiteness features; it is formally solved in terms of the node values of the displacement gradient, the idrostatic pressure, the initial strain and stress fields, a long with the unknown boundary displacements and tractions. Details regarding the fundamental solutions and the stationarity principle are reported in three appendices at the end of the paper.

A new locking-free brick element technique for large deformation problems in elasticity

In the present contribution, an innovative brick element formulation for large deformation problems in finite elasticity is discussed. The new formulation can be considered as a reduced integration plus stabilization concept with the stabilization factors being computed on the basis of the enhanced strain method. Such an idea has not been applied yet in the context of large deformation 3D problems and leads to a surprisingly well-behaved locking-free element formulation. Crucial to the method is the notion of the so-called equivalent parallelepiped. The major advantages of this element technology are its simplicity and robustness. Since the element quantities are evaluated only in the center of the element, the approach is also very efficient from the numerical point of view.

Aspects of Adaptive Remeshing Techniques in Finite Elasticity

AbstractOne of the major difficulties in applying the finite element method to boundary value problems of finite elasticity is the design of a suitable mesh in order to maximize the accuracy of the solution and to minimize the numerical cost. The basic problem in this context is the error estimation of the nonlinear problem. Our aim is to present possible a posteriori error estimators for finite elasticity which operate in the linear tangential space of the nonlinear problem.