Nonlinear Solid Mechanics Problems Research Articles

NURBS-based three-dimensional analysis of geometrically nonlinear elastic structures

A comprehensive kinematical description with a corotational approach, which is based on exact polar decomposition for solid elements using isogeometric analysis is the main objective of this work. A numerical model using NURBS-based FEM is developed to deal with nonlinear problems in solid mechanics involving large rotations. From the theoretical viewpoint, the rate type form of state equation is formulated within the framework of isotropic infinitesimal elasticity. It is shown in particular that the state equation relates the Jaumann derivative of the Cauchy stress to the Eulerian strain rate. Illustrative examples are analyzed to verify the efficiency in solving a wide variety of strong nonlinear problems. • NURBS-based analysis is employed to deal with nonlinear problems in solid mechanics. • Kinematical description with a corotational-based approach is proposed. • The rate form of state equation is formulated within framework of large rotations. • Several examples of shells structures are analyzed numerically.

On the efficiency of the iterative coupling FEM–BEM for solving the elasto-plastic problems

In this paper an interface relaxation finite element-boundary element coupling method has been developed to analyze the non-linear elastoplastic problems of solid mechanics. While finite element method (FEM) is used to model the nonlinear sub-region, the remainder which is linear elastic is discretized with boundary element method (BEM). In order to perform the coupling of the two numerical methods, separate computing for each sub-region and successive renewal of the variables on the interface between the two sub-regions is performed through an iterative procedure until the final convergence is achieved. To speed up the rate at which the algorithm converges, a relaxation of the displacement data at the interface is employed. This iterative scheme preserves the nature of the FEM and the BEM and there is no need to combine their coefficient matrices, which allows us to use different software packages for solving respective sub-problems. Hence, it is a physically based method and easy to implement. Furthermore, since the analysis of elastoplastic problems are non-linear and involves iterative solution, the reduced size of the final system of equations introduced by combining the two methods is very advantageous. The coupling FEM–BEM approach was implemented by using readily computer programs for an elastoplastic analysis using the FEM and an elastic analysis using the BEM and was tested through several examples. The obtained results were compared to the experimental data, the conventional methods FEM and BEM and, if available, analytical solutions to establish their accuracy.

Two-scale models of polycrystals: Macroscale motion decomposition

The paper considers the application of multiscale mathematical models to description of severe inelastic deformation of single- and polycrystals for which geometrically and physically nonlinear constitutive relations are required. For definiteness, the discussion is given on the example of two-scale models. One of the unsolved problems in nonlinear solid mechanics (and in construction of multiscale models in particular) is how to single out the component of material motion responsible for geometric nonlinearity (for example, when rigid rotations are imposed on strain-induced motion) or, in other words, how to decompose the macroscale motion into quasi-rigid motion and strain-induced motion. In multiscale constitutive models, this problem is added complexity by the necessity to match the constitutive relations with “affined” parameters of different scales due to different definitions of quasi-rigid motion on different scales. We consider three possible methods of representing the quasi-rigid motion on the macroscale. For each method, consistency conditions are determined; from these conditions also follows a relation between macroscale quasi-rigid rotation and mesoscale rotation of elements. It is shown that determination of rotations on the scales “from the top down” makes it impossible to select an arbitrary rotation model on the lower scales, and this considerably limits the applicability of physically substantiated rotation models on the lower scales. It is shown that the proposed method of defining the macroscale quasi-rigid motion as an averaged mesoscale spin makes it possible to obtain consistent constitutive relations of neighbor scales with no limitations on the choice of a lower scale rotation model.

Solution bound and nearly exact solution to nonlinear solid mechanics problems based on the smoothed FEM concept

Node-based and edge-based smoothed FEM (NS-FEM and ES-FEM), and α-FEM are extended to solve nonlinear problems. The nonlinear strain field is smoothed using the gradient smoothing. The continuous scalar scaling factor α enables the α-FEM continuously transforming from overestimated model to underestimated model. Numerical examples reveal that ES-FEM is a robust and stable method with high accuracy and computational efficiency for nonlinear problems. The exact solution in strain energy of force driven problems can be bounded by NS-FEM and FEM. The α-FEM can also be “tuned” to find nearly exact solution to nonlinear mechanics problems of solids of complicated geometry.

A Comparison of EFGM and FEM for Nonlinear Solid Mechanics Problems

In this paper, element free Galerkin method (EFGM) has been applied to solve nonlinear solid mechanics problems using updated Lagrangian approach. The nonlinear equations have been solved using Newton Raphson method. An associative J2 flow rule and isotropic hardening has been used for the modelling of elasto-plastic material behaviour. Elastic predictor and plastic corrector algorithm has been used for the integration of incremental stress-strain relation. Few test problems involving large deformations have been simulated and the results obtained by EFGM have been compared with those obtained by FEM.

On the spatial formulation of discontinuous Galerkin methods for finite elastoplasticity

In this paper, we present a consistent spatial formulation for discontinuous Galerkin (DG) methods applied to solid mechanics problems with finite deformation. This spatial formulation provides a general, accurate, and efficient DG finite element computational framework for modeling nonlinear solid mechanics problems. To obtain a consistent formulation, we employ the Incomplete Interior Penalty Galerkin (IIPG) method. Another requirement for achieving a fast convergence rate for Newton’s iterations is the consistent formulation of material integrators. We show that material integrators that are well developed and tested in continuous Galerkin (CG) methods can be fully exploited for DG methods by additionally performing stress returning on element interfaces. Finally, for problems with pressure or follower loading, stiffness contributed from loaded surfaces must also be consistently incorporated. In this work, we propose the Truesdell objective stress rate for both hypoelastoplastic and hyperelastoplastic problems. Two formulations based on the co-rotational and multiplicative decomposition-based frameworks are implemented for hypoelastoplasticity and hyperelastoplasticity, respectively. Two new terminologies, the so-called standard surface geometry stiffness and the penalty surface geometry stiffness, are introduced and derived through consistently linearizing the virtual work contributed from interior surface integrals. The performance of our DG formulation has been demonstrated through solving a cantilever beam problem undergoing large rotations, as well as a bipolar void coalescence problem where the voids grow up to several hundred times of their original volumes. Fast convergence rates for Newton’s iterations have been achieved in our IIPG implementation.

Solving nonlinear solid mechanics problems with the jacobian-free newton krylov method

The solution of the equations governing solid mechanics is often obtained via Newton's method. This approach can be problematic if the determination, storage, or solution cost associated with the Jacobian is high. These challenges are magnified for multiphysics applications with many coupled variables. Jacobian-free Newton-Krylov (JFNK) methods avoid many of the difficulties associated with the Jacobian by using a finite difference approximation. BISON is a parallel, object-oriented, nonlinear solid mechanics and multiphysics application that leverages JFNK methods. We overview JFNK, outline the capabilities of BISON, and demonstrate the effectiveness of JFNK for solid mechanics and solid mechanics coupled to other PDEs using a series of demonstration problems.

An analysis of the stress formula for energy-momentum methods in nonlinear elastodynamics

The energy-momentum method, a space---time discretization strategy for elastic problems in nonlinear solid, structural, and multibody mechanics relies critically on a discrete derivative operation that defines an approximation of the internal forces that guarantees the discrete conservation of energy and momenta. In the case of nonlinear elastodynamics, the formulation for general hyperelastic materials is due to Simo and Gonzalez, dating back to the mid-nineties. In this work we show that there are actually infinite second order energy-momentum methods for elastodynamics, all of them deriving from a modified midpoint integrator by an appropriate redefinition of the stress tensor at equilibrium. Such stress tensors can be interpreted as the solutions to local convex projections, whose precise definitions lead to different methods. The mathematical requirements of such projections are identified. Based on this geometrical interpretation several conserving methods are examined.

The hp‐d‐adaptive finite cell method for geometrically nonlinear problems of solid mechanics

SUMMARYThe finite cell method (FCM) combines the fictitious domain approach with the p‐version of the finite element method and adaptive integration. For problems of linear elasticity, it offers high convergence rates and simple mesh generation, irrespective of the geometric complexity involved. This article presents the integration of the FCM into the framework of nonlinear finite element technology. However, the penalty parameter of the fictitious domain is restricted to a few orders of magnitude in order to maintain local uniqueness of the deformation map. As a consequence of the weak penalization, nonlinear strain measures provoke excessive stress oscillations in the cells cut by geometric boundaries, leading to a low algebraic rate of convergence. Therefore, the FCM approach is complemented by a local overlay of linear hierarchical basis functions in the sense of the hp‐d method, which synergetically uses the h‐adaptivity of the integration scheme. Numerical experiments show that the hp‐d overlay effectively reduces oscillations and permits stronger penalization of the fictitious domain by stabilizing the deformation map. The hp‐d‐adaptive FCM is thus able to restore high convergence rates for the geometrically nonlinear case, while preserving the easy meshing property of the original FCM. Accuracy and performance of the present scheme are demonstrated by several benchmark problems in one, two, and three dimensions and the nonlinear simulation of a complex foam sample. Copyright © 2011 John Wiley & Sons, Ltd.

The finite cell method for geometrically nonlinear problems of solid mechanics

The Finite Cell Method (FCM), which combines the fictitious domain concept with high-order p-FEM, permits the effective solution of problems with very complex geometry, since it circumvents the computationally expensive mesh generation and guarantees exponential convergence rates for smooth problems. The present contribution deals with the coupling of the FCM approach, which has been applied so far only to linear elasticity, with established nonlinear finite element technology. First, it is shown that the standard p-FEM based FCM converges poorly in a nonlinear formulation, since the presence of discontinuities leads to oscillatory solution fields. It is then demonstrated that the essential ideas of FCM, i.e. exponential convergence at virtually no meshing cost, can be achieved in the geometrically nonlinear setting, if high-order Legendre shape functions are replaced by a hierarchically enriched B-spline patch.

Strict error bounds for linear and nonlinear solid mechanics problems using a patch-based flux-free method

We discuss, in this paper, a common flux-free method for the computation of strict error bounds for linear and nonlinear finite-element computations. In the linear case, the error bounds are on the energy norm of the error, while, in the nonlinear case, the concept of error in constitutive relation is used. In both cases, the error bounds are strict in the sense that they refer to the exact solution of the continuous equations, rather than to some FE computation over a refined mesh. For both linear and nonlinear solid mechanics, this method is based on the computation of a statically admissible stress field, which is performed as a series of local problems on patches of elements. There is no requirement to solve a previous problem of flux equilibration globally, as happens with other methods.

Mixed stabilized finite element methods in nonlinear solid mechanics: Part I: Formulation

This paper exploits the concept of stabilized finite element methods to formulate stable mixed stress/displacement and strain/displacement finite elements for the solution of nonlinear solid mechanics problems. The different assumptions and approximations used to derive the methods are exposed. The proposed procedure is very general, applicable to 2D and 3D problems. Implementation and computational aspects are also discussed, showing that a robust application of the proposed formulation is feasible. Numerical examples show that the results obtained compare favorably with those obtained with the corresponding irreducible formulation.

Alternative stress-integration schemes for large-deformation problems of solid mechanics

In solving nonlinear problems of solid mechanics by the finite-element method, stresses at integration points are usually obtained by integrating nonlinear constitutive equations, given known incremental strains. In a large-deformation analysis, stress–strain relationships must be frame independent such that any rigid-body motion does not induce strain within the material. This principle is generally satisfied by introducing an objective stress rate, such as the Jaumann or Truesdell stress rates, into the constitutive equations. This paper investigates three alternative algorithms for integrating stress–strain relationships in a large-deformation analysis. It is shown that the effect of rigid-body motion is equivalent to a stress transformation and this transformation can be introduced before, after or during integration of the stress–strain constitutive equations. Although there is no theoretical advantage, in terms of accuracy, for selecting one of these strategies over the others, in terms of efficiency of algorithms one is more advantageous than the others. Performance of the proposed algorithms is studied and compared by means of numerical examples. The results of this study can be used in the development of fast and robust algorithms for stress integration of constitutive equations in nonlinear finite-element analysis.

A sub-stepping scheme for multi-scale analysis of solids

This paper proposes a sub-stepping procedure aimed at saving overall computing time in the finite element solution of non-linear solid mechanics problems where homogenization-based multi-scale constitutive models are used to describe the material response. The procedure obtains better initial guesses for the Newton–Raphson scheme adopted in the solution of the incremental equilibrium problem of the representative volume element (RVE) associated with each Gauss quadrature point of the discretised macroscopic continuum. The resulting gain in robustness of the RVE Newton–Raphson iteration allows larger time/load steps to be prescribed at the macroscopic scale, leading to substantial savings in computing time. The proposed sub-stepping preserves the quadratic rate of asymptotic convergence of the macroscopic scale Newton–Raphson scheme, without compromising the accuracy of the solution. Numerical examples demonstrate that the achieved improvement in efficiency is particularly noticeable under the assumption of finite stains, where overall speed-up factors of up to four are observed.

Stochastic upscaling in solid mechanics: An excercise in machine learning

This paper presents a consistent theoretical and computational framework for upscaling in random microstructures. We adopt an information theoretic approach in order to quantify the informational content of the microstructural details and find ways to condense it while assessing quantitatively the approximation introduced. In particular, we substitute the high-dimensional microscale description by a lower-dimensional representation corresponding for example to an equivalent homogeneous medium. The probabilistic characteristics of the latter are determined by minimizing the distortion between actual macroscale predictions and the predictions made using the coarse model. A machine learning framework is essentially adopted in which a vector quantizer is trained using data generated computationally or collected experimentally. Several parallels and differences with similar problems in source coding theory are pointed out and an efficient computational tool is employed. Various applications in linear and non-linear problems in solid mechanics are examined.

On a composite implicit time integration procedure for nonlinear dynamics

Transient analysis of nonlinear problems in structural and solid mechanics is mainly carried out using direct time integration of the equations of motion. For reliable solutions, a stable and efficient integration algorithm is desirable. Methods that are unconditionally stable in linear analyses appear to be a natural choice for use in nonlinear analyses, but unfortunately may not remain stable for a given time step size in large deformation and long time range response solutions. A composite time integration scheme is proposed and tested in some example solutions against the trapezoidal rule and the Wilson θ-method, and found to be effective where the trapezoidal rule fails to produce a stable solution. These example results are indicative of the merits of the composite scheme.

On the orthogonal subgrid scale pressure stabilization of finite deformation J2 plasticity

The use of stabilization methods is becoming an increasingly well-accepted technique due to their success in dealing with numerous numerical pathologies that arise in a variety of applications in computational mechanics. In this paper a multiscale finite element method technique to deal with pressure stabilization of nearly incompressibility problems in nonlinear solid mechanics at finite deformations is presented. A J2-flow theory plasticity model at finite deformations is considered. A mixed formulation involving pressure and displacement fields is used as starting point. Within the finite element discretization setting, continuous linear interpolation for both fields is considered. To overcome the Babuška–Brezzi stability condition, a multiscale stabilization method based on the orthogonal subgrid scale (OSGS) technique is introduced. A suitable nonlinear expression of the stabilization parameter is proposed. The main advantage of the method is the possibility of using linear triangular or tetrahedral finite elements, which are easy to generate and, therefore, very convenient for practical industrial applications. Numerical results obtained using the OSGS stabilization technique are compared with results provided by the P1 standard Galerkin displacements linear triangular/tetrahedral element, P1/P1 standard mixed linear displacements/linear pressure triangular/tetrahedral element and Q1/P0 mixed bilinear/trilinear displacements/constant pressure quadrilateral/hexahedral element for 2D/3D nearly incompressible problems in the context of a nonlinear finite deformation J2 plasticity model.

Iterative triangularization of updated finite element stiffness matrices

In many problems in nonlinear solid mechanics, the finite element method is executed incrementally, with the positive definite stiffness matrix updated after one or more load (or time) increments. In solving the resulting large linear perturbed systems, it is often attractive to use Cholesky triangularization, followed by forward and backward substitution. The present investigation introduces and demonstrates an iterative procedure for updating the triangular factors of the updated stiffness matrix. An approximate convergence criterion is formulated. Simple examples are presented indicating rapid convergence. In the scalar case this method exactly tracks the Taylor series.

A numerical analysis for some non-linear constitutive problems in solid mechanics

The paper deals with the application FEM for solving nonlinear constitutive problems in solid mechanics. The basis equations and algorithms of iterative processing are presented. Some programs written by languages Gibian and special operators in Castem 2000 are established. The problem for the spherical shell made of elasto-plastic material subjected to monotone increasing pressures is solved and calculated results are compared with the theoretical solution and give a good agreement. The influence of the pressure values on the plastic regions of sphere is investigated. The stress, displacement and plastic deformation states for spherical shell and plate with hollow acted on by complex cyclic loads are considered. These given programs can be applied in other problem with more complex geometry, load and material conditions.

Recent trends in ALE formulation and its applications in solid mechanics

This paper presents a complete derivation and implementation of the arbitrary Lagrangian Eulerian (ALE) formulation for the simulation of nonlinear static and dynamic problems in solid mechanics. The ALE equations are presented using a fully coupled implicit approach. Full expression for the ALE virtual work equations and finite element matrices are given. The matrix equations are presented in a level of details that facilitates straightforward implementation. Time integration relations for the dynamic equations are also derived. Special techniques are presented to adapt the formulation for simulating crack propagation problems. Several applications in the areas of metal forming, metal cutting and crack propagation are solved and presented.