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

Chapter 2 - Galerkin Method of Approximation: Irreducible and Mixed Forms

A new approach to parallelization of materially non‐linear problems in solid mechanics is developed. It is based on approximating generalized models of subdomains. The procedure does not retain the same substructuring technique used in a linear version. The convergence proof of the single‐ and multilevel‐domain decomposition algorithms uses the principle of minimum potential energy dissipation and investigated properties of the substructural models. The high efficiency of the approach introduced is shown through the study of several examples. The method developed in this paper for steady creep can be used without modification to solve non‐linear elasticity problems and, at active loading, plasticity problems for bodies of the power‐law strain–stress diagrams. Copyright © 2001 John Wiley & Sons, Ltd.

This paper presents the solution of geometrically nonlinear problems in solid mechanics by the Domain-Boundary Element Method. Because of the Total-Lagrange approach, the arising domain and boundary integrals are evaluated in the undeformed configuration. Therefore, the system matrices remain unchanged during the solution procedure, and their time-consuming computation needs to be performed only once. While the integral equations for axisymmetric finite deformation problems will be derived in detail, the basic ideas of the formulation in two and three dimensions can be found in [1]. The present formulation includes torsional problems with finite deformations, where additional terms arise due to the curvilinear coordinate system. A Newton–Raphson scheme is used to solve the nonlinear set of equations. This involves the solution of a large system of linear equations, which has been a very time-consuming task in former implementations, [1, 2]. In this work, an iterative solver, i.e. the generalized minimum residual method, is used within the Newton–Raphson algorithm, which leads to a significant reduction of the computation time. Finally, numerical examples will be given for axisymmetric and two/three-dimensional problems.

If a naturally straight thin rod, such as a plastic or metal ruler, is subjected to a small compressive thrust applied to its ends, it remains straight. If the thrust is slowly increased beyond a certain critical value, called the buckling load, the rod assumes a configuration, called a buckled state, that is not straight. See Fig. 1.1. This process is called buckling. Depending on the precise mode of loading and the nature of the rod, the transition to a buckled state can be very rapid. If the thrust is further increased, the deflection of the rod from its straight state is likewise increased. If this entire process is repeated, the rod may well buckle into another configuration such as the reflection of the first through a plane of symmetry. The performance of a whole series of such experiments on different rods would lead to the observation that the buckling loads and the nature of buckled states depend upon the material and shape of the rod and upon the manner in which it is supported at its ends. It can also be observed that the results of experiments are highly sensitive to slight deviations of the rod from perfect straightness or of the thrust from perfect symmetry. The study of buckling for different bodies is one of the richest sources of important problems in nonlinear solid mechanics.

On the Gauss–Legendre quadrature rule of deep energy method for one-dimensional problems in solid mechanics

Examination of flux projection-type error estimators in nonlinear finite element analysis

This paper presents a comparative study of a series of flux projection type error estimators for elasto-plastic and viscoplastic materials undergoing large strains and large rotations. The error estimators are: L2 norms of stress error, total strain error, equivalent strain error, incremental total strain error per load step, and energy rate norm error, respectively. Numerical examples are presented in both two and three dimensions. The information provided by these error estimators may be used for adaptive mesh refinement and for subsequent data transfer (rezoning) in a large deformation analysis. The error estimators based on the energy rate and the incremental total strain were found to be consistently accurate in the prediction of the discretization error for the two dimensional test cases considered. The L2 norm of incremental strain error estimator consistently gave the most conservative estimate of error in two dimensions. The overall behavior of these error estimators in three dimensions was acceptable, although, not as intuitive as it was in two dimensions.

In the field of solid mechanics it is often necessary to use constitutive (physical) relations in the rate form, for example, for formulation of the boundary value problem in the rate form with contact conditions when a contact area is a priori unknown and changes in a deformation process.The paper considers some questions of formulating geometrically nonlinear constitutive relations of elastic material in the rate form and the relationship of these equations and the constitutive relations in the finite form.In many existing elastic and elasto-plastic models Hooke's law (written in terms of the actual configuration) is used as the constitutive relation.As a rule, the rate measure of the strain is deformation rate tensor (the symmetrical part of velocity gradient) and the stress rate measure is some objective derivative (convective or corotational) of weighted Kirchhoff stress tensor.The use of the convective derivative leads to certain difficulties, for example, analysis of evolution stress state in a deformable basis is complicated; consequently, convective derivatives are excluded from consideration.Using a stress tensor corotational derivative instead of the material derivative (derivative by time) allows satisfying the principle of material indifference (the principle of independence of constitutive relation from the choice of reference frame), however a choice of the derivative type can be implemented by many ways.An arbitrary selection of stress corotational derivative leads to undesirable effects: stress oscillations for the simple shear monotonic loading (for example, for Jaumann derivative), "not closed" stress trajectories and nonzero stress work for a closed deformation trajectory.In papers of A. Meyers, H. Xiao, O. Bruhns corotational derivative (with logarithmic spin or logspin) was proposed.The derivative of the right Hencky's strain tensor is exactly equal to deformation rate tensor.When using this derivative in the constitutive relation the described effects are missing, on the basis of this remarked logspin authors state the logspin exclusivity and recommend it only for using in rate form constitutive relations.The article shows full compliance between Hooke's law in the rate and finite form under conditions: existence of the material basis in which properties of the body (in this case -elastic) remain unchanged and the use of the same type corotational derivative for stress and strain measures.Computational experiments for illustration of the proved assertion about the equivalence of different Hooke's law forms were conducted: various

Integration of the modified Cam-Clay model in non-linear finite element analysis

Approximate Solutions to Nonlinear Problems of Solid Mechanics by Quasilinearization Method

Problems in solid mechanics constitute perhaps the largest field of application of finite element methods. The vast majority of solid mechanics problems involve the standard momentum balance equation, posed in a Lagrangian setting, with different models distinguished by the choice of nonlinear or linearized kinematics, and the constitutive model for determining the stress. For some common models, the constitutive relationships are rather complex. This chapter addresses a number of canonical solid mechanics models in the context of automated modeling, and focuses on some pertinent issues that arise due to the nature of the constitutive models. The solution of equations with second-order time derivatives, which characterizes dynamic problems, is also considered.

Effect of rigid body motion in phase-field models of solid-state sintering

In this work, a new homogenization method of elasto-viscoplastic composites is developed. Using the fully implicit backward Euler’s integration algorithm, the nonlinear ordinary differential equations in the constitutive laws of the matrix and inclusion phases are discretized. Three classical incremental linearization approaches, i.e., direct, secant and tangent ones are adopted and an affine relationship between the stress and strain increments is deduced. In order to reduce the interaction between the inclusion and matrix phases, a second-ordered mapping tensor is introduced and a new mapping-tangent linearization approach is proposed. Different linearization approaches are implemented by the incremental self-consistent scheme to predict the overall stress-strain response of particle-reinforced composites. It is shown that the predicted stress-strain curves given by the proposed mapping-tangent linearization approach are softer than that by other three classical ones, and are much closer to that from a fullfield finite element simulation. Moreover, the linearized elasto-viscoplastic constitutive equation based on the proposed mapping- tangent approach has the same mathematical structure as the rate-independent elasto-plastic constitutive law. In this sense, the homogenization problems faced in the elasto-plastic and elasto-viscoplastic heterogeneous materials can be unified.

This work proposes a physics-informed feature engineering approach to improve the predictive accuracy of surrogate models for problems in solid mechanics. The prediction of the maximum deflection of a thin square plate under a uniform load is considered as a case study. A dataset of 5000 samples was generated using the finite element method in Ansys Mechanical. This was followed by a physical validation based on the theory of small deflections, and invalid samples were discarded. A random forest regressor algorithm was used for the surrogate model, and its hyperparameters were optimized using RandomizedSearchCV. Two input feature architectures were compared: a baseline architecture (using fundamental physical parameters) and a physics-informed architecture (using complex engineering features, specifically relative flexibility K1 and cylindrical rigidity D). The results showed a significant increase in accuracy when using physics-informed features compared to the baseline approach. An analysis of feature importance confirmed the dominant role of K1 and the load p, which is fully consistent with theoretical mechanics. The obtained results demonstrate that feature engineering based on physical principles improves both the accuracy and interpretability of machine learning surrogate models.

In the first part of this chapter, the basic algorithmic structure and performance characteristics of thep‐version of the finite element method (FEM) are surveyed with reference to elliptic problems in solid mechanics. For this class of problems, the theoretical basis of thep‐version is fully established, and a very substantial amount of engineering experience covering linear and nonlinear applications is available. It is shown thatp‐extensions on properly designed meshes make realization of exponential rates of convergence in practical computations possible and provide for the estimation and control of relative errors in terms of any quantity of interest. In the second part, thep‐version of the FEM is extended to a high‐order fictitious domain approach, the finite cell method (FCM). While the FCM inherits the advantages of thep‐version with respect to accuracy and robustness, it relieves analysts from the necessity of generating finite element meshes. Thus, it strongly supports the analysis of problems with highly complicated geometry, for which meshing with finite elements would be very difficult. Given the growing demand for verified numerical solutions, thep‐version of the finite element and FCMs are expected to play an increasingly important role.