A parallel computational method in steady power‐law creep

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.

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

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

Approximate Solutions to Nonlinear Problems of Solid Mechanics by Quasilinearization Method

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

Chapter 8 - Basic Equations and Solution Procedure

Exact Dirichlet boundary Physics-informed Neural Network EPINN for solid mechanics

The paper discusses some of the relevant computational advances which permit the simulation of large-scale problems involving nonlinear solids within realistic time frames and computational resources. The need for rigorous consideration of both theoretical and algorithmic issues is emphasized, particularly in relation to the computational treatment of finite-strain elasto-plastic (viscoplastic) deformation, the modelling of frictional contact conditions and element technology capable of dealing with material incompressibility. Practically important aspects such as adaptive mesh refinement procedures are discussed and attention is given to choice of appropriate error estimators for elasto-plastic materials and the transfer of solution parameters between successive meshes. The role of explicit solution techniques in the simulation of large-scale nonlinear problems is also discussed. The concept of discrete elements is briefly described and their applications to a wide range of solid mechanics problems illustrated. Some advances in the field of iterative equation solution methods are reviewed and their potential advantages in the simulation of large-scale nonlinear solid mechanics problems are demonstrated.

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.

On the spatial formulation of discontinuous Galerkin methods for finite elastoplasticity

See the attached PDF file In this paper, the Coefficient Inverse Problems (I.P.), the Geometrical I.P. and Microstructure (I.P.) in elasticity, plasticity, hardening, fatigue, fracture and microstructure are formulated and solved. The retrospective I.P. and the boundary I.P. are commented. It was prepared equations for evaluation of the velocities for propagation of longitudinal and transversal ultrasonic wave. The relationships between following mechanical properties: modulus of elasticity, Henki’s coefficient, parameters in hardening law, parameters in Wohler ‘s curve, fatigue limit, coefficient in fracture mechanics and acoustics characteristics of materials are obtained. The geometrical I.P. for reflected area and crack depth are solved by means reflected and diffraction methods respectively.The microstructure I.P. is solved by means mathematical - ultrasonic dualism. KEY WORDS: Inverse Problems, Solid Mechanics, Ultrasonic.

A centroid-enriched strain-smoothed radial point interpolation method for nearly incompressible elastoplastic problems in solid mechanics

The technique of digital simulation of Gaussian random fields (which are nonhomogeneous in space or are part of homogeneous random fields in space) is presented to a solution of stochastic boundary-value problems. The method consists of expanding the simulated field, with known mean and autocorrelation function, in series in terms of the structural ’’natural’’ mode shapes, and the Fourier coefficients of the truncated series are then simulated as random normal vectors. The method is applicable to static or dynamic stochastic two-point boundary-value problems in mechanics of solids.

Adaptive methodology for meshless finite point method