Applications in 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

Granular materials: constitutive equations and strain localization

A new technique for the computer solution of problems in solid mechanics is introduced. The total potential energy of the system is expressed in terms of displacements that are expanded into trigonometric series, each of whose terms satisfies the boundary and symmetry conditions. Powers of these series from one to four occur in the potential energy expression, and it is necessary to multiply series by series before minimizing the energy with respect to the undetermined coefficients to find the equilibrium equations. Techniques are developed to perform this multiplication on the digital computer and to store the results on tape for use by a subroutine in which the energy is minimized. This technique, in which the exponents and subscripts of the undetermined coefficients and the arguments of the trigonometric terms are coded in integer or bit form, is not limited to applications involving minimization of the potential energy, but can also be used to aid in the solution of nonlinear differential equations with variable coefficients, where the dependent variables are expanded in trigonometric series. The method has been applied successfully to problems involving one and two independent variables.

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

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

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

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

Discontinuous Petrov–Galerkin method with optimal test functions for thin-body problems in solid mechanics

We consider a history-dependent variational inequality in a real Hilbert space, for which we recall an existence and uniqueness result. We associate this inequality with a gap function, together with two additional problems: a nonlinear equation and a minimization problem. Then, we prove that solving these problems is equivalent to solving the original history-dependent variational inequality. Next, we state and prove a convergence criterion, i.e., we provide necessary and sufficient conditions which guarantee the convergence of a sequence of functions to the solution of the considered inequality. Based on the equivalence above, we deduce various consequences that present some interest on their own, and, moreover, we obtain convergence results for the two additional problems considered. Finally, we apply our abstract results to the study of an inequality problem in solid mechanics. It concerns the study of a viscoelastic constitutive law with long memory and unilateral constraints, for which we deduce a convergence result and provide the corresponding mechanical interpretations.

The temperature and thermal flux fields are one of the important problems in solid mechanics, especially when the material properties change with temperature, such as in the process of heat conduction. Using the generalized complex variable method, the problem with circular boundary obtained by conformal transformation have been discussed. However, it is not suitable for the model whose cross-sectional shape is invariable. In this paper, using semi-inverse method we solve the three-dimensional temperature-dependent heat conduction problem of strip flake crack whose cross-sectional shape is invariable. The temperature-thermal flux fields have been obtained. The results show that the thermal flux field of three-dimensional temperature dependent problem is the same as the classical result, while the temperature field is different from the classical result.

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.

Jacobian weighted element differential method for solid mechanics