Quasilinearization technique for solving nonlinear problems of solid mechanics

2021 ◽  
Author(s):  
L. V. Stepanova ◽  
R. M. Zhabbarov
Keyword(s):  
Solid Mechanics ◽  
Nonlinear Problems ◽  
Quasilinearization Technique

Mechanics of Growing Solids: New Track in Mechanical Engineering

2014 ◽  
Author(s):  
Alexander V. Manzhirov
Keyword(s):  
Mathematical Theory ◽  
Solid Mechanics ◽  
Volcanic Rocks ◽  
Nonlinear Problems ◽  
Biological Tissues ◽  
Growth Processes ◽  
Continuous Growth ◽  
Engineering Sciences ◽  
Space Objects ◽  
Classical Boundary

A vast majority of objects around us arise from some growth processes. Many natural phenomena such as growth of biological tissues, glaciers, blocks of sedimentary and volcanic rocks, and space objects may serve as examples. Similar processes determine specific features of many industrial processes which include crystal growth, laser deposition, melt solidification, electrolytic formation, pyrolytic deposition, polymerization and concreting technologies. Recent researches indicates that growing solids exhibit properties dramatically different from those of conventional solids, and the classical solid mechanics cannot be used to model their behavior. The old approaches should be replaced by new ideas and methods of modern mechanics, mathematics, physics, and engineering sciences. Thus, there is a new track in solid mechanic that deals with the construction of adequate models for solid growth processes. The fundamentals of the mathematical theory of growing solids are under consideration. We focus on the surface growth when deposition of a new material occurs at the boundary of a growing solid. Two approaches are discussed. The first one deals with the direct formulation of the mathematical theory of continuous growth in the case of small deformations. The second one is designed for the solution of nonlinear problems in the case of finite deformations. It is based on the ideas of the theory of inhomogeneous solids and regards continuous growth as the limit case of discrete growth. The constitutive equations and boundary conditions for growing solids are presented. Non-classical boundary value problems are formulated. Methods for solving these problems are proposed.

Coupling BEM and the Local Point Interpolation for the Solution of Anisotropic Elastic Nonlinear, Multi-Physics and Multi-Fields Problems

2019 ◽  
Vol 17 (09) ◽  
pp. 1950067
Author(s):  
Richard Kouitat Njiwa ◽  
Gael Pierson ◽  
Arnaud Voignier
Keyword(s):  
Solid Mechanics ◽  
Interpolation Method ◽  
Nonlinear Problems ◽  
Numerical Approach ◽  
Local Point ◽  
Problem Dimension ◽  
Point Interpolation Method ◽  
Elasticity Problems ◽  
Point Interpolation ◽  
Anisotropic Elastic

The pure boundary element method (BEM) is effective for the solution of a large class of problems. The main appeal of this BEM (reduction of the problem dimension by one) is tarnished to some extent when a fundamental solution to the governing equations does not exist as in the case of nonlinear problems. The easy to implement local point interpolation method applied to the strong form of differential equations is an attractive numerical approach. Its accuracy deteriorates in the presence of Neumann-type boundary conditions which are practically inevitable in solid mechanics. The main appeal of the BEM can be maintained by a judicious coupling of the pure BEM with the local point interpolation method. The resulting approach, named the LPI-BEM, seems versatile and effective. This is demonstrated by considering some linear and nonlinear elasticity problems including multi-physics and multi-field problems.

The finite cell method for geometrically nonlinear problems of solid mechanics

2010 ◽  
Vol 10 ◽  
pp. 012170 ◽  
Author(s):  
Dominik Schillinger ◽  
Stefan Kollmannsberger ◽  
Ralf-Peter Mundani ◽  
Ernst Rank
Keyword(s):  
Solid Mechanics ◽  
Nonlinear Problems ◽  
Geometrically Nonlinear ◽  
Finite Cell Method ◽  
Cell Method ◽  
Finite Cell

Nonlinear Dynamical Systems

1985 ◽  
Vol 38 (10) ◽  
pp. 1284-1286 ◽  
Author(s):  
F. C. Moon
Keyword(s):  
Dynamical Systems ◽  
Solid Mechanics ◽  
Nonlinear Dynamical Systems ◽  
Fractal Dimensions ◽  
Nonlinear Problems ◽  
Nonlinear Dynamical ◽  
Time Dynamic ◽  
Long Time ◽  
New Ideas ◽  
Initial Starting

New discoveries have been made recently about the nature of complex motions in nonlinear dynamics. These new concepts are changing many of the ideas about dynamical systems in physics and in particular fluid and solid mechanics. One new phenomenon is the apparently random or chaotic output of deterministic systems with no random inputs. Another is the sensitivity of the long time dynamic history of many systems to initial starting conditions even when the motion is not chaotic. New mathematical ideas to describe this phenomenon are entering the field of nonlinear vibrations and include ideas from topology and analysis such as Poincare´ maps, fractal dimensions, Cantor sets and strange attractors. These new ideas are already making their way into the engineering vibrations laboratory. Further research in this field is needed to extend these new ideas to multi-degree of freedom and continuum vibration problems. Also the loss of predictability in certain nonlinear problems should be studied for its impact on the field of numerical simulation in mechanics of nonlinear materials and structures.

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