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.

Mathematical Theory of Transversally Isotropic Shells of Arbitrary Thickness at Static Load

2019 ◽  
Vol 968 ◽  
pp. 496-510
Author(s):  
Anatoly Grigorievich Zelensky
Keyword(s):  
Mathematical Theory ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Theory Of Elasticity ◽  
Elastic Plates ◽  
Dimensional Problem ◽  
Boundary Problems ◽  
Complex Boundary ◽  
Plates And Shells ◽  
Basic Equations

Classical and non-classical refined theories of plates and shells, based on various hypotheses [1-7], for a wide class of boundary problems, can not describe with sufficient accuracy the SSS of plates and shells. These are boundary problems in which the plates and shells undergo local and burst loads, have openings, sharp changes in mechanical and geometric parameters (MGP). The problem also applies to such elements of constructions that have a considerable thickness or large gradient of SSS variations. The above theories in such cases yield results that can differ significantly from those obtained in a three-dimensional formulation. According to the logic in such theories, the accuracy of solving boundary problems is limited by accepted hypotheses and it is impossible to improve the accuracy in principle. SSS components are usually depicted in the form of a small number of members. The systems of differential equations (DE) obtained here have basically a low order. On the other hand, the solution of boundary value problems for non-thin elastic plates and shells in a three-dimensional formulation [8] is associated with great mathematical difficulties. Only in limited cases, the three-dimensional problem of the theory of elasticity for plates and shells provides an opportunity to find an analytical solution. The complexity of the solution in the exact three-dimensional formulation is greatly enhanced if complex boundary conditions or physically nonlinear problems are considered. Theories in which hypotheses are not used, and SSS components are depicted in the form of infinite series in transverse coordinates, will be called mathematical. The approximation of the SSS component can be adopted in the form of various lines [9-16], and the construction of a three-dimensional problem to two-dimensional can be accomplished by various methods: projective [9, 14, 16], variational [12, 13, 15, 17]. The effectiveness and accuracy of one or another variant of mathematical theory (MT) depends on the complex methodology for obtaining the basic equations.

scholarly journals Numerical Solution of Fractional Diffusion Equation Model for Freezing in Finite Media

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
R. S. Damor ◽  
Sushil Kumar ◽  
A. K. Shukla
Keyword(s):  
Diffusion Equation ◽  
Interface Velocity ◽  
Biological Tissues ◽  
Fractional Diffusion ◽  
Equation Model ◽  
Fractional Diffusion Equation ◽  
Heat Transfer Analysis ◽  
Engineering Sciences ◽  
Space Fractional Diffusion Equation ◽  
Fractional Differential

Phase change problems play very important role in engineering sciences including casting of nuclear waste materials, vivo freezing of biological tissues, solar collectors and so forth. In present paper, we propose fractional diffusion equation model for alloy solidification. A transient heat transfer analysis is carried out to study the anomalous diffusion. Finite difference method is used to solve the fractional differential equation model. The temperature profiles, the motion of interface, and interface velocity have been evaluated for space fractional diffusion equation.

scholarly journals Fundamentals of Continuous Growth Processes in Technology and Nature

2017 ◽  
Vol 23 ◽  
pp. 1-12 ◽  
Author(s):  
Alexander V. Manzhirov ◽  
Narinder K. Gupta
Keyword(s):  
Growth Processes ◽  
Continuous Growth

Mechanics of Angioplasty: Wall, Balloon and Stent

2000 ◽  
Author(s):  
Gerhard A. Holzapfel ◽  
Christian A. J. Schulze-Bauer ◽  
Michael Stadler
Keyword(s):  
Solid Mechanics ◽  
Three Dimensional ◽  
Element Model ◽  
Biological Tissues ◽  
Interventional Treatment ◽  
Strong Nonlinearity ◽  
Three Dimensional Model ◽  
Plaque Disruption ◽  
Geometrical Data ◽  
Elastoplastic Constitutive Model

Abstract Studying the solid mechanics of angioplasty provides essential insight in the mechanisms of angioplasty such as overstretching the disease-free tissue, plaque disruption or dissection, redistribution inside the wall and lipid extrusion etc. We desribe our current understanding of the mechanics of angioplasty based on the example of a human iliac artery with an eccentric stenosis. We outline a new approach which has the potential to improve interventional treatment planning, to predict the balloon and stent-induced wall stresses as well as the dilation success. In particular, we use MRI to obtain accurate geometrical data for the vessel wall and plaque architecture and to identify their different types of soft (biological) tissues and calcifications. One issue is to characterize the quasistatic stress-strain response of these components in both axial and circumferential directions. We present new experimental results showing strong nonlinearity and anisotropy. Another issue is to identify predominant directions of each component by analyzing orientations of cellular nuclei. The morphological and mechanical information is used for the elastoplastic constitutive model designed to capture the finite strains of the stenotic artery during angioplasty. The three-dimensional model is fitted to the experimental data. Associated material parameters, corresponding to the different tissues of the stenosis, are presented. The numerical part outlines briefly the concept of the finite element model and, based on a computational structural analysis, discusses the mechanism of angioplasty for the considered type of stenosis.

Efficient Residual and Matrix-Free Jacobian Evaluation for Three-Dimensional Tri-Quadratic Hexahedral Finite Elements With Nearly-Incompressible Neo-Hookean Hyperelasticity Applied to Soft Materials on Unstructured Meshes in Parallel, With PETSc and libCEED

2020 ◽  
Author(s):  
Arash Mehraban ◽  
Jed Brown ◽  
Valeria Barra ◽  
Henry Tufo ◽  
Jeremy Thompson ◽  
...  
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Optimization Design ◽  
Solid Mechanics ◽  
Soft Materials ◽  
Biological Tissues ◽  
Higher Order ◽  
Mixed Formulation ◽  
Element Formulation ◽  
Matrix Free

Abstract Soft materials such as rubber, elastomers, and soft biological tissues mechanically deform at large strain isochorically for all time, or during their initial transient (when a pore fluid, typically incompressible such as water, does not have time to flow out of the deforming polymer or soft tissue porous skeleton). Simulating these large isochoric deformations computationally, such as with the Finite Element Method (FEM), requires higher order (typically quadratic) interpolation functions and/or enhancements through hybrid/mixed methods to maintain stability. Lower order (linear) finite elements with hybrid/mixed formulation may not perform stably for all mechanical loading scenarios involving large isochoric deformations, whereas quadratic finite elements with or without hybrid/mixed formulation typically perform stably, especially when large bending or folding deformations are being simulated. For topology-optimization design of soft robotics, for instance, the FEM solid mechanics solver must run efficiently and stably. Stability is ensured by the higher order finite element formulation (with possible enhancement), but efficiency for higher order FEM remains a challenge. Thus, this paper addresses efficiency from the perspective of computer science algorithms and programming. The proposed efficient algorithm utilizes the Portable, Extensible Toolkit for Scientific Computation (PETSc), along with the libCEED library for efficient compiler optimized tensor-product-basis computation to demonstrate an efficient nonlinear solution algorithm. For preconditioning, a scalable p-multigrid method is presented whereby a hierarchy of levels is constructed. In contrast to classical geometric multigrid, also known as h-multigrid, each level in p-multigrid is related to a different approximation polynomial order, p, instead of the element size, h. A Chebyshev polynomial smoother is used on each multigrid level. Algebraic MultiGrid (AMG) is then applied to the assembled Q1 (linear) coarse mesh on the nodes of the quadratic Q2 (quadratic) mesh. This allows low storage that can be efficiently used to accelerate the convergence to solution. For a Neo-Hookean hyperelastic problem, we examine a residual and matrix-free Jacobian formulation of a tri-quadratic hexahedral finite element with enhancement. Efficiency estimates on AVX-2 architecture based on CPU time are provided as a comparison to similar simulation (and mesh) of isochoric large deformation hyperelasticity as applied to soft materials conducted with the commercially-available FEM software program ABAQUS. The particular problem in consideration is the simulation of an assistive device in the form of finger-bending in 3D.

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