scholarly journals Small-on-large geometric anelasticity

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160659 ◽  
Author(s):  
Souhayl Sadik ◽  
Arash Yavari
Keyword(s):  
Exact Solutions ◽  
Stress Fields ◽  
Symmetric Distributions ◽  
Small Perturbations ◽  
Systematic Analysis ◽  
Nonlinear Solids ◽  
Isotropic Solids ◽  
Geometric Formulation ◽  
Finite Eigenstrains ◽  
Small On Large

In this paper, we are concerned with finding exact solutions for the stress fields of nonlinear solids with non-symmetric distributions of defects (or more generally finite eigenstrains) that are small perturbations of symmetric distributions of defects with known exact solutions. In the language of geometric mechanics, this corresponds to finding a deformation that is a result of a perturbation of the metric of the Riemannian material manifold. We present a general framework that can be used for a systematic analysis of this class of anelasticity problems. This geometric formulation can be thought of as a material analogue of the classical small-on-large theory in nonlinear elasticity. We use the present small-on-large anelasticity theory to find exact solutions for the stress fields of some non-symmetric distributions of screw dislocations in incompressible isotropic solids.

scholarly journals Physical modelling of failure in composites

2016 ◽  
Vol 374 (2071) ◽  
pp. 20150280 ◽  
Author(s):  
Ramesh Talreja
Keyword(s):  
Composite Materials ◽  
Failure Modes ◽  
Structural Integrity ◽  
Failure Mechanisms ◽  
Local Stress ◽  
Physical Modelling ◽  
Stress Fields ◽  
Systematic Analysis ◽  
Composite Failure ◽  
Failure Theories

Structural integrity of composite materials is governed by failure mechanisms that initiate at the scale of the microstructure. The local stress fields evolve with the progression of the failure mechanisms. Within the full span from initiation to criticality of the failure mechanisms, the governing length scales in a fibre-reinforced composite change from the fibre size to the characteristic fibre-architecture sizes, and eventually to a structural size, depending on the composite configuration and structural geometry as well as the imposed loading environment. Thus, a physical modelling of failure in composites must necessarily be of multi-scale nature, although not always with the same hierarchy for each failure mode. With this background, the paper examines the currently available main composite failure theories to assess their ability to capture the essential features of failure. A case is made for an alternative in the form of physical modelling and its skeleton is constructed based on physical observations and systematic analysis of the basic failure modes and associated stress fields and energy balances. This article is part of the themed issue ‘Multiscale modelling of the structural integrity of composite materials’.

scholarly journals New Exact Solutions for an Oldroyd-B Fluid in a Porous Medium

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
I. Khan ◽  
M. Imran ◽  
K. Fakhar
Keyword(s):  
Porous Medium ◽  
Shear Stress ◽  
Exact Solutions ◽  
Strong Influence ◽  
Stress Fields ◽  
Porous Space ◽  
New Exact Solutions ◽  
Laplace Transform Technique ◽  
Similar Solutions ◽  
Analytical Expressions

New exact solutions for unsteady magnetohydrodynamic (MHD) flows of an Oldroyd-B fluid have been derived. The Oldroyd-B fluid saturates the porous space. Two different flow cases have been considered. The analytical expressions for velocity and shear stress fields have been obtained by using Laplace transform technique. The corresponding solutions for hydrodynamic Oldroyd-B fluid in a nonporous space appeared as the limiting cases of the obtained solutions. Similar solutions for MHD Newtonian fluid passing through a porous space are also recovered. Graphs are sketched for the pertinent parameters. It is found that the MHD and porosity parameters have strong influence on velocity and shear stress fields.

The Engineering of Polymers for Mechanical Behavior

1969 ◽  
Vol 42 (4) ◽  
pp. 1175-1185 ◽  
Author(s):  
F. N. Kelley ◽  
M. L. Williams
Keyword(s):  
Relaxation Modulus ◽  
Engineering Properties ◽  
Stress Fields ◽  
Interaction Matrix ◽  
Systematic Analysis ◽  
Bulk Compressibility ◽  
Starting Point ◽  
Engineering Parameters ◽  
Broad Subject ◽  
Rupture Properties

Abstract The broad subject matter in this paper is treated in only cursory fashion. Our intent was to present an approach to the systematic analysis of the relationships between engineering properties and molecular variables for viscoelastic materials. Literature omissions will undoubtedly be numerous, but the basic concept of the Interaction Matrix allows for continual upgrading and refinement. The examples chosen were based on the stress relaxation modulus which provided an ideal starting point due to the wide use of this property in viscoelastic stress analysis and fracture mechanics. Other important engineering parameters which might be selected for similar examination include bulk compressibility and dilation for example; and, of course, the rupture properties under various imposed stress fields.

A REVIEW ON THE GEOMETRIC FORMULATION OF TOLMAN–BONDI EQUATIONS IN GENERAL RELATIVITY

2009 ◽  
Vol 06 (04) ◽  
pp. 595-617 ◽  
Author(s):  
IVANA BOCHICCHIO ◽  
MAURO FRANCAVIGLIA ◽  
ETTORE LASERRA
Keyword(s):  
General Relativity ◽  
Symmetric Space ◽  
Exact Solutions ◽  
Structural Properties ◽  
Spherically Symmetric ◽  
Principal Curvatures ◽  
Spherical Shells ◽  
Geometric Formulation ◽  
Dust Universe

This work is focused on spherically symmetric space-times. More precisely, geometric and structural properties of spatially spherical shells of a dust universe are analyzed in detail considering recent results of our research. Moreover, exact solutions, obtained for constant Ricci principal curvatures, are inferred and qualitatively analyzed through suitable classic analogies.

On a summation formula for the Appell function F2

1967 ◽  
Vol 63 (4) ◽  
pp. 1087-1089 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Quantum Mechanics ◽  
Exact Solutions ◽  
Double Series ◽  
Summation Formula ◽  
Systematic Analysis ◽  
Image Position ◽  
Appell Function

1. In the course of a systematic analysis of certain problems in quantum mechanics it has been observed that their exact solutions can be expressed in terms of the Appell function F2 defined by means of (see e.g. (8), p. 211)where, as usual,and for convergence of the double series,

scholarly journals Geometric formulation of quantum stress fields

2002 ◽  
Vol 65 (22) ◽  
Author(s):  
Christopher L. Rogers ◽  
Andrew M. Rappe
Keyword(s):  
Stress Fields ◽  
Geometric Formulation

A Method for Exact Series Solutions in Structural Mechanics

1999 ◽  
Vol 66 (2) ◽  
pp. 380-387 ◽  
Author(s):  
J. T.-S. Wang ◽  
C.-C. Lin
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Integral Form ◽  
Structural Mechanics ◽  
Series Solutions ◽  
Analysis Method ◽  
Weighting Functions ◽  
Systematic Analysis ◽  
Unified Method

A systematic analysis method for solving boundary value problems in structural mechanics is presented. Euler-Lagrange differential equations are transformed into integral form with respect to sinusoidal weighting functions. General solutions are represented by complete sets of functions without being concerned with boundary conditions in advance while all boundary conditions are satisfied in the process. The convergence of results is assured, and the procedure leads to pointwise exact solutions. A number of simple structural mechanics problems of stress, buckling, and vibration analyses are presented for illustrative purposes. All results have verified the exactness of solutions, and indicate that this unified method is simple to use and effective.

scholarly journals Nonlinear elastic inclusions in isotropic solids

2013 ◽  
Vol 469 (2160) ◽  
pp. 20130415 ◽  
Author(s):  
Arash Yavari ◽  
Alain Goriely
Keyword(s):  
Residual Stress ◽  
Spherical Inclusion ◽  
The Body ◽  
Model Systems ◽  
Stress Fields ◽  
Elastic Solids ◽  
Geometric Framework ◽  
Spherical Ball ◽  
Nonlinear Elastic ◽  
Nonlinear Solids

We introduce a geometric framework to calculate the residual stress fields and deformations of nonlinear solids with inclusions and eigenstrains. Inclusions are regions in a body with different reference configurations from the body itself and can be described by distributed eigenstrains. Geometrically, the eigenstrains define a Riemannian 3-manifold in which the body is stress-free by construction. The problem of residual stress calculation is then reduced to finding a mapping from the Riemannian material manifold to the ambient Euclidean space. Using this construction, we find the residual stress fields of three model systems with spherical and cylindrical symmetries in both incompressible and compressible isotropic elastic solids. In particular, we consider a finite spherical ball with a spherical inclusion with uniform pure dilatational eigenstrain and we show that the stress in the inclusion is uniform and hydrostatic. We also show how singularities in the stress distribution emerge as a consequence of a mismatch between radial and circumferential eigenstrains at the centre of a sphere or the axis of a cylinder.

scholarly journals An Algorithm: Optimal Homotopy Asymptotic Method for Solutions of Systems of Second-Order Boundary Value Problems

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Muhammad Rafiq Mufti ◽  
Muhammad Imran Qureshi ◽  
Salem Alkhalaf ◽  
S. Iqbal
Keyword(s):  
Exact Solutions ◽  
Boundary Value Problems ◽  
Asymptotic Method ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Small Perturbations ◽  
Forcing Function ◽  
Optimal Homotopy Asymptotic Method ◽  
Linear And Nonlinear Systems

Optimal homotopy asymptotic method (OHAM) is proposed to solve linear and nonlinear systems of second-order boundary value problems. OHAM yields exact solutions in just single iteration depending upon the choice of selecting some part of or complete forcing function. Otherwise, it delivers numerical solutions in excellent agreement with exact solutions. Moreover, this procedure does not entail any discretization, linearization, or small perturbations and therefore reduces the computations a lot. Some examples are presented to establish the strength and applicability of this method. The results reveal that the method is very effective, straightforward, and simple to handle systems of boundary value problems.

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