Preface

Although Continuum Mechanics belongs to a traditional topic, the research in this field has never been stopped. The goal of this book is to introduce the latest progress in the fundamental aspects and the applications in various engineering areas. The first three chapters are on the fundamentals of Continuum Mechanics. Chapter 1 introduces the Spencer Operator and presents the applications of this useful operator in solving Continuum Mechanics problems. The authors extend the ideas for tackling general Mathematical Physics problems. Chapter 2 is on Transversality Condition. The author clearly defines the transversality and provides a rigorous derivation for the problem. In Chapter 3, fluid is treated as the continuum media. Related mechanics analysis is given with the emphasis on non-Newtonian fluid.

The rest five chapters are on the applications of continuum mechanics in emerging engineering fields. Chapter 4 uses Continuum Mechanics concepts to analyze the structure-performance relation of solid oxide fuel cells. Three-dimensional reconstructed microstructures are proposed based on both analytical solutions and simulations. In Chapter 5, the mechanical responses are examined in hydraulic piping systems. Noise and vibration related to such systems are presented. Chapter 6 deals with the mechanics associated with the precision machining process. Finite element method (FEM) was used to analyze the mechanistic aspect of materials removal at small scales. Chapter 7 applies Fracture Mechanics approach to predict the progressive stiffness loss of symmetric laminated plates. Specifically, transverse cracks are treated in the studies. Finally, Chapter 8 is on the surface damage analysis. The energy dissipation criteria based on Continuum Mechanics and Micromechanics are proposed to evaluate the surface contact damage evolution. Each chapter is self-contained. The book should be a good reference for researchers in Applied Mechanics.

Ms. Maja Bozicevic, the Publishing Process Manager is acknowledged for her effort on collecting the chapters and assistance in editing. Without her help, the publication of this book would not be possible.

**Dr. Yong X. Gan**  University of Toledo, Member of American Society of Mechanical Engineers, Member of Sigma Xi Scientific Society, USA

**1. Introduction**

*we have to vary*

Let us revisit briefly the foundation of n-dimensional elasticity theory as it can be found today in any textbook, restricting our study to *n* = 2 for simplicity. If *x* = (*x*1, *x*2) is a point in the plane and *ξ* = (*ξ*1(*x*), *ξ*2(*x*)) is the displacement vector, lowering the indices by means of the Euclidean metric, we may introduce the "small" deformation tensor *�* = (*�ij* = *�ji* = (1/2)(*∂iξ<sup>j</sup>* + *∂jξi*)) with *n*(*n* + 1)/2 = 3 (independent) *components* (*�*11, *�*<sup>12</sup> = *�*21, *�*22). If we study a part of a deformed body, for example a thin elastic plane sheet, by means of a variational principle, we may introduce the local density of free energy *ϕ*(*�*) = *ϕ*(*�ij*|*i* ≤

**Spencer Operator and Applications:** 

**From Continuum Mechanics** 

*CERMICS, Ecole Nationale des Ponts et Chaussées,* 

**to Mathematical Physics** 

Accordingly, the "decision" to define the stress tensor *σ* by a symmetric matrix with *σ*<sup>12</sup> = *σ*<sup>21</sup> is purely artificial within such a variational principle. Indeed, the usual Cauchy device (1828) assumes that each element of a boundary surface is acted on by a surface density of force *σ* with a linear dependence *σ* = (*σir*(*x*)*nr*) on the outward normal unit vector *n* = (*nr*) and does not make any assumption on the stress tensor. It is only by an equilibrium of forces and couples, namely the well known *phenomenological static torsor equilibrium*, that one can "prove" the symmetry of *σ*. However, even if we assume this symmetry, *we now need the different summation σijδ�ij* = *σ*11*δ�*<sup>11</sup> + 2*σ*12*δ�*<sup>12</sup> + *σ*22*δ�*<sup>22</sup> = *σir∂rδξi*. An integration

equations *∂rσir* = 0. *The classical approach to elasticity theory, based on invariant theory with respect to the group of rigid motions, cannot therefore describe equilibrium of torsors by means of a variational*

There is another equivalent procedure dealing with a *variational calculus with constraint*. Indeed, as we shall see in Section 7, the deformation tensor is not any symmetric tensor as it must satisfy *<sup>n</sup>*2(*n*<sup>2</sup> <sup>−</sup> <sup>1</sup>)/12 compatibility conditions (CC), that is only *<sup>∂</sup>*22*�*<sup>11</sup> <sup>+</sup> *<sup>∂</sup>*11*�*<sup>22</sup> <sup>−</sup> 2*∂*12*�*<sup>12</sup> = 0 when *n* = 2. In this case, introducing the *Lagrange multiplier* −*φ* for convenience,

by parts now provides the parametrization *<sup>σ</sup>*<sup>11</sup> <sup>=</sup> *<sup>∂</sup>*22*φ*, *<sup>σ</sup>*<sup>12</sup> <sup>=</sup> *<sup>σ</sup>*<sup>21</sup> <sup>=</sup> <sup>−</sup>*∂*12*φ*, *<sup>σ</sup>*<sup>22</sup> <sup>=</sup> *<sup>∂</sup>*11*<sup>φ</sup>* of the stress equations by means of the Airy function *φ* and the *formal adjoint* of the CC, *on the condition to observe that we have in fact* <sup>2</sup>*σ*<sup>12</sup> <sup>=</sup> <sup>−</sup>2*∂*12*<sup>φ</sup>* as another way to understand the deep meaning of the factor "2" in the summation. In arbitrary dimension, it just remains to notice

(*ϕ*(*�*) − *φ*(*∂*22*�*<sup>11</sup> + *∂*11*�*<sup>22</sup> − 2*∂*12*�*12))*dx for an arbitrary �*. A double integration

*<sup>ϕ</sup>*(*�*)*dx* with *dx* <sup>=</sup> *dx*<sup>1</sup> <sup>∧</sup> *dx*<sup>2</sup> by

**1**

J.F. Pommaret

*France* 

(*σ*11*δ�*<sup>11</sup> + *σ*12*δ�*<sup>12</sup> + *σ*22*δ�*22)*dx*.

(*∂rσir*)*δξidx* leading to the stress

*<sup>j</sup>*) = *<sup>ϕ</sup>*(*�*11, *�*12, *�*22) and vary the total free energy *<sup>F</sup>* =

introducing *<sup>σ</sup>ij* <sup>=</sup> *∂ϕ*/*∂�ij* for *<sup>i</sup>* <sup>≤</sup> *<sup>j</sup>* in order to obtain *<sup>δ</sup><sup>F</sup>* <sup>=</sup>

by parts and a change of sign produce the volume integral

*principle where the proper torsor concept is totally lacking*.
