Introduction to Continuum Mechanics II doc

[...]... the general approach to the subject matter of continuum mechanics We are pleased that many instructors of continuum mechanics have found this little book serves that purpose well However, we have also understood that many instructors have used this book as one of the texts for a beginning graduate course in continuum mechanics It is this latter knowledge that has motivated us to write this new edition... understanding of the continuum theory, the point to be made is simply that whether the continuum theory is justified in a given situation is a matter of experimental test, not of philosophy Suffice it to say that more than a hundred years of experience have justified such a theory in a wide variety of situations 1.2 Contents of Continuum Mechanics Continuum mechanics studies the response of materials to different... and then, in order to substitute the bi's in (ii) into (i) we first change the free index in (ii) from i to m and the dummy index m to some other letter, say n so that Now, (i) and (iii) give Note (iv) represents three equations each having the sum of nine terms on its right-hand side Part A Manipulations with the Indicia! Notation 9 (b) Multiplication If and then, It is important to note that pq #... base vectors as follows: Tex = 4CJ+C2 Te2 = 2e!+3e3 Te3 = e1+3e2+e3 Solution By Eq (2B2.1a) it is clear that: [T]= 4 2 -1~ 1 0 3 [0 3 1 Example 2B2.2 Let T transform every vector into its mirror image with respect to a fixed plane If ej is normal to the reflection plane (e2 and 63 are parallel to this plane), find a matrix of T Solution Since the normal to the reflection plane is transformed into its... Krempl July, 1993 xii Preface to the First Edition This text is prepared for the purpose of introducing the concept of continuum mechanics to beginners in the field Special attention and care have been given to the presentation of the subject matter so that it is within the grasp of those readers who have had a good background in calculus, some differential equations, and some rigid body mechanics For pedagogical... property: where a and b are two arbitrary vectors and a and ft are arbitrary scalars If two tensors, T and S, transform any arbitrary vector a in an identical way, then these tensors are equal to each other, i.e., Ta=Sa -» T=S Let T be a transformation which transforms every vector into a fixed vector n Is this transformation a tensor? Solution Let a and b be any two vectors, then by the definition of T, Ta... 520 523 526 532 537 Answer to Problems 543 References 550 Index 552 Preface to the Third Edition The first edition of this book was published in 1974, nearly twenty years ago It was written as a text book for an introductory course in continuum mechanics and aimed specifically at the junior and senior level of undergraduate engineering curricula which choose to introduce to the students at the undergraduate... enough to provide for a better understanding of later courses in the various branches of continuum mechanics and related fields The major portion of the material has been successfully class-tested at Rensselaer Polytechnic Institute for undergraduate students However, the authors believe the text may also be suitable for a beginning graduate course in continuum mechanics We take the liberty to say... Transformation 11 Part B Tensors 281 Tensor - A Linear Transformation Let T be a transformation, which transforms any vector into another vector If T transforms a into c and b into d, we write Ta = c and Tb = d If T has the following linear properties: where a and b are two arbitrary vectors and a is an arbitrary scalar then T is called a linear transformation It is also called a second-order tensor or... Chairman of Mechanics Division at RPI, to whom the authors wish to express their heartfelt thanks Gratefully acknowledged is the financial support of the Ford Foundation under a grant which is directed by Dr S W Yerazunis, Associate Dean of Engineering The authors also wish to thank Drs V C Mow and W B Browner, Jr for their many useful suggestions Special thanks are given to Dr H A Scarton for painstakingly . America. Contents Preface to the Third Edition xii Preface to the First Edition xiii The Authors xiv Chapter 1 Introduction 1 1.1 Continuum Theory 1 1.2 Contents of Continuum Mechanics 1 Chapter .