By Mase, G. Thomas; Mase, George E.; Smelser, Ronald E.
Continuum TheoryContinuum MechanicsStarting OverNotationEssential MathematicsScalars, Vectors and Cartesian TensorsTensor Algebra in Symbolic Notation - Summation ConventionIndicial NotationMatrices and DeterminantsTransformations of Cartesian TensorsPrincipal Values and relevant DirectionsTensor Fields, Tensor CalculusIntegral Theorems of Gauss and StokesStress PrinciplesBody and floor Forces, Mass DensityCauchy tension PrincipleThe tension TensorForce and second Equilibrium; tension Tensor SymmetryStress Transformation LawsPrincipal Stresses; relevant pressure DirectionsMaximum and minimal St. Read more...
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Additional info for Continuum Mechanics for Engineers, 3rd Edition
At the undergraduate level students have studied linear elastic, fluid and gas behavior. But the topics generally are taught in separate courses, and often the common, underlying theory is not noticed. Also, the methods used to determine the relationship between stress and strain are not considered. Continuum Theory 3 So continuum mechanics is not a new or challenging topic. Rather it is a chance to start over and put all that was studied previously under a single umbrella. A course in continuum mechanics is a chance to synthesize what was learned during an undergraduate education into a coherent structure.
2) 8 Continuum Mechanics for Engineers x3 x3 v3 ^3 e v x2 O ^2 e O v1 ^1 e x1 v2 x2 x1 (a) Unit vectors in the coordinate directions x1 , x2 and x3 . (b) Rectangular Cartesian components of the vector v. 1 Base vectors and components of a Cartesian vector. This vector and its coordinate components are pictured in Fig. 1(b). For the symbolic description, vectors will usually be given by lower-case Latin letters in bold-faced print, with the vector magnitude denoted by the same letter. Thus v is the magnitude of v.
55) Note that this is simply the inner product of the i row with the j row of the matrix A. 56) ^i = aji e ^j , e which in matrix form is ^1j [aij ] or [^ e1i ] = e ^1 e ^2 e ^3 e = ^1 e ^2 e ^3 e a11 a21 a31 a12 a22 a32 a13 a23 . 58) ^i and e ^i . 59) which is the inner product of the jth column with the kth column of A. Consider next an arbitrary vector v having components vi in the unprimed system, and vi in the primed system. 60) which is the transformation law expressing the primed components of an arbitrary vector in terms of its unprimed components.