Introduction to Continuum Mechanics 3 Episode 1 ppsx

[...]... of 3 terms on the right-hand side In fact, T 11 = A1mA1m = A11A 11 + A 12 A 12 +A 13 A l3 T12 =A1mA2m =A11A 21 +A12A22 +A13A 23 T 13 = A1mA3m = A11A 31 + A12A32 + A13A 33 T 33 = A3mA3m = A31A 31+ A32A32 + A33A 33 Again, equations such as Tij = Tik have no meaning, 2A3 Kronecker Delta The Kronecker delta, denoted by dij, is defined as That is, Part A Permutation Symbol 7 d 11 = d22 = d 33 = 1 d12 =d 13 =d 21 =d 23. .. of Linear Momentum 7.7 Moving Frames 7.8 Control Volume Fixed with Respect to a Moving Frame 7.9 Principle of Moment of Momentum 7 .10 Principle of Conservation of Energy Problems Chapter 8 Part A 8 .1 8.2 8 .3 8.4 Non-Newtonian Fluids 38 3 38 4 38 7 39 0 39 1 39 4 39 6 39 9 4 01 402 404 408 412 419 427 427 430 433 435 437 440 447 449 4 51 454 458 462 Linear Viscoelastic Fluid 464 Linear Maxwell Fluid Generalized... 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... Tej = T^i+7* 216 2+ 7 31 63, etc If we had adopted the convention Te^ = 7ne1+7t1262+^I3e3' etc-' tnen we would have obtained 7* [b]=[T] [a] for the tensorial equation b = Ta, which would not be as natural Example 2B3 .1 Given that a tensor T which transforms the base vectors as follows: Tej = 2e1-6e2+4e3 T02 = 3ej+462- 63 Te3 = -26J+62+2 63 How does this tensor transform the vector a = ej+262 +36 3? Solution... = -ej Re3 = e3 so that, 0 -1 0~ [R]= 1 0 0 0 0 1 (b)In a similar manner to (a) the transformation of the base vectors is given by Se1 = e1 Se2 = e3 Se3 = -e2 so that, "l 0 0" [S]= 0 0 - 1 [0 1 0 20 Tensors (c)Since S(Ra) = (SR)a, the resultant rotation is given by the single transformation SR whose components are given by the matrix "l 0 Ol [0 -1 0] |~0 -1 0" [SR]= 0 0 - 1 1 0 0 = 0 0 - 1 [0 1 OJ [0... 486 496 497 498 5 03 5 03 506 511 Viscometric Flow Of Simple Fluid 516 8.20 Viscometric Flow 8. 21 Stresses in Viscometric Flow of an Incompressible Simple Fluid 8.22 Channel Flow 8. 23 Couette Flow Problems Appendix: Matrices 4 91 4 93 494 516 520 5 23 526 532 537 Answer to Problems 5 43 References 550 Index 552 Preface to the Third Edition The first edition of this book was published in 19 74, nearly twenty... we shall always take n to be 3 so that, for example, aixi = amxm = a1x1 + a2x2 + a3x3 aii = amm = a 11 + a22 + a 33 aiei = a1 ei1 + a2 e2 + a3 e3 The summation convention obviously can be used to express a double sum, a triple sum, etc For example, we can write simply as Expanding in full, the expression (2A1.8) gives a sum of nine terms, i.e., For beginners, it is probably better to perform the above... Maxwell Fluid with a Continuous Relaxation Spectrum 464 4 71 4 73 474 Contents xi Part B 8.5 8.6 8.7 8.8 8.9 8 .10 8 .11 8 .12 8. 13 8 .14 8 .15 8 .16 8 .17 8 .18 8 .19 Part C Nonlinear Viscoelastic Fluid 476 Current Configuration as Reference Configuration Relative Deformation Gradient Relative Deformation Tensors Calculations of the Relative Deformation Tensor History of Deformation Tensor Rivlin-Ericksen Tensors Rivlin-Ericksen... (2B3.1b) b i\ [ 2 3 - 2 ] fll [ 2 " b2 = -6 4 1 2 = 5 b3 [ 4 -1 2J [3J [8 or b = 2e1+5e2+8e3 2B4 Sum of Tensors Let T and S be two tensors and a be an arbitrary vector The sum of T and S, denoted by T + S, is defined by: 18 Tensors It is easily seen that by this definition T + S is indeed a tensor To find the components of T + S, let Using Eqs (2B2.2) and (2B4 .1) , the components of W are obtained to. .. Under a transformation T, these vectors, el5 62, e3 become Tels Te2, and Te3 Each of these Te/ (/= 1, 2 ,3) , being a vector, can be written as: or It is clear from Eqs (2B2.1a) that or in general The components TJJ in the above equations are defined as the components of the tensor T These components can be put in a matrix as follows: TII n T 13 T [T] = 7^! r22 r 23 Til 732 ^33 This matrix is called the matrix . Deformation Components 11 9 3. 18 Deformation Gradient 12 0 3. 19 Local Rigid Body Displacements 12 1 3. 20 Finite Deformation 12 1 3. 21 Polar Decomposition Theorem 12 4 3. 22 Calculation of the . and the Angular Velocity Vector 11 1 3. 15 Equation of Conservation Of Mass 11 2 3. 16 Compatibility Conditions for Infinitesimal Strain Components 11 4 3. 17 Compatibility Conditions for. Deformation Gradient 12 6 3. 23 Right Cauchy-Green Deformation Tensor 12 8 3. 24 Lagrangian Strain Tensor 13 4 3. 25 Left Cauchy-Green Deformation Tensor 13 8 3. 26 Eulerian Strain Tensor 14 1 3. 27 Compatibility