Introduction to Continuum Mechanics 3 Episode 1 ppsx

... (2A1.2) 3 etc. x Contents 6 .17 Dissipation Functions for Newtonian Fluids 38 3 6 .18 Energy Equation for a Newtonian Fluid 38 4 6 .19 Vorticity Vector 38 7 6.20 Irrotational Flow 39 0 6. 21 ... Spectra 4 73 8.4 Generalized Linear Maxwell Fluid with a Continuous Relaxation Spectrum 474 Part A Permutation Symbol 7 d 11 = d 22 = d 33 = 1 d 12 =d 13 =d...

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... of the tensor T. (b) If 11 1 ,11 2 ,1 13 are the principal directions, write [T] n (c) Could the following matrix represent the tensor T in respect to some basis? 2B 41. Do the previous ... the motion of a continuum can be described either by the pathlines equation Eq. (3. 1. 1) or by its displacement vector field as given by Eq. (3. 5 .1) . Example...

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... Q 23 TII T^ 2 7 *33 ^ 13 0 23 033 T 31 T 32 T 33 0 31 032 033 or PartB Symmetric and Antisymmetric Tensors 35 This is the transformation law for the components of a vector. Thus, fy ... Eq. (2B 13 . 1a) reads Tn T'n T^\ \Q n Q 2l 0 31 1 |"r n T n T 13 \ \Q n Q n Qi3\ < 2B 13 - lb ) T 2l T 22 T 23 ~ Ql2 Qn 032 T 21...

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... existence of single-valued continuous solutions #1, #2 an ^ U 3 °f the six equation Eq. (3. 16 .1) to Eq. (3. 16 .6) are 11 8 Kinematics of a Continuum and Thus, the equation is satisfied, ... from Eq. (3. 23. 4), we have That is That is similarly, Kinematics of a Continuum 13 3 thus, for this material element (d) For dX = dS^ and dX = dS 2 e...

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... (3. 30.9d). 3. 76. Derive Eqs. (3. 30 .10 ). 3. 77. Using Eqs. (3. 30 .10 ) derive Eqs. (3. 30 .12 a) and (3. 30 .12 d). 3. 78. Verify Eqs. (3. 30. 13 a) and (3. 30 .13 d). Kinematics of a Continuum 15 1 where e 0 / ... respect to the basis at the reference configuration 3. 73. From Eqs. (3. 30.4a), obtain Eqs. (3. 30.5). 3. 74. Verify Eq. (3. 30.8b) and (3....

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... distribution of the stress vector acting on the square in thex\ = 0 plane with vertices located at (0 ,1, 1), (0, -1, 1), (0 ,1, -1) , (0, -1, -1) . (b) Find the total resultant force and ... Sect. 3. 12 , we obtained [see Eq. (3. 12 .4)], Since dx - F dX [ see Eq. (3. 7.2)], therefore Equation (i) is to be true for all dX, thus or, Using Eqs. (4 .1...

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... strength M t , the state of stress inside the bar is given by Eq, (5. 13 . 11 ). Example 5. 13 . 1 For a circular bar of radius a in torsion (a) find the magnitude and location of the greatest normal ... 227 Example 5 .3. 1 Find the components of stress at a point if the strain matrix is and the material is steel with A = 11 9.2 GPa (17 .3 xl()...

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... Eqs. (4.8 .1) ], (noting that there is no z dependence). The Elastic Solid 279 From the compatibility equations, Eqs. (3. 16 .8), (3. 16 .9) and (3. 16 .7), we have Thus, G(XI 1) = a ... — 0 is automatically satisfied together with On the other hand, since Q& = 1, we have This requirement leads to That is, Similarly, the equation Ci 233 = 0 leads...

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... (5 .33 .9) and (5 .33 .11 ), as those cor- responding to a change of rectangular Cartesian basis, then we come to the conclusion that the constitutive equation given by Eq. (5 .33 .6) ... stress field? 5 .15 . Repeat Problem 5 .14 , except that the displacement components are ui = kX 2 X 3 , u 2 = kX 1 X 3 , u 3 = kX 1 X 2t k = 10 ~ 4 5 .16 . R...

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... Navier-Stokes equations, Eqs. (6.7.2) yield: Equations (6 .12 .1b) and (6 .12 .1c) state that p does not depend on* 2 and * 3 . If we differentiate Eq. (6 .12 . la) with respect to ... given by Eq. (6. 13 . 3), however, the driving force now is Newtonian Viscous Fluid 36 1 These are known as the Navier-Stokcs Equations of motion for incompressible...

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