CONTINUUM MECHANICS for ENGINEERS

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Determine (a) the transformation matrix between the two sets of axes, and show that it is a proper orthogonal transformation. (b) the equation of the plane x1 + x2 + x3 = 1/ 2 in its primed axes form, that is, in the form + + = b. Answer: bx 1 1 ′ bx 2 2 ′ bx 3 3 ′ ⎡1/ 2 0 1/ 2⎤ ⎢ ⎥ (a) [ aij]= ⎢ 1/ 2 1/ 2 −1/ 2⎥, ⎢ ⎣ −1/ 2 1/ 2 1/ 2⎥ ⎦ (b) ′ + ′ + = 1 ′ 2.20 Making use of Eq 2.4-11 of the text in the form det A = ε ijk A 1iA 2jA 3k write Eq 2.6-6 as 2x 1 and show by expansion of this equation that to verify Eq 2.6-8 of the text. = 0 2.21 For the matrix representation of tensor B shown below, determine the principal values (eigenvalues) and the principal directions (eigenvectors) of the tensor. Answer: λ1 = 17, λ2 = 26, λ3 = –39 (2) , nˆ = 4eˆ + 7eˆ / 65 , 2.22 Consider the symmetrical matrix x 2 T − λδ = ε T −λδ T λδ T λδ ij ij ijk 1i1i2j2j3k3k x 3 ( ) − ( ) ( ) − ( ) 3 2 ⎡ 1 ⎤ λ − Tiiλ + TiiTjj −TijTji λ− εijkT1iT2jT3k= 0 ⎣⎢ 2 ⎦⎥ ˆ ˆ n e (1) = 1 ⎡17 0 0⎤ ⎥ ⎢ 0 23 28 ⎥ ⎣ ⎢ 0 28 10⎦ ⎥ ⎢ [ Bij]= − [ Bij]= ( 2 3 ) ⎡ 5 3 0 ⎤ 2 2 ⎢ ⎥ ⎢0 4 0⎥ ⎢ 3 5⎥ ⎣ 0 2 2⎦ ( 2 3) (3) nˆ = − 7eˆ + 4eˆ / 65
(a) Show that a multiplicity of two occurs among the principal values of this matrix. (b) Let λ 1 be the unique principal value and show that the transformation matrix * gives B* according to B = a a B . * (c) Taking the square root of and transforming back to Ox1x2x3 axes show that [ aij]= (d) Verify that the matrix is also a square root of . ⎡1/ 2 0 −1/ 2 ⎤ ⎢ ⎥ ⎢ 0 1 0 ⎥ ⎢ ⎥ ⎣ 1/ 2 0 1/ 2 ⎦ 2.23 Determine the principal values of the matrix ij [ Bij ] = ⎡ ⎢ [ Cij]= ⎢ ⎢ ⎣ [ Bij] iq jm qm B [ ij] ⎡ 3 1 0 ⎤ 2 2 ⎢ ⎥ ⎢0 2 0⎥ ⎢ 1 3⎥ ⎣ 0 2 2⎦ 1 3 0 ⎤ 2 2⎥ 0 2 0⎥ 3 1 0 ⎥ 2 2⎦ − − − − ⎡4 0 0⎤ ⎥ ⎢ 0 11 3 ⎥ ⎣ ⎢0 − 3 9⎦ ⎥ ⎢ [ Kij]= − and show that the principal axes Ox* 1 x2* x3* are obtained from by a rotation of 60° about the x1 axis. Answer: λ1 = 4, λ2 = 8, λ3 = 12. Oxxx 1 2 3
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CONTINUUM MECHANICS for ENGINEERS S
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Library of Congress Cataloging-in-P
Page 5 and 6: University, for helpful comments on
Page 7 and 8: Reference Section for constitutive
Page 9 and 10: Nomenclature x 1, x 2, x 3 or x i o
Page 11 and 12: W Strain energy per unit volume, or
Page 13 and 14: 4.5 The Material Derivative 4.6 Def
Page 15 and 16: 1 Continuum Theory 1.1 The Continuu
Page 17 and 18: 2 Essential Mathematics 2.1 Scalars
Page 19 and 20: FIGURE 2.1A Unit vectors in the coo
Page 21 and 22: 1. Addition of vectors: 2. Multipli
Page 23 and 24: and, furthermore, that for repeated
Page 25 and 26: A sum of dyads such as is called a
Page 27 and 28: (b) Start with the first equation i
Page 29 and 30: Solution By definition of symmetric
Page 31 and 32: Addition of matrices is commutative
Page 33 and 34: A A A det A = Aij = A A A A A A (2.
Page 35 and 36: which is identical to Eq 2.4-9 and
Page 37 and 38: FIGURE 2.2A Rectangular coordinate
Page 39 and 40: Consider next an arbitrary vector v
Page 41 and 42: FIGURE E2.5-1 Vector νννν with
Page 43 and 44: or in expanded form ( Tij − λδi
Page 45 and 46: The transformation matrix here is o
Page 47 and 48: have a multiplicity of two, and det
Page 49 and 50: we write φ for ; vi,j for for ; an
Page 51 and 52: FIGURE 2.5B Bounding space curve C
Page 53 and 54: (b) the trace of A is expressed in
Page 55: 2.15 Using the square matrices belo
Page 59 and 60: 2.29 Transcribe the left-hand side
Page 61 and 62: 3 Stress Principles 3.1 Body and Su
Page 63 and 64: FIGURE 3.2A Typical continuum volum
Page 65 and 66: where S I and S II are the bounding
Page 67 and 68: FIGURE 3.5 Free body diagram of tet
Page 69 and 70: FIGURE 3.6 Cartesian stress compone
Page 71 and 72: (b) The equation of the plane ABC i
Page 73 and 74: or But xj,q = δjq and by Eq 3.4-3,
Page 75 and 76: FIGURE E3.5-1 Rotation of axes x 1
Page 77 and 78: FIGURE 3.9A Traction vector at poin
Page 79 and 80: this system the shear stresses, σ
Page 81 and 82: FIGURE 3.10B Table displaying direc
Page 83 and 84: 1 Obviously, n from the second of t
Page 85 and 86: FIGURE 3.12 Normal and shear compon
Page 87 and 88: 1 1 1 n1 = 0 , n2 = ± , n3 = ± ;
Page 89 and 90: exterior to circle C1. Thus, combin
Page 91 and 92: FIGURE 3.15A Reference angles φ an
Page 93 and 94: FIGURE E3.8-1 Three-dimensional Moh
Page 95 and 96: FIGURE 3.16A Mohr’s circle for pl
Page 97 and 98: FIGURE 3.18A Representative rotatio
Page 99 and 100: Solution For this stress state, the
Page 101 and 102: which demonstrates that ( q) is als
Page 103 and 104: Example 3.11-1 Determine directly t
Page 105 and 106: FIGURE P3.6 Stress vectors represen
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FIGURE P3.9 Cylinder of radius r an
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(b) Project each of the stress vect
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Determine (a) the principal stress
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(b) Verify the result determined in
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⎡ −12 − 12 + 3 2 −12 −32
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A motion of body B is a continuous
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emphasize, however, that the materi
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Example 4.2-1 Let the motion of a b
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and v = v(X,t) = v[χ -1 (x,t),t] =
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Additionally, with regard to the ma
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In this equation, the first term on
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Therefore, substituting u i for P i
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upon X, in which case the deformati
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form consistent with deformation an
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For diagonal BH, and for diagonal O
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which, upon factoring the left-hand
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FIGURE 4.4 A rectangular parallelep
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FIGURE 4.6A Rotated axes for plane
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Consider once more the two neighbor
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where dx is the deformed magnitude
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In a similar fashion, from dX (1)
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Thus from Eq 4.8-4 the principal st
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and as is obvious, the inverse (U*)
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Finally, from Eq 4.9-11, [ RAB]= =
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which becomes (after dividing both
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This rate of decrease in the angle
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FIGURE 4.8 Area dS° between vector
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FIGURE 4.9 Volume of parallelepiped
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(a) Show that the Jacobian determin
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4.5 The Lagrangian description of a
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u 2 = - x 2 - (x 1 + x 2)e -t /2 +
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and from it the longitudinal (norma
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FIGURE P4.27 Unit square OBCD in th
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Answer: (a) (b) 2 2 (c) (1) a1a 2a
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FIGURE P4.35 Circular cylinder in t
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(a) the components of acceleration
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FIGURE P4.47A Unit cube having diag
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5 Fundamental Laws and Equations 5.
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which upon application of the diver
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which is known as the continuity eq
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FIGURE 5.1 Material body in motion
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where ∆f is the resultant force a
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which reduces to where we have used
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Ε ABδ iAδ jB = e ij = 0(ε) o Ex
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this text we consider only mechanic
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outward normal is n i (hence the mi
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θ = η ∂u ∂ (5.8-2) Furthermor
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The entropy production is always po
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One of the uses for the Clausius-Du
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Furthermore, assume the temperature
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In the first, a continuum body’s
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FIGURE 5.2 Reference frames Ox 1x 2
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Recall the definition t + = t + a f
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where the last substitution comes f
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With the use of Eqs 5.10-29 and 5.1
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This being the case, it is clear fr
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as the invariance requirement on th
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i ∂ {P} = p (5.12-7a) i i ∂ t i
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* 5.2 Let the property P in Eq 5.2-
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5.12 Determine the form which the e
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σ = - ij pδ + τ ij ij σ ij = -
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a. b. 5.30 Show that the Jacobian t
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FIGURE 6.1 Uniaxial loading-unloadi
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where the factor of two on the shea
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ased on the strain energy function.
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( q) ( q) σ = (λδijεkk + 2µε
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FIGURE 6.2 Simple stress states: (a
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FIGURE 6.3 For plane stress: (a) ro
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x 1x 2 plane to be one of elastic s
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FIGURE 6.4 (continued) From the def
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A most important feature of the fie
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FIGURE 6.5 (a) Plane stress problem
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For the plane strain situation (Fig
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By inserting Eq 6.6-2 into Eq 6.6-1
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FIGURE E6.7-1 (a) Rectangular regio
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Solution It is easily verified, by
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(6.7-8b) (6.7-8c) in which φ = φ
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These conditions result in the foll
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Solution From Eq 6.7-8 the stress c
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FIGURE E6.7-5 Uniaxial loaded plate
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FIGURE 6.7 (a) Cylinder with self-e
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where ψ(x 1,x 2) is called the war
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Thus, By eliminating ψ from this p
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where λ is a constant. Thus, Φ is
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Equilibrium equations, Eq 6.4-1 σi
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It should be pointed out that while
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This equation may be written in a m
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where A 1 and A 2 are constants of
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Upon setting the off-diagonal terms
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6.8 Show that the distortion energy
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6.16 For an elastic body whose x 3
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and B are constants, determine the
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Thus, for the beam shown the stress
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Using the sketch on the facing page
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For a fluid in motion the shear str
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The first of this pair relates the
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posed in this section are relevant
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FIGURE E7.4-1A Flow down an incline
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and thus which describes the pressu
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where dx i is a differential tangen
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7.9 Show that for an incompressible
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8 Nonlinear Elasticity 8.1 Molecula
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strain. Highly cross-linked and fil
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FIGURE 8.3 A freely connected chain
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proportional to the probability per
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The deformation is volume preservin
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Note that is essentially the same a
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where dependence on I 3 has been in
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where P 11 is the 11-component of t
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This simplifies to where and ∂p
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σ yy ⎡⎛ ∂X ⎞ ⎛ ∂Y ⎞
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q( x)= Ae + Be kx −kx Eqs 8.4-19a
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FIGURE 8.6 Stresses on deformed rub
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8.2 Derive the following relationsh
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Answer: 8.9 Show F B iA = ij = g
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9 Linear Viscoelasticity 9.1 Introd
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˙ε ii ≈ Dii so that now, from E
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FIGURE 9.2 Mechanical analogy for s
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FIGURE 9.5 Three-parameter standard
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FIGURE 9.7 Graphic representation o
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Development of details relative to
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FIGURE 9.9 Stress history with an i
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γ ()= t γ ( cos ωt+ isinωt)= γ
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FIGURE 9.10 Shear lag in viscoelast
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But by Eq 9.6-7, σ 12 (t) = γ o [
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Depending upon the particular state
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() (9.7-9a) (9.7-9b) (9.7-9c) From
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and so on for higher derivatives. N
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This expression may be inverted wit
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9.4 Develop the constitutive equati
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9.7 For the model shown the stress
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9.11 For the model shown determine
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9.16 Let the stress relaxation func
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9.24 For the rather complicated mod
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9.28 A viscoelastic body in the for
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9.32 The deflection at x = L for an
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CONTINUUM MECHANICS for ENGINEERS

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