CONTINUUM MECHANICS for ENGINEERS

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6.8 Show that the distortion energy density W for a linear elastic ( 2) medium may be expressed in terms of (a) the principal stresses, σ , σ , σ , and (b) the principal strains, ε , ε , ε , in the form ( 1) ( 2) ( 3) 2 2 ( ) + ( − ) + − ( ) (a) (b) 6.9 Beginning with the definition for W (Eq 6.1-21a), show for a linear elastic material represented by Eq 6.2-2 or by Eq 6.2-7 that and W( 2) = σ( 1) − σ( 2) σ( 2) σ( 3) 12G σ( 3) σ( 1) 1 2 2 2 ⎡ ⎤ W( G 2) = ( ε 1 2 2 3 3 1 3 ( ) −ε( ) ) + ( ε( ) −ε( ) ) + ( ε( ) −ε( ) ) ⎣⎢ ⎦⎥ ∂ ∂ε = σ ∂ ∂σ = ε . (Note that ∂ε / ∂ε = δ δ .) W / ij ij ij mn im jn ( 1) ( 2) ( 3) 6.10 For an isotropic, linear elastic solid, the principal axes of stress and strain coincide, as was shown in Example 6.2-1. Show that, in terms of engineering constants E and v this result is given by ε ( q) [ ] ( ) − + + 1+ v σ v σ σ σ = E ( q) ( 1) ( 2) ( 3) , (q = 1, 2, 3). Thus, let E = 10 6 psi and v = 0.25, and determine the principal strains for a body subjected to the stress field (in ksi) [ σ ij]= ⎡12 0 4⎤ ⎢ ⎥ ⎢ 0 0 0 ⎥ ⎣ ⎢ 4 0 6⎦ ⎥ Answer: ε (1) = –4.5 × 10 –6 , ε (2) = 0.5 × 10 –6 , ε (3) = 13 × 10 –6 6.11 Show for an isotropic elastic medium that 1 2 (a) , (b) , (c) , 1+ 3 2 (d) 2µ(1+ν) = 3K(1–2ν) 6.12 Let the x1x3 plane be a plane of elastic symmetry such that the transformation matrix between Ox1x2x3 and is = ( λ + µ ) v v λ + µ 1− v 2 = λ 2µν 3Kν = λ + µ 1−2ν1+ ν O xxx ′ ′ ′ ⎡1 0 0⎤ ⎥ ⎢ 0 1 0 ⎥ ⎣ ⎢0 0 1⎦ ⎥ ⎢ [ aij]= − 1 2 3 2 W / ij ij
Show that, as the text asserts, this additional symmetry does not result in a further reduction in the elastic constant matrix, Eq 6.3-7. 6.13 Let the x 1 axis be an axis of elastic symmetry of order N = 2. Determine the form of the elastic constant matrix C αβ, assuming C αβ = C βα. Answer: C C C C C C [ Cαβ ] C C C C C C C C C C = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎣⎢ 0 0 0 0 0 0 0 11 12 13 14 12 22 23 24 13 23 33 34 14 24 34 44 0 0 ⎤ 0 0 ⎥ ⎥ 0 0 ⎥ ⎥ 0 0 ⎥ C C ⎥ 55 56 ⎥ C56 C66⎦⎥ 6.14 Assume that, by the arguments of elastic symmetry, the elastic constant matrix for an isotropic body has been reduced to the form C C C C C C C C C [ Cαβ ] C C C = ⎡ 11 12 12 0 0 0 ⎤ ⎢ ⎥ ⎢ 12 11 12 0 0 0 ⎥ ⎢ ⎥ 12 12 11 0 0 0 ⎢ ⎥ ⎢ 0 0 0 44 0 0 ⎥ ⎢ 0 0 0 0 ⎥ 44 0 ⎢ ⎥ ⎣⎢ 0 0 0 0 0 44⎦⎥ Show that, if the x 1 axis is taken as an axis of elastic symmetry of any order (θ is arbitrary), C11 = C12 + 2 C44. (Hint: Expand σ = a2q a3m σqm and = a 23 ′ ε23 ′ 2qa3mεqm.) 6.15 If the axis which makes equal angles with the coordinate axes is an axis of elastic symmetry of order N = 3, show that there are twelve independent elastic constants and that the elastic matrix has the form C C C C C C C C [ Cαβ ] C C C C C C C C C C C C C C C C C C C C C C C C C C C C = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣⎢ 11 12 13 14 15 16 13 11 12 16 14 15 12 13 11 15 16 14 41 42 43 44 45 46 43 41 42 46 44 45 42 43 41 45 46 44 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦⎥
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CONTINUUM MECHANICS for ENGINEERS S
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Library of Congress Cataloging-in-P
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University, for helpful comments on
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Reference Section for constitutive
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Nomenclature x 1, x 2, x 3 or x i o
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W Strain energy per unit volume, or
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4.5 The Material Derivative 4.6 Def
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1 Continuum Theory 1.1 The Continuu
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2 Essential Mathematics 2.1 Scalars
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FIGURE 2.1A Unit vectors in the coo
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1. Addition of vectors: 2. Multipli
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and, furthermore, that for repeated
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A sum of dyads such as is called a
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(b) Start with the first equation i
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Solution By definition of symmetric
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Addition of matrices is commutative
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A A A det A = Aij = A A A A A A (2.
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which is identical to Eq 2.4-9 and
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FIGURE 2.2A Rectangular coordinate
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Consider next an arbitrary vector v
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FIGURE E2.5-1 Vector νννν with
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or in expanded form ( Tij − λδi
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The transformation matrix here is o
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have a multiplicity of two, and det
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we write φ for ; vi,j for for ; an
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FIGURE 2.5B Bounding space curve C
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(b) the trace of A is expressed in
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2.15 Using the square matrices belo
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(a) Show that a multiplicity of two
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2.29 Transcribe the left-hand side
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3 Stress Principles 3.1 Body and Su
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FIGURE 3.2A Typical continuum volum
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where S I and S II are the bounding
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FIGURE 3.5 Free body diagram of tet
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FIGURE 3.6 Cartesian stress compone
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(b) The equation of the plane ABC i
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or But xj,q = δjq and by Eq 3.4-3,
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FIGURE E3.5-1 Rotation of axes x 1
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FIGURE 3.9A Traction vector at poin
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this system the shear stresses, σ
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FIGURE 3.10B Table displaying direc
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1 Obviously, n from the second of t
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FIGURE 3.12 Normal and shear compon
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1 1 1 n1 = 0 , n2 = ± , n3 = ± ;
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exterior to circle C1. Thus, combin
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FIGURE 3.15A Reference angles φ an
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FIGURE E3.8-1 Three-dimensional Moh
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FIGURE 3.16A Mohr’s circle for pl
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FIGURE 3.18A Representative rotatio
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Solution For this stress state, the
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which demonstrates that ( q) is als
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Example 3.11-1 Determine directly t
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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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Page 237 and 238: ( q) ( q) σ = (λδijεkk + 2µε
Page 240 and 241: FIGURE 6.2 Simple stress states: (a
Page 242 and 243: FIGURE 6.3 For plane stress: (a) ro
Page 244 and 245: x 1x 2 plane to be one of elastic s
Page 246 and 247: FIGURE 6.4 (continued) From the def
Page 248 and 249: A most important feature of the fie
Page 250 and 251: FIGURE 6.5 (a) Plane stress problem
Page 252 and 253: For the plane strain situation (Fig
Page 254 and 255: By inserting Eq 6.6-2 into Eq 6.6-1
Page 256 and 257: FIGURE E6.7-1 (a) Rectangular regio
Page 258 and 259: Solution It is easily verified, by
Page 260 and 261: (6.7-8b) (6.7-8c) in which φ = φ
Page 262 and 263: These conditions result in the foll
Page 264 and 265: Solution From Eq 6.7-8 the stress c
Page 266 and 267: FIGURE E6.7-5 Uniaxial loaded plate
Page 268 and 269: FIGURE 6.7 (a) Cylinder with self-e
Page 270 and 271: where ψ(x 1,x 2) is called the war
Page 272 and 273: Thus, By eliminating ψ from this p
Page 274 and 275: where λ is a constant. Thus, Φ is
Page 276 and 277: Equilibrium equations, Eq 6.4-1 σi
Page 278 and 279: It should be pointed out that while
Page 280 and 281: This equation may be written in a m
Page 282 and 283: where A 1 and A 2 are constants of
Page 284 and 285: Upon setting the off-diagonal terms
Page 288 and 289: 6.16 For an elastic body whose x 3
Page 290 and 291: and B are constants, determine the
Page 292 and 293: Thus, for the beam shown the stress
Page 294 and 295: Using the sketch on the facing page
Page 296 and 297: For a fluid in motion the shear str
Page 298 and 299: The first of this pair relates the
Page 300 and 301: posed in this section are relevant
Page 302 and 303: FIGURE E7.4-1A Flow down an incline
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Page 306 and 307: where dx i is a differential tangen
Page 308 and 309: 7.9 Show that for an incompressible
Page 310 and 311: 8 Nonlinear Elasticity 8.1 Molecula
Page 312 and 313: strain. Highly cross-linked and fil
Page 314 and 315: FIGURE 8.3 A freely connected chain
Page 316 and 317: proportional to the probability per
Page 318 and 319: The deformation is volume preservin
Page 320 and 321: Note that is essentially the same a
Page 322 and 323: where dependence on I 3 has been in
Page 324 and 325: where P 11 is the 11-component of t
Page 326 and 327: This simplifies to where and ∂p
Page 328 and 329: σ yy ⎡⎛ ∂X ⎞ ⎛ ∂Y ⎞
Page 330 and 331: q( x)= Ae + Be kx −kx Eqs 8.4-19a
Page 332 and 333: FIGURE 8.6 Stresses on deformed rub
Page 334 and 335: 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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