CONTINUUM MECHANICS for ENGINEERS

Recommendations

Info

9.28 A viscoelastic body in the form of a block is elastic in dilatation and obeys the Maxwell law in distortion. The block is subjected to a pressure impulse σ 11 = –p oδ(t) distributed uniformly over the x 1 face. If the block is constrained so that ε 22 = ε 33 = 0, determine ε 11(t) and σ 22(t). Answer: σ 22(t) = p o[(2G–3K)δ(t)/(3K+4G)–(6G/(3K+4G))e [–3K/(3K+4G)τ]t ]U(t) ε 11(t) = p o[3δ(t)/(3K+4G)–[4G/(3K+4G)K]e [–3K/(3K+4G)τ]t ]U(t) 9.29 A viscoelastic cylinder is inserted into a snug fitting cavity of a rigid container. A flat, smooth plunger is applied to the surface x1 = 0 of the cylinder and forced downward at a constant strain rate ˙ε . 11 ε If the material is modeled as the three-parameter solid shown in shear and as elastic in dilatation, determine σ11(t) and σ22(t) during the downward motion of the plunger. = o Answer: σ 11(t) = –ε o[(4Gτ/3)(1–e –t/τ )+(K+4G/3)t]U(t) σ 22(t) = ε o[(2Gτ/3)(1–e –t/τ )+(–K+2G/3)t]U(t)
9.30 For a thick-walled elastic cylinder under internal pressure po, the stresses are σr = A – B/r2 ; σθ = A + B/r2 and the radial displacement is 1+ ν given by u = [A(1 – 2ν)r + B/r] where A and B are constants E involving po, E is Young’s modulus, and ν is Poisson’s ratio. Determine σr, σθ , and u for a viscoelastic cylinder of the same dimensions that is Kelvin in shear and elastic dilatation if p = po U(t). Answer: σ r = same as elastic solution but with p o now p oU(t) σ θ = same as elastic solution but with p o now p oU(t) u(t) = (3Ar/(6K+2G))(1–e –(3K+G)t/Gτ )U(t) + (B/2Gr)(1–e –t/τ )U(t) 9.31 A viscoelastic half-space is modeled as Kelvin in shear, elastic in dilatation. If the point force P = Poe –t is applied at the origin of a stress free material at time t = 0, determine σrr(t) knowing that for an elastic half-space the radial stress is σ rr = P o[(1–2ν)A–B]/2π where A and B are functions of the coordinates only. Answer: σ rr = (P o/2π)[(3(G – η)e –t + 9Ke –(3K+G)t/η ]A/(3K + G – η) – e –t B
Page 1 and 2:
CONTINUUM MECHANICS for ENGINEERS S
Page 3 and 4:
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 and 56:
2.15 Using the square matrices belo
Page 57 and 58:
(a) Show that a multiplicity of two
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
Page 107 and 108:
FIGURE P3.9 Cylinder of radius r an
Page 109 and 110:
(b) Project each of the stress vect
Page 111 and 112:
Determine (a) the principal stress
Page 113 and 114:
(b) Verify the result determined in
Page 115 and 116:
⎡ −12 − 12 + 3 2 −12 −32
Page 117 and 118:
A motion of body B is a continuous
Page 119 and 120:
emphasize, however, that the materi
Page 121 and 122:
Example 4.2-1 Let the motion of a b
Page 123 and 124:
and v = v(X,t) = v[χ -1 (x,t),t] =
Page 125 and 126:
Additionally, with regard to the ma
Page 127 and 128:
In this equation, the first term on
Page 129 and 130:
Therefore, substituting u i for P i
Page 131 and 132:
upon X, in which case the deformati
Page 133 and 134:
form consistent with deformation an
Page 135 and 136:
For diagonal BH, and for diagonal O
Page 137 and 138:
which, upon factoring the left-hand
Page 139 and 140:
FIGURE 4.4 A rectangular parallelep
Page 141 and 142:
FIGURE 4.6A Rotated axes for plane
Page 143 and 144:
Consider once more the two neighbor
Page 145 and 146:
where dx is the deformed magnitude
Page 147 and 148:
In a similar fashion, from dX (1)
Page 149 and 150:
Thus from Eq 4.8-4 the principal st
Page 151 and 152:
and as is obvious, the inverse (U*)
Page 153 and 154:
Finally, from Eq 4.9-11, [ RAB]= =
Page 155 and 156:
which becomes (after dividing both
Page 157 and 158:
This rate of decrease in the angle
Page 159 and 160:
FIGURE 4.8 Area dS° between vector
Page 161 and 162:
FIGURE 4.9 Volume of parallelepiped
Page 163 and 164:
(a) Show that the Jacobian determin
Page 165 and 166:
4.5 The Lagrangian description of a
Page 167 and 168:
u 2 = - x 2 - (x 1 + x 2)e -t /2 +
Page 169 and 170:
and from it the longitudinal (norma
Page 171 and 172:
FIGURE P4.27 Unit square OBCD in th
Page 173 and 174:
Answer: (a) (b) 2 2 (c) (1) a1a 2a
Page 175 and 176:
FIGURE P4.35 Circular cylinder in t
Page 177 and 178:
(a) the components of acceleration
Page 179 and 180:
FIGURE P4.47A Unit cube having diag
Page 181 and 182:
5 Fundamental Laws and Equations 5.
Page 183 and 184:
which upon application of the diver
Page 185 and 186:
which is known as the continuity eq
Page 187 and 188:
FIGURE 5.1 Material body in motion
Page 189 and 190:
where ∆f is the resultant force a
Page 191 and 192:
which reduces to where we have used
Page 193 and 194:
Ε ABδ iAδ jB = e ij = 0(ε) o Ex
Page 195 and 196:
this text we consider only mechanic
Page 197 and 198:
outward normal is n i (hence the mi
Page 199 and 200:
θ = η ∂u ∂ (5.8-2) Furthermor
Page 201 and 202:
The entropy production is always po
Page 203 and 204:
One of the uses for the Clausius-Du
Page 205 and 206:
Furthermore, assume the temperature
Page 207 and 208:
In the first, a continuum body’s
Page 209 and 210:
FIGURE 5.2 Reference frames Ox 1x 2
Page 211 and 212:
Recall the definition t + = t + a f
Page 213 and 214:
where the last substitution comes f
Page 215 and 216:
With the use of Eqs 5.10-29 and 5.1
Page 217 and 218:
This being the case, it is clear fr
Page 219 and 220:
as the invariance requirement on th
Page 221 and 222:
i ∂ {P} = p (5.12-7a) i i ∂ t i
Page 223 and 224:
* 5.2 Let the property P in Eq 5.2-
Page 225 and 226:
5.12 Determine the form which the e
Page 227 and 228:
σ = - ij pδ + τ ij ij σ ij = -
Page 229 and 230:
a. b. 5.30 Show that the Jacobian t
Page 231 and 232:
FIGURE 6.1 Uniaxial loading-unloadi
Page 233 and 234:
where the factor of two on the shea
Page 235 and 236:
ased on the strain energy function.
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 286 and 287:
6.8 Show that the distortion energy
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
Page 304 and 305:
and thus which describes the pressu
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
Page 336 and 337: Answer: 8.9 Show F B iA = ij = g
Page 338 and 339: 9 Linear Viscoelasticity 9.1 Introd
Page 340 and 341: ˙ε ii ≈ Dii so that now, from E
Page 342 and 343: FIGURE 9.2 Mechanical analogy for s
Page 344 and 345: FIGURE 9.5 Three-parameter standard
Page 346 and 347: FIGURE 9.7 Graphic representation o
Page 348: Development of details relative to
Page 352 and 353: FIGURE 9.9 Stress history with an i
Page 354 and 355: γ ()= t γ ( cos ωt+ isinωt)= γ
Page 356 and 357: FIGURE 9.10 Shear lag in viscoelast
Page 358 and 359: But by Eq 9.6-7, σ 12 (t) = γ o [
Page 360 and 361: Depending upon the particular state
Page 362 and 363: () (9.7-9a) (9.7-9b) (9.7-9c) From
Page 364 and 365: and so on for higher derivatives. N
Page 366 and 367: This expression may be inverted wit
Page 368 and 369: 9.4 Develop the constitutive equati
Page 370 and 371: 9.7 For the model shown the stress
Page 372 and 373: 9.11 For the model shown determine
Page 374 and 375: 9.16 Let the stress relaxation func
Page 376 and 377: 9.24 For the rather complicated mod
Page 380: 9.32 The deflection at x = L for an
show all

CONTINUUM MECHANICS for ENGINEERS

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?