CONTINUUM MECHANICS for ENGINEERS

Recommendations

Info

6 Linear Elasticity 6.1 Elasticity, Hooke’s Law, Strain Energy Elastic behavior is characterized by the following two conditions: (1) where the stress in a material is a unique function of the strain, and (2) where the material has the property for complete recovery to a “natural” shape upon removal of the applied forces. If the behavior of a material is not elastic, we say that it is inelastic. Also, we acknowledge that elastic behavior may be linear or non-linear. Figure 6-1 shows geometrically these behavior patterns by simple stress-strain curves, with the relevant loading and unloading paths indicated. For many engineering applications, especially those involving structural materials such as metals and concrete, the conditions for elastic behavior are realized, and for these cases the theory of elasticity offers a very useful and reliable model for design. Symbolically, we write the constitutive equation for elastic behavior in its most general form as = G() ε (6.1-1) where G is a symmetric tensor-valued function and ε is any one of the various strain tensors we introduced earlier. However, for the response function G in this text we consider only that case of Eq 6.1-1 for which the stress is a linear function of strain. Also, we assume that, in the deformed material, the displacement gradients are everywhere small compared with unity. Thus, the distinction between the Lagrangian and Eulerian descriptions is negligible, and following the argument of Eq 4.7-3 we make use of the infinitesimal strain tensor defined in Eq 4.7-5, which we repeat here: 1 ⎛ ∂ u ⎝ ∂ u ⎞ ⎠ 1 ⎛ ∂ u ⎝ ∂ u ⎞ ⎠ i j i j εij = (6.1-2) 2 ⎜ + u u ∂ X ∂ X ⎟ = j i 2 ⎜ + i,j j,i ∂ x j ∂ x ⎟ = + i 2 1 ( )
FIGURE 6.1 Uniaxial loading-unloading stress-strain curves for (a) linear elastic; (b) nonlinear elastic; and (c) inelastic behavior. Within the context of the above assumptions, we write the constitutive equation for linear elastic behavior as σij = Cijkm εkm or = Cε (6.1-3) where the tensor of elastic coefficients C ijkm has 3 4 = 81 components. However, due to the symmetry of both the stress and strain tensors, it is clear that C ijkm = C jikm = C ijmk which reduces the 81 possibilities to 36 distinct coefficients at most. (6.1-4)
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: a. b. 5.30 Show that the Jacobian t
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 378 and 379:
9.28 A viscoelastic body in the for
Page 380:
9.32 The deflection at x = L for an
show all

CONTINUUM MECHANICS for ENGINEERS

Create successful ePaper yourself

Delete template?

Save as template?