CONTINUUM MECHANICS for ENGINEERS

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Authors G. Thomas Mase, Ph.D. is Associate Professor of Mechanical Engineering at Kettering University (formerly GMI Engineering & Management Institute), Flint, Michigan. Dr. Mase received his B.S. degree from Michigan State University in 1980 from the Department of Metallurgy, Mechanics, and Materials Science. He obtained his M.S. and Ph.D. degrees in 1982 and 1985, respectively, from the Department of Mechanical Engineering at the University of California, Berkeley. Immediately after receiving his Ph.D., he worked for two years as a senior research engineer in the Engineering Mechanics Department at General Motors Research Laboratories. In 1987, he accepted an assistant professorship at the University of Wyoming and subsequently moved to Kettering University in 1990. Dr. Mase is a member of numerous professional societies including the American Society of Mechanical Engineers, Society of Automotive Engineers, American Society of Engineering Education, Society of Experimental Mechanics, Pi Tau Sigma, Sigma Xi, and others. He received an ASEE/NASA Summer Faculty Fellowship in 1990 and 1991 to work at NASA Lewis Research Center. While at the University of California, he twice received a distinguished teaching assistant award in the Department of Mechanical Engineering. His research interests include design with explicit finite element simulation. Specific areas include golf equipment design and vehicle crashworthiness. George E. Mase, Ph.D., is Emeritus Professor, Department of Metallurgy, Mechanics, and Materials Science (MMM), College of Engineering, at Michigan State University. Dr. Mase received a B.M.E. in Mechanical Engineering (1948) from the Ohio State University, Columbus. He completed his Ph.D. in Mechanics at Virginia Polytechnic Institute and State University (VPI), Blacksburg, Virginia (1958). Previous to his initial appointment as Assistant Professor in the Department of Applied Mechanics at Michigan State University in 1955, Dr. Mase taught at Pennsylvania State University (instructor), 1950 to 1951, and at Washington University, St. Louis, Missouri (assistant professor), 1951 to 1954. He was appointed associate professor at Michigan State University in 1959 and professor in 1965, and served as acting chairperson of the MMM Department, 1965 to 1966 and again in 1978 to 1979. He taught as visiting assistant professor at VPI during the summer terms, 1953 through 1956. Dr. Mase holds membership in Tau Beta Pi and Sigma Xi. His research interests and publications are in the areas of continuum mechanics, viscoelasticity, and biomechanics.
Nomenclature x 1, x 2, x 3 or x i or x Rectangular Cartesian coordinates x 1 * , x2 * , x3 * Principal stress axes ˆ , ˆ , ˆ e e e 1 2 3 δ ij ε ijk ∂ t ∂ x φ = grad φ = φ ,j v = ∂ jv i = v i,j ∂ jv j = v j,j ε ijkv k,j Unit vectors along coordinate axes Kronecker delta Permutation symbol Partial derivative with respect to time Spatial gradient operator Scalar gradient Vector gradient Divergence of vector v Curl of vector v b i or b Body force (force per unit mass) p i or p Body force (force per unit volume) f i or f Surface force (force per unit area) V Total volume V .o Referential total volume ∆V Small element of volume dV Infinitesimal element of volume S Total surface S o Referential total surface ∆S Small element of surface dS Infinitesimal element of surface ρ Density ni or ˆn Unit normal in the current configuration NA or ˆN Unit normal in the reference configuration ( ) ( ) nˆ nˆ ti or t Traction vector
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: Reference Section for constitutive
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
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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
Page 85 and 86:
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
Page 91 and 92:
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
Page 101 and 102:
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
Page 113 and 114:
(b) Verify the result determined in
Page 115 and 116:
⎡ −12 − 12 + 3 2 −12 −32
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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] =
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Additionally, with regard to the ma
Page 127 and 128:
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
Page 143 and 144:
Consider once more the two neighbor
Page 145 and 146:
where dx is the deformed magnitude
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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]= =
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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
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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 +
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and from it the longitudinal (norma
Page 171 and 172:
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
Page 187 and 188:
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
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
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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
Page 217 and 218:
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-
Page 225 and 226:
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
Page 231 and 232:
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
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
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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 ⎞
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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
Page 336 and 337:
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
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
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γ ()= t γ ( cos ωt+ isinωt)= γ
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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
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