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Unformatted text preview: CHAPTER 16 ONE DIMENSIONAL WAVE MOTION IN
VISCOELASTIC MEDIA 161 I o UCTIO TO VISCO 3 1c MA ERI s We now consider wave motion in noneﬂabiic media where the stress is no
longer directly proportional to the strain. We will limit the discussion
to small deformations and consider only one dimensional problems. The
non—elastic behavior will be described as uéécoeﬂaéxic. Viscoelasticity
theory attempts to describe material behavior where there is both eZaAiLc
and UiécouA responses. Viscoelastic materials exhibit time dependent behavior
such as creep and stress relaxation; and in addition, manifest some strain
rate and frequency dependence during dynamic loading. For linear viscoelasticity, which we will be using, the superposition
of elastic and viscous responses may be done in two ways: 1. Mechanical SpringDashpot Models 2. Hereditary Integral Constitutive Relations
Each method produces the stress—strain or constitutive relation . In this brief presentation, we shall only follow the first method. 16.2 MECHANICA! Mongls The standard Lama/L eta/sac Zap/ting and Linea/lily \M'Acows daAhpO/t are represented as E
Elastic _<___..___/\/ij__4 w
Spring US as n Viscous .é~_.r_~lnem4 F»—_mm¢—ma—
Dashpot O O (16.1) 162 where Us, 0d, 83, 8d are the applied stress and strain response of the spring and dashpot, E is the spring (elastic) moduﬂué and n is dashpot ULAQOALIy.
By combining the previous two elements in various ways, many useful
compound models which approximately describe real material behavior may be constructed. Consider the following two elementary examples: Maxwell Fluid Consider the combination of a spring and dashpot in series. E n W This model is usually referred to as a Mawvctﬂ ﬁKuLd. The stress in each element must be the same, i.e.,
o = o = o . (16.2) In addition the total strain e is equal to the sum of the strains of each element, so
e=es+€d. (16.3) Using (16.1) in (16.3) along with (16.2) gives G+%8=n€ , (16.4) which is the stress—strain response (constitutive relation) for the Maxwell fluid. Note the behavior of a Maxwell fluid: 5i
J
i
i
5 163 constant =$> S a t ll i.) For C E
_.._.t
constant =;. O a e 11 ii.) For 8 Kelvin—Voigt Solid Next consider the case of a spring and dashpot connected in parallel. This model is normally called a KQKV£nVOLgi éOZLd. The strain in each element is now the same, i.e., e = 8d = e . (16.5) The individual stresses now add, so we get o=os+od . (16.6) Equations (16.1), (16.5) and (16.6) then yield 0 = E8 + né , (16.7) 0 — t constant =¢~ E = —9~(1 — e ). Relations (16.4) and (16.7) are differential E 31111 E
i
I
which is the response of the Kelvin—Voigt solid. For this model, 0 = 00 = I
l
E
l
relations among the stress and strain, and this is typical of viscoelastic : l behavior. More complicated models are built—up and analyzed in a similar 1 manner 0 16—4 163 LONGITUDINAL WAVES IN VISCOELASTIC RODS Consider now one—dimensional elementary longitudinal wave motion down a viscoelastic rod. Recall from Newton's Law we had, equation (3.3), i.e., 80 = p azu
3X 3t2 This result is valid regardless of the constitutive model used. Maxwell Fluid Using (16.4) with 8 =43: to eliminate the displacement in the above
equation, we get
2
3 g _ _1§ 3 _.Q G = 0 (16.8)
8X c n where, c0 = ”Dim Consider a semi—infinite rod with the following initial and boundary conditions
0(X’O) = O
u(x,0) = 0
(16.9)
ﬁ(0,t) = VH(t)
O(°°,t) = 0 Taking the Laplace transform of (16.8) with respect to time gives 2 Q! 2
d 1
____ __§ (3 2 _ +.§ S)3'= 0 , . (16.10)
. n
dx c0 where we have used the fact of zero initial conditions, (16.9) The 1,2.
bounded solution of (16.10) satisfying (16.9)4 is 15(x,s) = A exp { §—¢ s2 +‘% s } , (16.11)
0 where A is an arbitrary constant. Now from the governing equation (3.3) and (16.9)3 we get &§(O,s) dX = 0V . (16.12) Combining (16.11) and (16.12) gives the constant A to be so (16.11) then becomes *4 ch
0(X,s) = — ———Jl*——— exp {— §'V s + E's} . (16.13)
0 82 +§ s
T] The inverse transform of (16.13) follows from tables to be o(x,t) = —Vpc e 10(5— V 7)H(t — *) , (16.14)
C0 0 Where 10 is the modified Bessel function of order zero. Note for the e£aAiZc caAe, n + w and the solution is 0(S)(X,t) = —VpcOH(t — X—) . (16.15)
C0
XE tE ,
Introducing the notation B = ——ﬁand T = ——, equation (16.14) becomes
CO
0 _ T/2 l 2 _ 2
0(5) — e 10(2 6 6) . (16.16) (0123458709101? 14.15 m 20 * E. H. Lee. and L Kamer. Wave propagation in ﬁnite rods of viscoelastic material J. Appl. Phy..s
V. 24.9 (1953). 16—5 Has/fr: waves W ' T"“"T“"T*1ir"”T““T“’T"'1
' I r=2 r=3 r=4 r=5 r=6‘:1'=7 [7:8 I
x
l
l
I
I
l
l
V Kelvin—Voigt Solid Equations (3.3) and (16.7) yield Again consider the semi—infinite rod problem with ﬁ(0,t) = VH(t) u(x,0) = O
ﬁ(x,0) = O
u(m,t) = 0 Taking the Laplace transform of (16.17) gives d 2; n 82
—_2(1+ES) —— dx C02 The bounded solution to (16.19) is E=o .E(X,S) = A exp f~ ————~—~*— , , (16.17) (16.18) (16.19) (16.20) 16—6 1
4
16—7 } where again A is a constant. Applying the transformed boundary condition (l6.l8)l, i.e., EKO,S) = y§~, S 1
implies that _ 3L.
A ‘ 2
S Hence (16.20) becomes E(x,s) = 1’3 exp { ———3‘—S—— } . (16.21) j
s /C 2 +‘ﬂ s :
0 p f
s
s
The inverse transform of (16.21) gives the solution g
t co /——~){l1_ 32 u(X,t) — vfo I: 10(2 gIS—g ﬁf exp( Z? — T)
(16.22) g
+l [eS erch‘§_ — V?) — eS erfc (“g—‘+ VE)]}deT i 2 2/? 2/? ’ where erfc( ) is the complementary error function defined by 2 m _u2
erfc(z) = 3% [z e du and
KC p m
[I ...
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