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Unformatted text preview: 40 .' 2hr Menuhin nl 1h: t 'illlallltllivc Equation which is one of the equations of {4]. Other t‘tlllllltlll.'i III trll can he explained
in a similar manner. For shear, the stress a” lllltl Ihe strain a” ll sell are
directly proportional. 2.9 The Effect of Temperature In the preceding sections, the constitutive equations are stated at a given
temperature. The viscosity of a ﬂuid, however, varies with temperature
(think of the motor oil in your car) as does the elastic modulus of a solid. In
other words, 93W in Eq. (23:1) and Cu“ in Eq. [2.2M] are functions of
temperature and are coeﬂicicnts determined under an isothermal experiment
(with temperature kept uniform and constant). If the temperature ﬁeld is variable, Hooke‘s law must be modiﬁed into the
Dahamei—Neamann form. Let the elastic constants C£me be measured at a
uniform constant temperature To. Then if the temperature changes to T,
we put on = Cameo * ﬁniT — To): (1) in which 13,, is a symmetric tensor, measured at zero strains. For an isotropic
material, the second order tensor [in must also be isotropic. It follows that
ﬁb— must be of the form [35”, hence rm = deikétj + 268;} "— MT T Tomi!" (2) Here it and G are Lame constants measured at constant temperature. The
inverse can be written as
l + v v ij— E a'Ij—Eaursu+ot(T— Toldﬁ. The constant a is the coefficient of linear expansion. 2.10 Materials with More Complex Mechanical Behavior Nonviscous ﬂuids, Newtonian ﬂuids, and lIookean elastic solids are abstrac
tions. No real material is known to behave exactly as one of these, although
in limited ranges of temperature, stress, and strain, some materials may
follow one of these laws accurately. They are the simplest laws we can devise
to relate the stress and strain, or strain rate. They are not expected to be
exhaustive. Almost any real material has a more complex behavior than these simple
laws describe. Among Iluids, blood is nonNewtonian lluust‘llnltl paints and
varnish, wct clay and mttd, and most colloidal solutions ttl't‘ nunNewtonian.
litn' solids. most structural materials are, liu'llnuttnly. llunlutult ill the tIHcl‘ttl
range ol‘ stresses and strains; httt hcyund col'lnhl lllllill. llnulto'n lhw no i ZJ l Viscoelasticity 41 longer applies. For example, virtually every known solid material can be
broken (fractured) one way or another, under sufﬁciently large stresses or
strains; but to break is to disobey Hooke’s law. Few biological tissues obey
Hooke‘s law. Their properties will be discussed in the following chapters. 2.11 Viseoelasticity When a body is suddenly strained and then the strain is maintained constant
afterward, the corresponding stresses induced in the body decrease with
time. This phenomena is called stress relaxation. or relaxation for short. If
the body is suddenly stressed and then the stress is maintained constant
afterward, the body continues to deform, and the phenomenon is called
great). If the body is subjected to a cyclic loading, the stress—strain relationship
in the loading process is usually somewhat different from that in the un
loading process, and the phenomenon is called hysteresis.
The features of hﬁtfresis, relaxation, and creep are found in many
materials. Collectively, they am called features of aiseoelasrieiry.
Mechanical models are often used to discuss th?ﬁ§cﬁaastic behavior
of materials. In Fig. 2.11:1 are shown three mechanical models of material
behavior, namely, the Maxwell model, the Voigt model. and the Kelvin model [also called the standard linear solid). all of which are composed ofcombina (a) A Maxwell body '1
films [1
F VIII”. F
m
h——ut “3—H (b) A Voigt body_ '1 Fl a F; {c} A Kelvin bodylta standard linear solid) ’l'l tut Fl irllurc ll I: l 'l‘hree mechanical mmlelu ul' vlutsnelnntic Imilcrhtl. [a] A Maxwell hotly.
bill Vela! hotly. and [cl a Kelvin hotly [a standard llucttr unlitll. done of linear uprlnus with sprlllu uuntttlntt Jt tll‘lll tllltll'tl'ltlltl wlth eoelttetent
of viscosity n. A “new spring is supposed to produce Instnntnneotmly n
dcl'tn'n'nttinn proput'tlmutl to the loud. A dashpot is supposed to produces
velocity proportional to the load in any instttnt. Thus il' l" is the force acting
in :I spring and n is its extension. then P“! tili."ll'lhe l'orce t'":tt:lsnn :1 dashpot.
it will produce it velocity ul‘ delleetion ti. and l'  nth' ‘1e shock absorber
on an airplnne‘s landing gear is an example ol'n dashpot ., ow, in u Maxwell
model. shown in Fig. 2.1 l : Hit). the same force is transmitted from the spring
to the dasl'ipot. This force produces a displacement t‘"/,u in the spring and
:1 velocity FM in the dashpot. The velocity of the spring extension is F/p
it" we denote a differentiation with respect to time by a clot. The total velocity
is the sum oflhese two: _ F F
u=_~+— .s(Maxwell model}. (1)
ﬂ 7? li‘urthermore, if the force is suddenly applied at the instant of time t = 0,
the spring will be suddenly deformed to MD) = F{0)/,u, but the initial dashpot
dclicclion would be zero, because there is no time to deform. Thus the initial
condition for the differential equation (1) is F(0) 0=—# 2
at} # (J is For the Voigt modet, the spring and the dashpot have the same diSplace ment. 1f the displacement is u, the velocity is it, and the spring and dashpot
will produce forces put and ml, respectively. The total force F is therefore F = int +nti (Voigt model). (3)
If F is suddenly applied, the appropriate initial condition is
n(0) = 0. (4) For the Kelvin model {or standard linear model), let us break clown the
displacement it into ul of the dashpot and it} for the spring, whereas the total force F is the sum of the force F0 from the spring and F1 from the Maxwell
element. Thus = + ’, b F=F +F,
(a) ” I"1 “1“\ ( ) a . 1 f (5)
(0} F0 = #0“: ‘ {(1) F1 = ’311‘12 #131 From this we can verify by substitution that F = #0“ + Mi“’1=(#o + #1} H “’ Pitui
Hence F+ﬂF=(#D+ul)u_—#lu1+1£ (#o‘i‘ﬂiiﬁ—ﬂtﬁl #1 H1 21" f Ilplﬂutttﬂ tltu ItIIIt tol‘tlt try pin] tutu llllllﬂ ml. 1.1:”. wt. mu...” t I ”' t5‘= it.“ I M1 l “")n. {6)
t”: .“I
Tltll nqnlnion can be written in the form
II ‘4 II— '" (Kelvin model), (7’)
Will‘lt‘
Ti. 2 I‘ll—ll? To = 3—1 + E)! ER z “0 ttl o #1
Lot it suddenly applied force F(0) and displacement u(0), the initial c::diti0r;” “(mME“
1217(0) = Ekrgum). (9)343) For reasons that will become clear below, the constant 1,. is called the
relaxation time for constant strain. whereas 1:cr is called the relaxation time
)‘tn' t'mtsttmr stress. II' we solve Eqs. (I), (3), and ('F) for u{t) when Flt) is a unitstep junction
lit). the results are called creep functions, which represent the elongation
n odneed by a sudden application at t = 0 ofa constant force of magnitude
nutty. They are: Mnswell solid: (’0‘) = + l r) Ill); {10)
.1” ll Vitth solid:
c('t}':= —e""”f‘%)l_(t}_, (11) J ntnndnrd linear solid: . _ + where the unitstep function 10:) is deﬁned as [see Fig. 2.11 :2(a)] 1;: [Fitill”. (12) 1 when t > 0,
HI) = s when r = 0, (l3)
0 when t «r. 0. A body that obeys a loaddeflection relation like that given by Maxwell’s
Inmch is said to be a Maxwell solid. Since a dashpot behaves as a piston
moving in a viscous ﬂuid, the abovenamed models are called models of
viscoelasticity. Interchanging the roles of F and u. we obtain the relaxation function as
n response Flt) = ktt} corresponding to an elongation ntt) = 1(t). The re
laxation function ktt) is the force that must be applied in order to produce (n) A unlliluplhnullon Ill) 4————oo 0 Time: +00 (b) A unitimpulse function 5(1) With height tending to no
but area under the curve = l 0 Time t Figure 2.11 :2 (a) A unitstep function 1(r). (b) A unitimpulse function 5(t). The central
spike has a height tending to so but the area under the curve remains to be unity. an elongation that changes at r = 0 from zero to unity and remains unity
thereafter. They are Maxwell solid: k0) = tie—menu}, (14)
Voigt solid: k0) = n56} + #10). (15} standard linear solid: k(t) = 5,,[1 # (l — (“Jun (16) Here we have used the symbol 6(r) to indicate the unitimpulse function or
Diracdelta function, which is deﬁned as a function with a singularity at the
origin (see Fig. 2.11:2(bl): 5(t) = 0 (for r < 0, and t > 0),
remand: = no) (a > 0), where ﬁt) is an arbitrary function, continuous at I: 0. These functions,
e(t) and Mt), are illustrated in Figs. 2.11 :3 and 2.11:4, respectively, for which
we add the following comments. For the Maxwell solid, the sudden application of a load induces an
immediate deﬂection by the elastic spring, which is followed by “creep”
of the dashpot. 0n the other hand, a sudden deformation produces an
immediate reaction by the spring, which is followed by stress relaxation
according to an exponential law [see Eq. (14)]. The factor n/u, with the dimension of time, may be called the relaxation time: it characterizes the
rate of decay of the force. (17) b S?
‘r" E
a E
2% E;
l 3 '3
' m a L?
'l ime Time Time
{U} ibl (c)
I tutor .'. l l 2.1 Creep functions of (a) a Maxwell. (b) a Voigt, and (c) a standard linear
mlhl A negative phase is superposed at the time of unloading.
1'}8(.f‘.fo)
II: 8 g
IE1». E 8
t s ’° s‘
i O O J: “‘5 “a
I I C1 0
Time ’0 Time Time
(a) (bl (cl 1 more 2.1 l :4 Relaxation functions of (a) a Maxwell, (b) a Voigt, and (c) a standard
IIIH'HI' solid. For the Voigt Solid, a sudden application of force will produce no im
mediate deﬂection, because the dashpot, arranged in parallel with the spring,
will not move instantaneously. Instead, as shown by Eq. (1 l) and Fig. 2.11 :3,
.1 deformation will be gradually built up, while the spring takes a greater
and greater share of the load. The dashpot displacement relaxes expo
nentially. Here the ratio n/n is again a relaxation time: it characterizes the
tale of relaxation of the dashpot. For the standard linear solid, a similar interpretation is applicable. The
constant I, is the time of relaxation of load under the condition of constant
ilelleetion [see Eq. (16)], whereas the constant 1:, is the time of relaxation
of deflection under the condition of constant load [see Eq. (12)]. As I —> 00.
the dashpot is completely relaxed, and the loaddeﬂection relation becomes
Ihal of the springs, as is characterized by the constant E R in Eqs. (12) and
(In). Therefore, E R is called the relaxed elastic modulus. Maxwell introduced the model represented by Eq. (3), with the idea that
all fluids are elastic to some extent. Lord Kelvin showed the inadequacy of
the Maxwell and Voigt models in accounting for the rate of dissipation of
energy in various materials subjected to cyclic loading. Kelvin’s model is ...
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 Fall '10
 Various
 Tissue Engineering

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