viscoelastic+materials - 40 .' 2hr Menuhin nl 1h:- t...

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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‘t|lllllltlll.'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'I-j—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 l-Iookean 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 non-Newtonian lluust‘llnltl paints and varnish, wct clay and mttd, and most colloidal solutions ttl't‘ nun-Newtonian. litn' solids. most structural materials are, liu'llnuttnly. llunlutult ill the tIHcl‘ttl range ol‘ stresses and strains; httt hcyund col'lnhl lllllill. l-lnulto'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;” “(m-ME“ 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 unit-step 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 unit-step function 10:) is defined as [see Fig. 2.1-1 :2(a)] 1;: [Fit-ill”. (12) 1 when t > 0, HI) = s when r = 0, (l3) 0 when t «r. 0. A body that obeys a load-deflection 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 above-named 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 unll-iluplhnullon Ill) 4————oo 0 Time: +00 (b) A unit-impulse function 5(1) With height tending to no but area under the curve = l 0 Time t Figure 2.11 :2 (a) A unit-step function 1(r). (b) A unit-impulse 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 unit-impulse function or Dirac-delta 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‘ 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 load-deflection 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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viscoelastic+materials - 40 .' 2hr Menuhin nl 1h:- t...

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