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A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

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22 H. T. Banks, S. H. Hu <strong>and</strong> Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51<br />

Figure 4: Stress <strong>and</strong> strain histories in the stress relaxation test.<br />

Figure 5: Stress <strong>and</strong> strain histories in the creep test.<br />

while <strong>for</strong> viscoelastic fluids the stress vanishes to zero, i.e.,<br />

Creep<br />

lim G(t) = 0.<br />

t→∞<br />

In a creep test, a constant stress σ 0 acts as ”input” to the material from time t 0 , the<br />

resulting time-dependent strain is increasing as depicted in Fig. 5.<br />

The strain function J(t) resulting from the unit step stress (i.e., σ 0 = 1) is called the<br />

creep compliance.<br />

In a creep test, the resulting strain <strong>for</strong> viscoelastic solids increases until it reaches a<br />

nonzero equilibrium value, i.e.,<br />

lim J(t) = J ∞ > 0,<br />

t→∞<br />

while <strong>for</strong> viscoelastic fluids the resulting strain increases without bound as t increases.<br />

Hysteresis<br />

Hysteresis can be seen from the stress-strain curve which reveals that <strong>for</strong> a viscoelastic<br />

material the loading process is different than in the unloading process. For example,<br />

the left plot in Fig. 6 illustrates the associated stress-strain curve <strong>for</strong> the Hookean elastic<br />

solid, <strong>and</strong> that in the right plot <strong>of</strong> Fig. 6 is <strong>for</strong> the Kelvin-Voigt model (a linear<br />

viscoelastic model discussed below in Section 3.2.3). From this figure, we see that we<br />

can differentiate between the loading <strong>and</strong> unloading <strong>for</strong> the Kelvin-Voigt material, but<br />

we cannot do this <strong>for</strong> Hookean elastic material. Thus the Kelvin-Voigt material ”remembers”<br />

whether it is being loaded or unloaded, hence exhibiting ”hysteresis” in<br />

the material.

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