A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

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30 H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51 Figure 11: Schematic representation of the Kelvin-Voigt model. Because the two elements are subject to the same strain, the model is also known as an iso-strain model. The total stress is the sum of the stress in the spring and the stress in the dashpot, so that σ = κε + η dε dt . (3.17) We first consider the stress relaxation function and creep function for the Kelvin-Voigt model (3.17). The stress relaxation function corresponds to the solution of (3.17) when We find ε(t) = ε 0 H(t − t 0 ), and σ(0) = 0. σ(t) = κε 0 H(t − t 0 ) + ηε 0 δ(t − t 0 ). This stress relaxation function for the Kelvin-Voigt model (3.17) is illustrated in Fig. 12. The creep function again is the solution ε(t) to (3.17) corresponding to σ(t) = σ 0 H(t − t 0 ) and ε(0) = 0. We find κε + η dε dt = σ 0H(t − t 0 ). Let ˆε(s) = L {ε(t)}(s), or in terms of the Laplace transform we have e ˆε(s) = −t 0s σ 0 s(κ + ηs) = σ [ 0 e −t 0 s − e−t 0s ] . κ s s + κ/η Using the inverse Laplace transform we obtain ε(t) = 1 κ [ ( 1 − exp − κ )] η (t − t 0) σ 0 H(t − t 0 ). The corresponding creep function for the Kelvin-Voigt model (3.17) is illustrated in Fig. 13 (compare with Fig. 5). Thus we find that the Kelvin-Voigt model is extremely accurate in modelling creep in many materials. However, the model has limitations in its ability to describe the commonly observed relaxation of stress in numerous strained viscoelastic materials.
H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51 31 Figure 12: Stress relaxation function for the Kelvin-Voigt model. Figure 13: Creep function of Kelvin-Voigt model. Finally we may obtain the storage modulus G ′ and loss modulus G” for the Kelvin- Voigt model (3.17). Letting ε(t) = ε 0 exp(iωt), and σ(t) = σ 0 exp ( i(ωt + δ) ) , and substituting ε and σ into (3.17), we find Hence, by (3.14) we have G ∗ = σ 0 ε 0 exp(iδ) = κ + iηω. G ′ = κ, G” = ηω. Remark 3.4. Because of one of our motivating applications (discussed in Section 4 below), we are particularly interested in the elastic/viscoelastic properties of soil. Hardin in [31] presented an analytical study of the application of the Kelvin-Voigt model to represent dry soils for comparison with test results. From this study he found that the Kelvin-Voigt model satisfactorily represented the behavior of sands in these small-amplitude vibration tests if the viscosity η in the model was treated as varying inversely with the frequency ω to maintain the ratio ηω/κ constant. Since Hardin’s work, describing soils as a Kelvin-Voigt material has become accepted as one of the best ways in soil dynamics of calculating wave propagation and energy dissipation. The Kelvin-Voigt model also governs the analysis of standard soil tests, including consolidation and resonant-column tests. The interested reader can refer to [40, 41, 47] as well as the references therein for more information of the model’s historical backgrounds and examples of its application in soil dynamics. It is interesting to note that the author in [13] showed that the Kelvin-Voigt model can be used to describe the dynamic response of the saturated poroelastic materials
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A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

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