A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

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34 H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51 Finally the usual arguments for the standard linear model lead to G ∗ = σ 0 ε 0 exp(iδ) = κ r(1 + τ σ τ ε ω 2 ) 1 + (τ ϵ ω) 2 + i κ r(τ σ − τ ε )ω 1 + (τ ϵ ω) 2 . By definition of the storage and loss modulus, we thus have G ′ = κ r(1 + τ σ τ ε ω 2 ) 1 + (τ ϵ ω) 2 , G” = κ r(τ σ − τ ε )ω 1 + (τ ϵ ω) 2 . Remark 3.5. It was demonstrated in [45] that the relaxation behavior of compressed wood can be adequately described by the standard-linear-model. The Boltzmann superposition model A general approach widely used to model linear viscoelastic materials is due to Boltzmann (1844-1906) and is called the Boltzmann superposition model or simply the Boltzmann model. If the origin for time is taken at the beginning of motion and loading (i.e., σ(0) = 0 and ε(0) = 0), then the stress-strain law is given by ∫ t σ(t) = κ r ε(t) + 0 K(t − s) dε(s) ds, (3.19) ds where κ r represents an instantaneous relaxation modulus, and K is the ”gradual” relaxation modulus function. The relaxation modulus function G(t) for the Boltzmann model (3.19) is given by G(t) = κ r + K(t). Note that ϵ(0) = 0. Hence, (3.19) can be rewritten as follows σ(t) = ∫ t 0 G(t − s) dε(s) ds. (3.20) ds If the strain ϵ(t) has a step discontinuity at t = 0, then by integration of the resulting delta function for its derivative we obtain the following representation from (3.20) ∫ t σ(t) = G(t)ε(0) + 0 G(t − s) dε(s) ds. (3.21) ds This results in a decaying stress if the strain is held constant after a step discontinuity. The interested reader can refer to [16, pp. 5–6] for more information on the connection of several different forms of the Boltzmann superposition model. We find that when model (3.19) is incorporated into force balance laws (equation of motion), it results in integro-partial differential equations which are most often phenomenological in nature as well as being computationally challenging both in simulation and control design.
H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51 35 Remark 3.6. Any constitutive equation of form (3.15), along with the appropriate initial conditions, can be expressed in either the form (3.19) (or (3.21)) or its corresponding compliance form (i.e., the strain in terms of the stress and stress rate). On the other hand, a constitutive equation of form (3.19) (or (3.21)) can be reduced to the form (3.15) if and only if the stress relaxation modulus satisfy specific conditions. A detailed discussion of this can be found in [28]. For example, the Maxwell, Kelvin-Voigt, and standard linear models can be expressed in the Boltzmann formulation. The choice of parameters in (3.19) to yield these models are readily verified to be: 1. The Maxwell model: κ r = 0, and K(t − s) = κ exp [ − (t − s)/τ ] with τ = η/κ; 2. The Kelvin-Voigt model: κ r = κ, and K(t − s) = ηδ(t − s); 3. The standard linear model: K(t − s) = ( τ σ /τ ε − 1 ) κ r exp [ − (t − s)/τ ε ] , which can be simplified as K(t − s) = κ 1 exp [ − (t − s)/τ ε ] with τε = η 1 /κ 1 . Internal variable approach Other special cases of (3.19) that are often encountered include a generalization of the single spring-dashpot paradigm of Fig. 14 (the standard linear model) to one with multiple spring-dashpot systems in parallel as depicted in Fig. 17. This is a generalized standard linear model (also referred to as generalized Maxwell model or Wiechert model or Maxwell-Wiechert model in the literature) that results in the relaxation response kernel K(t − s) = n ( ∑ κ j exp − t − s ) , (3.22) τ j=1 j where τ j = η j /κ j , j = 1, 2, · · · , n. We observe that the Boltzmann formulation with relaxation response kernel (3.22) is equivalent to the formulation σ(t) = κ r ε(t) + where the ε j satisfy the following ordinary differential equations dε j (t) dt n ∑ κ j ε j (t), (3.23) j=1 + 1 ε j (t) = dε(t) , ε j (0) = 0, j = 1, 2, · · · , n. (3.24) τ j dt The formulation (3.23) with (3.24) is sometimes referred to as an internal variable model (e.g., see [7,12]–for other discussions of internal state variable approaches, see also [35, 36, 51–53]) because the variables ε j can be thought of as ”internal strains” that are driven by the instantaneous strain according to (3.24) and that contribute to the total stress via (3.23). Hence, we can see that this internal variable modeling leads to an efficient computational alternative for the corresponding integro-partial differential equation models involving (3.19). In addition, we note that this approach provides a ”molecular” basis for the models as it can be thought of as a model for a heterogeneous material containing multiple types of molecules [7,12,19,32], each possessing a
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A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

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