56 S. Taguchi et al. 6.2 Objective This study proposes a method for evaluating stresses from measured in-plane strains of the viscoelastic body under a plane stress condition. A through-thickness strain is obtained from in-plane strains based on pseudoelasticity in the linear viscoelastic theory. That is, the in-plane strains as well as the through-thickness strain is computed in a Laplace domain. The Laplace transformation is performed numerically using a method developed by the authors [8]. The stresses are computed from the through-thickness strains and in-plane strains and by numerical integration. The effectiveness of the proposed method is demonstrated by applying it to viscoelastic stress analysis. 6.3 Constitutive Equation of Viscoelasticity The three-dimensional constitutive equations of viscoelasticity are expressed by [9]. σkk(t) =3K(t)εkk(0) +3 t 0 K(t −τ) ∂εkk (τ) ∂τ dτ (6.1) sij(t) =2G(t)eij(0) +2 t 0 G(t −τ) ∂eij (τ) ∂τ dτ (6.2) where σkk(t) is the sum of normal stress components, sij(t) is the deviatoric stress, eij(t) is the deviatoric strain, and K(t) is the relaxation bulk modulus and G(t) are relaxation shear modulus. It can be recognized from Eqs. (6.1) and (6.2) that the through-thickness strain εzz(t) is required for computing stresses from strains even if an object is under the plane stress condition. Because the Poisson’s ratio of a viscoelastic material usually depends on time and temperature, it is difficult to obtain through-thickness strains. In the present study, the through-thickness strains are obtained based on pseudoelasticity in the linear viscoelastic theory. This is the property that the constitutive equation of a linear viscoelastic material is represented in the same form as an elastic body in Laplace domain. Therefore, the through-thickness strain in Laplace domain is expressed as ∼εzz(s) =− s∼υ(s) 1−s∼υ(s) ∼εxx(s) + ∼εyy(s) (6.3) where ∼εzz(s), ∼υ(s), ∼εxx(s),and ∼εyy(s) are the through-thickness strain, the Poisson’s ratio, and the normal strains in the Laplace domain, respectively. The through-thickness strain εzz(t) is obtained from ∼εzz(s) by Laplace inverse transformation. Then, the stresses can be computed from obtained values and in-plane strains by the numerical integration using Eqs. (6.1) and (6.2). 6.4 Numerical Laplace Transformation Using FFT Taking the variable s in the Laplace transformation and the inverse transform equations as s =γ +iωand considering that f (t) =0 at t < 0, the Laplace transformation can be expressed as [9] F (ω) = ∞ −∞ f(t) · e−γt · e−iωt dt (6.4) f(t) = eγt 2π γ+i∞ γ−i∞ F (ω) · eiωt dω (6.5)
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