Chapter 6 Evaluating Stresses from Measured Strains in Viscoelastic Body Using Numerical Laplace Transformation S. Taguchi, K. Takeo, and S. Yoneyama Abstract This study proposes a method for computing stresses from measured values of in-plane strains in viscoelastic body under plane stress condition. Since Poisson’s ratio depends on time and temperature, it is difficult to evaluate stresses from in-plane strains unless Poisson’s ratio is treated as a constant. This research focuses on pseudoelasticity in the linear viscoelasticity, and the stresses are computed using a numerical Laplace transformation. Since the relation between a throughthickness strain and in-plane strains is expressed in Laplace domain, Poisson’s ratio can be treated as time- and temperaturedependence. The Laplace transformation is performed numerically using a method developed by the authors. The stresses are computed from the through-thickness strains and in-plane strains by numerical integration. Therefore, the stresses are computed ignoring the effect caused by using the constitutive equations in Laplace domain. The effectiveness of the proposed method is demonstrated by computing stresses from strains. Results show that the stresses can be evaluated from in-plane strains even if the Poisson’s ratio exhibits time- and temperature-dependence more accurate than the developed method. Keywords Viscoelasticity · Plane stress · Numerical Laplace transformation · Pseudoelasticity 6.1 Introduction Not only in-plane strains but a through-thickness strain exists in an object under plane stress condition even if a throughthickness stress is vanishingly small. In the case of an elastic body, in-plane stresses can be computed from in-plane strains because a through-thickness strain is obtained from in-plane strains through a relation with a constant Poisson’s ratio. A similar situation can be seen in the case of a plastic body which exhibits incompressibility. In the case of a viscoelastic body, on the other hand, a through-thickness strain cannot be obtained from in-plane strains because the Poisson’s ratio depends on time and temperature. Therefore, it is difficult to evaluate stresses from measured values of in-plane strains in a viscoelastic body under plane stress condition. Conventionally, the Poisson’s ratio of a viscoelastic body is often treated as a constant [1, 2]. In this case, in-plane stresses can be obtained from in-plane strains as with elastic and plastic bodies. However, the Poisson’s ratio actually exhibits time and temperature-dependence [3–7], and thus, the assumption of a constant Poisson’s ratio can induce an error in stress analysis. The stress-strain relation of a viscoelastic body is represented by a higher-order differential equation. The constitutive equation of a linear viscoelastic material is represented in the same form as an elastic body in Laplace domain. This relation is called the correspondence law. Since the relation between a through-thickness strain and in-plane strains is expressed in Laplace domain, Poisson’s ratio can be treated as time- and temperature-dependence. The authors have proposed a method for evaluating stresses from measured in-plane strains of the viscoelastic body under a plane stress condition using numerical Laplace transformation [8]. By using this method, stresses can be evaluated from in-plane strains even if Poisson’s ratio exhibits time- and temperature-dependence. However, the accuracy of calculation by the method for computing stresses using numerical Laplace transformation depends on the constitutive equation in Laplace domain. Therefore, it is necessary that stresses can be evaluated more accurately. S. Taguchi ( ) · K. Takeo · S. Yoneyama Department of Mechanical Engineering, Aoyama Gakuin University, Sagamihara-shi, Kanagawa, Japan e-mail: c5619123@aoyama.jp; takeo@me.aoyama.ac.jp; yoneyama@me.aoyama.ac.jp © The Society for Experimental Mechanics, Inc. 2021 M.-T. Lin et al. (eds.), Advancement of Optical Methods & Digital Image Correlation in Experimental Mechanics, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-59773-3_6 55
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