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Finite Element Based Improved Characterization of Viscoelastic Materials

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Finite Element Based Improved Characterization of Viscoelastic Materials 1 Finite Element Based Improved Characterization of Viscoelastic Materials Xianfeng Song1*, Stefan Dircks1, Daniil Mirosnikov1, and Benny Lassen2 1Mads Clausen Institute, SDU, Sønderborg, Denmark 2DONG Energy, Fredericia, Denmark *SDU, Alsion 2, 6400, Sønderborg, Denmark, [email protected] Abstract: The objective of this study is to acquire II. Theory a full characterization of a hyperelastic material. The For the given material, linear elastic models (e.g. Neo- process is realized by performing a Dynamic Mechanical Hookean model) can not accurately describe the observed Analysis (DMA) while simultaneously extending it by behavior, so in order to overcome this problem hyperelastic advanced image processing algorithms in order to measure material models are introduced. Hyperelasticity provides the changing distance in the contracting direction. Using a means of modeling the stress-strain behavior of such COMSOL’s weak form PDE physics interface, three materials [4]. mathematical models are developed, where the strain- In this section a derivation of the governing equation is energy density function depends on different hyperelastic given. This theoretical part is required in order to perform constitutive equations. The chosen constitutive equations a finite element analysis. are Mooney-Rivlin, Yeoh and Arruda-Boyce. The results of the numerical study demonstrate that the all three models exhibit the correct trend of the non-linear behavior of the material, while the Arruda-Boyce model shows the best fitting performance. x3 B Keywords: finite element method, COMSOL, vis- −→ z3 u Bt coelastic material, image processing, dynamic mechanical −→ −→ analysis r R i3 I. Introduction z2 x2 i1 i2 z1 Viscoelastic materials have been used in many different applications, for example isolating vibration, dampening x1 noise and absorbing shock. Synthetic polymers, wood, and human tissue, as well as metals at high temperature, display significant viscoelastic effects [1]. Viscoelastic ma- Fig. 1. General coordinate system with deformed and terials exhibit several non-trivial effects such as highly undeformed bodies. non-linear dynamic behavior. When an elastic material is deformed due to an applied external force, internal Figure 1 graphically represents the general coordinate resistance counters the deformation and restores it to its −→ −→ −→ system where i1 , i2 and i3 are the three unit normal original state if the external force is no longer applied vectors. The projection of the undeformed body with [2], [3]. In general, it is inherently complex to model volume B is represented by positions x1, x2 and x3 on behaviors of such materials using a finite element software the general coordinate system, while the projection of the due to its non-linear properties. This paper describes deformed body with volume Bt is represented by positions the full characterization of a viscoelastic material. The z1, z2 and z3. acquired experimental results, from DMA and advanced The position vector from the origin (0, 0, 0) to a certain −→ image processing, are subsequently implemented in the point in the undeformed body is denoted by r : finite element software for developing an optimized model ∑3 of the material. −→ −→ −→ The image processing algorithms are developed using r = xi ii = xi ii , (1) MATLAB version R2015a. The optimization and devel- i=1 opment of the final model of the material is performed in where the latter term denotes Einsteins summation con- the finite element analysis, solver and simulation software vention, used throughout the entire paper. The corre- - COMSOL Multiphysics version 5.1. sponding position after deformation of the body is given Excerpt from the Proceedings of the 2016 COMSOL Conference in Munich 2 −→ by R. 2) Yeoh Model: −→ −→ R = zi ii . (2) ∑n ∑n ¯ i 2k −→ W = Ci0(I1 − 3) + Ck1(J − 1) , (12) In addition the displacement u is given as i=1 k=1 −→ −→ −→ u = R − r . (3) where Ci0 and Ck1 are material parameters, and we choose n = 2 in order to reduce the time to perform Hamilton’s Principle: parameter estimation. Moreover, by increasing n will not Hamilton’s principle is used to derive the governing equa- give significantly more accurate results. tion for the deformation of the viscoelastic material. The action integral is defined as 3) Arruda-Boyce Model: ∫ t2 ( ) 2 − ∑5 I = Ldt, (4) J 1 − i−1 ¯i − i W = D1 ln J + C1 αiβ (I1 3 ), (13) t1 2 i=1 where L is the Lagrangian which is defined as the differ- where D1, C1 and β are the optimized parameters and ence between the kinetic and potential energy: 1 1 1 11 19 β := N , α1 := 2 , α2 := 20 , α3 := 1050 , α4 := 7000 and 519 L = T − W, (5) α5 := 673750 . where T is the kinetic energy and W is the potential energy. In the undeformed configuration (reference con- Note that figuration) we have: I¯ = J −2/3I , (14) ∫ 1 1 1 −→2 −4/3 T = ρR v dB, (6) I¯2 = J I2, (15) B 2 −→ where B is the undeformed volume, v is the velocity and J = det(F ), (16) ρ the reference mass density. R where I and I are the first and the second invariant The potential energy W is given by: 1 2 ∫ of the unimodular component of the left Cauchy-Green deformation tensor. W = ρRedB, (7) B In order to compare with the DMA experiments results, it is needed to calculate the generated force on the lower where e is the deformation energy density, which depends surface, shown in Figure 2. on the deformation gradient F : e = e(F ), (8) with ∂zi ∂ui Fij = = + δij. (9) ∂xj ∂xj Finding the first variation of the action integral leads to the governing equation ∫ ∫ t2 ( ) ∂ui ∂δui δI = ρR − δW dBdt. (10) t1 B ∂t ∂t Equation (7) is the accurate representation of the potential energy, but due to its non-linearity, it is too complex to find the expression for the strain energy density function. Therefore the potential energy W is expressed as one of the hyperelastic constitutive models. In this case the Fig. 2. The time-dependent lower boundary condition. potential energy is expressed by Mooney-Rivlin [5] [6], Yeoh [7] [8] and Arruda-Boyce [9] [10] hyperelastic models. The force is given by ∫ ∫ 1) Mooney-Rivlin Model: F = TijnjdSt = SijNjdS, (17) A A 2 t W = C01(I¯2 − 3) + C10(I¯1 − 3) + D1(J − 1) , (11) where At is the surface area in the deformed configuration where C01, C10 and D1 are material parameters. and A is the surface area in the undeformed configuration. It is of less complexity to calculate the force in the Excerpt from the Proceedings of the 2016 COMSOL Conference in Munich 3 undeformed configuration. According to the first Piola Kirchhoff stress ∂e Sij = ρR , (18) ∂Fij since W = ρRe (19) equation (18) can be re-written into ∂W Sij = . (20) ∂Fij Combining equation (17) and (20) will give the force F . Then it is possible to calculate the error between the Fig. 4. The result of force vs. displacement, where one measured and simulated forces. can see from this hysteresis that the displacement and the force are non-linear. III. Experimental Results stretched sample (a) On this figure the (b) The final frame detection aims the left where the sample is and the right dot, re- stretched. Again the camera spectively. The sample code detects the left is unstretched. and the right dot, re- spectively. Fig. 5. The results of the image processing. The two figures on the left represents the unstretched sample and the two figure on the right represents the stretched sample. Fig. 3. Schematic setup of the dynamic mechanical analysis with image processing extension. The mounted Go Pro Hero 4 Black aims the sample on the right side. After performing the test, graphically shown in Figure 3, the acquired results are illustrated in Figure 4. The resultant force-displacement graph demonstrates the ma- terial non-linearity. Therefore it is essential to implement a hyperelastic model of higher order. Figure 5 illustrates the obtained results from image processing. The Fig. 5 (a) shows the initial position of the detected dots where the material is undeformed. On the other side Figure 5 (b) shows the final position, when the sample is under deformation. Figure 6 represents the distance between the left and right dots and how it changes over the time of the video. As it can be seen, the distance between the dots Fig. 6. Distance between the left and right dots vs. decreases as the material is being stretched. This behavior Time. As the material is being stretched the distance is expected due to the fact that the middle part contracts between the dots reduces. when the material is being stretched. Excerpt from the Proceedings of the 2016 COMSOL Conference in Munich 4 IV. COMSOL Implementation In order to perform optimization, the error between the measured data and modeled data needs to be calculated: ∫ Model Building T 2 This model is built in a three dimensional space using E1 = (F − Fmeasured) dt, (21) 0 time-dependent weak form PDE. The geometry of the ∫ T model, graphically represented in Figure 7, is based on 2 E2 = (D − Dimage processing) dt, (22) sample dimensions, including the painted dots which are 0 placed by using domain point probes. E = α1E1 + α2E2, (23) where E1, E2 and E are the error between the simulated and measured force, error between the simulated and image processing distances and the total error, respectively. In addition, α1 and α2 are weighting values. The challenge of solving time dependent integrals in COMSOL is circumvented by manually differentiating equation (23) and solving it by using the physics Global ODEs and DAEs: dE dE dE − α 1 − α 2 = 0. (24) dt 1 dt 2 dt The Nelder-Mead method is used to minimize the in- tegral of the total error, which is done by optimizing the control variables, which depend on the hyperelastic model.
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