Determination of the Relaxation Modulus of a Linearly Viscoelastic Material

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Determination of the Relaxation Modulus of a Linearly Viscoelastic Material Joonas Sorvari and Matti Malinen Department of Physics, University of Kuopio, PO Box 1627, FIN-7211 Kuopio, Finland; E-mail

1 Determination of the Relaxation Modulus of a Linearly Viscoelastic Material Joonas Sorvari and Matti Malinen Department of Physics, University of Kuopio, PO Box 1627, FIN-7211 Kuopio, Finland; joonas.sorvari@uku.fi Abstract. In this paper, a numerical method for computing the relaxation modulus of a linearly viscoelastic material is presented. The method is valid for relaxation tests where a constant strain rate is followed by a constant strain. The method is similar to the procedure suggested by Zapas and Phillips. Unlike Zapas-Phillips approach, this new method can be also applied for times shorter than t 1/2, where t 1 denotes time when the maximum strain is achieved. Therefore this method is very suitable for materials that experiences fast relaxation. The method is verified with numerical simulations. Results from the simulations are compared with analytical solution and Zapas-Phillips method. Results indicate that the presented approach is suitable for estimating the relaxation modulus. Keywords: numerical algorithm, relaxation test, finite ramp time, relaxation modulus, linear viscoelasticity 1. Introduction The fundamental behavior of a linearly viscoelastic material depends on the relaxation modulus. The relaxation modulus can be determined by applying a step-strain. As it is well known, in practice the stepstrain test cannot be performed due to a infinite short ramp time. In this work it is assumed that the constant strain level ǫ is applied with the constant rate of strain ǫ. The ramp time is taken to be t 1, thus ǫ = ǫ t 1. Zapas and Phillips (1971) developed a method where the true relaxation time is t t 1 /2. The Zapas-Phillips method is simple to use, but it cannot be applied times shorter than t 1 /2. For materials with significant stress decay in the beginning of the relaxation test it is also necessary that relaxation modulus can be determined in the time period t < t 1 /2. Lee and Knauss (2) derived a forward and backward recursive procedures for the determination of relaxation modulus from ramp test. These methods are very accurate if the following requirements are fulfilled. For the backward computation technique the factor-of-1 rule (Meissner, 1978) must hold, that is, at t 1t 1 relaxation modulus can be determined from the step-strain case, i.e. E(t) = σ(t)/ǫ, where E(t) is the relaxation modulus. Since the backward method is recursive c 26 Kluwer Academic Publishers. Printed in the Netherlands. DetOfRel2.tex; 14/7/26; 12:36; p.1

2 2 and contains numerical differentiation of stress, noise level should be rather low. For the forward computation technique the stress has to be measured with good accuracy at t t 1. The Zapas-Phillips and the backward Lee-Knauss method was compared by Flory and McKenna (24). They concluded that the Zapas- Phillips method provides a better approximation of the relaxation modulus than the Lee-Knauss method. Various other methods have been also proposed, such as Kelchner and Aklonis (1971) and Smith (1979) for obtaining the relaxation modulus from non-ideal relaxation test. Procedure proposed by Kelchner and Aklonis (1971) requires that the factor-of-1 is valid and the method developed by Smith (1979) uses the stress history in the time interval t < t 1 (Flory and McKenna, 24). The aim of this paper is to derive an alternative method to approximate relaxation modulus of linear viscoelastic systems. The proposed method is a simple nonrecursive method that avoids the factor-of-1 rule described earlier. Moreover, this method can be also applied times shorten than t 1 without using the stress history in the time interval t < t 1. The method can be easily derived from the linear viscoelastic constitutive equation and it is tested with numerical simulations. Results from the simulations indicate that the accuracy for estimating the relaxation modulus is in the same level as with the Zapas-Phillips method RELAXATION TEST 2. Methods of Analysis The integral representation of viscoelastic constitutive equation takes the form (Findley et al., 1989) σ(t) = t E(t τ) ǫ(τ)dτ, (1) where σ is the stress, t is the time, E is the relaxation modulus and ǫ is the strain rate. The strain in the relaxation test is shown in Figure 1 and it can be written as { ǫ t t < t ǫ(t) = 1. (2) ǫ t t 1 Equation (1) then becomes { t ǫ σ(t) = E(t τ)dτ t < t 1 ǫ t1 E(t τ)dτ t t 1. (3) DetOfRel2.tex; 14/7/26; 12:36; p.2

3 Determination of the Relaxation Modulus ZAPAS-PHILLIPS METHOD Zapas and Phillips (1971) derived their method using the incompressible isothermal form of the BKZ theory (Bernstein et al., 1963) as the constitutive equation for viscoelastic material. However, the Zapas- Phillips method can be also derived from Equation (3) as follows. The stress at time t t 1 is given by t1 σ(t) = ǫ E(t τ)dτ. (4) Using a simple numerical integration rule, midpoint rule, Equation (4) becomes σ(t) = ǫ t 1 E(t t 1 /2) = ǫ E(t t 1 /2). (5) Then the relaxation modulus is given by or (Flory and McKenna, 24) E(t t 1 /2) = σ(t) ǫ t t 1, (6) E(t) = σ(t + t 1/2) ǫ t t 1 /2. (7) Error estimate for the used midpoint rule is (Bakhvalov, 1977) 2.3. PROPOSED METHOD ε = t E. (8) We differentiate Equation (4) with respect to time. This yields t1 t1 σ(t) = ǫ t E(t τ)dτ = ǫ τ E(t τ)dτ (9) = ǫ (E(t) E(t t 1 )). (1) Then the relaxation modulus at time t is given by E(t) = σ(t) ǫ + E(t t 1 ). (11) This is also the backward method introduced by Lee and Knauss (2). On the other hand, we use two point trapezoidal rule to integrate Equation (4) numerically, giving σ(t) = 1 2 ǫ t 1 (E(t t 1 ) + E(t)) = 1 2 ǫ (E(t t 1 ) + E(t)). (12) DetOfRel2.tex; 14/7/26; 12:36; p.3

4 4 Substituting Equation (11) in Equation (12) gives or E(t t 1 ) = σ(t) ǫ σ(t) 2 ǫ t t 1. (13) E(t) = σ(t + t 1) ǫ σ(t + t 1) 2 ǫ t, (14) where ǫ = ǫ /t 1. For the stress rate we can use for example following numerical differentiation σ(t) = σ(t + h) σ(t h) 2h, (15) where h is the length of the time step. Error estimate for the used trapezoidal rule is (Bakhvalov, 1977) ε = t E. (16) 3. Numerical studies The error estimate for the numerical integration methods was presented in the last section. Here we take a closer look at the Zapas-Phillips and the proposed method by evaluating error estimate in terms of stress. The stress at time t 3t 1 /2 1 in the Zapas-Phillips method is given by t1 σ zp (t) = ǫ E zp (t τ)dτ (17) = 1 t1 σ(t + t 1 /2 τ)dτ (18) t 1 = 1 t1 σ (n) (t) (t 1 /2 τ) n dτ (19) t 1 n! n= = 1 [σ(t)t ] t 1 24 t3 1σ (t) + (2) σ(t) t2 1σ (t) (21) 1 Note that we cannot choose t t 1. In the Zapas-Phillips method relaxation modulus is known for times larger than t 1/2 and since we integrate E zp(t τ) from to t 1, it follows that t 3t 1/2. DetOfRel2.tex; 14/7/26; 12:36; p.4

5 Determination of the Relaxation Modulus 5 and stress at time t t 1 for the proposed method is t1 σ p (t) = ǫ E p (t τ)dτ (22) t1 [ 1 = σ(t + t 1 τ) 1 ] t1 2 σ(t + t 1 τ) dτ (23) = 1 t 1 t1 n= t1 thus the error estimates are σ (n) (t) (t 1 τ) n dτ (24) n! 1 σ (n+1) (t) (t 1 τ) n dτ (25) 2 n! n= [ = σ(t) t 1σ (t) + 1 ] 6 t2 1σ (t) + (26) [ 1 2 t 1σ (t) + 1 ] 4 t2 1σ (t) + (27) σ(t) 1 12 t2 1σ (t), (28) σ zp (t) σ(t) 1 24 t2 1 σ (t), t 3 2 t 1 (29) σ p (t) σ(t) 1 12 t2 1 σ (t), t t 1 (3) for the Zapas-Phillips and the proposed method, respectively. Based on the error estimation we can conclude that both of these methods are second-order accurate. To illustrate the accuracy of the proposed method, we simulate several relaxation tests. For materials A and B we use relaxation modulus given in Flory and McKenna (24) E(t) = E e (t/τ)β, (31) where E = 1 9 Pa, β =.5, τ = 3 s (material A) and τ = 1 s (material B). For material C we use relaxation modulus ( ) t d E(t) = a b, (32) c where a =.8 MPa, b =.2 MPa, c = 1 s and d =.3 The rate of strain is ǫ =.1 s 1 and test termination time is t end = 2 s. Other test parameters, ramp time t 1, noise level and time step h for five case studies are given in Table I. In case 1 we consider relative small ramp time and in case 2 large ramp time. In cases 3, 4 and 5 DetOfRel2.tex; 14/7/26; 12:36; p.5

6 6 random Gaussian noise with variance of ±5% of the mean value of stress is added to original stress. In these cases we investigate the effect of noise and the size of time step to the approximation of the relaxation modulus. For the numerical differentiation rule we use Equation (15) when t > and σ(t 1 ) = σ(t 1 + h) σ(t 1 ), (33) h when t =. The relative error between the analytical solution and numerical methods were computed as error = E Ẽ, (34) E where E is the exact solution of the relaxation modulus and Ẽ is the relaxation modulus computed from numerical method, i.e. proposed method or Zapas-Phillips method. The results from the error estimation are shown in Tables II, III and IV for materials A, B and C, respectively. The simulations with noise free data shows that the relative errors are always smaller for the proposed method than for the Zapas-Phillips method. Figure 2 show a good agreement between the analytical solution and the proposed method even if the ramp time is large. The Zapas-Phillips method gives also accurate results in the time interval of validity. However, if the ramp time is large, it is evident that the accuracy of the Zapas-Phillips method is reduced in the estimation of relaxation modulus, see Figure 2. When noise is added to data, results indicate that the proposed method produces larger error than the Zapas-Phillips method. This is due to instability of numerical differentiation. However, as the time step is increased the relative error decreases for the proposed method as show in Figure 3. Therefore, the instability of the proposed method can be kept under control. Moreover, in case 5 for material A the relative error is smaller as compared to the Zapas-Phillips method. As a conclusion, the relative errors with the proposed method is at the same levels than with the Zapas-Phillips method in all simulated cases. However, the main advantage of the proposed method is that it can be used to compute the relaxation modulus during the whole time interval of a ramp test. 4. Conclusions In this paper, an alternative method for determining the relaxation modulus of a linearly viscoelastic material was presented. The method DetOfRel2.tex; 14/7/26; 12:36; p.6

7 Determination of the Relaxation Modulus 7 avoids factor-of-1 rule and it can be applied to the whole time interval of the relaxation test. The method was tested with numerical simulations. Results from the simulations indicate that a good agreement with an analytical solution can be obtained. Due to the fact that the relaxation modulus can be approximated in the time interval t [, t 1 ], the proposed method offers a convenient tool for the determination of the relaxation modulus for materials in which the rate of relaxation is fast. In addition, the method is very suitable for cases where the ramp time is relatively large. The error estimate formulation shows that the error is somewhat the same than in the Zapas-Phillips method. However, the simulation results indicate that smaller relative error is obtained in noise free cases with the proposed method. In the proposed method numerical differentiation of the stress produces slight oscillation when noisy data is used. Nevertheless, it was shown that instability can be kept under control by adjusting time step. In addition, the effect of the noise can be decreased with regularization or by fitting an additional function to the data. However, the purpose of this study was to derive a convergent method to compute the relaxation modulus in the whole time interval of ramp test. Simulations show that the relative error is at reasonable level with noisy data even without regularization. Although the Zapas-Phillips method is valid for nonlinear viscoelastic systems, unlike the proposed method, it has not been extensively studied for large deformations (Flory and McKenna, 24). Moreover, it is not valid for the widely used Schapery s nonlinear viscoelastic material model (Schapery, 1969). Acknowledgements We are grateful to Metso Paper, Inc. for the financial support of our research project. References Bakhvalov, N. S., Numerical methods: analysis, algebra, ordinary differential equations, MIR, Moscow, Bernstein, B., Kearsley, E. A and Zapas, L. J., A study of stress relaxation with finite strain, Trans. Soc. Rheol. VII, 1963, Findley, W. N., Lai, J. S. and Onaram, K., Creep and relaxation of nonlinear viscoelastic material, with an introduction to linear viscoelasticity, Dover, New York, DetOfRel2.tex; 14/7/26; 12:36; p.7

8 8 Flory, A. and McKenna, G. B., Finite step rate corrections in stress relaxation experiments: A Comparison of Two Methods, Mech. Time Mater. 8, 24, Kelchner, R. E. and Aklonis, J. J., Measurements of the stress-relaxation modulus in the primary transition region, J. Polym. Sci. A-2 9, 1971, Lee, S. and Knauss, W. G., A note on the determination of the relaxation and creep data from ramp tests, Mech. Time Mater. 4, 2, 1-7. Meissner, J., Combined constant strain rate and stress relaxation test for linear viscoelastic studies, J. Polym. Sci: Polym. P. Educ. 16, 1978, Schapery, R. A., On the characterization of nonlinear viscoelastic materials, Polymer Eng. Sci. 9 (4), 1969, Smith, T. L., Evaluation of the relaxation modulus from the response to a constant rate of strain followed by a constant strain, J. Polym. Sci: Polym. P. Educ. 17, 1979, Zapas, L. J. and Phillips, J. C., Simple shearing flows in polyisobutylene solutions, J. Res. Nat. Bur. Stds. 75A(1), 1971, DetOfRel2.tex; 14/7/26; 12:36; p.8

9 Determination of the Relaxation Modulus 9 ǫ ǫ t 1 Figure 1. Strain in the relaxation test. t 1 DetOfRel2.tex; 14/7/26; 12:36; p.9

10 1 1 x 19 9 analytical proposed method Zapas Phillips method 8 relaxation modulus [Pa] time [s] Figure 2. The true relaxation modulus and the approximated relaxation modulus for Zapas-Phillips and proposed method for material A in case 2. DetOfRel2.tex; 14/7/26; 12:36; p.1

11 Determination of the Relaxation Modulus 11 9 x 15 8 analytical proposed method Zapas Phillips method relaxation modulus [Pa] time [s] 8 x 15 7 analytical proposed method Zapas Phillips method relaxation modulus [Pa] time [s] Figure 3. The true relaxation modulus and the approximated relaxation modulus for Zapas-Phillips and proposed method for material C in case 3 (upper) and 5 (lower). DetOfRel2.tex; 14/7/26; 12:36; p.11

12 12 Table I. Parameters for different simulation cases. case t 1 [s] noise level % h [s] Table II. Relative errors for material A. Error 1 is the error for Zapas-Phillips method and Error 2 is the error for proposed method for time interval t [t 1/2, t end ] s. Error 2,t<t1 /2 is the error for proposed method when t < t 1/2. case Error 1(%) Error 2(%) Error 2,t<t1 /2 (%) Table III. Relative errors for material B. Case Error 1(%) Error 2(%) Error 2,t<t1 /2 (%) DetOfRel2.tex; 14/7/26; 12:36; p.12

13 Determination of the Relaxation Modulus 13 Table IV. Relative errors for material C. Case Error 1(%) Error 2(%) Error 2,t<t1 /2 (%) DetOfRel2.tex; 14/7/26; 12:36; p.13

14 DetOfRel2.tex; 14/7/26; 12:36; p.14

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