Investigation of Simple and Pure Shear from Hyperelastic Material Based on Digital Image Correlation Method

INVESTIGATION OF SIMPLE AND PURE SHEAR FROM HYPERELASTIC MATERIAL BASED ON DIGITAL IMAGE CORRELATION METHOD

A.L. Pereira, [email protected] D.C. Moreira, [email protected] L.C.S Nunes, [email protected]

1Laboratory of Opto-Mechanics (LOM/LMTA), Department of Mechanical Engineering (TEM/PGMEC), Universidade Federal Fluminense, Rua Passo da Pátria, 156, Bloco E, Niteroi-RJ, CEP: 24210-240, Brazil.

Abstract. This paper describes and analyzes the mechanical behavior of a hyperelastic material in two different conditions, i.e., simple and pure shear deformations. The experimental procedure is carried out using the Digital Image Correlations (DIC) method, which is an available optical-numerical approach to estimate full-field displacements. The simple shear deformations are obtained by single lap joints testing, while the pure shear is carried out by means of planar tension testing. Moreover, classical constitutive models are employed to describe the mechanical behavior of the hyperelastic material. The main goal is to analyze the difference between simple and pure shear behaviors.

Keywords: Simple shear, pure shear, hyperelasticity, Digital Image Correlation

1. INTRODUCTION

Several experimental tests have been developed to investigate mechanical behavior and properties of polymeric materials (Ward and Sweeney, 2004; Brown, 2002). Guélon et al. (2009) proposed a new characterization method for rubber, which consists of performing only one heterogeneous mechanical test. The first experiment for providing pure shear deformation on a thin sheet of rubber was proposed by Treloar (1945). Moreover, experiments on the pure shear of large elastic deformations of incompressible isotropic material was developed by Rivlin and Saunders (1951). Nunes (2010 and 2011) have studied the mechanical behavior of Polydimethylsiloxane under small and large simple shear deformations. For some authors, there is no essential difference between simple and pure shear deformations. The simple shear configuration is assumed to be superposition of pure shear deformation associated with simple rigid body rotation. However, for large deformation this concept is not well defined. Many studies have been produced over the last decade to explain these two deformation states (Tikoff and Fossen, 1993; Destrade et al; Segal, 2002; Holzapfel, 2008; Ogden, 1997). Hyperelastic behavior is commonly observed in some polymers, mainly the long chain polymers, like as elastomers (or rubbers) that are characterized by flexibility and stability. Polydimethylsiloxane (PDMS) is a silicone rubber, which has a wide range of applications in mechanical sensors (Kim et al., 2008; Lin et al., 2009), electronic components (Tiercelin et al., 2006; Lee et al., 2009) and medical devices (Lawrence et al., 2009). The main goal of the present work is to compare the mechanical behavior of Polydimethylsiloxane under large simple and pure shear deformations. In order to do that, two experimental approaches are performed: single lap joints under tensile for simple shear and thin sheet under tensile for pure shear. The displacement fields are estimated by means of DIC method. Using this information, principal stretches are evaluated to investigate both cases.

2. DIGITAL IMAGE CORRELATION

The Digital Image Correlation (DIC) method is a powerful optical-numerical method developed to estimate full- field surface displacement, being well documented in the literature (Dally and Riley; 2005; Sutton et al., 2009). This method has been considerably improved over the last years. The well-known principle of the DIC method is to match maximum correlation between small zones (or subsets) of the specimen in the undeformed and deformed states. The VII Congresso Nacional de Engenharia Mecânica, 31 de julho a 03 de Agosto 2012, São Luis - Maranhão

specimen surface is coated by a random pattern in order to provide a grayscale distribution with sufficient contrast. To determine the displacement of each point, a square reference subset, f, of (2M+1) x (2M+1) pixels from undeformed image is chosen and it is used to find the corresponding on the target subset of (2N+1) x (2N+1) pixels from deformed image, g. M and N are positive integers, being M < N. For this purpose, the minimization of the correlation coefficient is taken into account. From a given image-matching procedure, the in-plane displacement fields designated by u(x,y) and v(x,y) associated with x- and y-coordinates can be computed. Figure 2 illustrates the scheme of the DIC method.

Figure 1. Scheme of the Digital Image Correlation method.

3. DEFORMATION STATE

In this section, two states of deformation are presented, i.e., simple and pure shear. The idea is to describe the behavior associated with those two different states. In order to do that, a material element defined by dX in the reference configuration can be transformed into a material element dx in the current configuration, using the deformation gradient tensor F and taking into account the relation given by dX = Fdx. Two stretches, defined by λ1 and λ2, are taken at the plane of interest and the third stretch λ3 is determined by the condition λ1λ2λ3 = 1. This relationship is assumed when the volume does not change during the deformation, which is a characteristic of incompressible materials.

3.1. Simple shear € Firstly, Let us consider the case of simple shear deformation, as illustrated in Fig. 2. This case is characterized by an angular distortion of a rectangular block, in which the horizontal line elements remain fixed in length and direction. In this way, the rectangular Cartesian coordinate of any point of deformed element in the current configuration can be written as a function of reference configuration,

x1 = X1 + γX2 ; x2 = X2 ; x3 = X3 (1)

where γ is the amount of shear. Using Eq. (1), the deformation gradient tensor for simple shear, F, can be expressed € € € as

⎡ 1 γ 0⎤ ⎢ ⎥ F = ⎢ 0 1 0⎥ (2) ⎣⎢ 0 0 1⎦⎥

From Eq. (2), the right Cauchy-Green deformation tensor for simple shear, C, can be written as

€ ⎡ 1 γ 0⎤ ⎢ ⎥ C FT F 2 = = ⎢γ γ +1 0⎥ (3) ⎣⎢ 0 0 1⎦⎥

Considering λ = 1, the characteristic equation det C− λ2I = 0 reduces to λ − λ−1 = γ , since we may take 3 ( ) −1 λ2 ≡ λ1 = λ ≥ 1 without loss of generality.€ Consequently, the principal stretches can be given by

€ € € γ 2 γ 2 γ 2 γ 2 € λ = 1+ + γ 1+ and λ = 1+ − γ 1+ (4) 1 2 4 2 2 4

€ € VII Congresso Nacional de Engenharia Mecânica, 31 de julho a 03 de Agosto 2012, São Luis - Maranhão

Figure 2. Simple shear deformation

3.2. Pure shear (planar)

One way to obtain the pure shear deformation state is to consider a thin sheet under uniaxial extension. A state of pure shear exists in the rectangular sheet at an angle of 45o to the stretching direction, assuming that the volume remains constant. The pure shear deformation occurs only in the central part of the sheet. Figure 3 illustrates a small region at central part of rectangular sheet of material along a parallel pair of clamped edges. A pure shear deformation may be achieved using the configuration of plane deformation, in which

1 x1 = λ1X1; x2 = X2 ; x3 = X3 (5) λ1

The associated principal stretches are defined as a function of initial and final length (L0 and L) in the stretched direction. These expressions are given as € € L € −1 λ1 = , λ2 = 1 and λ3 = λ1 (6) L0

€ € €

Figure 3. Pure shear deformation

4. EXPERIMENTAL SETUP

VII Congresso Nacional de Engenharia Mecânica, 31 de julho a 03 de Agosto 2012, São Luis - Maranhão

Two different experimental tests were performed to investigate the mechanical behavior of Polydimethylsiloxane under shear deformation. In this way, full-field displacements were measured by means of the digital image correlation method, which basic principle was previously presented in section 2. A single lap joint was made with adherends of steel A36 and adhesive of Polydimethylsiloxane, being a suitable configuration to provide a simple shear deformation. It is important to remark that the adherends stiffness is much greater than the adhesive, in order to guarantee that the adherends do not deform and the adhesive only deform in shear. Figure 4 shows the experimental arrangement of single lap joint. The associated dimensions are as follows: the length of restraint against transversal motion of 25 mm; segment of length, D = 50 mm; joint length, L, equal to 51 mm; joint width, w = 25.4 mm; adherend and adhesive thicknesses, t = 1.6 mm and ta = 1.6 mm, respectively.

Figure 4. Experimental arrangement for simple shear deformation

To provide pure shear deformation, a planar shear test was carried out. This test is based on a rectangular sheet of Polydimethylsiloxane under tensile in its plane normal to the clamped edges. Figure 5 illustrates the rectangular specimen under tensile and the DIC system. In the experiment, a sheet of Polydimethylsiloxane with dimensions of 170x70x5 mm3 was employed. It is important to emphasize that the width of specimen was at least 10 times wider than the length in the stretching direction. As a result, the specimen must remain perfectly constrained in the lateral direction and all specimen thinning occurs in the thickness direction.

Figure 5. Experimental arrangement for pure shear deformation VII Congresso Nacional de Engenharia Mecânica, 31 de julho a 03 de Agosto 2012, São Luis - Maranhão

5. RESULTS AND DISCUSSION

Figure 6 illustrates full-field displacements associated with horizontal and vertical direction, i.e., u(x,y) and v(x,y), respectively, for simple shear deformation. These results were obtained under a shear stress equal to 0.21MPa. It is important to remark that full-field displacements were taken on surface area at central region of adhesive. The results were obtained by means of DIC method, considering the experimental arrangement for simple shear deformation. Note that u-displacement varies linearly along the vertical direction, while the v-displacement remains practically in the same value. The upper adherend was fixed and, as a consequence, the superior displacement of u(x,y) tends to zero.

Figure 6. Full-field displacement for simple shear deformation: τ = 0.21 MPa

The full-field displacements of a small rectangle on the surface at central region of the PDMS sheet were also estimated through the DIC method. In this case, results were obtained using the experimental arrangement for pure shear deformation (see Figure 5). The u- and v-displacements of the selected region for an applied stress of 0.27 MPa are illustrated in Fig. 7. In pure shear results, u-displacement field does not present significant variation and v- displacement field varies linearly with the vertical direction.

Figure 7. Full-field displacement for pure shear deformation: σ = 0.27 MPa

In order to determine the principal stretches, full-field displacements for simple and pure shear deformations were used. Moreover, Eqs. (4) and (6) were also considered. Firstly, to estimate the principal stretches for simple shear case, the angular distortion was calculated taking u- and v-displacement fields. In this way, the amount of shear was determined and, by means of Eq. (4), the principal stretches were achieved. For evaluating the principal stretches for pure shear case, the initial and final size of the small area at central region of PDMS sheet were taken into account. The size variations were determined by DIC program. Using these results and Eq. (6), the principal stretches were achieved. It is important to emphasize that the results present in Figs (6) and (7) are for only one applied load. Figure 8 shows the comparison of principal stretches for simple and pure cases, considering several loads. The idea is to analyze the variance in both stretches for the two cases. Clearly, it can be noted that the orthogonal stretch λ2 does not vary significantly with an extension of the principal stretch λ1 in plane normal to the edges for pure shear deformation. VII Congresso Nacional de Engenharia Mecânica, 31 de julho a 03 de Agosto 2012, São Luis - Maranhão

However, in simple shear case, the stretch λ2 decreases monotonically with increasing principal stretch λ1 under shear load condition.

1.05

1 Simple shear: 0.95 2 Simple shear: 0.9 3 Pure shear: 3 0.85 2 Pure shear: 0.8 3 and 2

0.75

0.7

0.65

0.6

0.55 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1 Figure 8. Principal stretches comparison: simple shear and pure shear

6. CONCLUSION

The aim here was to compare simple and pure shear configurations under large deformation. In this way, the mechanical behavior of Polydimethylsiloxane under shear deformation was investigated by means of two different experimental approaches: a single lap joint under tensile and a thin sheet under normal tensile, providing simple and pure shear deformation, respectively. Full-field displacements were estimated through the Digital Image Correlation method and the principal stretches were estimated in both cases. The results of simple shear case show that the stretch λ2 decreases with increasing principal stretch λ1, while, for pure shear, the stretch λ2 remains constant with increasing principal stretch λ1. Therefore, it is clear that there are some differences between the two shear cases, which are clearly seen in this work. Finally, it should be mentioned that these are preliminary results and further investigations will follow this work.

7. ACKNOWLEDGEMENTS

The authors would like to express their gratitude to the Ministry of Science and Technology. The present paper received financial support from Brazilian agencies CNPq and FAPERJ.

8. REFERENCES

Brown, R., 2002, Handbook of Polymer Testing, Rapra Technology Limited, Shropshire, UK. Dally, J.W. and Riley, W.F., Experimental Stress Analysis, 4th ed. McGraw Hill, 2005. Destrade, M., Murphy, J.G., Saccomandi, G., “Simple shear in not simple”, International Journal of Non-Linear Mechanics, Available online 27 May 2011 Guélon, T., Toussaint, E., Le Cam, J.B., Promma, N., Grédiac,M., 2009, “A new characterization method for rubber”, Polymer Testing, Vol. 28, pp. 715-723. Holzapfel, G.A., 2008, Nonlinear Solid Mechanics: A continuum approach for engineering, John Wiley & Sons Ltd. Kim, J.H., Lau, K.T., Shepherd, R., Wu, Y., Wallace, G., Diamond, D., Performance characteristics of a polypyrrole modified polydimethylsiloxane (PDMS) membrane based microfluidic pump. Sensors and Actuators A 148 (2008) 239–244. Lawrence, B.D., Marchant, J.K., Pindrus, M.A., Omenetto, F.G., Kaplan, D.L., Silk film biomaterials for cornea tissue engineering. Biomaterials 30 (2009) 1299–1308. Lee, D., Mekaru, H., Hiroshima, H., Matsumoto, S., Itoh, T., Takahashi, M., Maeda, R., 3D replication using PDMS mold for microcoil. Microelectronic Engineering 86 (2009) 920–924. Lin, Y. H., Kang, S.W., Wu, T.Y., Fabrication of polydimethylsiloxane (PDMS) pulsating heat pipe. Applied Thermal Engineering 29 (2009) 573–580. Nunes, L.C.S., 2010,”Shear modulus estimation of the polymer polydimethylsiloxane (PDMS) using digital image VII Congresso Nacional de Engenharia Mecânica, 31 de julho a 03 de Agosto 2012, São Luis - Maranhão

correlation”, Materials and Design, Vol. 31, pp. 583-588. Nunes, L.C.S., 2011, “Mechanical characterization of hyperelastic polydimethylsiloxane by simple shear test”, Materials Science and Engineering A, Vol. 528, pp. 1799-1804. Ogden R.W., Non-Linear Elastic Deformations, Halsted Press, Wiley, New York, 1984, Dover Publications, Mineola, NY, 1997. Rivlin, R.S. and Saunders, D.W., 1951, “Large elastic deformations of isotropic materials. VII. Experiments on the deformation of rubber”, Philosophical Transactions of the Royal Society of London, Vol. A243, pp. 251-288. Segal, V.M., 2002,“Severe plastic deformation: simple shear versus pure shear”, Materials Science and Engineering, Vol. A338, pp. 331-344. Sutton, M.A., Orteu, J.J., Schreier, H.W., Image Correlation for Shape, Motion and Deformation Measurements, Springer Science and Business Media LCC 2009. Treloar, L.R.G., 1944, “Stress-strain data for vulcanized rubber under various types of deformation”. Transactions of the Faraday Society, Vol. 40, pp. 59-70. Tikoff, B. and Fossen,H., 1993, “Simultaneous pure and simple shear: the unifying deformation matrix”, Tectonophysics, Vol. 30; pp. 267-283. Tiercelin, N., Coquet, P., Sauleau, R., Senez, V. and Fujita, H., Polydimethylsiloxane membranes for millimeter-wave planar ultra ﬂexible antennas. J. Micromech. Microeng. 16 (2006) 2389–2395. Ward, I.M. and Sweeney, J., 2004, An Introduction to the Mechanical Properties of Solid Polymers, 2nd Ed. John Wiley & Sons Ltd.

9. RESPONSIBILITY NOTICE

The authors are the only responsible for the printed material included in this paper.