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Comparison of Simple and Pure Shear for an Incompressible Isotropic Hyperelastic Material Under Large Deformation

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Comparison of Simple and Pure Shear for an Incompressible Isotropic Hyperelastic Material Under Large Deformation CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Polymer Testing 32 (2013) 240–248 Contents lists available at SciVerse ScienceDirect Polymer Testing journal homepage: www.elsevier.com/locate/polytest Test method Comparison of simple and pure shear for an incompressible isotropic hyperelastic material under large deformation D.C. Moreira, L.C.S. Nunes* Laboratory of Opto-Mechanics (LOM/LMTA), Department of Mechanical Engineering, TEM/PGMEC, Universidade Federal Fluminense, UFF, Rua Passo da Patria 156, 24210-240 Niterói, RJ, Brazil article info abstract Article history: The concepts of simple and pure shear are well known in continuum mechanics. For small Received 28 September 2012 deformations, these states differ only by a rotation. However, correlations between them Accepted 8 November 2012 are not well defined in the case of large deformations. The main goal of this study is to compare these two states of deformation by means of experimental and theoretical Keywords: approaches. An incompressible isotropic hyperelastic material was used. The experimental Nonlinear elasticity procedures were performed using digital image correlation (DIC). The simple shear Rubber material deformation was obtained by single lap joint testing, while the pure shear was achieved by Large deformations Digital image correlation method means of planar tension testing. Classical hyperelastic constitutive equations available in the literature were used. As a consequence, the results indicate that simple shear cannot be considered as pure shear combined with a rotation when large deformation is assumed, as widely considered in literature. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction incompressible isotropic material under large elastic deformations. In 1964, Mooney reported a series of The shear stress-strain response has been extensively measurements using a thin-walled hollow cylinder instead studied in recent years. Basically, there are two ways to of a flat sheet [10]. The characterization of hyperelastic interpret shear deformation, which are defined by simple rubber-like materials by means of planar testing has also shear and pure shear. In the case of small deformations, been performed by Sasso et al. [11]. pure shear may be considered as simple shear followed by Recently, there has been growing interest in the state of rigid rotation. In addition, the two states of deformation simple shear [12–14]. Rajagopal and Wineman [15] have have often been assumed identical in classical literature developed a study of new universal relations for a nonlinear [1,2]. Despite the fact that these concepts are also well isotropic elastic block subjected to simple shear deforma- known in continuum mechanics [3,4], the correlation tion superposed on triaxial extension. Some investigators between them is not well defined. Jones and Treolar [5,6] have concluded that: “simple shear is not so simple” [16,17]. assumed that “Simple shear differs from pure shear only by Nunes [18,19] has studied the nonlinear mechanical a rotation” in an investigation of a rubber sheet under behavior of a hyperelastic material under small and large biaxial strain. simple shear deformations. According to Horgan and The first experiment for providing pure shear on a thin Murphy [17], “A simple shear deformation reflects essentially sheet of rubber-like material was proposed by Treloar [7]. a conceptual experiment that would be extremely difficult to Rivlin and Saunders [8,9] developed an experimental and replicate in a laboratory”. Despite all the contributions, there theoretical investigation on pure and simple shear states of is a need to clarify both simple and pure shear states when large deformations are taken into account. * Corresponding author. Tel.: þ55 21 2629 5588. The purpose of this paper is to investigate simple and E-mail address: [email protected] (L.C.S. Nunes). pure shear states on a hyperelastic material under large 0142-9418/$ – see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.polymertesting.2012.11.005 D.C. Moreira, L.C.S. Nunes / Polymer Testing 32 (2013) 240–248 241 deformations. The two states of shear deformation were compared by means of experimental [5,18,19] and theo- retical [8,9] analyses, used to verify the validity of the assumption made by Jones and Treloar. 2. Material and methods: experimental setup Two different experimental tests were performed to investigate the states of simple and pure shear under large deformations. Simple shear was obtained by single lap joint testing, while pure shear was carried out by means of planar tension testing [20]. The material used for manufacturing the single lap joint and thin sheet speci- mens was an adhesive based on silane modified polymer Ò (Flextec FT 101). This adhesive is suitable for many types of construction materials and presents high flexibility and elasticity. The single lap joint specimens were made with adher- ends of steel A36 and FT 101 adhesive, being a suitable configuration to provide a simple shear deformation [18– 20]. They had an overlap length of 50 mm and a joint width of 25.4 mm. The adherend and adhesive thicknesses were 1.6 mm. It is important to note that the adherend stiffness is much greater than the adhesive, in order to guarantee that the adherends do not deform and the adhesive only deforms in shear. Fig. 1 shows the experi- mental arrangement with the single lap joint specimen Fig. 2. Experimental arrangement for pure shear. mounted on the load apparatus. The apparatus was used to ensure that the adherends remained parallel as the load was applied. documented in the literature [21,22]. The basic principle of In order to provide pure shear, a planar shear test DIC is to match maximum correlation between small zones was carried out [6,7,20]. This test is based on a rectangular (or subsets) of the specimen in the undeformed and sheet of FT 101 adhesive under tension in its plane deformed states. normal to the clamped edges. Fig. 2 illustrates the The specimens were sprayed with black paint to obtain rectangular specimen under tension. Thin sheets with a random black and white speckle pattern in order to dimensions of 150 Â 70 Â 3.4 mm3 were employed, the perform the correlation procedure. A CCD camera (Sony effective area being 150 Â 10 mm2 due to the clamped XCD-SX910) set perpendicularly to the specimen was used edges. It is important to emphasize that the width of the for capturing the images. All images were acquired using effective area was at least 10 times greater than the length a10Â Zoom C-Mount lens. It is important to emphasize in the stretching direction. As a result, the specimen must that the experiments were carried out in quasi-static remain perfectly constrained in the lateral direction while conditions and at room temperature, i.e., 25 C. The specimen thinning occurs only in the thickness direction. images of the undeformed and deformed specimen were The digital image correlation (DIC) method was captured and processed using a DIC program (home-made employed for measuring the displacements of the polymer. DIC code), in order to estimate the displacement fields. The DIC is a powerful optical-numerical method developed to size of the measurement field was 1280 Â 960 pixels and estimate full-field surface displacements, being well the reference and target subsets equal to 31 Â 31 and Fig. 1. Experimental arrangement for simple shear deformation. 242 D.C. Moreira, L.C.S. Nunes / Polymer Testing 32 (2013) 240–248 71 Â 71 pixels, respectively. The accuracy of this method is l ¼ L ; l ¼ l ¼ L0 approximately equal to 0.01 pixels. 1 2 1 and 3 (4) L0 L Using Eqs. (1) and (3), the deformation gradient tensors 3. Hyperelastic constitutive relationships for simple and pure shear Fss and Fps, respectively, can be expressed as Consider a rectangular block of material in the central 2 3 2 3 region of the adhesive from a single lap joint specimen, as 1 k 0 l 00 illustrated in Fig. 3. If this element is subjected only to 4 5 4 5 Fss ¼ 010 and Fps ¼ 01 0 (5) À simple shear deformation, the deformed configuration as 001 00l 1 a function of a reference configuration can be written as where the subscripts ss and ps denote simple and pure x1 ¼ X1 þ kX2; x2 ¼ X2; x3 ¼ X3 (1) shear, respectively. The right Cauchy-Green tensor, which is also known as where k is the amount of shear. Green’s deformation tensor, is an important strain measure The relations between principal stretches and amount in material coordinates. The deformation tensors for simple of shear, considering a plane strain state, i.e. l ¼ 1 [23], 2 and pure shear, C and C , can be written as functions of may be given by ss ps the deformation gradient tensors, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 u 2 2 1 k 0 À t k k 1 T 4 2 5 T l1 À l ¼ k; l1 ¼ 1 þ þ k 1 þ and l3 C ¼ F F ¼ kkþ 10 and C ¼ F F 1 2 4 ss ss ss ps ps ps vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 001 u sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 u 2 t 2 2 l 00 k k 4 5 ¼ 1 þ À k 1 þ (2) ¼ 01 0 (6) 2 4 À 00l 2 Different to simple shear, the pure shear state is obtained There are many forms for expressing the three scalar in a thin sheet under uniaxial extension. The pure shear invariants of a second-order tensor. They can be written in occurs only in the central part of the sheet. Fig. 4 illustrates terms of the metric tensor in the undeformed and a small region at the central part of a rectangular sheet of deformed states, as well as the relative stretches.
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