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

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Polymer Testing 32 (2013) 240–248

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Polymer Testing

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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 theoretical [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 specimens 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 adherends 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 experimental 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 70 3.4 mm3 were employed, the perform the correlation procedure. A CCD camera (Sony effective area being 150 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 a10 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 960 pixels and estimate full-field surface displacements, being well the reference and target subsets equal to 31 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 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) 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 t k k 1 T 4 2 5 T l1 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 k 1 þ (2) ¼ 01 0 (6) 2 4 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. The material stretched along a parallel pair of clamped edges. principal scalar invariants of the right Cauchy-Green Assuming that the material is incompressible, i.e. deformation tensor can be determined as l1l2l3 ¼ 1, the principal stretches can be expressed as 1 l1 ¼ l, l2 ¼ 1 and l3 ¼ l . Thus, the deformed configu- ss 2 ps 2 2 I ¼ tr Css ¼ k þ 3 and I ¼ tr Cps ¼ l þ l þ 1 (7a) ration is given by 1 1

1 h i l ss 1 2 2 2 x1 ¼ X1; x2 ¼ X2 and x3 ¼ X3 (3) I ¼ ðtr C Þ tr C ¼ k þ 3 and l 2 2 ss ss h i 1 À Á The associated principal stretches are deﬁned as a func- Ips ¼ tr C 2tr C2 ¼ l2 þ l 2 þ 1 ð7bÞ 2 2 ps ps tion of initial and ﬁnal lengths (L0 and L) in the stretched direction. These expressions are given as

ss ¼ ¼ ¼ ps ¼ ¼ ¼ I3 J detCss 1 and I3 J detCps 1 (7c) Numerous stress tensors have been defined in the literature. The first Piola-Kirchhoff stress tensor P is called the engineering stress (or nominal stress) because it is defined as the force per unit unstrained area. This tensor is in general not symmetric. The Cauchy stress tensor s, also known as true stress, is defined as the force per unit strained area. The relation between these stress tensors is given by

s ¼ J1PFT (8)

For large deformations, the constitutive equation that describes the relation between the Cauchy stress tensor and the strain-energy function is given by

vWðCÞ s ¼ 2J1F FT : (9) vC Considering an incompressible isotropic hyperelastic Fig. 3. Simple shear deformation. material, the strain-energy function may be expressed as D.C. Moreira, L.C.S. Nunes / Polymer Testing 32 (2013) 240–248 243

Fig. 4. Pure shear.

1 sheared by an amount k along the X1-direction, which is W ¼ W½I1ðCÞ; I2ðCÞ pðI3 1Þ (10) 2 associated with the applied load. The stress components s11 and s22 are considered because it was reported in literature where p is the arbitrary hydrostatic pressure. that simple shear deformation produces not only shear There are several forms of strain-energy function in the stresses. The tangential and normal components to one of literature to describe the elastic properties of hyperelastic the inclined surfaces are also shown in this figure. The first materials [23,24]. An available and sophisticated model for Piola-Kirchhoff and the Cauchy shear stresses are given by finding the stress response was proposed by Ogden [23]. This model that associates the strain-energy with the Fss P12 ¼ and s12 ¼ P12 (12) principal stretches is given by Ass

XN where F and A are the applied load and the unstrained m À a a a Á ss ss Wðl ; l ; l Þ¼ p l p þ l p þ l p 3 (11) 1 2 3 a 1 2 3 area, respectively. p ¼ 1 p As developed by Rivlin [8] and assuming that P c ss ~ N WðI Þ¼WðI1; I1; 1Þ¼WðI1; I2; 1Þ, the stress components where the shear modulus is m ¼ 1=2ð ¼ m apÞ with p 1 p from state of simple shear (Eq. (9)) may be expressed by, mpap > 0. 0 1 c vW 3.1. Simple shear deformation sss ¼ 4@ Ak (13a) 12 vIss At the beginning of this section, the state of simple shear deformation was described. Fig. 5 illustrates the stress 0 1 components to support the present analysis. The element is c vW sss ¼p þ 2@ Ak2 (13b) 11 vIss

0 1 c vW sss ¼p 2@ Ak2 (13c) 22 vIss

sss ¼ 33 p (13d) There are two approaches that are commonly used to solve the problem of simple shear. The ﬁrst approach is based on a plane stress condition (in the case, s33 ¼ 0) and the second is to consider the normal component of the traction on the inclined face equal to zero, i.e., N ¼ 0. These Fig. 5. Stress components for simple shear deformation. analyses may be found in literature [12,17]. According to the 244 D.C. Moreira, L.C.S. Nunes / Polymer Testing 32 (2013) 240–248

experimental condition described in Section 2, it is more s1 ¼ s11 ¼ l1P11 (20) suitable to assume that the surface tractions are zero on the In order to compare simple and pure shear, the simple surfaces X3 ¼w/2, where w is equal to the joint width [8], such that shear stress is also analyzed. In the present work, the statement assumed by Jones and Treloar [5,6] was consid- c “ 2k2 À Á vW ered. They have assumed that Simple shear differs from p ¼ 3 þ k2 (14) 1 þ k2 vIss pure shear only by a rotation; the strain energy W is, thus, the same as for the equivalent pure shear”. Taking this assump- With the substitution of Eq. (14) into Eq. (13a–d), this tion into account, leads to dW dW vl l2 s ¼ ¼ 1 ¼ 1 Fss 12 P11 2 (21) s12 ¼ P12 ¼ (15a) dk vl1 dk l þ 1 1 Ass with k ¼ l l 1 À Á 1 1 k 2 þ k2 s11 ¼ s12 (15b) 1 þ k2 4. Results and discussion

k Fig. 6(a) and (b) illustrates full-field displacements s ¼ s (15c) 22 1 þ k2 12 associated with horizontal and vertical directions, respectively, for simple shear deformation, as indicated in Fig. 3. À Á These results were obtained under an applied load equal to 2 1 k 3 þ k 948 N. It is important to note that full-field displacements s33 ¼ s12 (15d) 2 1 þ k2 were taken on surface area at the central region of the adhesive. The results were obtained by means of the DIC As a result, the principal stresses are method, considering the experimental arrangement for qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À Á simple shear deformation (see Fig. 1). Note that X1- 2 2 k 3 þ k þ k2ð3 þ k2Þ þ4 displacement varies linearly along the vertical direction s1 ¼ s12 (16a) (X ), while the X -displacement remains practically at the 2ð1 þ k2Þ 2 2 same value. This indicates that an angular distortion was À Á generated and the vertical distribution remained almost 1 k 3 þ k2 undeformed. s ¼ s (16b) 2 2 1 þ k2 12 The full-field displacements of a small rectangle on the surface at the central region of the thin sheet were also estimated using the DIC method. In this case, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À Á results were obtained using the experimental arrangement k 3 þ k2 k2ð3 þ k2Þ2þ4 for pure shear (see Figs. 2 and 4). The X1-andX2- s ¼ s (16c) 3 2ð1 þ k2Þ 12 displacements of the selected region for an applied load of 455 N are illustrated in Fig. 7(a) and (b). In pure shear results, X2-displacement field does not present significant 3.2. Pure shear variation, and X1-displacement field varies linearly with the vertical direction (X1). Due to the low magnitude of X2- Considering a thin sheet under tensile load, as described displacements, deformation in the horizontal direction can by Jones and Treloar [5], it is possible to calculate the work be neglected; therefore it can be considered that the done by the external applied force taking into account the deformation occurs only in the vertical direction. unstrained block, which gives To determine the principal stretches, full-field displacements for simple and pure shear deformations Fps dW ¼ P11dl1; with P11 ¼ (17) were used. Moreover, Eqs. (1)–(4) were also considered. A ps Firstly, to estimate the principal stretches for simple shear where Fps and Aps are the applied load and the unstrained case, the angular distortion was calculated taking X1- and area, respectively. X2-displacement fields. In this way, the amount of shear The principal Cauchy stresses [23] are given by was determined and the principal stretches were achieved. More information about the estimation process may be vW found in the literature [18,19]. For evaluating the principal si ¼ li p (18) vli stretches for the pure shear case, the initial and final sizes of the small area at the central region of thin sheet were or in the present case, being s3 ¼ 0 taken into account. The size variations were determined by vc the DIC program. Using these results and Eq. (4), the s ¼ l W 1 1 vl (19) principal stretches were achieved. For each applied load, an 1 image was captured and the procedure was repeated. c ~ 1 where WðlÞ¼Wðl; l Þ¼Wðl1; l2; l3Þ Fig. 8 illustrates the true shear stress as a function of the From Eqs. (17) and (19) amount of shear. Three tests were performed by means of D.C. Moreira, L.C.S. Nunes / Polymer Testing 32 (2013) 240–248 245

Fig. 6. Full-field displacements for simple shear deformation, considering an applied load of 948 N: (a) horizontal and (b) vertical directions. the experimental setup for simple shear configuration. The simple and pure shear are essentially of the same order of data obtained from repeated measurements show suitable magnitude. However, for large deformation this concept is repeatability. One can see that the relationship between not well-known. In order to understand these states for large shear stress and amount of shear is nonlinear. The results deformations, the true stresses as functions of the principal also indicate that the adhesion failure occurred for values of stretches were taken into consideration. The data were ob- shear larger than 2.5. tained by considering Eqs. (15a) and (21). The results are Results achieved using the experimental setup for pure shown in Fig. 11. It is possible to note that, for small defor- shear are shown in Fig. 9. In this case, the response of the mations, stress-stretch response for simple and pure shear is normal stress as a function of the principal stretch was the same. Nevertheless, a divergence between simple and investigated. The data present good repeatability and pure shear occurs for values of stretch larger than 1.3. nonlinear behavior can also be observed. It is clear that simple shear cannot be considered as pure In order to compare both nominal and true stresses shear combined with a rotation when large deformation is from pure shear, Eqs. (17) and (20) were used. The mean assumed. An alternative way of comparing the states of results are illustrated in Fig. 10. It is possible to observe the simple and pure shear is shown in Fig. 12. In this illustra- linear relationship between true stress and principal tion, the principal stresses as functions of the principal stretch. The stress components from simple shear are not stretch are plotted. The data were achieved considering the presented because the nominal and true stresses are the Baker-Ericksen inequalities [25], which establishes that the same. This is true in cases with no significant variation of greater principal stress always occurs in the direction of the strained area. greater principal stretch. The results of s1 indicate that the It is clear that, for homogeneous isotropic materials stress-stretch relations present a divergence between under small deformation, the stress-strain relations of simple and pure shear, mainly for large stretch values.

Fig. 7. Full-ﬁeld displacement for pure shear, considering an applied load of 455 N: (a) vertical and (b) horizontal directions. 246 D.C. Moreira, L.C.S. Nunes / Polymer Testing 32 (2013) 240–248

1.2 P : First Piola−Kirchhoff (Nominal) Test 1 2 11 Test 2 σ : Cauchy (True) 11 1 Test 3

1.5 0.8

0.6 1 (MPa) 12 σ 0.4

Normal stress (MPa) 0.5 0.2

0 0 0 0.5 1 1.5 2 2.5 3 3.5 1 1.2 1.4 1.6 1.8 2 2.2 k λ

Fig. 8. True shear stress versus amount of shear for simple shear Fig. 10. Nominal and true stresses versus stretch for pure shear. deformation.

Fig. 12 shows the principal stresses s and s as func- 1 a a 2 3 Pure shear : P ¼ m l l 1 l 1 (23b) tions of the principal stretch for the states of simple shear. 11 1 The principal stress s represents the hydrostatic pressure 2 and increases with increasing stretch, while the values of s a a 3 Pure shear : s ¼ m l 1 l 1 (23c) tend to zero. 1 1 In order to compare the predictions of the resulting theoretical stress-stretch relations with experimental data with the shear modulus equal to m ¼ m1a1/2. over simple and pure shear, the strain-energy function was Equations (23a–c) were ﬁtted to the experimental data, taken into account. It is clear that the material properties as illustrated in Fig. 13. The estimated parameters for must be invariant under changes of observer and load simple and pure shear states are summarized in Table 1.As m conditions. For that reason, the values of shear modulus ( ) one can observe, the estimates of shear modulus are ﬁ for simple and pure shear states were estimated. To nd approximately equal to 0.5 MPa. It should be noted that this material property, the Ogden model (See Eq. (11))was those results support the proposed analysis. used in the following form: Overall, the results of stress-stretch response indicate that simple and pure shear must not be assumed as the m a a ¼ 1 l 1 þ l 1 W a 2 (22) same when large deformation is considered. It is 1 important to emphasize that the statement proposed by Using Eq. (22) into Eqs. (13a), (17) and (19), Treloar was assumed. However, it was observed that the state of simple shear is not so simple, as was stated by l a a : s ¼ m l 1 l 1 Destrade. Simple shear 12 2 1 (23a) 1 þ l2

1.2 Test 1 0.9 Test 2 Simple shear 0.8 Pure shear 1 Test 3 0.7 0.8 0.6

0.5 0.6 (MPa) (MPa) 0.4 11 12 P σ 0.4 0.3

0.2 0.2 0.1

0 0 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 2.2 λ λ

Fig. 9. Normal stress versus stretch for pure shear. Fig. 11. Shear stress versus stretch. D.C. Moreira, L.C.S. Nunes / Polymer Testing 32 (2013) 240–248 247

σ (Pure shear) relations for simple and pure shear were compared 2 1 assuming the statement proposed by Treloar. In the case of σ (Simple shear) 1 tangential stress, the divergence between simple and pure σ (Simple shear) 2 shear occurs for deformation larger than 30%. Furthermore, 1.5 σ (Simple shear) the relationship between the principal stresses and prin- 3 cipal stretches was also investigated, showing a divergence between simple and pure shear, mainly for large stretch 1 values. In order to support the results, the shear modulus was evaluated by means of the Ogden model, showing the same value for both cases. The results indicate, overall, that 0.5 simple shear cannot be considered as pure shear combined

Principal stress (MPa) with a rotation when large deformation is assumed, as widely considered in literature. As a closing remark, one 0 should mention that the present work might be used to support recent investigation about simple shear, which 1 1.2 1.4 1.6 1.8 2 2.2 λ have already evidenced that this state of deformation is not so simple. Fig. 12. Principal stress versus stretch. Acknowledgements

P : Nominal stress (Pure shear) The ﬁnancial support of Rio de Janeiro State Funding, 11 2 σ : True stress (Pure shear) FAPERJ, and Research and Teaching National Council, CNPq, 11 are gratefully and acknowledged. σ : Shear stress (Simple shear) 12 Curve fitting 1.5 References

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