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Simple shear under large deformation: Experimental and theoretical analyses
Accepted Manuscript Simple shear under large deformation: experimental and theoretical analyses L.C.S. Nunes, D.C. Moreira PII:
S0997-7538(13)00077-6
DOI:
10.1016/j.euromechsol.2013.07.002
Reference:
EJMSOL 2954
To appear in:
European Journal of Mechanics / A Solids
Received Date: 2 February 2013 Revised Date:
18 June 2013
Accepted Date: 7 July 2013
Please cite this article as: Nunes, L.C.S., Moreira, D.C., Simple shear under large deformation: experimental and theoretical analyses, European Journal of Mechanics / A Solids (2013), doi: 10.1016/ j.euromechsol.2013.07.002. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Graphical Abstract (for review)
Experimental Lopez−Pamies model
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*Highlights (for review)
ACCEPTED MANUSCRIPT Research highlights
Experimental and theoretical analyses on simple shear deformation were developed
Simple shear deformation was experimentally achieved using a single lap joint test Nonlinear relation between shear stress and amount of shear was obtained
Normal stress components were obtained from the experimental data
The Lopez-Pamies model was fitted to the experimental data
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Manuscript Click here to view linked References
ACCEPTED MANUSCRIPT Simple shear under large deformation: experimental and theoretical analyses L.C.S. Nunes* and D.C. Moreira
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Laboratory of Opto-Mechanics / Laboratory of Theoretical and Applied Mechanics (LOM/LMTA) Department of Mechanical Engineering, TEM/PGMEC Universidade Federal Fluminense, UFF Rua Passo da Patria 156, 24210-240 Niterói, RJ, Brazil
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Abstract
The aim of the present work is to analyze the simple shear state of an incompressible
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hyperelastic solid under large deformation by experimental and theoretical approaches. The experimental procedure was performed using a single lap shear test and the displacement fields were determined by the Digital Image Correlation method. The applied force and the measured angular distortion were used to evaluate the shear stress and the amount of shear. Hence, a nonlinear stress-strain response was achieved. In addition, the normal stress components were obtained from the experimental data by
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assuming two hypotheses: the first one was based on a plane stress condition, while the normal component of the traction on the inclined surfaces was assumed to be zero on the second. Finally, to verify the presented results, the initial shear modulus of the hyperelastic material was estimated and compared with the value obtained using the
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data from a planar tensile test. The Lopez-Pamies strain energy function was used in the inverse analysis in order to estimate the material property, which was similar for both
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experimental tests.
Keywords: Simple shear, large deformation, hyperelasticity, full-field displacement method
*Corresponding author. Tel.: +55 21 2629-5588 E-mail address: [email protected]
ACCEPTED MANUSCRIPT 1. Introduction Recently, there has been growing interest in the state of simple shear. In the linear theory of elasticity, simple shear deformation is well defined [1,2]. This state is also reported in many textbooks on nonlinear theory [3-6], however, some points concerning stress components remain unclear. In fact, in an isotropic nonlinear elastic solid, pure
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shear stress is not accompanied by simple shear strain [3]. The first analysis of simple shear under finite deformation, considering a block of incompressible material, was proposed by Rivlin [7]. In addition, universal relations for nonlinear isotropic elastic materials under simple shear superposed on triaxial extension
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have been widely investigated [8-10]. Based on those works, recent investigations have been developed. Mihai and Goriely [11] showed that for a homogeneous isotropic
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hyperelastic material subject to pure (uniform) shear stress, the resulting deformation consists of a triaxial stretch (pure strain deformation) combined with a simple shear in the direction of the shear force if and only if the BE inequalities hold. Furthermore, Destrade et al. [12] have concluded that a pure shearing stress field produces a deformation field that is not the classical simple shear deformation, but a simple shear deformation superimposed upon a triaxial stretch. Gent et al. [13] and Destrade et al.
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[14] have investigated the stress distributions of a nonlinear elastic block under large shear deformations by finite element analysis. They considered two rigid plates bonded with elastic material subjected to shear deformations, keeping the distance between the plates. Motivated by this work, Horgan and Murphy [15] conducted an investigation
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into three different formulations of simple shear deformation.
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Simple shear test is not easy to perform experimentally [16]. Therefore, few experimental works on simple shear have been reported in the literature. Mooney reported a series of measurements on soft rubber using a thin-walled hollow cylinder [17]. More recently, Nunes [18,19] studied the nonlinear mechanical behavior of a hyperelastic material under small and large simple shear deformations by a single lap shear test. Some points concerning mechanical behavior on shear state are still open to discussion. For instance, it is known that most materials have a tendency to expand in the direction perpendicular to the applied shear load. Nevertheless, Janmey et al [20] showed experimentally that networks of semiflexible biopolymers exhibit the opposite tendency. For such materials, the empirical inequalities must not be imposed [11].
ACCEPTED MANUSCRIPT Despite all previous contributions, it is evident that further investigation based on experimental and theoretical studies is essential in order to understand simple shear. In this context, the present work investigates large simple shear deformations by experimental and theoretical approaches. The experimental procedure was performed using a single lap shear test; in which specimens manufactured with two rigid plates
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bonded by a hyperelastic material were submitted to shear deformation. In this test, the full-field displacements were evaluated employing a powerful and noncontacting method, known as Digital Image Correlation (DIC). The applied force and the measured angular distortion were used to evaluate the shear stress and the amount of shear. In
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addition, the normal stress components were obtained from the experimental data by assuming two hypotheses: a plane stress condition was considered in the first one, while
on the second.
2. Material and methods
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the normal component of the traction on the inclined surfaces was assumed to be zero
For the purpose of this study, simple shear deformation of large amount was
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obtained by a single lap shear test. This test was chosen due to its simplicity. Although quadruple-lap shear is a standard test method for determining shear modulus [21], single- and double-lap shear testing can also be considered for this purpose. However, the supporting plates may present a tendency of moving out of parallel under load in
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both single and double-lap test methods [16]. To overcome this limitation, the stiffness of the plates (adherends) must be much greater than the stiffness of the adhesive
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[18,19].
The single lap joint (SLJ) specimens were manufactured with two rigid plates (or adherends of steel A36) bonded by an adhesive, which was the analyzed sample. This adhesive is based on silane modified polymer (FlextecFT 101), and was chosen because it presents high flexibility and elasticity. The SLJ specimens had overlap lengths of 21, 34 and 38 mm and a joint width of 25.4 mm, with the adherend and adhesive thicknesses equal to 1.6 mm. It is important to emphasize that the stiffness of the adherends is much greater than that of the adhesive, such that the adherends do not deform and the sample only deforms in shear.
ACCEPTED MANUSCRIPT The experimental arrangement was composed of a SLJ specimen mounted on the load apparatus and the DIC system, as illustrated in Fig. 1. A special apparatus was used in order to ensure that one of the adherends moved parallel towards the applied load, keeping the distance between plates. Accordingly, the lower adherend was kept fixed, while the upper adherend could move in the horizontal direction. A CCD camera (Sony
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XCD-SX910) set perpendicularly to the specimen was used for capturing the images. All images were acquired using a 10x Zoom C-Mount lens. It is important to remember that the experiments were carried out in quasi-static condition and at room temperature, i.e., 25o C.
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The Digital image correlation (DIC) method was employed to measure full-field displacements of the SLJ specimens, mainly the distortion of the sample. DIC is a optical-numerical
method
developed
to
estimate
full-field
surface
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powerful
displacements, being well reported in the literature [22,23]. Displacement fields are obtained through image comparison. Correlation process is performed by comparing small subsets from the undeformed image to subsets from each of the deformed images. 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.
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The SLJ specimens were sprayed with black paint to obtain a random black and white speckle pattern to perform the correlation procedure. In order to estimate the displacement fields, images of the undeformed and deformed states were captured and
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processed using a home-made DIC code.
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3. Constitutive equation for simple shear Figure 2 illustrates schematically a single lap joint specimen under applied load F. A rectangular block of specimen may be associated to a small rectangle on the surface at the central region of the sample (adhesive). By considering that this element is subjected only to simple shear deformation and that there is no triaxial extension, i.e.,
1 2 3 1, the deformed configuration is written as a function of the reference configuration, as follows:
x1 X1 kX2 x2 X 2 x3 X 3
(1)
ACCEPTED MANUSCRIPT where k is the amount of shear. Assuming a plane strain condition, the principal stretches can be expressed as functions of the amount of shear,
k k2 4 ( 1) 2 1 2 1 3 1
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1
(2)
in which 1 3 2 , where 3 is in the X3-direction. As noted by Ogden [4] the
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k 1 11 for 1 1. amount of shear is given by
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Using Eq. (1), the deformation gradient tensor for simple shear can be expressed as 1 k 0 F 0 1 0 0 0 1
(3)
In this case, the left Cauchy-Green strain tensor is written as k 2 1 k 0 B FF T k 1 0 0 1 0
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(4)
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and the principal invariants of B are given by
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I1 trB k 2 3 1 2 I2 trB trB 2 k 2 3 2 I3 1
(5)
The constitutive relation for an incompressible, homogeneous and isotropic material at finite strains can be derived from a strain energy density function W . This function
can be expressed as functions of I1 and I2 , i.e., W W I1,I2 . As obtained by Rivlin [7], the Cauchy stresses associated with simple shear are given by
dW dW 1 pI 2 B 2 B dI1 dI2
(6)
ACCEPTED MANUSCRIPT where p is the Lagrange multiplier, which is associated with incompressibility. It has been reported in the hyperelastic materials literature that the stored energy is strongly dependent on first invariant. Materials characterized by such behavior are called generalized neo-Hookean materials. Several strain-energy density functions
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defined by W W I1 have been proposed [24-28]. In the present work it is assumed a strain energy density function that depends only onfirst invariant, i.e., W W I1. In this particular case, the Cauchy stress tensor expressed in (6) can be rewritten as pI 2
dW B dI1
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(7)
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The stress components for simple shear deformation are schematically shown in Fig. be expressed by, 3. By using Eq (7), they can
11 p 2k 2 1 dW dI1 dW 33 p 2 dI1 dW 12 2k dI1
dW dI1
22 p 2
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(8)
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In the present investigation, the Cauchy shear stress is defined by 12 F A , where F and A are the applied load and the unstrained cross-sectional area, respectively. Two approaches are commonly used in order to solve theproblem of simple shear: the first one is based on a plane stress condition, in this case 33 0; the second hypothesis assumes that the normal component of the traction on inclined surfaces is equal to zero, i.e. N 0 , [7,13,15]. The tangential S and normal N components on inclined surfaces are illustrated in Fig. 3. As pointed out by Rivlin [7] and used by Horgan and Murphy [15], these tractions can be expressed by S
12 and N 22 kS 1 k 2
Employing relations (8), this yields
(9)
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2k dW 2 dW and N p 2 1 k dI1 1 k 2 dI1
(10)
Assuming the hypothesis of plane stress condition, the Cauchy stress tensor components are given by:
dW dI1 22 33 0 dW 12 2k dI1
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(11)
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and using the relation 12 F A , (11) becomes: 11 k12 22 33 0 F 12 A
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11 2k 2
12 2 dW k N p 2 2 12 and p 1 k k 1 k dI1
(12)
(13)
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In contrast with the previous approach, if one considers that the normal component inclined surfaces is equal to zero, of the traction on the equations for the Cauchy
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stresses are:
11
2k 2 2 k 2 dW
1 k 2 dI1 2 2k dW 22 33 1 k 2 dI1 dW 12 2k dI1
(14)
and by substituting 12 F A , (14) turns into:
k 2 k 2 11 12 1 k 2 k 22 33 1 k 2 12 F 12 A
(15)
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12 k 1 k 2
(16)
It should be mentioned that relations (11) and (14) are in agreement with universal for isotropic elastic solid may be described by relations. A class of universal relations the coaxiality of the left Cauchy-Green strain tensor and Cauchy stress tensor, i.e.
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B B , as presented by Beatty [32]. Substituting (4) and (7) into this universal rule,
the trivial relations given by 13 23 31 32 0 and the usual one 11 22 k12 were obtained. Furthermore, in the present work a strain energy that depends only on first invariant is used; consequently, the relation 22 33 was achieved by u Bu u 0 , where u is an arbitrary vector [33,34].
4. Results and discussion
Full-field displacements of a small rectangle on the surface at the central region of the sample (see Fig. 2) were estimated using the DIC method. The X1- and X2displacement fields of the selected region, related with horizontal and vertical directions for an applied load of 0.948 kN, are illustrated in Figs. 4(a) and 4(b). It can be noted
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that the X1-displacement field varies linearly along the vertical direction (X2), while the X2-displacement field does not present significant variation. This indicates that an angular distortion was generated and the vertical distribution remained almost
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underformed.
Displacement fields data were used to determine the amount of shear. Moreover, the values of horizontal and vertical extensions from the selected region were estimated. By
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using such data, the lateral and diagonal stretches were also evaluated. The lateral and diagonal stretches are defined as the lateral surface and the diagonal of rectangular element previously presented in Fig 2. Figure 5 shows the mean values of horizontal and vertical extensions, i.e. 1 and 2 respectively (see Fig.2), as a function of amount of shear. Three tests were performed using the experimental setup for simple shear configuration. The standard deviation of the experimental data is also illustrated in Fig.5. It is possible to observe that the values are approximately equal to 1 with an error of approximately 5% for 1 and 2.5% for 2. This is a reasonable uncertainty and can be associated to the experimental procedure
ACCEPTED MANUSCRIPT and DIC measurement. Moreover, if these values are assumed, the simple shear deformation is achieved and, consequently, Eq.(1) is suitable. The values of 2 indicate that the upper plate moved parallel under load; moreover, no significant deformation was generated in vertical direction. Overall, the results indicate that the performed single lap shear test is appropriated to generate simple shear state.
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Comparison of lateral, diagonal and principal stretches from the selected region is illustrated in Fig. 6. As physically expected, the principal stretch presents the same value of the diagonal stretch for small values of amount of shear, while it tends to
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lateral stretch for large deformations.
Figure 7 shows the experimental results of applied force against the amount of shear for three SLJ specimens with different cross-sectional areas. Considering the applied
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loads per cross-sectional area of each specimen, the correlation between shear stress and amount of shear was also found. The mean values of shear stress and standard deviation from three repeated measurements are shown in Fig. 8. These data are summarized in Appendix A. One can notice that the relation between shear stress and amount of shear is nonlinear, as reported in the literature [18,19]. According to the results, the sample is softening in shear. Therefore, this relation rules out neo-Hookean and Mooney-Rivlin
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models and also rules out the one-term Ogden material at 1 2 . As previously described in section 3, the normal stress components can be expressed
as functions of shear stress assuming two approaches, i.e. N = 0 or 33 0 . Thus, by
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substituting the experimental data of shear stress into (12) and (15), the normal stress
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components were achieved. Results of normal stress components as functions of the amount of shear are presented in Fig. 9. Values of the first normal stress are close. However, the plots for the second normal stress present significant differences between both approaches. It is known that most materials have a tendency to expand in the perpendicular direction to the applied shear stress, yielding a positive normal stress, i.e., positive 22 [11]. It can be seen from equations (11)2 and (14)2 that 33 22 for both hypotheses.
normal components of the traction on As additional information, the tangential and obtained using Eqs. (9) and (13). Figures 10(a) and 10(b) the inclined faces were illustrate the components tangential S and normal N on the inclined faces as functions of
ACCEPTED MANUSCRIPT amount of shear and lateral stretch, respectively. It is possible to notice that the maximum value of tangential traction in Fig 10(a) corresponds to the maximum value of difference between principal and lateral stretches, at k 0.8, which can be observed in Fig 6. Moreover, Fig 10(b) indicates that, for lateral stretch greater than 1.2, both
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components have the same order, showing that the deformation is homogeneous [12].
Strain-energy density analysis
The values of material parameters that govern the constitutive relations can vary significantly, depending on the type of experimental test. However, the initial shear
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modulus remains almost identical for all the loading cases and sample configurations. For that reason, the initial shear modulus of the present material was estimated using
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single lap shear test.
In the experimental procedures, time-dependent effects were neglected. Hence, elastic behavior under large deformations has been assumed and it can be described in terms of a strain-energy function that depends only on first invariant. Gent model [24] is one of the most used. Another well-known model is the Arruda-Boyce model [25],
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which is essentially identical to Gent model. Both models were initially tested, however, they were unable to characterize the behavior of the experimental data of simple shear. On the other hand, Lopez-Pamies model [28] presented good results. This
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model is defined as
31 r r I1 r 3 r 2 r r 1 M
W I1
(17)
where M indicates the number of terms, while r and r are material parameters being
M
r 0 and. r 1 2 . The initial shear modulus is defined as r .
r 1
Substituting (17), considering only one term, into (11)3 or (14)3 yields k 2 3 1 12 k 3
(18)
In the present work, eq. (18) was fitted to the measured data of stress versus amount the initial shear modulus. The parameters were estimated of shear in order to determine
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0.52 0.03 and 0.69 0.05. Consequently, the value of initial shear modulus was equal to 0.52 MPa. Appendix B presents a brief description of the parameter estimation.
The same material was characterized by Moreira and Nunes [30], employing a pure
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shear test. For comparison purpose, Lopez-Pamies strain energy density was also applied in the theoretical pure shear analysis and the first Piola-Kirchhoff stress for pure shear [31] was fitted to experimental data obtained from our previous work [30] (see Appendix B). Thus, the estimated value of initial shear modulus was equal to 0.53 MPa,
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which is approximately the same value estimated using simple shear analysis.
5. Conclusions
In this work, experimental and theoretical analyses on large simple shear deformation were carried out. Simple shear deformation was experimentally achieved using a single lap shear test. In this configuration, only simple shear was obtained and no significant triaxial extension was observed. Full-field displacements were measured
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by the Digital Image Correlation method. In addition, experimental data were used in the hyperelastic constitutive equations, assuming the strain energy density as a function of the first invariant. Two approaches were investigated to compare the normal stresses: the first one was based on plane stress, while the second assumed that the normal
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component of the traction on the inclined surfaces was zero. As a closing remark, it is important to mention that a nonlinear relation between shear stress and amount of shear was observed, which ruled out classical models such as neo-Hookean and Mooney-
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Rivlin. Therefore, the Lopez-Pamies model was used. As observed and expected, the normal stress components do not vanish in simple shear under finite strain. Finally, the Lopez-Pamies model was fitted to the experimental data and the initial shear modulus was estimated and compared to that achieved by pure shear test in a previous study, presenting good agreement between both values. The main contribution of this work is to provide experimental and theoretical analyses taking into account only simple shear. Accordingly, the obtained results, which were based on experimental data, can provide additional information for comparative evaluation of various recent works reported in literature.
ACCEPTED MANUSCRIPT Acknowledgements The financial support of Rio de Janeiro State Funding, FAPERJ, and Research and Teaching National Council, CNPq, are gratefully acknowledged. Moreover, the authors
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would like to thank Christina Otto for her assistance in manuscript revision.
Appendix A. Experimental data for simple shear
Table A1 presents experimental results of the amount of shear associated with mean
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shear stress obtained from simple shear test. These data were based on three
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experimental tests.
Appendix B. Parameter estimation
The classical Levenberg-Marquardt method for nonlinear parameter estimation was employed in order to estimate the parameters and . For simple shear case, expression
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(18) was used. Moreover, experimental data of a thin sheet of FlextecFT 101 polymer under planar tension were also utilized [30]. In this last case, the first Piola-Kirchhoff stress for pure shear [31], which was derived from the Lopez-Pamies strain energy
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density (see Eq.(17)), can be expressed as a function of stretch by
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P11 31
4 1 2 2 3 5 1
(B1)
In order to characterize that material, the experimental datum set was fitted using (B1). The estimated parameters were 0.53 0.01 and 0.94 0.08 . The results of
experimental data and fitted model for simple and pure shear are illustrated in Figs. B1(a) and B1(b), respectively.
The residue was defined as the difference between the measured (EXP) and estimated (T) stresses. The values of residue, for simple and pure shear, are presented in Figs B2(a) and B2(b). Note that, the results shown in those figures exhibit a random signature, which corroborates the adequacy of the mathematical model. Local sensitivity analyses for estimated shear and first Piola-Kirchhoff stresses with
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illustrate
and J1 12
sensitivity
coefficients
J1 12
( J1 P11 )
( J2 P11 ), as well as the ratio between them, i.e., J1 J2 .
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According to Figs. B3(a) and B3(b), for the chosen sets of parameters, one may conclude that there is no linear dependence between these sensitivity coefficients; therefore, the two parameters may be estimated simultaneously.
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Reference
1. S.H. Crandall, N.C. Dahl, T.J. Lardner. An Introduction to the Mechanics of
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Solids: Second Edition with SI Units. 1999
2. A.L. Lurie. Theory of Elasticity, Springer, Berlin. 2005 3. M.E. Gurtin, E. Fried, L. Anand. The mechanics and thermodynamics of continua. Cambridge, UK: Cambridge University Press. 2010 4. R.W. Ogden. Non-Linear Elastic Deformations, Dover Publications, Mineola,
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NY. 1997
5. G.A. Holzapfel. Nonlinear Solid Mechanics: A continuum approach for engineering, John Wiley & Sons Ltd. 2008
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6. C. Truesdell and W. Noll. The Non-Linear Field Theories of Mechanics, 2nd ed. Springer, Berlin. 1965
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the
7. R.S. Rivlin. Large Elastic Deformations of Isotropic Materials. IV. Further Developments of the General Theory. Phil. Trans. R. Soc. Lond. A 241 (1948) 379-397.
8. A. Wineman and M. Gandhi. On local and global universal relations in elasticity. J. Elast. 14 (1984) 97-102. 9. K.R. Rajagopal and A.S. Wineman. New universal relations for nonlinear isotropic elastic materials. J. Elast. 17 (1987) 75-83. 10. M. Destrade, R.W. Ogden, Surface waves in a stretched and sheared
ACCEPTED MANUSCRIPT incompressible elastic material, Int. J. Non-Linear Mech. 40 (2005) 241–253. 11. L.A. Mihai and A. Goriely. Positive or negative Poynting effect? The role of adscititious inequalities in hyperelastic materials. Proc. R. Soc. A 467/2136 (2011) 3633-3646
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12. M. Destrade, J.G. Murphy, G. Saccomandi. Simple shear is not so simple. Int. J. Nonlinear Mech. 47 (2012) 210–214
13. A.N. Gent, J.B. Suh, S.G. Kelly III. Mechanics of rubber shear springs. Int. J.
14. M.
Destrade,
M.D.
Gilchrist,
J.
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NonLinear Mech. 42 (2007) 241 – 249. Motherway,
J.G.
Murphy.
Slight
Engng, 90 (2012) 403-411.
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compressibility and sensitivity to changes in Poisson's ratio, Int. J. Numer. Meth.
15. C.O. Horgan and J.G. Murphy. Simple Shearing of Incompressible and Slightly Compressible Isotropic Nonlinearly Elastic Materials. J. Elast. 98 (2010) 205– 221.
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16. R. Brown, Physical Testing of Rubber, Springer, New York, 2006. 17. M. Mooney. Stress-strain curves of rubbers in simple shear. J. Appl. Phys. 35 (1944) 23–26.
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18. L.C.S. Nunes. Mechanical characterization of hyperelastic polydimethylsiloxane by simple shear test. Mat Sci Eng A 528 (2011) 1799–1804.
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19. L.C.S. Nunes. Shear modulus estimation of the polymer polydimethylsiloxane (PDMS) using digital image correlation. Mater Design 31 (2010) 583–588.
20. P.A. Janmey, M.E. McCormick, S. Rammensee, J.L. Leight, P.C. Georges & F.C. MacKintosh. Negative normal stress in semiflexible biopolymer gels. Nat. Mater. 6 (2006) 48–51. 21. ISO 1827:2011-11(E). Rubber, vulcanized or thermoplastic – Determination of shear modulus and adhesion to rigid plates – Quadruple-shear methods 22. J.W. Dally and W.F. Riley. Experimental Stress Analysis, 4th ed. McGraw Hill. 2005
ACCEPTED MANUSCRIPT 23. Sutton, M.A., Orteu, J.J., Schreier, H.W. Image Correlation for Shape, Motion and Deformation Measurements, Springer Science and Business Media LCC. 2009. 24. A.N. Gent, A new constitutive relation for rubber, Rubber Chem. Technol. 69
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(1996) 56–61. 25. E.M. Arruda, M.C. Boyce, A three-dimensional constitutive model for the large stretch behavior of elastic materials, J.Mech. Phys. Solids 41 (1993) 389–412.
Nonlinear Mech.40 (2005) 271 – 279
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26. A. Wineman. Some results for generalized neo-Hookean elastic materials. Int. J.
27. C.O. Horgan and G. Saccomandi, Antiplane shear deformations for non-
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Gaussian isotropic, incompressible hyperelastic materials. Proc. Roy. Soc. London A 457 (2001) 1999–2017.
28. O. Lopez-Pamies, A new I1-based hyperelastic model for rubber elastic materials. C. R. Mecanique 338 (2010) 3–11
29. M.N. Özisik and H.R.B. Orlande, Inverse Heat Transfer: Fundamentals and
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Applications, Taylor and Francis, 2000. 30. D.C. Moreira and L.C.S Nunes. Comparison of simple and pure shear for an incompressible isotropic hyperelastic material under large deformation, Polym.
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Test, 32(2013) 240-248.
31. L.R.G. Treloar. The physics of rubber elasticity, 3rd ed., Clarendon Press,
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Oxford. 2005.
32. M.F. Beatty, A class of universal relations in isotropic elasticity theory. J. Elasticity 17 (1987)113–121.
33. E. Pucci and G. Saccomandi, On universal relations in continuum mechanics. Continuum Mech. Thermodyn. 9 (1997) 61–72. 34. [Y.B. Fu and R.W. Ogden. Nonlinear Elasticity: Theory and Applications (London Mathematical Society Lecture Note Series). Cambridge University Press, 2001
ACCEPTED MANUSCRIPT Figure captions Fig. 1. Experimental arrangement with detailed SLJ specimen Fig. 2. Configurations in the reference and deformed states Fig. 3. The stress components for simple shear deformation
Fig. 5. Horizontal and vertical extensions Fig. 6. Comparison of lateral, diagonal and principal stretches
Fig. 8. Shear stress versus amount of shear
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Fig. 7. Applied force versus amount of shear
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Fig. 4. Full-field displacements: (a) horizontal and (b) vertical displacements
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Fig. 9. Normal stresses versus amount of shear for plane stress and N = 0 approaches Fig. 10. Normal and tangential components of the traction on the inclined faces versus amount of shear (a) and lateral stretch (b) Fig. B1. Comparison between experimental data and Lopez-Pamies model: (a) Simple shear stress versus amount of shear and (b) The first Piola-Kirchhoff stress versus stretch for pure shear. Fig. B2. Residue: (a) Simple shear and (b) Pure shear.
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Tables
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Fig. B3. Sensitivity coefficients and sensitivity coefficients ratio: (a) Simple shear and (b) Pure shear.
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Table A1. Experimental data for simple shear
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X −Displacement (mm)
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1
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4
2
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3
1 −0.6 1
−0.8
0.8
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Table
ACCEPTED MANUSCRIPT Table A1. Experimental data for simple shear
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Standard deviation (MPa) 0.00 0.02 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.03 0.04 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.05 0.05
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Mean shear stress (MPa) 0.00 0.09 0.14 0.19 0.23 0.28 0.32 0.36 0.40 0.44 0.48 0.51 0.55 0.58 0.61 0.65 0.68 0.71 0.74 0.76 0.79 0.82 0.85 0.87 0.90 0.93 0.96 0.97 0.99 1.00 1.03
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Amount of shear 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00
S0997-7538(13)00077-6
DOI:
10.1016/j.euromechsol.2013.07.002
Reference:
EJMSOL 2954
To appear in:
European Journal of Mechanics / A Solids
Received Date: 2 February 2013 Revised Date:
18 June 2013
Accepted Date: 7 July 2013
Please cite this article as: Nunes, L.C.S., Moreira, D.C., Simple shear under large deformation: experimental and theoretical analyses, European Journal of Mechanics / A Solids (2013), doi: 10.1016/ j.euromechsol.2013.07.002. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Graphical Abstract (for review)
Experimental Lopez−Pamies model
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*Highlights (for review)
ACCEPTED MANUSCRIPT Research highlights
Experimental and theoretical analyses on simple shear deformation were developed
Simple shear deformation was experimentally achieved using a single lap joint test Nonlinear relation between shear stress and amount of shear was obtained
Normal stress components were obtained from the experimental data
The Lopez-Pamies model was fitted to the experimental data
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Manuscript Click here to view linked References
ACCEPTED MANUSCRIPT Simple shear under large deformation: experimental and theoretical analyses L.C.S. Nunes* and D.C. Moreira
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Laboratory of Opto-Mechanics / Laboratory of Theoretical and Applied Mechanics (LOM/LMTA) Department of Mechanical Engineering, TEM/PGMEC Universidade Federal Fluminense, UFF Rua Passo da Patria 156, 24210-240 Niterói, RJ, Brazil
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Abstract
The aim of the present work is to analyze the simple shear state of an incompressible
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hyperelastic solid under large deformation by experimental and theoretical approaches. The experimental procedure was performed using a single lap shear test and the displacement fields were determined by the Digital Image Correlation method. The applied force and the measured angular distortion were used to evaluate the shear stress and the amount of shear. Hence, a nonlinear stress-strain response was achieved. In addition, the normal stress components were obtained from the experimental data by
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assuming two hypotheses: the first one was based on a plane stress condition, while the normal component of the traction on the inclined surfaces was assumed to be zero on the second. Finally, to verify the presented results, the initial shear modulus of the hyperelastic material was estimated and compared with the value obtained using the
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data from a planar tensile test. The Lopez-Pamies strain energy function was used in the inverse analysis in order to estimate the material property, which was similar for both
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experimental tests.
Keywords: Simple shear, large deformation, hyperelasticity, full-field displacement method
*Corresponding author. Tel.: +55 21 2629-5588 E-mail address: [email protected]
ACCEPTED MANUSCRIPT 1. Introduction Recently, there has been growing interest in the state of simple shear. In the linear theory of elasticity, simple shear deformation is well defined [1,2]. This state is also reported in many textbooks on nonlinear theory [3-6], however, some points concerning stress components remain unclear. In fact, in an isotropic nonlinear elastic solid, pure
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shear stress is not accompanied by simple shear strain [3]. The first analysis of simple shear under finite deformation, considering a block of incompressible material, was proposed by Rivlin [7]. In addition, universal relations for nonlinear isotropic elastic materials under simple shear superposed on triaxial extension
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have been widely investigated [8-10]. Based on those works, recent investigations have been developed. Mihai and Goriely [11] showed that for a homogeneous isotropic
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hyperelastic material subject to pure (uniform) shear stress, the resulting deformation consists of a triaxial stretch (pure strain deformation) combined with a simple shear in the direction of the shear force if and only if the BE inequalities hold. Furthermore, Destrade et al. [12] have concluded that a pure shearing stress field produces a deformation field that is not the classical simple shear deformation, but a simple shear deformation superimposed upon a triaxial stretch. Gent et al. [13] and Destrade et al.
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[14] have investigated the stress distributions of a nonlinear elastic block under large shear deformations by finite element analysis. They considered two rigid plates bonded with elastic material subjected to shear deformations, keeping the distance between the plates. Motivated by this work, Horgan and Murphy [15] conducted an investigation
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into three different formulations of simple shear deformation.
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Simple shear test is not easy to perform experimentally [16]. Therefore, few experimental works on simple shear have been reported in the literature. Mooney reported a series of measurements on soft rubber using a thin-walled hollow cylinder [17]. More recently, Nunes [18,19] studied the nonlinear mechanical behavior of a hyperelastic material under small and large simple shear deformations by a single lap shear test. Some points concerning mechanical behavior on shear state are still open to discussion. For instance, it is known that most materials have a tendency to expand in the direction perpendicular to the applied shear load. Nevertheless, Janmey et al [20] showed experimentally that networks of semiflexible biopolymers exhibit the opposite tendency. For such materials, the empirical inequalities must not be imposed [11].
ACCEPTED MANUSCRIPT Despite all previous contributions, it is evident that further investigation based on experimental and theoretical studies is essential in order to understand simple shear. In this context, the present work investigates large simple shear deformations by experimental and theoretical approaches. The experimental procedure was performed using a single lap shear test; in which specimens manufactured with two rigid plates
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bonded by a hyperelastic material were submitted to shear deformation. In this test, the full-field displacements were evaluated employing a powerful and noncontacting method, known as Digital Image Correlation (DIC). The applied force and the measured angular distortion were used to evaluate the shear stress and the amount of shear. In
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addition, the normal stress components were obtained from the experimental data by assuming two hypotheses: a plane stress condition was considered in the first one, while
on the second.
2. Material and methods
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the normal component of the traction on the inclined surfaces was assumed to be zero
For the purpose of this study, simple shear deformation of large amount was
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obtained by a single lap shear test. This test was chosen due to its simplicity. Although quadruple-lap shear is a standard test method for determining shear modulus [21], single- and double-lap shear testing can also be considered for this purpose. However, the supporting plates may present a tendency of moving out of parallel under load in
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both single and double-lap test methods [16]. To overcome this limitation, the stiffness of the plates (adherends) must be much greater than the stiffness of the adhesive
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[18,19].
The single lap joint (SLJ) specimens were manufactured with two rigid plates (or adherends of steel A36) bonded by an adhesive, which was the analyzed sample. This adhesive is based on silane modified polymer (FlextecFT 101), and was chosen because it presents high flexibility and elasticity. The SLJ specimens had overlap lengths of 21, 34 and 38 mm and a joint width of 25.4 mm, with the adherend and adhesive thicknesses equal to 1.6 mm. It is important to emphasize that the stiffness of the adherends is much greater than that of the adhesive, such that the adherends do not deform and the sample only deforms in shear.
ACCEPTED MANUSCRIPT The experimental arrangement was composed of a SLJ specimen mounted on the load apparatus and the DIC system, as illustrated in Fig. 1. A special apparatus was used in order to ensure that one of the adherends moved parallel towards the applied load, keeping the distance between plates. Accordingly, the lower adherend was kept fixed, while the upper adherend could move in the horizontal direction. A CCD camera (Sony
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XCD-SX910) set perpendicularly to the specimen was used for capturing the images. All images were acquired using a 10x Zoom C-Mount lens. It is important to remember that the experiments were carried out in quasi-static condition and at room temperature, i.e., 25o C.
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The Digital image correlation (DIC) method was employed to measure full-field displacements of the SLJ specimens, mainly the distortion of the sample. DIC is a optical-numerical
method
developed
to
estimate
full-field
surface
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powerful
displacements, being well reported in the literature [22,23]. Displacement fields are obtained through image comparison. Correlation process is performed by comparing small subsets from the undeformed image to subsets from each of the deformed images. 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.
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The SLJ specimens were sprayed with black paint to obtain a random black and white speckle pattern to perform the correlation procedure. In order to estimate the displacement fields, images of the undeformed and deformed states were captured and
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processed using a home-made DIC code.
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3. Constitutive equation for simple shear Figure 2 illustrates schematically a single lap joint specimen under applied load F. A rectangular block of specimen may be associated to a small rectangle on the surface at the central region of the sample (adhesive). By considering that this element is subjected only to simple shear deformation and that there is no triaxial extension, i.e.,
1 2 3 1, the deformed configuration is written as a function of the reference configuration, as follows:
x1 X1 kX2 x2 X 2 x3 X 3
(1)
ACCEPTED MANUSCRIPT where k is the amount of shear. Assuming a plane strain condition, the principal stretches can be expressed as functions of the amount of shear,
k k2 4 ( 1) 2 1 2 1 3 1
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1
(2)
in which 1 3 2 , where 3 is in the X3-direction. As noted by Ogden [4] the
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k 1 11 for 1 1. amount of shear is given by
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Using Eq. (1), the deformation gradient tensor for simple shear can be expressed as 1 k 0 F 0 1 0 0 0 1
(3)
In this case, the left Cauchy-Green strain tensor is written as k 2 1 k 0 B FF T k 1 0 0 1 0
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(4)
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and the principal invariants of B are given by
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I1 trB k 2 3 1 2 I2 trB trB 2 k 2 3 2 I3 1
(5)
The constitutive relation for an incompressible, homogeneous and isotropic material at finite strains can be derived from a strain energy density function W . This function
can be expressed as functions of I1 and I2 , i.e., W W I1,I2 . As obtained by Rivlin [7], the Cauchy stresses associated with simple shear are given by
dW dW 1 pI 2 B 2 B dI1 dI2
(6)
ACCEPTED MANUSCRIPT where p is the Lagrange multiplier, which is associated with incompressibility. It has been reported in the hyperelastic materials literature that the stored energy is strongly dependent on first invariant. Materials characterized by such behavior are called generalized neo-Hookean materials. Several strain-energy density functions
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defined by W W I1 have been proposed [24-28]. In the present work it is assumed a strain energy density function that depends only onfirst invariant, i.e., W W I1. In this particular case, the Cauchy stress tensor expressed in (6) can be rewritten as pI 2
dW B dI1
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(7)
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The stress components for simple shear deformation are schematically shown in Fig. be expressed by, 3. By using Eq (7), they can
11 p 2k 2 1 dW dI1 dW 33 p 2 dI1 dW 12 2k dI1
dW dI1
22 p 2
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(8)
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In the present investigation, the Cauchy shear stress is defined by 12 F A , where F and A are the applied load and the unstrained cross-sectional area, respectively. Two approaches are commonly used in order to solve theproblem of simple shear: the first one is based on a plane stress condition, in this case 33 0; the second hypothesis assumes that the normal component of the traction on inclined surfaces is equal to zero, i.e. N 0 , [7,13,15]. The tangential S and normal N components on inclined surfaces are illustrated in Fig. 3. As pointed out by Rivlin [7] and used by Horgan and Murphy [15], these tractions can be expressed by S
12 and N 22 kS 1 k 2
Employing relations (8), this yields
(9)
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2k dW 2 dW and N p 2 1 k dI1 1 k 2 dI1
(10)
Assuming the hypothesis of plane stress condition, the Cauchy stress tensor components are given by:
dW dI1 22 33 0 dW 12 2k dI1
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(11)
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and using the relation 12 F A , (11) becomes: 11 k12 22 33 0 F 12 A
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11 2k 2
12 2 dW k N p 2 2 12 and p 1 k k 1 k dI1
(12)
(13)
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In contrast with the previous approach, if one considers that the normal component inclined surfaces is equal to zero, of the traction on the equations for the Cauchy
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stresses are:
11
2k 2 2 k 2 dW
1 k 2 dI1 2 2k dW 22 33 1 k 2 dI1 dW 12 2k dI1
(14)
and by substituting 12 F A , (14) turns into:
k 2 k 2 11 12 1 k 2 k 22 33 1 k 2 12 F 12 A
(15)
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12 k 1 k 2
(16)
It should be mentioned that relations (11) and (14) are in agreement with universal for isotropic elastic solid may be described by relations. A class of universal relations the coaxiality of the left Cauchy-Green strain tensor and Cauchy stress tensor, i.e.
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B B , as presented by Beatty [32]. Substituting (4) and (7) into this universal rule,
the trivial relations given by 13 23 31 32 0 and the usual one 11 22 k12 were obtained. Furthermore, in the present work a strain energy that depends only on first invariant is used; consequently, the relation 22 33 was achieved by u Bu u 0 , where u is an arbitrary vector [33,34].
4. Results and discussion
Full-field displacements of a small rectangle on the surface at the central region of the sample (see Fig. 2) were estimated using the DIC method. The X1- and X2displacement fields of the selected region, related with horizontal and vertical directions for an applied load of 0.948 kN, are illustrated in Figs. 4(a) and 4(b). It can be noted
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that the X1-displacement field varies linearly along the vertical direction (X2), while the X2-displacement field does not present significant variation. This indicates that an angular distortion was generated and the vertical distribution remained almost
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underformed.
Displacement fields data were used to determine the amount of shear. Moreover, the values of horizontal and vertical extensions from the selected region were estimated. By
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using such data, the lateral and diagonal stretches were also evaluated. The lateral and diagonal stretches are defined as the lateral surface and the diagonal of rectangular element previously presented in Fig 2. Figure 5 shows the mean values of horizontal and vertical extensions, i.e. 1 and 2 respectively (see Fig.2), as a function of amount of shear. Three tests were performed using the experimental setup for simple shear configuration. The standard deviation of the experimental data is also illustrated in Fig.5. It is possible to observe that the values are approximately equal to 1 with an error of approximately 5% for 1 and 2.5% for 2. This is a reasonable uncertainty and can be associated to the experimental procedure
ACCEPTED MANUSCRIPT and DIC measurement. Moreover, if these values are assumed, the simple shear deformation is achieved and, consequently, Eq.(1) is suitable. The values of 2 indicate that the upper plate moved parallel under load; moreover, no significant deformation was generated in vertical direction. Overall, the results indicate that the performed single lap shear test is appropriated to generate simple shear state.
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Comparison of lateral, diagonal and principal stretches from the selected region is illustrated in Fig. 6. As physically expected, the principal stretch presents the same value of the diagonal stretch for small values of amount of shear, while it tends to
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lateral stretch for large deformations.
Figure 7 shows the experimental results of applied force against the amount of shear for three SLJ specimens with different cross-sectional areas. Considering the applied
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loads per cross-sectional area of each specimen, the correlation between shear stress and amount of shear was also found. The mean values of shear stress and standard deviation from three repeated measurements are shown in Fig. 8. These data are summarized in Appendix A. One can notice that the relation between shear stress and amount of shear is nonlinear, as reported in the literature [18,19]. According to the results, the sample is softening in shear. Therefore, this relation rules out neo-Hookean and Mooney-Rivlin
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models and also rules out the one-term Ogden material at 1 2 . As previously described in section 3, the normal stress components can be expressed
as functions of shear stress assuming two approaches, i.e. N = 0 or 33 0 . Thus, by
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substituting the experimental data of shear stress into (12) and (15), the normal stress
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components were achieved. Results of normal stress components as functions of the amount of shear are presented in Fig. 9. Values of the first normal stress are close. However, the plots for the second normal stress present significant differences between both approaches. It is known that most materials have a tendency to expand in the perpendicular direction to the applied shear stress, yielding a positive normal stress, i.e., positive 22 [11]. It can be seen from equations (11)2 and (14)2 that 33 22 for both hypotheses.
normal components of the traction on As additional information, the tangential and obtained using Eqs. (9) and (13). Figures 10(a) and 10(b) the inclined faces were illustrate the components tangential S and normal N on the inclined faces as functions of
ACCEPTED MANUSCRIPT amount of shear and lateral stretch, respectively. It is possible to notice that the maximum value of tangential traction in Fig 10(a) corresponds to the maximum value of difference between principal and lateral stretches, at k 0.8, which can be observed in Fig 6. Moreover, Fig 10(b) indicates that, for lateral stretch greater than 1.2, both
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components have the same order, showing that the deformation is homogeneous [12].
Strain-energy density analysis
The values of material parameters that govern the constitutive relations can vary significantly, depending on the type of experimental test. However, the initial shear
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modulus remains almost identical for all the loading cases and sample configurations. For that reason, the initial shear modulus of the present material was estimated using
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single lap shear test.
In the experimental procedures, time-dependent effects were neglected. Hence, elastic behavior under large deformations has been assumed and it can be described in terms of a strain-energy function that depends only on first invariant. Gent model [24] is one of the most used. Another well-known model is the Arruda-Boyce model [25],
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which is essentially identical to Gent model. Both models were initially tested, however, they were unable to characterize the behavior of the experimental data of simple shear. On the other hand, Lopez-Pamies model [28] presented good results. This
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model is defined as
31 r r I1 r 3 r 2 r r 1 M
W I1
(17)
where M indicates the number of terms, while r and r are material parameters being
M
r 0 and. r 1 2 . The initial shear modulus is defined as r .
r 1
Substituting (17), considering only one term, into (11)3 or (14)3 yields k 2 3 1 12 k 3
(18)
In the present work, eq. (18) was fitted to the measured data of stress versus amount the initial shear modulus. The parameters were estimated of shear in order to determine
ACCEPTED MANUSCRIPT using the Levenberg-Marquardt method [29] and the obtained values were
0.52 0.03 and 0.69 0.05. Consequently, the value of initial shear modulus was equal to 0.52 MPa. Appendix B presents a brief description of the parameter estimation.
The same material was characterized by Moreira and Nunes [30], employing a pure
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shear test. For comparison purpose, Lopez-Pamies strain energy density was also applied in the theoretical pure shear analysis and the first Piola-Kirchhoff stress for pure shear [31] was fitted to experimental data obtained from our previous work [30] (see Appendix B). Thus, the estimated value of initial shear modulus was equal to 0.53 MPa,
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which is approximately the same value estimated using simple shear analysis.
5. Conclusions
In this work, experimental and theoretical analyses on large simple shear deformation were carried out. Simple shear deformation was experimentally achieved using a single lap shear test. In this configuration, only simple shear was obtained and no significant triaxial extension was observed. Full-field displacements were measured
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by the Digital Image Correlation method. In addition, experimental data were used in the hyperelastic constitutive equations, assuming the strain energy density as a function of the first invariant. Two approaches were investigated to compare the normal stresses: the first one was based on plane stress, while the second assumed that the normal
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component of the traction on the inclined surfaces was zero. As a closing remark, it is important to mention that a nonlinear relation between shear stress and amount of shear was observed, which ruled out classical models such as neo-Hookean and Mooney-
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Rivlin. Therefore, the Lopez-Pamies model was used. As observed and expected, the normal stress components do not vanish in simple shear under finite strain. Finally, the Lopez-Pamies model was fitted to the experimental data and the initial shear modulus was estimated and compared to that achieved by pure shear test in a previous study, presenting good agreement between both values. The main contribution of this work is to provide experimental and theoretical analyses taking into account only simple shear. Accordingly, the obtained results, which were based on experimental data, can provide additional information for comparative evaluation of various recent works reported in literature.
ACCEPTED MANUSCRIPT Acknowledgements The financial support of Rio de Janeiro State Funding, FAPERJ, and Research and Teaching National Council, CNPq, are gratefully acknowledged. Moreover, the authors
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would like to thank Christina Otto for her assistance in manuscript revision.
Appendix A. Experimental data for simple shear
Table A1 presents experimental results of the amount of shear associated with mean
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shear stress obtained from simple shear test. These data were based on three
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experimental tests.
Appendix B. Parameter estimation
The classical Levenberg-Marquardt method for nonlinear parameter estimation was employed in order to estimate the parameters and . For simple shear case, expression
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(18) was used. Moreover, experimental data of a thin sheet of FlextecFT 101 polymer under planar tension were also utilized [30]. In this last case, the first Piola-Kirchhoff stress for pure shear [31], which was derived from the Lopez-Pamies strain energy
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density (see Eq.(17)), can be expressed as a function of stretch by
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P11 31
4 1 2 2 3 5 1
(B1)
In order to characterize that material, the experimental datum set was fitted using (B1). The estimated parameters were 0.53 0.01 and 0.94 0.08 . The results of
experimental data and fitted model for simple and pure shear are illustrated in Figs. B1(a) and B1(b), respectively.
The residue was defined as the difference between the measured (EXP) and estimated (T) stresses. The values of residue, for simple and pure shear, are presented in Figs B2(a) and B2(b). Note that, the results shown in those figures exhibit a random signature, which corroborates the adequacy of the mathematical model. Local sensitivity analyses for estimated shear and first Piola-Kirchhoff stresses with
ACCEPTED MANUSCRIPT respect to changes in the parameters and are shown in Figs B3(a) and B3(b). These figures
illustrate
and J1 12
sensitivity
coefficients
J1 12
( J1 P11 )
( J2 P11 ), as well as the ratio between them, i.e., J1 J2 .
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According to Figs. B3(a) and B3(b), for the chosen sets of parameters, one may conclude that there is no linear dependence between these sensitivity coefficients; therefore, the two parameters may be estimated simultaneously.
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Reference
1. S.H. Crandall, N.C. Dahl, T.J. Lardner. An Introduction to the Mechanics of
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Solids: Second Edition with SI Units. 1999
2. A.L. Lurie. Theory of Elasticity, Springer, Berlin. 2005 3. M.E. Gurtin, E. Fried, L. Anand. The mechanics and thermodynamics of continua. Cambridge, UK: Cambridge University Press. 2010 4. R.W. Ogden. Non-Linear Elastic Deformations, Dover Publications, Mineola,
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NY. 1997
5. G.A. Holzapfel. Nonlinear Solid Mechanics: A continuum approach for engineering, John Wiley & Sons Ltd. 2008
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6. C. Truesdell and W. Noll. The Non-Linear Field Theories of Mechanics, 2nd ed. Springer, Berlin. 1965
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7. R.S. Rivlin. Large Elastic Deformations of Isotropic Materials. IV. Further Developments of the General Theory. Phil. Trans. R. Soc. Lond. A 241 (1948) 379-397.
8. A. Wineman and M. Gandhi. On local and global universal relations in elasticity. J. Elast. 14 (1984) 97-102. 9. K.R. Rajagopal and A.S. Wineman. New universal relations for nonlinear isotropic elastic materials. J. Elast. 17 (1987) 75-83. 10. M. Destrade, R.W. Ogden, Surface waves in a stretched and sheared
ACCEPTED MANUSCRIPT incompressible elastic material, Int. J. Non-Linear Mech. 40 (2005) 241–253. 11. L.A. Mihai and A. Goriely. Positive or negative Poynting effect? The role of adscititious inequalities in hyperelastic materials. Proc. R. Soc. A 467/2136 (2011) 3633-3646
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12. M. Destrade, J.G. Murphy, G. Saccomandi. Simple shear is not so simple. Int. J. Nonlinear Mech. 47 (2012) 210–214
13. A.N. Gent, J.B. Suh, S.G. Kelly III. Mechanics of rubber shear springs. Int. J.
14. M.
Destrade,
M.D.
Gilchrist,
J.
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NonLinear Mech. 42 (2007) 241 – 249. Motherway,
J.G.
Murphy.
Slight
Engng, 90 (2012) 403-411.
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compressibility and sensitivity to changes in Poisson's ratio, Int. J. Numer. Meth.
15. C.O. Horgan and J.G. Murphy. Simple Shearing of Incompressible and Slightly Compressible Isotropic Nonlinearly Elastic Materials. J. Elast. 98 (2010) 205– 221.
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16. R. Brown, Physical Testing of Rubber, Springer, New York, 2006. 17. M. Mooney. Stress-strain curves of rubbers in simple shear. J. Appl. Phys. 35 (1944) 23–26.
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18. L.C.S. Nunes. Mechanical characterization of hyperelastic polydimethylsiloxane by simple shear test. Mat Sci Eng A 528 (2011) 1799–1804.
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19. L.C.S. Nunes. Shear modulus estimation of the polymer polydimethylsiloxane (PDMS) using digital image correlation. Mater Design 31 (2010) 583–588.
20. P.A. Janmey, M.E. McCormick, S. Rammensee, J.L. Leight, P.C. Georges & F.C. MacKintosh. Negative normal stress in semiflexible biopolymer gels. Nat. Mater. 6 (2006) 48–51. 21. ISO 1827:2011-11(E). Rubber, vulcanized or thermoplastic – Determination of shear modulus and adhesion to rigid plates – Quadruple-shear methods 22. J.W. Dally and W.F. Riley. Experimental Stress Analysis, 4th ed. McGraw Hill. 2005
ACCEPTED MANUSCRIPT 23. Sutton, M.A., Orteu, J.J., Schreier, H.W. Image Correlation for Shape, Motion and Deformation Measurements, Springer Science and Business Media LCC. 2009. 24. A.N. Gent, A new constitutive relation for rubber, Rubber Chem. Technol. 69
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(1996) 56–61. 25. E.M. Arruda, M.C. Boyce, A three-dimensional constitutive model for the large stretch behavior of elastic materials, J.Mech. Phys. Solids 41 (1993) 389–412.
Nonlinear Mech.40 (2005) 271 – 279
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26. A. Wineman. Some results for generalized neo-Hookean elastic materials. Int. J.
27. C.O. Horgan and G. Saccomandi, Antiplane shear deformations for non-
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Gaussian isotropic, incompressible hyperelastic materials. Proc. Roy. Soc. London A 457 (2001) 1999–2017.
28. O. Lopez-Pamies, A new I1-based hyperelastic model for rubber elastic materials. C. R. Mecanique 338 (2010) 3–11
29. M.N. Özisik and H.R.B. Orlande, Inverse Heat Transfer: Fundamentals and
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Applications, Taylor and Francis, 2000. 30. D.C. Moreira and L.C.S Nunes. Comparison of simple and pure shear for an incompressible isotropic hyperelastic material under large deformation, Polym.
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Test, 32(2013) 240-248.
31. L.R.G. Treloar. The physics of rubber elasticity, 3rd ed., Clarendon Press,
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Oxford. 2005.
32. M.F. Beatty, A class of universal relations in isotropic elasticity theory. J. Elasticity 17 (1987)113–121.
33. E. Pucci and G. Saccomandi, On universal relations in continuum mechanics. Continuum Mech. Thermodyn. 9 (1997) 61–72. 34. [Y.B. Fu and R.W. Ogden. Nonlinear Elasticity: Theory and Applications (London Mathematical Society Lecture Note Series). Cambridge University Press, 2001
ACCEPTED MANUSCRIPT Figure captions Fig. 1. Experimental arrangement with detailed SLJ specimen Fig. 2. Configurations in the reference and deformed states Fig. 3. The stress components for simple shear deformation
Fig. 5. Horizontal and vertical extensions Fig. 6. Comparison of lateral, diagonal and principal stretches
Fig. 8. Shear stress versus amount of shear
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Fig. 7. Applied force versus amount of shear
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Fig. 4. Full-field displacements: (a) horizontal and (b) vertical displacements
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Fig. 9. Normal stresses versus amount of shear for plane stress and N = 0 approaches Fig. 10. Normal and tangential components of the traction on the inclined faces versus amount of shear (a) and lateral stretch (b) Fig. B1. Comparison between experimental data and Lopez-Pamies model: (a) Simple shear stress versus amount of shear and (b) The first Piola-Kirchhoff stress versus stretch for pure shear. Fig. B2. Residue: (a) Simple shear and (b) Pure shear.
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Fig. B3. Sensitivity coefficients and sensitivity coefficients ratio: (a) Simple shear and (b) Pure shear.
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Table A1. Experimental data for simple shear
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Figure4a
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X −Displacement (mm)
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1
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4
2
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3
1 −0.6 1
−0.8
0.8
−1
X (mm) 2
0.6 0.4
X (mm) 1
Figure4b
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X −Displacement (mm)
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4
1
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3
0 −0.6
1.2 1
−0.8
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−1
X (mm) 2
0.6 0.4
X (mm) 1
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1.05
δ
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1.05
1
1.5
2
2.5
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1
1.5
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2.5
3
δ2
1
0.5
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0.9
EP
0.95
0.95 0.9 0
0.5
k
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1
2
EP AC C
Stretches
2.5
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Lateral Diagonal λ
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1 0
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1.5
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2.5
3
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2
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0.4
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Applied force (kN)
1
Area = 533.4 mm 2 Area = 863.6 mm 2 Area = 965.2 mm
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1.5
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Figure8
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σ (MPa)
0.8
0.2
0 0
0.5
1
1.5
k
2
2.5
3
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σ
(N = 0)
σ
=σ
σ
(Plane stress)
σ
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11
2.5
22 11
2 1.5 1
33
(N = 0)
(Plane stress)
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33
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Normal stresses (MPa)
3
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Figure10a
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Tangential Normal
−0.3 −0.4 0
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1.5
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Figure10b
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Traction (MPa)
0.2 0.1
Tangential Normal
−0.2 −0.3 −0.4 1
1.5
2
2.5
Lateral stretch
3
3.5
FigureB1a
Experimental Lopez−Pamies model
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12
σ (MPa)
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0.2
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1.5
k
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3
FigureB1b
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Experimental Lopez−Pamies model
0.8
0.5 0.4 0.3
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P (MPa)
0.7
0.2 0.1 0 1
1.2
1.4
λ
1.6
1.8
2
FigureB2a
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σExp −σT (MPa) 12 12
0.01
−0.03
−0.04 0
0.5
1
1.5
k
2
2.5
3
FigureB2b
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x 10
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−2 −4 −6 −8
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PExp − PT (MPa) 11 11
2
−10 −12 −14 1
1.2
1.4
λ
1.6
1.8
2
FigureB3a
1
1
12
J = ∂σ /∂µ 2
12
0 0
1
1.5
2
2.5
3
2
2.5
3
k
1
J /J
2
1
0.5
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J = ∂σ /∂α
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Sensitivity coefficients
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0.5
0 0
0.5
1
1.5
k
FigureB3b
1
1
11
J = ∂P /∂µ 2
11
0.4
1.2
1.4
λ
1.6
1.8
2
1.6
1.8
2
0.2
1
J /J
2
0.3
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0.5 0 1
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J = ∂P /∂α
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Sensitivity coefficients
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0.1 0 1
1.2
1.4
λ
Table
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Standard deviation (MPa) 0.00 0.02 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.03 0.04 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.05 0.05
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Mean shear stress (MPa) 0.00 0.09 0.14 0.19 0.23 0.28 0.32 0.36 0.40 0.44 0.48 0.51 0.55 0.58 0.61 0.65 0.68 0.71 0.74 0.76 0.79 0.82 0.85 0.87 0.90 0.93 0.96 0.97 0.99 1.00 1.03
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Amount of shear 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00