Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`DUROMETER HARDNESS AND THE STRESS-STRAIN BEHAVIOR OF
`
`ELASTOMERIC MATERIALS
`
`H. J. QI, K. JOYCE, M. C. BOYCE*
`
`DEPARTMENT OF MECHANICAL ENGINEERING
`MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MA 02139
`
`ABSTRACT
`
`The Durometer hardness test is one of the most commonly used measurements to qualitatively assess and compare
`the mechanical behavior of elastomeric and elastomeric-like materials. This paper presents nonlinear finite element sim-
`ulations of hardness tests which act to provide a mapping of measured Durometer Shore A and D values to the stress-
`strain behavior of elastomers. In the simulations, the nonlinear stress-strain behavior of the elastomers is first represent-
`ed using the Gaussian (neo-Hookean) constitutive model. The predictive capability of the simulations is verified by com-
`parison of calculated conversions of Shore A to Shore D values with the guideline conversion chart in ASTM D2240. The
`simulation results are then used to determine the relationship between the neo-Hookean elastic modulus and Shore A and
`Shore D values.
`The simulation results show the elastomer to undergo locally large deformations during hardness testing. In order
`to assess the potential role of the limiting extensibility of the elastomer on the hardness measurement, simulations are
`conducted where the elastomer is represented by the non-Gaussian Arruda-Boyce constitutive model. The limiting exten-
`sibility is found to predict a higher hardness value for a material with a given initial modulus. This effect is pronounced
`as the limiting extensibility decreases to less than 5 and eliminates the one-to-one mapping of hardness to modulus.
`However, the durometer hardness test still can be used as a reasonable approximation of the initial neo-Hookean modu-
`lus unless the limiting extensibility is known to be small as is the case in many materials, such as some elastomers and
`most soft biological tissues.
`
`INTRODUCTION
`
`Durometer (Shore) hardness1 is one of the most commonly used hardness tests for elas-
`tomeric materials. Durometer hardness measurements, which assess the material resistance to
`indentation, are widely used in the elastomer industry for quality control and for quick and sim-
`ple mechanical property evaluation.1 The hardness value is primarily a function of the elastic
`behavior of the material. The nondestructive and relatively portable nature of the test enables
`property evaluation directly on elastomeric products or components. This feature has also led to
`the use of hardness tests in mechanical property evaluation of soft tissues such as skin3-4 and
`tumors surrounded by soft tissues.5
`Durometer hardness is related to the elastic modulus of elastomeric materials. Several theo-
`retical efforts have been conducted in the past to establish the relationship between hardness and
`elastic modulus.6-7 However, most of these efforts have been based on linear elasticity, even
`though the indentation in durometer hardness tests involves significant large-scale nonlinear
`deformation. Gent6 obtained a simple relation between the elastic modulus and durometer Shore
`A hardness by approximating the truncated cone indentor geometry as a cylinder and using the
`classic linear elastic solution for the flat punch contact problem. Briscoe and Sebastian7 consid-
`ered the actual shape of the Shore A indentor and linear elasticity theory to obtain a prediction
`using an iterative solution. The difference between the Gent and the Briscoe and Sebastian results
`was as large as 15% to 25% for durometer hardness values larger than 50A. In this paper, the
`ability of durometer Shore hardness tests to provide properties for the stress-strain behavior of
`elastomers for small to large deformation is assessed. Fully nonlinear finite element analyses are
`conducted to simulate the durometer hardness tests. The nonlinear stress-strain behavior of the
`materials is modeled using the Gaussian (neo-Hookean) model and the Arruda-Boyce eight-
`
`* Corresponding author. Ph: 1-617-253-2342; Fax: 1-617-258-8742; email: mcboyce@mit.edu
`
`419
`
`ClearCorrect Exhibit 1058, Page 1 of 17
`
`

`

`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`420
`
`RUBBER CHEMISTRY AND TECHNOLOGY
`
`VOL. 76
`
`chain non-Gaussian model. The latter constitutive model captures the limiting extensibility of
`elastomers (and also of soft tissues8) and thus permits evaluation of the relevance of correlating
`the durometer measurements to limiting aspects of material behavior. Durometer tests for Shore
`A and Shore D scales are simulated for the Gaussian (neo-Hookean) material. The ability of the
`model to predict the corresponding Shore D hardness for a given Shore A hardness material acts
`as a verification. A mapping of Shore A and D values to the elastic modulus predictions is then
`provided. Comparisons of the new model with prior models are also given. The influence of the
`limiting extensibility of elastomeric materials on this mapping is assessed.
`
`MODELS
`
`THE MODEL OF DUROMETER HARDNESS TESTS
`
`The durometer hardness test is defined by ASTM D 2240,1 which covers seven types of
`durometer: A, B, C, D, DO, O, and OO. Table I shows the comparison of different durometer
`scales.
`
`TABLE I
`COMPARISON OF DIFFERENT SCALES OF DUROMETER TESTS
`
`Most commercially available products for durometer tests consist of, according to ASTM D
`2240, four components: presser foot, indentor, indentor extension indicating device, and cali-
`brated spring, as shown in Figure 1. The scale reading is proportional to the indentor movement
`(Figure 2)
`
`(cid:54)
`L
`
`0 025.
`mm
`where H is the hardness reading; (cid:54)L is the movement of the indentor.
`
`H
`
`=
`
`(cid:54)
`=
`,
`L L
`0
`
`(cid:60)
`
`L
`
`(1)
`
`ClearCorrect Exhibit 1058, Page 2 of 17
`
`

`

`DUROMETER HARDNESS AND THE STRESS-STRAIN BEHAVIOR OF ELASTOMERIC MATERIALS 421
`
`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`FIG. 1. — A typical durometer.
`
`a)
`
`b)
`
`FIG. 2. — Schematics of the working mechanism of durometers.
`(a) Before the durometer is pressed down; (b) the durometer is pressed down.
`
`A durometer essentially measures the reaction force on the indentor through the calibrated
`spring when it is pressed into the material. The relation between the force measured and the
`movement of the indentor is
`
`for type A, B and O durometers; and
`
`F = 0.55 + 3(cid:54)L
`
`F = 17.78(cid:54)L
`
`(2a)
`
`(2b)
`
`ClearCorrect Exhibit 1058, Page 3 of 17
`
`

`

`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`422
`
`RUBBER CHEMISTRY AND TECHNOLOGY
`
`VOL. 76
`
`for type C, D and DO durometers.
`In durometer tests, as the durometer is pressed onto the specimen surface, the indentor pen-
`etrates into the specimen, and is simultaneously pressed up into the device as well. This process
`is depicted in Figure 2, where L0 is the free length of the calibrated spring; d0 is the distance
`between the indentor tip and the presser foot lower surface and according to ASTM D 2240, d0
`= 2.5 mm; d1 is the corresponding distance in the fully loaded condition. Since the lower surface
`of the presser foot is always in contact with the specimen surface when the reading is taken, it is
`straightforward to obtain (Figure 2)
`
`The indentor is in equilibrium, therefore
`
`(cid:54)L + (d1 - 0) = 2.5
`
`( )
`F h
`r
`
`== 1
`
`h d
`
`F
`
`(3)
`
`(4)
`
`where Fr is the reacting force of the elastomeric specimen due to the indentor penetration denot-
`ed by h. Therefore, the objective equations relating the hardness measurement to the stress-strain
`behavior of the elastomers consist of Equation (3) and Equation (4). The exact form of Fr(h)
`however is unknown. Gent6 used the linear elastic Hertz contact solution for the case of a sim-
`plified indentor shape. Briscoe and Sebastian7 considered the actual geometry of the indentor.
`This method however requires computationally cumbersome numerical methods for the solution.
`In this paper, we take advantage of developments in nonlinear finite element method (FEM) and
`numerically simulate the hardness tests to obtain Fr(h) in the form of a force vs indentation, F vs
`h, curve. The hardness scale reading is then obtained by finding the intercept point as shown in
`Figure 3.
`
`FIG. 3. — Schematic of the method to obtain durometer readings.
`
`ClearCorrect Exhibit 1058, Page 4 of 17
`
`

`

`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`DUROMETER HARDNESS AND THE STRESS-STRAIN BEHAVIOR OF ELASTOMERIC MATERIALS 423
`
`FEM MODELS FOR INDENTATION SIMULATIONS
`
`Geometry. — The force vs indentation curve is obtained using a fully nonlinear finite ele-
`ment simulation of the indentation test. Since the indentors have axially symmetric cross sec-
`tions, it is effective to model the problem as an axisymmetric one. As shown in Figure 4, the ver-
`tical boundary AD is subjected to the axisymmetric boundary condition, ur|AD = 0. In order to
`reduce the influence from the specimen boundary, ASTM D 2240 requires that the test specimen
`should be at least 6 mm in thickness and the locus of indentation should be at least 12 mm away
`from any edges. Therefore, in Figure 4, AB = 15mm, BC = 8 mm . Due to the existence of fric-
`tion, the lower surface AB of the specimen cannot move freely along the horizontal direction,
`ur|AB = uz|AB = 0.
`
`FIG. 4. — The finite element method model for simulations of indentation.
`
`The boundary value problem is solved using the finite element code ABAQUS.
`Axisymmetric 8-node, hybrid continuum elements with biquadratic interpolation of the dis-
`placement field and linear interpolation of pressure are used to model the elastomer. Figure 5
`shows the mesh used for the durometer A analyses. The indentor is modeled as a rigid surface
`since it is much stiffer than the elastomers being tested. The mesh is refined in the vicinity of the
`contact region where large gradients in stress and strain prevail. Several mesh densities were ana-
`lyzed and an optimal mesh was finally chosen for use in all simulations. For durometer D analy-
`ses, a similar mesh has been used with the exception of the shape of the indentor.
`
`ClearCorrect Exhibit 1058, Page 5 of 17
`
`

`

`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`424
`
`RUBBER CHEMISTRY AND TECHNOLOGY
`
`VOL. 76
`
`FIG. 5. — The mesh used for the FEM simulations.
`
`The effect of friction on the simulation was studied by simulating selected cases of A-type
`and D-type tests using a friction coefficient of 0.3 between the indentor and the elastomer.
`Friction was found to increase the F vs h curve by no more than 4% in most cases. A similar
`result has been reported by Chang et al.9 for indentation with a rigid ball. Therefore, friction was
`neglected with exception that in D type analyses a friction coefficient of 0.1 was used to reduce
`the large deformation experienced by the elements on the contact surface.
`
`Material Model. — The rubber elasticity constitutive laws in the simulations are the
`Gaussian (neo-Hookean) model10 and the Arruda-Boyce eight-chain model.11 Both models are
`based on the concept of an elastomer as a three-dimensional network of long chain molecules,
`linked together at points of cross-linkage. The Gaussian model assumes Gaussian chain statistics
`to apply and the strain energy density function is given by
`
`(5)
`
`(cid:60)(
`µ
`I
`1
`
`)
`3
`
`1 2
`
`(cid:60)
`
`) =
`3
`
`32
`
`(
`µ (cid:104) (cid:104) (cid:104)
`+
`+
`
`22
`
`12
`
`1 2
`
`G =
`W
`
`where µ = nk(cid:79); n is the number of chains per unit volume; k is Boltzmann’s constant; (cid:79) is
`absolute temperature; (cid:104)1, (cid:104)2, and (cid:104)3 are the three principal stretches; and I1 is the first invariant
`(cid:104) (cid:104) (cid:104)
`=
`+
`+
`of the stretch
`.
`I1
`The limitation of the Gaussian theory is that as chains become highly stretched, the stress
`level is under-predicted, necessitating a non-Gaussian statistical theory to depict the behavior of
`the chain deformations approaching their limited extensibility. The Arruda-Boyce model
`employs a representative description of the network containing eight non-Gaussian chains
`extending from the center of a cube to each corner (see Figure 6) to simulate the network struc-
`ture of the polymer. The initial chain length is, from random walk statistics, given by r0 = (cid:51)
`l,
`where N is the number of rigid links of length l between the points of cross linkage. The maxi-
`mum length of the chain is Nl and the maximum chain stretch, which is called the locking stretch
`
`32
`
`22
`
`12
`
`—N
`
`ClearCorrect Exhibit 1058, Page 6 of 17
`
`

`

`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`DUROMETER HARDNESS AND THE STRESS-STRAIN BEHAVIOR OF ELASTOMERIC MATERIALS 425
`
`or limiting extensibility, is
`
`(cid:104)L
`
`=
`
`Nl
`
`r
`0
`
`=
`
`N
`
`(6)
`
`The cube is deformed in the principal stretch space and the stretch of each chain in the network
`is
`
`(7)
`
`I
`
`1 3
`
`r
`chain
`r
`0
`
`=
`
`(cid:104)chain
`
`=
`
`Note that the concept of the effective chain stretch (cid:104)chain can also be applied to the Gaussian
`model. The strain energy density function for this 8-chain network is11
`
`(8)
`
`(cid:60)
`
`(cid:79)
`c
`
`(cid:165) (cid:166)(cid:180)
`
`(cid:96)
`
`(cid:96)
`sinh
`
`N
`
`(cid:96)
`
`+
`
`1n
`N
`
`(cid:163) (cid:164)(cid:178)
`
`µ (cid:104)
`
`chain
`
`W
`AB
`
`=
`
`where (cid:96) is the inverse Langevin function,
`stant.
`
`(cid:96)
`
`=
`
`1
`
`(cid:60)L
`
`[
`(cid:104)
`
`chain
`
`/
`
`N
`
`]
`
`, and
`
`[ ] =
`L (cid:96)
`
`coth
`
`(cid:96)
`
`(cid:60) (
`/1
`
`)
`(cid:96)
`
`, c is a con-
`
`a)
`
`b)
`
`FIG. 6. — The network for the eight chain model: (a) unstretched state; (b) stretched state.
`
`ClearCorrect Exhibit 1058, Page 7 of 17
`
`

`

`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`426
`
`RUBBER CHEMISTRY AND TECHNOLOGY
`
`VOL. 76
`
`It can be shown that the initial elastic modulus of the material from a uniaxial tensile test is
`
`E0 = 3µ
`
`(9a)
`
`for the Gaussian model and
`
`(9b)
`
`(cid:165)(cid:166)
`
`...
`
`+
`
`4
`
`42039
`
`67375
`
`N
`
`40425
`
`67375
`
`N
`
`+
`
`39501
`
`67375
`
`N
`
`+
`
`3
`
`1
`
`+
`
`(cid:163)(cid:164)
`
`µ
`3
`
`E
`0
`
`=
`
`for the Arruda-Boyce model using a power series expansion representative of the inverse
`Langevin function. Observe that the elastic modulus predicted by Gaussian model is independ-
`ent of chain limiting extensibility. The difference between the initial moduli for the Arruda-
`Boyce model and the Gaussian model increases as N decreases or the crosslinking density
`increases; when N is 6, the difference is 10% and when N is 4, the difference is 16%.
`
`RESULTS AND DISCUSSIONS
`
`SIMULATIONS ON DUROMETER A AND DUROMETER D
`
`Finite element simulations of representative durometer hardness tests are conducted for
`Gaussian materials with µ = 1.6MPa for a Shore A test and µ = 30MPa for a Shore D test, respec-
`tively.
`In Figure 7, the force-indentation curve from the simulation of the Shore A test is depicted
`together with the spring behavior described earlier in Equation (2a). The Shore A hardness value
`is obtained by finding the intersection of these two curves, which occurs at an indentation depth
`of 1.00 mm, giving a Shore A value of 60A. Figure 8(a) and (b) show the contours of principal
`strain and chain stretch ratio (cid:104)chain at the indentation depth of 1.00 mm, respectively. These con-
`tours reveal that the elastomer experiences modest to large strain for this representative case (the
`Shore A scale is commonly used to indicate the durometer of materials between 30A to 90A).
`The maximum chain stretch ratio is observed around the corner of the indentor tip and is 2.2. For
`elastomers with a low limiting extensibility, this result indicates that it may be necessary to con-
`sider this effect when relating durometer to nonlinear stress-strain behavior.
`
`ClearCorrect Exhibit 1058, Page 8 of 17
`
`

`

`DUROMETER HARDNESS AND THE STRESS-STRAIN BEHAVIOR OF ELASTOMERIC MATERIALS 427
`
`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`FIG. 7. — The solid line is the force-penetration curve for the calibrated spring following Equation (2a).
`Dashed line is the force-indentation curve from the Shore A test simulation for a Gaussian material with µ = 1.6 MPa .
`
`a)
`
`b)
`
`FIG. 8. — Results for Shore A simulation for the Gaussian material with µ = 1.6 MPa.
`(a) Contours of the maximum principal strain; (b) contours of the chain stretch ratio.
`
`Similar to the Shore A case, the Shore D hardness value for the Gaussian material of µ =
`30MPa is obtained by finding the intersection of the spring response of Equation (2b) and the
`force-indentation curve for Shore D hardness indentation. The Shore hardness for this material
`is 62D and the penetration is 0.95 mm. Figure 9(a) and (b) show the maximum principal strain
`contour and the chain stretch ratio contour, respectively, at the fully loaded indentation (0.95
`mm) for this representative Shore D test. The strains are highly localized for this cone shape
`indentor and are rather high. The maximum chain stretch ratio is 9.5, and maximum principal
`strain is 3.9, suggesting that the limiting extensibility of the material may play an important role
`in evaluating material Shore hardness.
`
`ClearCorrect Exhibit 1058, Page 9 of 17
`
`

`

`428
`
`RUBBER CHEMISTRY AND TECHNOLOGY
`
`VOL. 76
`
`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`a)
`
`b)
`
`FIG. 9. — Results for Shore D simulation for the Gaussian material with µ = 30 MPa.
`(a) Contours of the maximum principal strain; (b) Contours of the chain stretch ratio.
`
`COMPARISONS BETWEEN SHORE A AND SHORE D
`
`To verify the proposed method, the finite element simulations of durometer hardness tests
`for scale A and scale D were conducted for three Gaussian materials with µ = 3.3MPa, µ =
`6.0MPa, µ = 9.2MPa, respectively. The materials were chosen to provide durometer readings
`where the A and D scales overlap.
`For the material with µ = 3.3MPa, the predicted durometer hardness is 75A and 25D.
`According to the guideline comparison chart in ASTM D 2240,1 the material with durometer
`hardness 75 in scale A should give a hardness of about 24 in scale D. Similar comparisons are
`made for materials with µ = 6.0MPa, µ = 9.2MPa and are listed in Table II. It should be noted
`that the comparison chart in ASTM D 2240 is loosely defined and cannot be used for absolute
`comparison purposes. Indeed, conversions which differ slightly were found.12 Therefore,
`although a relatively large error exists for 90A, Table II shows generally good agreement
`between simulated values and values from the comparison chart. This verifies the capability of
`the current finite element simulation to predict both A scale and D scale durometer hardness. The
`regions where direct conversions are not as reliable will be shown to occur at the tail end of the
`A scale where the relationship between the hardness and modulus becomes highly nonlinear.
`
`TABLE II
`COMPARISON OF SHORE A AND SHORE D HARDNESS BY SIMULATIONS. BOTH SHORE A AND SHORE D VALUES
`ARE OBTAINED FOR THE SAME MATERIALS BY FINITE ELEMENT METHOD SO THAT A CONVERSION
`BETWEEN A SCALE AND D SCALE IS ESTABLISHED. THESE CONVERSIONS ARE COMPARED WITH
`THE COMPARISON CHART IN TABLE I, GIVEN AS ERRORS IN D VALUES.
`
`µ(MPa)
`
`3.30
`
`6.0
`
`9.20
`
`Duro A
`
`Duro D (Simulated) Duro D (from Table I)
`
`Error
`
`75
`
`85
`
`90
`
`25
`
`32
`
`40
`
`24
`
`29
`
`35
`
`4%
`
`10%
`
`14%
`
`CORRELATION BETWEEN GAUSSIAN ELASTIC MODULUS AND HARDNESS
`
`Figure 10 shows the relationship between the elastic modulus E0 by Equation (9a) for the
`Gaussian model vs shore hardness A obtained by finite element simulations. The Gent predic-
`tions, and the Briscoe/Sebastian predictions are also presented for comparison purposes. In
`Figure 10, all predictions give the same trend for the relation between E0 and shore A hardness.
`However, the fully nonlinear analysis generally gives a higher prediction of elastic modulus for
`a given hardness than the other three predictions. Figure 11 shows the difference between the
`
`ClearCorrect Exhibit 1058, Page 10 of 17
`
`

`

`DUROMETER HARDNESS AND THE STRESS-STRAIN BEHAVIOR OF ELASTOMERIC MATERIALS 429
`
`elastic modulus predicted by the fully nonlinear analyses and those predicted by the other two
`theories, given as deviations from the fully nonlinear analyses. The difference between the
`Briscoe/Sebastian theory and the fully nonlinear model ranges from 18% to 10% and shrinks as
`the hardness increases, since the deviation of their theory from the fully nonlinear analyses
`becomes smaller as the penetration of the indentor decreases due to the increasing stiffness of the
`elastomer. However, a different trend is observed in the difference between the Gent theory and
`the fully nonlinear analyses. The reason for this trend is because the Gent theory used the aver-
`age diameter of the upper and lower surface of the truncated cone indentor as the diameter of the
`equivalent flat punch, whose deviation from the reality is pronounced as the penetration decreases.
`
`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`FIG. 10. — Relations between Shore A hardness and initial elastic moduli of elastomeric materials.
`
`FIG. 11. — The differences of the elastic modulus for the given hardness between the finite element
`analyses and the other theories, given as deviation from the finite element analyses.
`
`ClearCorrect Exhibit 1058, Page 11 of 17
`
`

`

`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`430
`
`RUBBER CHEMISTRY AND TECHNOLOGY
`
`VOL. 76
`
`A first order estimation for type D durometer hardness can be obtained using the linear elas-
`tic solution.13 For the cone indentor, the normal force on the indentor is
`
`D =
`
`F
`r
`
`2
`E
`(cid:60)(
`(cid:47)
`1
`
`v
`
`2
`
`)
`
`(cid:101)tan
`h
`
`2
`
`(10)
`
`Combining with Equation (5b), the relation between elastic modulus and durometer D hard-
`ness for (cid:101) = 15°, v = 0.5, is
`
`(
`(cid:60)
`
`
`20 78 188.
`
`+
`
`H
`
`D =
`
`100
`
`(cid:60)
`
`+
`
`
`6113 36 781 88. .
`
`
`
`E
`
`E
`
`)
`
`(11)
`
`Figure 12 shows the comparison between simulated results and the first order approxima-
`tions to D scale hardness. The two predictions give the same trend between the relation of elas-
`tic modulus and shore hardness D. However, the fully nonlinear analysis generally gives lower
`elastic modulus prediction than the linear elastic theory for a given hardness. The difference
`increases as the hardness increases.
`
`FIG. 12. — The relation between Shore D hardness and initial elastic moduli of
`elastomeric and elastomeric-like materials.
`
`Figure 13 shows the relationship between elastic modulus and Shore A/Shore D hardness,
`given by finite element simulations. For both hardness scales, the logarithm of elastic modulus
`is proportional to the hardness values in the range of 20A(D) to 80A(D). The region where the
`two scales overlap corresponds to a linear region for Shore D, but mostly nonlinear region for
`Shore A. When the overlap is in the linear region for both scales, good conversions can be
`obtained, as for 75A to 25D; whereas when the overlap occurs in linear vs nonlinear regions, the
`conversions become worse, as for 90A to 40D, since the logarithm of the elastic modulus varies
`with Shore A values in a much faster rate than it does with Shore D values.
`
`ClearCorrect Exhibit 1058, Page 12 of 17
`
`

`

`DUROMETER HARDNESS AND THE STRESS-STRAIN BEHAVIOR OF ELASTOMERIC MATERIALS 431
`
`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`FIG. 13. — The relations between elastic modulus and Shore A/Shore D hardness for elastomeric and elastomeric-like
`materials, given by finite element simulations. The linear approximation is given by Equation (20).
`
`It is also noted that there exists an almost linear relation between the logarithm of elastic
`modulus and a hardness scale
`
`(12a)
`
`(12b)
`
`log
`
`E
`0
`
`=
`
`
`
`.0 0235
`
`S
`
`(cid:60)
`
`
`
`.0 6403
`
`20
`
`80
`
`< <
`A S
`< <
`A S
`
`80
`
`A
`
`85
`
`D
`
`
`Shore A
`
` + 50
`Shore D
`
`(cid:168)(cid:169)(cid:170)
`
`S
`
`=
`
`LIMITING EXTENSIBILITY EFFECT
`
`As seen from Figure 8(b), large chain stretches can develop during the indentation process.
`For materials having small limiting extensibilities, this large chain stretch could be close to its
`locking stretch. Therefore, it is important to evaluate the effect of chain extensibility on the
`durometer hardness. Since the Arruda-Boyce material model captures the limiting extensibility
`of elastomeric materials, it is used in this section and then compared to the results that had been
`obtained using the Gaussian model.
`Figure 14 depicts the tensile stress vs stretch curve for the Arruda-Boyce materials with µ =
`1.35MPa and with various N values ranging from N = 2 to N = 100. At small stretches, the ini-
`tial modulus for the material with smaller N is larger (Equation (9b)); At large stretches, the stiff-
`ness of the material with smaller N increases dramatically. Note that N = 100 corresponds to a
`limiting chain extensibility of (cid:104)L = 10 and the upturn of the stress stretch curve occurs at an axial
`stretch of about 17, and N = 4 corresponds to a limiting chain extensibility of (cid:104)L = 2 and the
`upturn occurs at an axial stretch of about 3.4. Note that the chain limiting extensibility is signif-
`icantly lower than the limiting extension of the sample observed in a uniaxial tensile test because
`the molecular chains in the underlying molecular network accommodate macroscopic deforma-
`tion through both chain stretching and chain rotation.
`
`ClearCorrect Exhibit 1058, Page 13 of 17
`
`

`

`432
`
`RUBBER CHEMISTRY AND TECHNOLOGY
`
`VOL. 76
`
`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`FIG. 14. — The stress-stretch curves for µ = 1.35MPa and different Ns of the Arruda-Boyce model.
`
`Figure 15 shows the dependence of the durometer hardness vs limiting chain extensibility,
`N, for µ = 1.35MPa, µ = 0.75MPa and µ = 0.35MPa. The corresponding Gaussian result with the
`same E0 is also presented. For N smaller than about 25 (corresponding to a limiting chain exten-
`sibility of (cid:104)L = 5 and uniaxial tensile stretch of (cid:104) = 8.5), the predictions which account for the
`non-Gaussian behavior of the elastomer begin to deviate from those using the Gaussian materi-
`al model and are substantially higher than those of the Gaussian material as N decreases. These
`higher hardness predictions are not purely a result of the effect of N on the initial modulus since
`the simulations using the Gaussian model with the same initial E0 give lower predictions of hard-
`ness. The higher hardness prediction also result from the durometer test applying stretches that
`approach the limiting chain extensibility for cases when N is less than 25.
`
`FIG. 15. — The solid lines show the dependence of the durometer A hardness vs N for µ =1.35MPa, µ = 0.75MPa
`and µ = 0.35MPa. The corresponding Gaussian results with the same E0 are also presented as the dashed lines.
`
`ClearCorrect Exhibit 1058, Page 14 of 17
`
`

`

`DUROMETER HARDNESS AND THE STRESS-STRAIN BEHAVIOR OF ELASTOMERIC MATERIALS 433
`
`The effect of the limiting extensibility is also influenced by µ. As shown in Figure 15, when
`µ is small, large chain stretch can develop during indentation, the effect of the limiting extensi-
`bility is pronounced. When µ increases, the penetration of the indentor becomes small. Hence,
`the molecular chains are unlikely to be stretched close to their locking stretch, and the effect of
`the limiting extensibility is reduced. As shown in Table III, when µ = 2.95MPa, the difference in
`Shore A hardness is less than 5% when N varies from 4 to 100. Therefore, it is more likely that
`when (cid:104)L is less than 5 and µ is less than 2MPa the effect of the chain limiting extensibility is
`important. Note that limiting extensibilities significantly less than 5 are quite common in soft
`biological tissues8,14 as well as in elastomers.
`
`TABLE III
`PREDICTIONS OF DUROMETER A HARDNESS FOR DIFFERENT COMBINATIONS OF µ AND N IN THE ARRUDA-BOYCE MODEL
`
`0.35
`
`0.55
`
`0.75
`
`0.95
`
`1.35
`
`1.75
`
`2.15
`
`2.55
`
`4
`
`32.9
`
`42.9
`
`49.8
`
`54.9
`
`62.4
`
`67.7
`
`71.8
`
`74.9
`
`9
`
`27.6
`
`37.9
`
`45.3
`
`50.9
`
`58.8
`
`64.6
`
`68.9
`
`72.4
`
`µ(MPa)
`
`N
`
`16
`
`26.0
`
`35.8
`
`43.5
`
`49.9
`
`58.1
`
`63.6
`
`68.2
`
`71.8
`
`36
`
`23.1
`
`31.6
`
`40.0
`
`48.5
`
`56.9
`
`63.1
`
`67.3
`
`70.9
`
`73.7
`
`64
`
`23.1
`
`31.6
`
`40.0
`
`48.5
`
`56.6
`
`62.5
`
`67.1
`
`70.7
`
`73.6
`
`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`2.95
`
`3.35
`
`3.75
`
`4.15
`
`4.45
`
`5.0
`
`5.0
`
`6.0
`
`9.0
`
`12.0
`
`77.5
`
`79.6
`
`81.4
`
`82.9
`
`83.8
`
`85.4
`
`86.6
`
`87.6
`
`91.4
`
`93.4
`
`75.2
`
`77.5
`
`79.4
`
`81.1
`
`82.1
`
`83.8
`
`85.1
`
`86.2
`
`90.5
`
`92.7
`
`74.5
`
`77.0
`
`78.9
`
`80.4
`
`81.6
`
`83.3
`
`84.7
`
`85.8
`
`90.2
`
`92.4
`
`76.3
`
`78.2
`
`80.0
`
`81.2
`
`83.0
`
`84.3
`
`85.5
`
`90.0
`
`92.3
`
`76.1
`
`78.2
`
`79.9
`
`81.0
`
`82.8
`
`84.2
`
`85.4
`
`89.9
`
`92.2
`
`Table III gives predictions of Shore hardness A for different combinations of µ and N in the
`Arruda-Boyce model. In Table III, different combinations of N and µ may lead to the same pre-
`dicted durometer hardness. Table IV gives two examples of N and µ pairs which lead to Shore
`73.6A. The corresponding initial elastic moduli are also listed in Table IV. Although the materi-
`als have the same durometer hardness, their elastic moduli are different by about 5%. It is there-
`fore important to notice that predicting elastic moduli from durometer tests sometimes gives
`ambiguous results due to the limiting extensibility of the chain. There does not exist a one-to-one
`mapping between the initial elastic modulus and durometer hardness.
`
`ClearCorrect Exhibit 1058, Page 15 of 17
`
`

`

`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`434
`
`RUBBER CHEMISTRY AND TECHNOLOGY
`
`VOL. 76
`
`TABLE IV
`MATERIALS WITH DIFFERENT MATERIAL PARAMETERS HAVE SAME DUROMETER HARDNESS
`µ(MPa)
`
`N
`
`Duro (A)
`
`5.76
`
`36
`
`2.55
`
`2.95
`
`73.6
`
`73.6
`
`E0 (MPa)
`8.59
`
`9.01
`
`In the applications of durometer hardness, many researchers assume that larger durometer
`hardness corresponds to larger elastic modulus but less extensibility. However, one should be
`cautious when making this inference. For instance, considering two combinations of µ and N in
`the Arruda-Boyce model in Table V, the second combination gives larger durometer hardness, but
`has higher extensibility as well, as it can be seen from the stress-stretch curves (Figure 16).
`
`TABLE V
`MATERIAL WITH HIGHER DUROMETER HARDNESS AND HIGHER ELASTIC MODULUS CAN HAVE HIGHER EXTENSIBILITY
`µ(MPa)
`
`N
`
`9
`
`16
`
`0.95
`
`2.55
`
`E0(MPa)
`3.06
`
`7.95
`
`Duro (A)
`
`Maximum extension
`
`50.9
`
`71.8
`
`480%
`
`690%
`
`FIG. 16. — The stress-stretch curves for two different Arruda-Boyce materials. The material
`with µ = 2.55MPa and N = 16 has larger initial elastic modulus and larger extensibility.
`
`CONCLUSIONS
`
`Durometer hardness tests were analyzed using fully nonlinear finite element simulations
`accounting for both material and geometrical nonlinearities. The simulation results are verified
`by matching the predicted durometer hardness in scale D with ASTM D 2240 for the Gaussian
`material with the given durometer hardness in scale A. The relations between elastic modulus and
`the durometer hardness A scale and D scale are provided. The fully nonlinear finite element
`analyses predict higher values of the elastic modulus than those given by the Gent theory and the
`Briscoe/Sebastian theory. The influence of the limiting extensibility of the elastomer is evaluat-
`ed using the non-Gaussian Arruda-Boyce eight-chain model. The chain extensibility eliminates
`the one-to-one mapping between the elastic modulus and the durometer hardness. The limiting
`
`ClearCorrect Exhibit 1058, Page 16 of 17
`
`

`

`Downloaded from http://meridian.allenpress.com/rct/article-pdf/76/2/419/1945045/1_3547752.pdf by guest on 07 April 2023
`
`DUROMETER HARDNESS AND THE STRESS-STRAIN BEHAVIOR OF ELASTOMERIC MATERIALS 435
`
`extensibility also increases the reaction force to the indentation and thus increases the hardness
`of the material. This effect is pronounced as the chain extensibility decreases to values of (cid:104)L less
`than 5, particularly when µ is less than 2MPa. These results indicate that the durometer hardness
`test can be used to provide a reasonable approximation (the error is generally