A Robust, Compressible, Hyperelastic Constitutive Model for the Mechanical Response of Foamed Rubber

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1 TECHNISCHE MECHANIK, 36, -2, (206), 88-0 submitted: July 27, 205 A Rbust, Cmpressible, Hyperelastic Cnstitutive Mdel fr the Mechanical Respnse f Famed Rubber M. Lewis An verview f the rles that cellular elastmers play in engineered systems and the phenmenlgy f these materials is presented. Available data n these materials and empirical mdeling appraches that have been used t apprximate the mechanical respnse f these materials is then cnsidered. The Cmpressible, Hyperelastic, Istrpic, Prsity-based Fam mdel (CHIPFam), a mdel develped at Ls Alams Natinal Labratry (LANL) with a simple micrmechanical basis, is then described, and its ability t fit available data is demnstrated. Finally, extensins and imprvements t the mdel and the additinal predictivity t be achieved with these extensins are discussed. Intrductin Cellular slids are bth useful and cmmn in everyday applicatins ranging frm baked bread t impact absrbers. These materials are used in engineered systems. Specifically, cellular plydimethylsilxane (PDMS) with different fillers in either a blwn fam frm (rm temperature vulcanized r RTV) r in a leached fam frm (cellular silicne r CS) is ften used in cushins and pads in engineered systems. These fams are ften the mst cmpliant materials used in such systems. As a result, displacements f and stresses in neighbring cmpnents are strngly affected by the thermmechanical behavir f these materials. PDMS is a silicn-based plymer. These filled PDMS materials are in the rubbery regime, well abve their glass transitin temperatures, ver a large temperature range. In this regime, the materials can sustain large elastic strains. This feature puts these materials in the class f materials that must be mdeled with hyperelastic mdels. Hyperelastic mdels are described in terms f strain energy functins frm which stress states are derived. These strain energy functins are analgus t ptential functins in three-dimensinal space, such as gravity ptential r electrstatic ptential fields, whse gradients in their respective functin spaces prduce cnjugate frces (gravitatinal r electrmtive in these cases). Slid elastmers are ften idealized as incmpressible. This is because their bulk mduli are typically tw t fur rders f magnitude larger than their shear mduli fr small strain respnse. Unlike slid elastmers, famed rubber is initially quite cmpressible because f its prsity, s mst f the strain energy functins used fr slid elastmers are nt apprpriate fr famed elastmers. Our ability t mdel these materials has been pr until very recently. Despite attempts t mdel fam structures using representative vlume element (RVE) appraches like thse used by Hhe and Becker (2003) and Bardenhagen et al. (2005), the nly mdel fr this class f materials that reprduced the limited extant test data had n basis in actual fam mechanics and, as such, culd nt be applied t predict the respnse f fams with different prsities r relative densities based n data frm fams f knwn prsities r relative densities. We prpse a mdel that has a basis in the respnse f an islated pre in a Ne-Hkean slid that addresses this deficiency and has been applied t represent fam cmpnents with varying prsities. In the current paper, the basis f the Cmpressible, Hyperelastic, Istrpic, Prsity-based Fam mdel (CHIPFam) develped by Lewis and Rangaswamy (205) is described and crrected, and its applicability fr relevant materials is demnstrated. In the next sectin f this paper, Phenmenlgy, the phenmenlgy f elastmeric fams in general and sme f the issues in btaining data frm thin fam samples are described. Fllwing that, a quick review f hyperelasticity is presented, and the previusly used mdel, Hyperfam, is described and discussed, alng with issues assciated with its use fr ur cmpnents in the Hyperfam sectin. In the CHIPFam Basis sectin, the basis fr the CHIPFam mdel is presented, and the purpse f each term in the strain energy is explained. Additinally, cases when sme f the terms may be mitted are identified. 88

2 The prcess fr fitting available data is described in the Data Fitting sectin, alng with examples f a fit t data frm a near uniaxial strain test. The mdel implementatin as a UHYPER subrutine in ABAQUS is presented in the Mdel Implementatin sectin. Finally, recmmendatins fr imprvements t the mdeling f these materials are presented in the Further Wrk sectin f this paper. 2 Phenmenlgy 2. Nnlinearity and Dependence n Prsity As mentined previusly, the mechanical respnse f famed elastmers is nnlinear and very dependent n the prsity f the fam. The prsity is directly related t the fam density as fllws: ρ f s ρr =, () ρs ϕ = where φ is the prsity f the fam, s is the slidity f the fam, ρ r is the relative density f the fam, ρ f is the mass density f the fam, and ρ s is the mass density f the parent material. The nnlinearity f the fams uniaxial mechanical respnse is demnstrated in Figure, as is the strng dependence f the mechanical respnse n the prsity f the material. Figure. Engineering stress vs. engineering strain lading curves frm uniaxial cmpressin tests f cushin material f densities ranging frm 0.57 g/cc t 0.86 g/cc at 70 C. Nte that cmpressive quantities are pltted as negative in this figure. The plt demnstrates the nnlinear mechanical respnse seen in these fams and the strng dependence f this respnse n the density f the fams, which is directly related t the fam prsity by equatin (). 2.2 Mullins Effect, Hysteresis, and Large Defrmatin Elasticity Additinal mechanical phenmena are seen in these materials when they are cycled in cmpressin. Figure 2 shws the resulting stress-strain behavir when a material with an estimated prsity smewhere between 20% and 25% is cycled t nearly 20% strain in cmpressin fur times. The first cycle lading curve stresses lie abve the lading curve stresses fr subsequent cycles, which all tend t verlay each ther. This effect, a cyclic sftening f the lading curve, is knwn in the elastmer literature as the Mullins effect (Mullins, 948), after Lenard Mullins, a rubber researcher the mid-20 th century. The unlading curve stresses in Figure 2 are nearly identical frm ne cycle t the next and all are lwer than the lading cycle stresses at the same strains. This mechanical hysteresis is typically bserved in elastmers. An additinal feature f the curves shwn in Figure 2 is that the curves unlad and recver t a strain near zer at 89

3 zer stress, indicating that the large defrmatin (nearly 20% strain) is almst cmpletely recverable. In rder t mdel this phenmenlgy, a thery capable f handling large elastic strains is needed. Figure 2. Cyclic engineering stress vs. engineering strain curves frm uniaxial cmpressin tests f a cushin material at 20 C. Hysteresis is apparent in that the unlading f the specimen ccurs at stresses lwer than thse seen in lading. Mechanical energy is being dissipated. Als, the difference between the first cycle lading curve and subsequent cycle lading curves is evidence f the Mullins effect, a cyclic sftening phenmenn. 2.3 State f Stress and Strain in Thin Specimens While the data presented in Figures and 2 are reprted as axial stress and strain frm cmpressin tests, it shuld be nted that they are nt frm uniaxial stress tests. The thicknesses f these specimens were apprximately mm, while the diameter f the specimens were apprximately 6 mm. As the platens in these tests were nt lubricated and the specimen lateral strain was nt measured, it is reasnable t assume that the strain state mst nearly apprximated uniaxial strain, and that lateral strains were zer while radial stresses in the specimen were prbably nt negligible. An additinal issue is that many f these materials have skins, r regins f very lw prsity near free surfaces. As a result, stress cushins have mre f a sandwich cmpsite structure. Fr the present time we ignre this effect, thugh finite element mdeling techniques fr mdeling this srt f fam cre structure are available. 2.4 Lw Strain Behavir in Cmpressin under Near-Uniaxial Stress Cnditins A cmpressin test was cnducted n a sample f S5370, a blwn PDMS fam, at LANL. In the test, the platensample interfaces were lubricated and lateral strain was manually measured with a micrmeter. The bserved sample lateral strain was nt linear in the applied axial strain, and a plt shwing the large strain Pissn s rati and the decrease in relative vlume vs. axial cmpressive strain (shwn as psitive in cmpressin) is presented in Figure 3, with an accmpanying stress-strain curve shwn in Figure 4. The large strain Pissn s rati used is defined as fllws: 90

4 e l ν NL, (2) ea where ν NL is the large strain Pissn s rati, e l is the engineering lateral strain, and e a is the engineering axial strain in a sample subjected t uniaxial stress. Additinally, the relative vlume is defined as the rati f the initial density t the current density f the sample at a pint f interest, typically the gauge sectin. The decrease in Pissn s rati is typically assciated with an inflectin pint in the stress-strain curve and bth are thught t be a cnsequence f buckling in the fam. Figure. 3. Measured large strain Pissn s rati and relative vlume decrease in a lubricated S5370 (blwn PDMS fam) specimen as a functin f axial engineering strain, shwn psitive in cmpressin. Nte the drp in Pissn s rati early in cmpressin, thught t be assciated with hyperelastic buckling in the fam structure in fams f lw relative density r high prsity. It is ften assciated with an inflectin pint in the stress-strain curve, as indicated in Figure 4. Figure 4. Lading stress-strain curve fr an S5370 (blwn PDMS fam) specimen in cmpressin. Cmpressive quantities are shwn as psitive. Nte an inflectin pint in the curve near an engineering strain f 0., near the strain at which the Pissn s rati data shwn in Figure 3 begins t decrease. 9

5 2.5 Viscelasticity and Temperature Effects It is well knwn that stress cushin materials exhibit stress relaxatin and ther viscelastic phenmena. These phenmena are effected by temperature in that increasing temperature tends t decrease relaxatin times. Sme wrk n characterizing and mdeling such behavir is described by Yang et al. (2000). Additinally, wrk by Trelar (958) has demnstrated that the shear stiffness f rubber is strngly dependent n temperature, and is characteristically linear in abslute temperature, i.e. increasing in stiffness as temperature is raised. 3 Hyperfam It is clear that the mechanical behavir f elastmeric fams is cmplicated. While the Mullins effect, hysteresis, and ther dissipative phenmena like viscelasticity and ptential gas flw effects are imprtant parts f that behavir, a crucial part f the respnse is large defrmatin elasticity. In rder fr a mechanical cnstitutive thery fr these materials t be useful ver a large set f pssible strain histries, it must be based n hyperelastic thery. 3. A Brief Review f Hyperelastic Thery Hyperelastic thery is based n the existence f a strain energy density functin, W, that is cmputable fr each strain state. The strain measures used are apprpriate fr large elastic strain. The strain energy density is the wrk required/ per unit initial material vlume, t reach the strain state. Thrugh wrk and energy arguments, ne can shw that the Cauchy stress may be cmputed frm the strain energy functin as fllws: W = J F F T σ, (3) where σ is the Cauchy stress tensr, J is the relative vlume, and F is the defrmatin gradient, r the gradient f the current lcatin f each pint with respect t the riginal cnfiguratin, r x F, (4) X where x is the vectr t a pint in the current cnfiguratin and X is the vectr t that pint in the initial cnfiguratin. A superscript T indicates the transpse f a tensr. W may be expressed as a functin f invariants f the Cauchy-Green tensrs r in terms f principal stretches, as Ogden argues fr. Typical invariants used fr strain energy functins are J, I, and I 2. J has already been defined but can als be calculated as the determinant f F. I is the trace f the ischric Cauchy-Green tensr (either left r right), as fllws: 2 ( ) T B = Tr J 3 F F I Tr. (5) Althugh we will nt use it here, it is wrth mentining that I 2 is the trace f the inverse f the ischric Cauchy-Green tensr. It is relatively easy t shw that the Cauchy stress fr a material with a strain energy functin expressed as a functin f J and I is as fllws: 92

6 2 W W 2 W I W σ = dev( B) + i= B i + i, (6) J I J J I 3 J where i is the secnd rder identity tensr. Similarly, fr cases where the strain energy functin is expressed in terms f principal stretches, λ i, and J, as in the case f the Hyperfam mdel, the Cauchy stress can be shwn t be as fllws: 3 W W σ = λi pi pi + i, (7) J J i= λi where p i are the principal directins assciated with the principal stretches. W has additinal requirements f cnvexity in the space f the defrmatin gradient, but these are beynd the scpe f the current paper. 3.2 Hyperfam Mdel Frm The Hyperfam mdel is a variatin n an Ogden (984) strain energy functin prpsed by Jemil and Turtletaub (2000), wh mved frm the ischric principal stretches used by Ogden t the full principal stretches. The strain energy functin fr the Hyperfam mdel is in terms f principal stretches and J and is as fllws: W H α iβi ( J ) N 2µ i α i αi αi = λ + λ2 + λ3 + 2 i= αi βi. (8) The expnents in equatin (8) are typically nt whle numbers and may be either psitive r negative. The Hyperfam mdel reprduces test data well, prvided the right test data are available. As mentined previusly, available test data are ften best characterized as uniaxial strain cmpressin data. As a result, it is nt pssible t differentiate between the cntributin f the axial principal strain term and that f the relative vlume term in the strain energy functin. Fitting Hyperfam typically invlves using a nnlinear least squares methd, usually inside ABAQUS, and the resulting mdel parameters have n discernible physical assciatin. When data are mdified slightly, the mdel parameters can change substantially. Additinally, as available data are frm tests which apprximate uniaxial strain cmpressin cnditins, it is cmmn t set the Pissn s rati t zer t disambiguate the effects f the principal strain and relative vlume terms. While this apprach is a way t use available data, it artificially turns uniaxial strain data int uniaxial stress data and ignres the effects f variable Pissn s rati seen in Figure 3. One wuld expect that as a famed rubber is cnslidated in ne directin, that its respnse in transverse directins wuld stiffen. Using a zer Pissn s rati decuples the respnses in rthgnal directins very effectively and prevents this expected stiffening phenmenn. 4 CHIP Fam basis The reader shuld be cnvinced f the many shrtcmings f the Hyperfam mdel at this pint. This next sectin describes the fundamental basis f the CHIPFam mdel, which was develped t have a micrmechanical basis and be predictive f the respnse f fams with varius prsities. CHIPFam is, as the acrnym indicates, an istrpic, hyperelastic mdel that includes cmpressibility effects, bth in terms f the fam as a material and the parent material cmpressibility as well. The mdel cnsists f fur cmpnents. First, a smaller defrmatin cmpnent which attempts t capture small strain linear behavir and buckling-type behavir within the fam is included. This term includes uncupled 93

7 deviatric r shear and vlumetric parts. Next, a term that captures the respnse f an incmpressible spherical shell is used t describe the stiffening at large defrmatins. This term cuples deviatric and vlumetric behavirs. A third term accunts fr matrix cmpressibility t allw a cnsistent respnse at large cmpressins. Finally, an ptinal term which is smewhat cupled with the secnd third terms can be included t represent the effect f gas cmpressin in clsed cell fams. Fr mst f ur applicatins, this term and its cmplicatins can be ignred. 4. Smaller Strain Cmpressive Behavir Including Buckling Effects T capture smaller strain near linear behavir, a simple Ne-Hkean strain energy functin fr shear respnse is intrduced. An accmpanying vlumetric strain energy functin is added t include initial cmpressibility f the fam. It is quadratic in J. This early part f the strain energy functin is as fllws: Gˆ Kˆ W ( ) ( ) 2 L = I 3 + J. (9) 2 2 This strain energy functin prvides small strain behavir that linearizes t infinitesimal strain linear elasticity with shear and bulk mduli f Ĝ and ˆK. In rder t capture a decreasing Pissn s rati and buckling behavir, the vlumetric strain energy functin has been mdified t instantaneusly change frm quadratic t linear at a predefine relative vlume, as fllws: Gˆ WLB = ( I 3) 2 ( ) [ ] ( ) (0) ˆ Jb J Jb K Jb J + Η J Jb ( Jb ) J Η J J b in equatin (0) is the Heaviside step functin, equal t zer fr negative arguments and unity fr nnnegative arguments. While equatin (0) is nt strictly cnvex in J, terms t be added later will guarantee cnvexity f the ttal strain energy functin in J. The expressin [ ] A plt f the large strain Pissn s rati as defined in (2) is presented in Figure 5 fr the strain energy functin f equatin (9) and that f equatin (0) with shear t bulk mdulus rati f 0.6 and a value f J b, the buckling relative vlume, f

8 Figure 5. A plt f the large strain Pissn s rati as defined in (2) fr the strain energy functins f equatins (9) and (0) demnstrating the drp achievable with the buckling mdificatin presented in equatin (0). Nte that a rati f shear t bulk mdulus f 0.6 was used t prvide an initial Pissn s rati f 0.25 and a vlumetric buckling relative vlume f 0.94 was chsen. It shuld be nted that the mdificatin f the vlumetric term presented in equatin (0) is in n means the nly mdificatin acceptable r useful. Instead f transitining t an essentially zer bulk tangent mdulus, ne culd transitin t different tangent bulk mduli at different pints using sets f tangent parablas at specified relative vlumes. Such an apprach wuld allw fr near exact matches f Pissn s ratis, althugh the challenge f cding necessary t supprt this in a material mdel is prbably nt justified by these gains in fidelity. 4.2 Pre Mechanics Basis and Strain Energy Functin In rder t capture the stiffening behavir seen as a famed rubber is cmpressed t a relative vlume appraching the slidity f the material, the respnse f the simplest pssible istrpic Representative Vlume Element (RVE), a spherical shell f incmpressible rubber, is cnsidered. If the rati f inner radius t uter radius is chsen t prvide the right slidity f the vlume inside the uter bundary f the sphere, then this RVE has the same initial prsity as the fam, the nly material structure statistic that is reprduced with this RVE chice. As this particular RVE chice is s simple, and des nt capture spatially varying r relevant length scales, it is mre apprpriate t refer t it as a mechanical surrgate. This apprach is smewhat similar tt that taken by Dienes and Slem (999) fr the respnse f a vid in a linear elastic material. By invking symmetry and incmpressibility, ne can slve fr the displacement field inside this surrgate shell subjected t unifrm radial cmpressin n its bundary fllwed by a vlume cnserving stretch field superpsed n the entire cmpacted shell. Once the kinematics have been slved, ne can then use a strain energy functin fr the parent material and integrate the strain energy density ver the entire initial vlume f material, nrmalize it by the initial vlume enclsed by the uter bundary f the shell, and calculate a strain energy functin apprpriate fr this simplest f micrmechanical mdels. We initially assumed a parent material with a Mney-Rivlin strain energy functin, linear in bth the first and secnd ischric invariants. The resulting strain energy functin is published (Lewis and Rangaswamy, 20). The generality added by the inclusin f the secnd invariant was nt necessary fr gd data fits, s the strain energy was simplified t include nly a Ne- Hkean parent material. The resulting strain energy functin term is as fllws: 95

9 ( ) 3 φ ( ) W D = C 0 I f J,φ, () where the functin multiplying the first ischric invariant is as fllws: f ( J,φ )= 2J φ 3 + ( 2 2J φ J ), (2) 3 J ( φ ) and φ is the initial prsity r vid vlume fractin f the material. We have chsen t refer t the strain energy functin () as the Danielssn functin, after Mats Danielssn et al. (2004), wh published a versin f it in 2004, althugh it was nt expressed in invariant frm as shwn here. The Danielssn functin explicitly cuples vlumetric and shear respnse. This leads t shear stiffening respnse with cmpressin in additin t the vlumetric stiffening expected. It shuld be nted that the Danielssn strain energy functin lses plycnvexity if the initial prsity is greater than 0.7 and the applied relative vlume is apprximately 0.4 r less. Fr the fams we have been interested in mdeling thus far, the initial prsities have been substantially less than this limiting prsity. Nte that this functin has a singularity as the relative vlume appraches the slidity f the material. This is sensible in that the derivatin f the Danielssn functin invlves the assumptin f incmpressibility in the parent material. This is gd if ne is trying t mdel cnditins f true lck-up in the material, hwever it is prblematic fr real finite element analyses. During an implicit slutin calculatin in a finite element analysis, many trial states may be passed t a material mdel subrutine. The likelihd f the subrutine receiving a value f J that is less than the initial slidity f the material is high unless the entire analysis invlves unifrm lw strains. A plt f the functin f is shwn in Figure 6 fr a material with a prsity f Figure 6. The functin f f J in (2) fr a material with a prsity f The singular behavir discussed in the text is clearly demnstrated. 4.3 Parent Material Cmpressibility Effects t Remve Singular Behavir In rder t avid the singularity prblem mentined in the previus sectin, it was necessary t cnsider cmpressibility f the parent material. Rather than d this in an ad hc manner, the prblem f cmpressin f the spherical shell assuming a cmpressible parent material was attempted. That prblem was nt fund t be tractable. An apprximatin is pssible, hwever and is described here. Mre detail is available in Lewis and Rangaswamy (203). 96

10 The first bservatin t make is that in the cmpressin f a spherical shell f an incmpressible material, nly the deviatric stresses are directly cmputable frm the cnstitutive mdel. The pressure field must be slved fr t satisfy equilibrium and bundary cnditins. The pressure field was slved fr and it was fund that it culd be reasnably apprximated (Lewis and Rangaswamy, 203) as a cnstant pressure in the spherical shell as fllws: 3( 4J 4 + 5ϕ ) 4 3 ( J + ϕ ) ( 4J )( 4J ) ~ ϕ + p = p g + C0, (3) 4 3 3J where p g is the pressure in the interir f the pre, ptentially frm any gas present. The effect f a unifrm pressure in a cmpressible shell is simple t calculate and is t unifrmly cmpact the entire shell. Fr tractability, a cmpressibility mdel fr the parent material that leads t a lgarithmic relatin between relative vlume f the material and the pressure was assumed. This implies a vlumetric strain energy functin fr the parent material as fllws: W = K( J m ln J m J m +), (4) where J m is the relative vlume in the parent material and K is the apparent bulk mdulus f the parent material. The slutin fr J m is necessary fr this part f the strain energy functin, and is simply as fllws: ~ p K J m = e. (5) All f the previus suggests that sme mdificatins need t be made t the Danielssn functin t be cmpatible with the assumptin f parent material cmpressibility. It is apprpriate and necessary t decmpse the macrscpic relative vlume int tw parts, namely a part due t the incmpressible respnse f the material in the RVE and a part due t the parent material cmpressibility. The apprpriate decmpsitin is a multiplicative ne as fllws: J = JJ m. (6) Figure 7. Schematics f the cmpressive defrmatins assciated with J m (left) and J (right). J m is as in (5), but the pressure is nw 3( 4J 4 + 5ϕ ) 4 3 ( J + ϕ ) ( 4J )( 4J ) ~ ϕ + p = p g + C J, (7) cnsistent with the riginal derivatin. Nte that J has a physical lwer limit f the slidity f the material, s. 97

11 The Danielssn functin must be mdified similarly t be cnsistent with this parent material cmpressibility inclusin. The mdified Danielssn functin is as fllws: W C D = C J dj ϕ. (8) dj J = [ I f ( J, ) 3( ϕ )] J ( I 3)( ϕ ) 0 m Nte that equatin (8) is mdified frm what appears in Lewis and Rangaswamy (205), t meet requirements that the pressure be zer when the relative vlume is unity. The energy stred in the cmpressin f the parent material is as fllws: W m = ( φ ) K( J m ln J m J m +). (9) 4.4 Effect f Gas Pressure in Vids Equatins (3) and (7) explicitly cntain a gas pressure term. In rder t capture that term prperly and als include a gas cmpressin term in the strain energy functin t represent energy stred in the gas in clsed cell fams r under lading cnditins that are rapid enugh that the gas cannt escape, it is necessary t cnsider the behavir f gas trapped in the inner cavity f the spherical shell. It shuld be nted that fr many f ur cnditins these gas terms may be neglected. Fr cmpleteness they are presented here. Fr ur purpses and t be cnsistent with zer energy at a reference state with a nnzer abslute pressure, we frmulate the gas pressure as a functin f the relative vlume f the gas as fllws: γ ( J ) p = p, (20) g g where p is the ambient pressure, J g is the relative vlume f the gas, and γis an expnent that may be chsen as unity fr isthermal defrmatins and as the rati f the gas specific heats at cnstant pressure and cnstant vlume fr adiabatic defrmatins. A strain energy functin, then, fr the gas is btained by integrating the pressure with respect t the relative vlume with a negative sign applied fr thermdynamic cnsistency. We then scale the resulting strain energy functin by the initial vlume f the inner cavity as fllws: W g = p φ J g ln( J g ) isthermal J g γ γ J g γ ( ) adiabatic. (2) The nly missing piece here is the cnnectin between the relative vlume f the gas and ur multiplicatively decmpsed vlume terms. T be cnsistent with the spherical shell mechanical surrgate fr vid respnse, the relatin needed is as fllws: J g = J m J + ϕ ϕ 5 Data Fitting. (22) The full strain energy functin is the sum f the strain energy functins intrduced previusly, as fllws: W W + W + W + W =. (23) LB C D M g As previusly mentined, the last three vlumetric strain energy terms in equatin (23) supplement the first term t maintain cnvexity in relative vlume. 98

12 The full set f mdel parameters that must be fit t available data is { G ˆ, Kˆ, J, C, ϕ, K, p,γ } b 0. If gas effects are negligible, as they are typically fr mderate t high prsity fams defrmed slwly, there are nly six parameters. The cntributins f each mdel parameter t the uniaxial strain cmpressin respnse are unambiguus enugh t allw gd fitting t data frm ur tests. The prcess f calculating the relative vlume decmpsitin, which is an iterative prcess, and the expressin f the derivatives f the relative vlume cmpnents with respect t the ttal relative vlume, are presented mre fully in Lewis and Rangaswamy (205). An example fit t recently btained uniaxial S5370 (blwn PDMS fam) data is shwn in Figure 8. Figure 8. A mdel fit t recently btained uniaxial cmpressin data frm an S5370 (blwn PDMS fam) sample with a density f g/cc. Nte that the fitting prcess fr the data shwn in Figure 8 was smewhat simplified as the prsity f the fam was relatively lw. As a result, the strain energy terms assciated with lw strain respnse in cmpressin, namely W LB, can be mitted and the parameters assciated with it, namely Ĝ, ˆK, and Jb, d nt need t be fitted. Gas effects are negligible fr the data shwn in Figure 8, as the cmpressive strain rate was 0.0 s - and the specimen diameter was apprximately 3 mm. As such, the parameters assciated with W g, namely p and γ d nt require fitting. Additinally, the test specimen was apprximately mm thick, s an assumptin f uniaxial strain was deemed apprpriate. The nly remaining parameters t fit were the prsity (φ ), the parent material effective Ne-Hkean mdulus (C 0 ), and the parent material effective bulk mdulus (K). The fitting prcess cnsisted f lping ver a range f pssible prsities and using a Levenberg-Marquardt algrithm t btain the best fits fr the tw mduli fr each prsity, and selecting the fit assciated with the prsity that had the minimum RMS errr relative t the measured stress-strain curve. 6 Mdel Implementatin A versin f the CHIPFam mdel that mitted the premultiplier f J m in the expressin fr J g in equatin (22) and in the mdified Danielssn functin in equatin (8) was implemented as a UHYPER subrutine in ABAQUS. While the UHYPER framewrk is nt as flexible fr mechanical respnse calculatin as the mre general UMAT subrutine in ABAQUS, it is well suited t quick cding and testing f invariant-based hyperelastic frmulatins such as CHIPFam. The UHYPER subrutine is passed mdel parameter values in the PROPS array, alng with any user-defined state variables in the SDV array. The implementer may update the state variables but must update the values f the energy functin and its partial derivatives with respect t the first and secnd ischric invariants and the relative vlume up t third rder. These values allw ABAQUS t calculate the Cauchy stress tensr and a 99

13 material tangent stiffness tensr t be used in calculatin and iteratin. A critical part f the slutin fr the energy functin and its derivatives is the multiplicative decmpsitin f J. T achieve this decmpsitin equatin (5) was used alng with equatins (6), (7), (20), and (22) and a bisectin algrithm. This scheme wrks efficiently and rbustly. 7 Further wrk While much wrk has been and much achieved in the develpment f CHIPFam, much remains t be dne t maximize its usefulness. The areas fr further wrk can be gruped int tpics f hyperelastic thery, dissipative behavir, and use in explicit dynamics. In terms f hyperelastic thery develpment, there are three main thrusts. First, the idea f establishing a set f bulk tangent mduli at varius relative vlumes t best match bserved lateral expansin data shuld be explred and the results dcumented. If bserved lw strain Pissn s effects can be matched with a few terms added t equatin (0), this shuld be pursued. Secnd, the dependence f the parameters in equatin (0) n prsity fr a given parent material shuld be studied t determine whether the treatment f spatial prsity distributins mentined in the previus sectin culd be imprved t include smaller strain dependence n prsity. Third, the inclusin f a third relative vlume term in the multiplicative decmpsitin f J shuld be investigated t determine if the lack f plycnvexity that can be seen at high prsities (greater than 70%) can be remved by intrducing a part f the vlume change that results frm buckling in the fam. In the area f dissipative behavir, the inclusin f viscelasticity, hysteresis, and Mullins effects shuld be investigated t determine the best theretical and algrithmic treatments fr mdeling these phenmena. Fr Mullins effect appraches, we are cnsidering the apprach taken by Ogden and Rxburgh (988) and generalized by Naumann and Ihlemann (205) and that taken by Rickaby and Sctt (202). Once the best appraches are determined, it is likely that the mdel will have t be recded as a mre general UMAT t include these effects. While this mdel has been implemented as a UHYPER subrutine and used fr implicit, quasistatic analyses, there are reasns t apply it t explicit dynamic prblems. ABAQUS/Explicit des nt have an analg f the UHYPER subrutine, but has an anistrpic versin that culd prbably be used effectively. This cding needs t be dne and verified. 8 Acknwledgments I hpe it is bvius that the wrk in the develpment f this mdel has taken a substantial effrt ver several years. In reality, it has been ver fifteen years in the ffing. The wrk has been supprted ff and n by DOE/NNSA funding. I am indebted t Tm Zcc and Mark Chadwick fr the initial time t research this prblem and develp an understanding f the RVE apprach that supprts the Danielssn functin, alng with Jhn Dienes and Jhndale Slem, whse linear elastic RVE wrk (Dienes and Slem, 999) encuraged me. I als wuld like t thank Tm Stephens, Seth Gleiman, Jim Cns, and Rick Mday fr taking the first really cmpelling fam data n ur elastmeric fams, and fr Carl Cady and Cheng Liu fr furthering this wrk, and having patience while we determined what data we needed. I als wuld like t thank Partha Rangaswamy fr heading up recent test effrts and serving as a sunding bard fr the last several years. I wuld like t thank Ted Lyman fr cding up the first versin f CHIPFam as a UHYPER with nly a little guidance frm me. Thanks als t Devin Shunk wh has dne the first real assembly and thermal cycle analyses with this mdel recently. References Bardenhagen, S.G., Brydn, A.D., and Guilkey, J.E., Insight int the physics f fam densificatin via numerical simulatin, JMPS, 53 (2005), Danielssn, M., Parks, D.M., and Byce, M.C., Cnstitutive mdeling f prus hyperelastic materials, Mech. f Mat., 36, (2004),

14 Dienes, J.K. and Slem, J.C., Nnlinear behavir f sme hydrstatically stressed istrpic elastmeric fams, Acta Mech., 38, (999), Hhe, J. and Becker, W., Effective mechanical behavir f hyperelastic hneycmbs and tw-dimensinal mdel fams at finite strain, Int J Mech Sci, 45 (2003), Jemil, S. and Turtletaub, S., Parametric mdel fr a class f fam-like istrpic hyperelastic materials, J. Appl. Mech., 67, n. 2, (2000), Lewis, M.W. and Rangaswamy, P., A stable hyperelastic mdel fr famed rubber, Eur. Cnf. n Cnstit. Mdels fr Rubber VII, Dublin, Ireland (20), Lewis, M.W. and Rangaswamy, P., Tward a hyperelastic mdel fr famed rubber with matrix cmpressibility and pre gas effects, Eur. Cnf. n Cnstit. Mdels fr Rubber VIII, San Sebastian, Spain (203), Lewis, M.W. and Rangaswamy, P., Vlume decmpsitin fr a rbust, mechanics-based, hyperelastic fam mdel, Eur. Cnf. n Cnstit. Mdels fr Rubber IX, Prague, Czech Republic (205). Mullins, L., Effects f Stretching n the Prperties f Rubber, Rubber Chemistry and Technlgy, 2, (948), Naumann, C., and Ihmlemann, J., On the thermdynamics f pseud-elastic material mdels which reprduce the Mullins effect, IJSS, (205), Ogden, R.W., Chapter 7: Elastic Prperties f Slid Materials, Nn-Linear Elastic Defrmatins, st ed., Dver Press, Minela, New Yrk (984). Ogden, R.W. and Rxburgh, D.G., A pseud-elastic mdel fr the Mullins effect in filled rubber, Prc. Ryal Sc. Lndn, A: Math, Phys, and Eng Sci, 455 (988), Rickaby, S.R., and Sctt, N.H., The Mullins effect, Cnst Mdels fr Rubber VII (202), CRC Press, Lndn (202), Trelar, L.R.G., The Physics f Rubber Elasticity, 2 nd ed., Clarendn Press, Oxfrd (958). Yang, L.M., Shim, V.P.W., and Lim, C.T., A visc-hyperelastic apprach t mdeling the cnstitutive behavir f rubber, Int J Impact Eng, 24 (2000), Address: M.W. Lewis, MS A42, Ls Alams Natinal Labratry, Ls Alams, NM mlewis@lanl.gv 0