International audienceStrain localization has been observed in polyurethane foams subjected to confined compression up to 70% of deformation. This phenomenon is described by a one-dimensional model, in which the foam is represented as a chain of non-linear elastic springs with non-convex strain energy density, and localization is attributed to progressive phase transition

del Piero, Gianpietro

Pampolini, Giampiero

HAL AMU

Strain localization in polyurethane foams. Experimentsand theoretical model.Giampiero Pampolini, Gianpietro Del PieroTo cite this version:Giampiero Pampolini, Gianpietro Del Piero. Strain localization in polyurethane foams. Exper-iments and theoretical model.. 11th EUROMECH-MECAMAT Conference, Mar 2008, Torino,Italy. 46, pp.29-38, 2009, Lecture Notes in Applied and Computational Mechanics. <hal-00462212>HAL Id: hal-00462212https://hal.archives-ouvertes.fr/hal-00462212Submitted on 27 Apr 2010HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Noname manuscript No.(will be inserted by the editor)Strain localization in polyurethane foams. Experimentsand theoretical model.Giampiero Pampolini · Gianpietro Del PieroReceived: date / Accepted: dateAbstract Strain localization has been observed in polyurethane foams subjected toconfined compression up to 70% of deformation. This phenomenon is described bya one-dimensional model, in which the foam is represented as a chain of non-linearelastic springs with non-convex strain energy density, and localization is attributed toprogressive phase transition.Keywords polyurethane foams · strain localization · non-convex strain energy,solid-solid phase transition.1 IntroductionIn the uniaxial compression of foam polymers, and of cellular materials in general,three regimes can be identified. The first regime is an almost linear response, the sec-ond is a stress plateau with large displacements occurring at almost constant load,and the last is a branch characterized by large stress increase under relatively moder-ate deformations. In the literature, two main approaches for modeling the mechan-ical response of foam polymers are used: a microstructural approach, reproducingdirectly the microstructure with a complex structure of beams, and a macroscopicapproach, based on a full three-dimensional representation of the material as a con-tinuum. There has been a wide spread of models based on the first approach (e.g.,[Gent and Thomas (1963)], [Gibson and Ashby (1997)], [Warren and Kraynik (1997)],[Gong et al. (2005)]). In particular, Gong et al. studied the response curves of open-cell foams, and [Gong and Kyriakides (2005)] proposed a model with a periodic struc-Giampiero PampoliniLaboratoire de Me´canique et d’Acoustique, CNRS, 31 chemin Joseph Aiguier, 13402 Marseille,France.Universita` di Ferrara, Via Saragat 1, 44100 Ferrara, Italy.E-mail: pampolini@lma.cnrs-mrs.fr; giampiero.pampolini@unife.itGianpietro Del PieroUniversita` di Ferrara, Via Saragat 1, 44100 Ferrara, Italy.E-mail: dlpgpt@unife.it2ture given by a 14-sided polyhedron, called the Kelvin cell. The ligaments are mod-eled as shear-deformable extensional beams, and the basic material is assumed to belinearly elastic. In this approach, the numerical simulation is demanding from thecomputational point of view, due to the complexity of the assumed microstructureand to the difficulty of modeling the contact between beams in the post-bucklingregime, see [Bardenhagen et al. (2005)]. In the continuum approach, the material isconsidered as homogeneous and hyperelastic, and subject to non-homogeneous defor-mations. Indeed, there is experimental evidence of strain localization, see for exam-ple [Lakes et al. (1993)], [Wang and Cuitinho (2002)]. In particular, strain localizationalong bands orthogonal to the loading direction was observed by Wang and Cuitio,and confirmed by experiments performed by the present authors on polyurethanefoam specimens, subjected to axial compression up to 70%. The same three stagesin the evolution of deformation have been observed by [Bastawros et al. (2000)] and[Bart-Smith et al. (1998)] in both open- and closed-cell aluminium alloy foams. In thesecases, the deformation localizes in narrow bands, whose width is of the order of acell diameter. These results support the interpretation, suggested in previous works[Wang and Cuitinho (2002)], [Gioia et al. (2001)], of the non-homogeneous deforma-tion as a phase transition. Here we present a model in which the foam is representedas a chain of non-linear elastic springs with non-convex strain energy, and the local-ization is interpreted as a progressive phase change. The first part of the paper dealswith the test procedure and the experimental results, and the second contains an out-line of the theoretical model. A complete description of the model can be found in[Pampolini and Del Piero (2008)].2 EXPERIMENTAL TESTSFor the compression tests we used the load frame INSTRON 4467 located at the Lab-oratorio di Materiali Polimerici of the University of Ferrara, with a load cell of 500 N.The tests were made on commercial polyurethane foams in confined compression. Forconfinement we used a polystyrene box, clamped to the load frame. The polystyrenebox is more rigid than the tested material and, being transparent, allowed us to controloptically the evolution of the deformation. At the upper basis, the specimen was in con-tact with a steel plate fixed to the moving crosshead. Five specimens of dimensions 100× 100 × 50 mm were tested. The crosshead speed was chosen to be 1 mm/min, whichwe believe small enough to neglect all rate-dependent effects. The test was stoppedat a relative elongation of 70%. As we will see later, this value is sufficiently large toinclude the significant portions of the response curves.The stress-strain curves shown in Figure 1 reveal the characteristic behavior ofpolyurethane foams, and of cellular materials in general. From the figure, we noticethat the plateau begins at a relative elongation of about 10% and ends at about 63%,and that the stress is close to 6,5 kPa. The slope of the second branch is approximatively25 kPa. These values are close to those obtained by [Gioia et al. (2001)].During the tests, the deformation was initially homogeneous (Figure 2a). Then asevere deformation occurred at the upper layer of the specimen (Figure 2b), and prop-agated to the underlying layers (Figures 2c, 2d). When all layers had reached the samedeformation the specimen began again to deform homogeneously (Figure 2e). That thelarge deformation regime starts at one of the ends of the specimen can be attributedto a local boundary effect due to contact between steel plate and specimen. The same3DeformationStress (kPa)Fig. 1 Experimental curves of polyurethane foams in confined compression tests.evolution mechanism was observed by [Wang and Cuitinho (2002)] in polyurethanelow-density foams, and by [Bastawros et al. (2000)] and [Bart-Smith et al. (1998)] inaluminium alloy foams. In Wang and Cuitio’s experiments, localization does not initiateat the upper part of the specimen as it does in our tests.(a) (b) (c)(d) (e)Fig. 2 Progressive deformation mechanism. Initial homogeneous deformation (a). Strain lo-calization at the upper end of the specimen (b). Propagation to the underlying layers (c), (d).Back to homogeneous deformation (e).43 THE THEORETICAL MODEL3.1 Constitutive assumptionsLet Ω0 be the bounded open region of the three-dimensional space occupied by thebody in the reference configuration, and let f be the deformation that maps the pointsX of Ω0 into the points x = f(X). Denote by Ω the region f (Ω0) occupied by thedeformed body, and by F the deformation gradient ∇f . Assume that the body is madeof an isotropic hyperelastic material, with a strain energy density of the formw(F ) =12α F · F + Γ (detF ) . (1)where α is a positive material constant, and Γ is a function which takes the value+∞ for both detF = 0 and detF = +∞limdetF→0Γ (detF ) = limdetF→∞Γ (detF ) = +∞. (2)The gradient of wS = wF (F ) = αF + detFΓ′ (detF )F−T , (3)is the Piola-Kirchhoff stress tensor. If we impose that the reference configurationF = I be stress-free, from the conditions w(I) = wF (I) = 0 we getα = −Γ ′(1), 32α+ Γ (1) = 0. (4)For the function Γ we assume the expressionΓ (detF ) = c (detF )n(1n+ 2(detF )2 − 1n)−µ ln (detF )+12β√pi√kerf(√k (detF − a))+γ,(5)where α, c, n, µ, β, k are positive constants, with a ≤ 1, erf(·) is the error functionerf(x) =∫ x02√pie−t2, (6)and γ is the constant determined by the condition 3Γ ′(1) = 2Γ (1). The expres-sion (5) of Γ consists of two parts. The first part is similar to that one proposed in[Ogden (1997)], and takes into account the long-range effects. The second part, theerror function, provides a local effect at values of detF close to a. In the particularcase of confined compression in the direction e, one has F = I + (λ − 1)e ⊗ e. Then ,for Γ as in (5), the energy density isw(λ) =12α(2 + λ2)+ Γ (λ), (7)and the normal component of S in the direction of loading isσ =[µ− β exp(−k(1− a)2)]λ+ cλn−1(λ2 − 1)− µλ−1 + β exp(−k(λ− a)2). (8)The energy has the non-convex form shown in Figure 3c, and the stress-strain curvehas two ascending branches separated by a descending branch, as shown in Figure 3d.53.2 The discrete modelConsider a chain of n springs connected in series as sketched in Figure 3a, each springrepresenting a layer of cells as shown in Figure 3b. We assume that all springs havethe same strain energy w given by (7), and that, accordingly, the stress-strain curveis given by (8). The total energy of the system is the sum of the strain energies of thesprings,E (ε1, ε2, ..., εn) =n∑i=1w(εi), (9)where εi is the elongation of the i-th spring. The bar is subjected to the hard deviceconditionn∑i=1εi = nε0, (10)where nε0 is the imposed displacement at the upper basis, the displacement at thelower basis being zero. By setting to zero the partial derivatives of the total energy,one gets the equilibrium conditionw′ (εi) = w′ (εn) , i = 1, 2, ..., n− 1. (11)which implies that the force is the same in all springs. The common value of theforce will be denoted by σ.Fig. 3 Subdivision of the body into cell layers (a), representation of each layer as a non linearelastic spring (b) with non convex energy (c), and non-monotonic stress-strain curve (d).6For sufficiently large n, a sufficient condition for stability is that all elongations εi lieon an ascending branch of the stress-strain curve, see [Del Piero and Truskinowsky (2008)]or [Puglisi and Truskinovsky (2000)]. Therefore, we confine our attention to the equi-librium configurations which satisfy this condition. For every such configuration, letm be the number of the springs whose elongations lie on the first ascending branch(phase A), so that (n −m) is the number of the springs whose elongations lie on thesecond ascending branch (phase B). In particular, for m = 0 and for m = n we havesingle-phase configurations, and for 0 < m < n we have two-phase configurations. In asingle-phase configuration, all springs have the same elongationε0 = εi, i = 1, 2, ..., n, (12)and the global response curve (σ, ε0) coincides with the response curve (σ, εi) ofeach spring. In a two-phase configuration, let σ¯ be the force in the springs, and let ε¯1and ε¯2 be the elongations corresponding to σ¯ in the first and second ascending branch,respectively. Then the total elongation of the chain isnε¯0 = mε¯1 + (n−m)ε¯2. (13)By varying σ¯, one can construct the equilibrium path (σ¯, ε¯0) corresponding to eachgiven m. By dividing the interval (ε¯1, ε¯2) , into n equal parts, all equilibrium paths fordifferent m are obtained. In Figure 4a, these paths are shown for a system of 4 springs.Fig. 4 Response curves for a system of 4 springs (a) and for a system of 20 springs (b).If one assumes that the system evolves along equilibrium curves made of local en-ergy minimizers, then the system, when loaded starting from the initial configuration,initially follows the first ascending branch. This branch ends when the stress reachesthe critical value σmax shown in Figure 4a. At this point, for further increasing defor-mation it is reasonable to assume that the system jumps to the closest stable branch,corresponding to the configuration with one spring in phase B and three springs inphase A. When this branch ends, the system jumps to the branch with two springs inphase B and so on, until all springs undergo the phase transition. At this point the sys-tem evolves following the second ascending branch, which corresponds to single-phaseconfigurations with all springs in phase B. If we now increase the number of springs7(Figure 4b), the number of the intermediate branches increases, and the amplitudes ofthe jumps at the end of the branches decrease. In this case when the stress reaches thecritical value σmax, the system follows a wavy, approximately horizontal line, succes-sively assuming configurations with p springs in the highly deformed phase B and n−psprings in the low-deformation phase A, with p gradually increasing from zero to n.At unloading the system exhibits a similar behavior, after reaching the critical stressσmin. The model therefore predicts a hysteresis loop, which has indeed been observedexperimentally [Gong and Kyriakides (2005)].3.3 Comparison with the experimental dataIn Figure 5 the model response is compared with the experimental results. The dottedline is the average loading curve, obtained as an average of the experimental curvesshown in Figure 2. The solid line is the response curve of the model for a system of asufficiently large number of springs with strain energy of the type (1), evaluated withthe constitutive data reported in Table 1.Fig. 5 Theoretical response curves and hysteresis loop at loading and unloading (solid lines),compared with experimental response at loading (dotted line).The model’s response is close to the experimental curve in the first ascending branchand at large strains (ε0 > 0, 3). Instead of a horizontal plateau, the experimental curveexhibits a branch with variable positive slope. This discrepancy has already been theobject of investigation [Puglisi and Truskinovsky (2001)], [Marzano et al. (2003)]. Inparticular, Puglisi and Truskinovsky reproduce the positive slope by considering achain of springs with the same ascending branches, but separated by different barriers.Finally, Figure 6 shows the experimental curve at unloading [Pampolini (2007)]. Itconfirms the presence of the hysteresis loop predicted by the proposed model.8Table 1 Values of the constitutive constants.Material Constantsα 2.92 kPaµ 2.92 kPac 69.9 kPam 6β 3.5 kPak 18a 0.72Fig. 6 Experimental response with hysteresis loop [Pampolini (2007)].Acknowledgements This work was supported by the PRIN 2005 ”Modelli Matematici perla Scienza dei Materiali” of the Italian Ministry for University and Research.References[Bardenhagen et al. (2005)] S. G. Bardenhagen, A. D. Brydon, and J. E. Guilkey. Insight intothe physics of foam densification via numerical simulation. J. Mech. Phys. Solids, 53:597–617, 2005.[Bart-Smith et al. (1998)] H. Bart-Smith, A.-F. Bastawros, D. R. Mumm, A. G. Evans, D. J.Sypeck, and H. N. G. Wadley. Compressive deformation and yielding mechanisms incellular al alloys determined using x-ray tomography and surface strain mapping. ActaMaterialia, 46:3583–3592, 1998.[Bastawros et al. (2000)] A.-F. Bastawros, H. Bart-Smith, and A. G. Evans. Experimentalanalysis of deformation mechanism in a closed-cell aluminium alloy. J. Mech. Phys. Solids,48:301–322, 2000.[Del Piero and Truskinowsky (2008)] G. Del Piero and L. Truskinowsky. Elastic bars withdecohesions. In preparation, 2008.[Gent and Thomas (1963)] A. N. Gent and A. G. Thomas. Mechanics of foamed elastic ma-terials. Rubber Chem. Technol, 36:597–610, 1963.[Gibson and Ashby (1997)] L. J. Gibson and M. F. Ashby. Cellular Solids: Structure andProperties. Cambridge University Press, second edition, 1997.9[Gioia et al. (2001)] G. Gioia, Y. Wang, and A. M. Cuitino. The energetics of heterogeneousdeformation in open-cell solid foams. Proc. R. Soc. London A, 457:1079–1096, 2001.[Gong and Kyriakides (2005)] L. Gong and S. Kyriakides. Compressive response of open-cellfoams. part ii: Initiation and evolution of crushing. Int. J. Solids Struct., 42:1381–1399,2005.[Gong et al. (2005)] L. Gong, S. Kyriakides, and W. Y. Jang. Compressive response of open-cell foams. part i: Morphology and elastic properties. Int. J. Solids Struct., 42:1355–1379,2005.[Lakes et al. (1993)] R. Lakes, P. Rosakis, and A. Ruina. Microbuckling instability in elas-tomeric cellular solids. J.Mater.Sci., 28:4667–4672, 1993.[Marzano et al. (2003)] S. Marzano, M. D. Piccioni, and G. Puglisi. Un modello di isteresi perfili di leghe a memoria di forma. Proc. 16th AIMETA National Congress of Theoreticaland Applied Mechanics., 2003.[Ogden (1997)] R.W. Ogden. Non-Linear Elastic Deformations. Dover Publications, NewYork, 1997.[Pampolini (2007)] G. Pampolini. Experimental tests and theoretical model for strain local-ization and cyclic damage of polyurethane foam cylinders in uniaxial compression. Com-munication at the VI Meeting Unilateral Problems in Structural Analysis, Siracusa, 2007.[Pampolini and Del Piero (2008)] G. Pampolini and G. Del Piero. Strain localization inpolyurethane foams. Experiments and theoretical model. in preparation, 2008.[Puglisi and Truskinovsky (2000)] G. Puglisi and L. Truskinovsky. Mechanics of a discretechain with bi-stable elements. J. Mech. Phys. Solids, 48:1–27, 2000.[Puglisi and Truskinovsky (2001)] G. Puglisi and L. Truskinovsky. Hardening and hysteresisin transformational plasticity. Proc. 15th AIMETA National Congress of Theoretical andApplied Mechanics, 2001.[Wang and Cuitinho (2002)] Y. Wang and Y. Cuitinho. Full-field measurements of heteroge-neous deformation patterns on polymeric foams using digital image correlation. Int. J.Solids Struct., 39:3777–3796, 2002.[Warren and Kraynik (1997)] W. E. Warren and A. M. Kraynik. Linear elastic behavior of alow-density kelvin foam with open cells. ASME J. Appl. Mech., 64:787–793, 1997.

Strain localization in polyurethane foams. Experiments and theoretical model.

Abstract

Similar works

Full text

Available Versions

HAL AMU