Finite Element Models for Viscoelasticity in Polyurethane Foams

N.J. Mills Finite Element Models for the Viscoelasticity of Open-Cell Polyurethane foam

___________________________________________________________________________________

____________________________________________________________________________________

As in Cellular Polymers, Vol. 25, No. 5, 2006 293

Finite Element Models for the viscoelasticity of open-cell polyurethane foam

N.J. Mills

Metallurgy and Materials, University of Birmingham, Birmingham B15 2TT, UK

Received: 17 May 2006 Accepted: 28 July 2006

ABSTRACT

Finite Element Analysis (FEA) was used to model viscoelastic effects in the compression of polyurethane (PU) foam. Modelling of the impact compression of foam cubes predicted that a compressed region propagated through the sample, causing step increases in the stress traces. Modelling of head impacts on gymnastic ‘crash’ mats, made of PU chip foam, revealed that the majority, but not all, of the energy losses could be attributed to the polymer viscoelasticity.

The Gibson-Ashby micromechanics model was used to show that the hysteresis in a viscoelastic foam could be higher than in the parent polymer. A wet Kelvin foam, with a better foam geometry, was then used for viscoelastic analysis. The lattice of uniform-sized cells, in a Body Centred Cubic array, were compressed in the [111] direction. With the PU treated as a linear viscoelastic material, the predicted hysteresis in the cyclic stress-strain curve was slightly smaller than that measured for a PU flexible foam.

INTRODUCTION

Polyurethane (PU) foams exhibit many types of viscoelastic phenomena, such as energy losses in cyclic loading and creep in static loading(1). These are important in automotive seating applications, since foam damping reduces the amplitude of resonant vibrations, while excessive creep causes the driver’s position to drop relative to mirrors etc. Low-resilience or ‘viscoelastic’ PU foams have been profitable additions to the bedding foam market. The viscoelasticity of solid PU (2) fits into the pattern of polymer behaviour, with a loss maximum near the glass transition temperature of the amorphous phase.


___________________________________________________________________________________

____________________________________________________________________________________


This paper attempts viscoelastic modelling on both the microscopic and macroscopic levels. The viscoelasticity option in dynamic FEA is used to revisit the modelling of impacts on foam products (3), which assumed a hyperelastic foam material. This, while explaining several features of the foam response, failed to simulate any loss processes. One aim of the research was to see whether large foam strains and a complex stress field, for a head impact on a large foam mat, produce more complex viscoelastic effects than small uniform strains in a compressed foam cube, in particular whether hysteresis is predicted to increase with the impact speed. A comparison of the predicted responses of a foam cube and a mat with experimental data should identify sound wave effects and show whether air-flow losses are important. Mills and Lyn (4) used an axisymmetric finite-difference method to model the air flow losses in PU chip foam, under uniform compressive strain. It is not currently possible to implement such a model in FEA.

There are only a few micromechanics models for the viscoelasticity of open-cell PU foams. Huang and Gibson (5) used a simple cubic cell model to suggest that the foam creep compliance is a scaled version of the polymer creep compliance. Zhu and Mills (6) considered creep in a Kelvin foam model with uniform edge cross-sections, compressed in the [001] direction, and came to a similar conclusion. Recent micromechanics modelling (7), using a linear elastic material, predicted a plateau in the compressive stress-strain curve of PU foam. The implicit FEA solution method, used for solving static problems, sought convergence to an equilibrium state at every increment of model deformation. The Kelvin foam model was used with geometry created by Surface Evolver (8) software. The model for compression in the [111] direction, which causes the cell edges to both twist and bend, was preferred to that for [001] direction compression, which only causes edge bending. The viscoelastic modelling here aims to simulate the hysteresis in compressive loading/unloading cycles on PU foam.

Dynamic FEA of the impact compression of closed-cell (metal) foams (9), using a Kelvin foam with a representative unit cell (RUC) two cells high, predicted that the upper impacted cell collapsed before the lower cell, i.e. that there was inhomogeneous strain in the model. Such strain inhomogeneity is sometimes reported for PU foams (10). A stress plateau or a fall in the compressive stress-strain curve makes strain inhomogeneity likely. Consequently, attempts were made to predict strain inhomogeneity in an open cell foam model. It would cause the cell strain histories to differ, hence produce a complex viscoelastic response.

Therefore this paper explores whether cell micromechanics phenomena amplify the viscoelasticity of the solid PU, and how the viscoelasticity of PU foam products can be treated during large deformations.


___________________________________________________________________________________

____________________________________________________________________________________


Viscoelastic model

Polymer stress relaxation data can be approximated by a set of Maxwell viscoelastic elements (a spring and a dashpot in series) in parallel, plus a single elastic element ∞G for the long-time response (Figure 1). This linear viscoelastic model is described in the FEA program ABAQUS (11) by a shear relaxation modulus given by

−∑−=

=)exp(1)(

10 i

n

ii tgGtG τ (1)

where G0 is the instantaneous shear modulus, and the gi are dimensionless shear relaxation moduli associated with relaxation times τi related to the viscosities ηi by

i

ii G

ητ ≡ (2)

The sum of the gi cannot exceed 1.0 since the long term modulus G∞, defined by

∑−==

∞n

iigGG

10 1 (3)

must be positive. G0 is calculated from the Young’s modulus E0 and Poisson ratio ν0 using G0 = E0/2(1+ν0). In ABAQUS this model is referred to as a Prony series, due to its mathematical form. There is a similar expression to equation (1) for the relaxation of the material bulk modulus. If the Poisson’s ratio of the material is 0.5, the bulk modulus is effectively infinite. If the dimensionless bulk moduli in the Prony series are set at zero, then the instantaneous and long term bulk moduli are equal.

Zhu and Mills (6) carried out stress relaxation measurements on solid PU moulded from PU foam. The relaxation Young’s modulus E(t) of the solid Bulpren PU at 19 oC, when plotted against log t, was nearly a straight line. This was approximated by the Young’s modulus equivalent of equation (1), with relaxation times of 0.01, 0.1, 1, 10, ... 105 s and moduli Ei = 9.3 MPa for i = 1 to 8, plus E∞

= 28 MPa. At time zero, the sum of the moduli is E0= 102.4 MPa. Such data was approximated to simulate solid PU in FEA; two relaxation times were used per decade of time to smooth the transition between the Maxwell element relaxation responses, so gi = 0.05 for the 7 relaxation times given in set 1 of Table 1, plus the instantaneous value, calculated from E0 = 100 MPa and ν = 0.45, of G0 = 34.5 MPa.


___________________________________________________________________________________

____________________________________________________________________________________


Fig. 1. Maxwell viscoelastic elements (a spring and a dashpot in series) in parallel.

No stress relaxation data has been reported for virgin or remoulded PU foam. However creep compliance data for virgin PU foam at low compressive stresses can be predicted from the stress relaxation data for solid PU (6); consequently the same values of gi were used for the foam. The values of gi were empirically increased for rebonded PU foam, for which no creep data is available. The micromechanics modelling in section 4 suggests that the hysteresis in PU foams can be considerably higher than that in solid PU.

Table 1 shows four sets of normalized shear moduli used to characterize the foam. For cyclic compression tests lasting the order of 1 s, there is no need for τi greater than 10s or less than 0.01 s, since such Maxwell elements either flow freely or respond elastically, so sets 1 or 3 were used. For simulation of impacts on a time scale of 10 ms, sets 2 or 4 were used.

Hyperfoam model

The Hyperfoam model in ABAQUS FEA is based on the theory of compressible hyperelastic solids (12). The Ogden strain energy function is expressed as

( ) ( )

−+−++∑= −

=1132

3211 2

iiiii JUi

N

i i

i βαααα

βλλλ

αµ

(4)


___________________________________________________________________________________

____________________________________________________________________________________


Table 1. Prony series of shear moduli for PU viscoelasticity modelling

Set times τi (s) 0.001 0.003 0.01 0.03 0.1 0.3 1 3 10

1 0.05 0.05 0.05 0.05 0.05 0.05 0.05

2 0.05 0.05 0.05 0.05 0.05 0.05 0.05

3 0.1 0.1 0.1 0.1 0.1 0.1 0.1

4

Dimension-less shear modulus

gi 0.1 0.1 0.1 0.1 0.1 0.1 0.1

where λI are the principal extension ratios, J = λ1 λ2 λ3 is a measure of the relative volume, the µi are shear moduli, N is an integer, and αi and βi curve-fitting non-integral exponents. The latter are related to Poisson’s ratio νi by

i

ii ν

νβ21−

=. (5)

A N = 2 model, that matched the experimental data, was used (3) with the parameters in Table 2 to simulate remoulded PU foam. The shapes of the tensile loading curves were matched, but not the initial peak in the compressive loading curve. There was a reasonable match to the lateral strain response in tension. The elastic anisotropy of the foams was checked and found to be the order of 10%. The hyperfoam model acts as a numerical approximation for the PU foam response. Eventually micromechanics modelling will provide the parameters directly as functions of the foam density and the solid PU mechanical properties.

Table 2. Parameters for the hyperelastic model of remoulded PU foam

N shear modulus

µ kPa

exponent

α

Poisson’s ratio

ν

1 18 8 0

2 1.2 -2 0.45

Approaches to viscoelasticity in ABAQUS FEA

There are two main methods of considering viscoelasticity in ABAQUS, the current version of which is 6.5. The first is recommended when the simulation of sound waves in the material is not required. A ‘visco’ step replaces a ‘static-general’ step in Standard (implicit) FEA, with a Prony series of relaxation times in the material model.

The second method, in Dynamic FEA, is recommended when sound wave effects are important. It also uses a Prony series of relaxation times. The density of the polyurethane controls the speed of sound in the material, and consequently the time


___________________________________________________________________________________

____________________________________________________________________________________


interval ∆t between the FEA explicit computations. The explicit method makes a single estimate of the stress state at the next time interval. ∆t must be smaller than the time for a sound wave to cross the smallest element. In the simulation of impact tests on large foam blocks, the overall simulation time is the order of 20 ms, and the ∆t values for the relative large elements are reasonable. The sound waves in these simulations are relevant to interpreting foam impact tests.

In standard FEA, the displacement of the top of the foam is linearly ramped during a step to create a series of foam strains, hence the strain rate is constant during loading or unloading. In dynamic FEA, a rigid mass can be used to impact the top surface of the foam. The mass is sufficient to cause a high compressive strain in the foam, while its initial velocity determines the initial strain rate for loading. As the mass decelerates to zero then rebounds, the foam is subjected to a compressive loading and unloading cycle, and the strain rate varies with time, in contrast with the standard FEA simulation.

For micromechanics analysis of foams with cell sizes less than 1 mm, using fine meshing of the cell geometry, the time interval ∆t can be extremely small, making the computation unacceptably slow. Elastic modelling (7) revealed that cell size does not affect the predicted compressive stress-strain response. As only the foam viscoelastic response is of interest, the diameter D of the foam cells can be increased; ∆t is directly proportional to D if the mesh is kept the same. The initial velocity of a rigid striker is increased in proportion to D, so that the strain rate in the foam is the same. However, if D > 10 m for the wet Kelvin model, the deformation is predicted to be limited to a small region near the striker.

Cyclic compression data for virgin and remoulded PU foam

A virgin block of PU foam of density 25.8 kg m-3, previously tested in (13) had a typical open cell microstructure. As the solid PU density is 1200 kg m-3, the foam relative density is 0.0215. A remoulded PU foam (4) of density 72 kg m-3 contains a range of densities of chips of size circa 8 mm, uniaxially compressed by varying amounts in the moulding process, and bonded together. An Instron compression testing machine was used on 50 mm cubes of foam, with the compression direction being the rise direction of the PU foam or the moulding compression direction of the remoulded foam. A constant slow compression rate of 20 mm/min meant that there was no gas flow contribution to the stress.

The compression cycles were taken to 25%, 50% or 75% compression, with a recovery period of 1 hour for the virgin foam, and 5 minutes for the remoulded foam. Figure 2 shows that there is an initial stress peak for the virgin foam, with some memory of the initial loading cycle in the 2nd cycle. The stress-strain graph


___________________________________________________________________________________

____________________________________________________________________________________


Fig. 2. Cyclic stress strain graphs for: a) virgin, b) remoulded PU foam.

of the remoulded foam is more linear, and the hysteresis smaller, than for the virgin foam. The hysteresis, defined as the energy lost in a cycle (or in the loading and unloading of an impact) divided by the energy input, increases with the compressive strain (Table 3).


___________________________________________________________________________________

____________________________________________________________________________________


Table 3. Hysteresis measured on foam cubes compressed slowly in an Instron

Foam and density max strain 25% max strain 50% max strain 75%

Virgin 27 kg m-3 53 56 60

Chip 72 kg m-3 35 43 49

2 UNIAXIAL IMPACT COMPRESSION OF FOAM

Previous test conditions (3) were replicated, with the aim of interpreting the stress-strain data. The model consisted of one quarter of a foam cube of side 100 mm, with two vertical symmetry planes running through the centre point, supported on a rigid flat surface, and impacted by one quarter of a rigid flat-faced striker of total mass 2 kg (Figure 3). The initial impact velocity of 3.5 ms-1 caused an initial strain rate for

Fig. 3. Model of one quarter of a foam cube, after 2.4 ms, with contours of the vertical compressive stress component (kPa).

loading of 35 s-1. The foam was meshed at 10, 5 or 2.5 mm intervals, to check the effect of mesh size. The predictions for mesh sizes of 5 and 2.5 mm were identical, and only slightly different from the prediction for a mesh size of 10 mm. Therefore a 5 mm mesh size was used for further simulations. There were frictionless ‘hard’ contact boundary


___________________________________________________________________________________

____________________________________________________________________________________


conditions on the support table and the flat striker/foam top interface. The hyperfoam data of Table 2, with the two moduli empirically increased by 20%, were used as instantaneous moduli, and the Prony series set 4 of Table 1 used as the viscoelastic data, for both this and the head impact simulations described in the next section. The ‘table stress’ was computed from both the reaction force on the support table and the ‘striker stress’ from the striker acceleration. The impact was predicted to last about 40 ms.

Figure 4 shows the striker and table stresses versus strain, predicted by dynamic FEA. The initial striker-to-foam-top impact creates a pressure pulse which travels down through the foam. Figure 3 shows that, after 2.4 ms, the upper part of the specimen is under a nearly uniform stress of 6 kPa, while the lower surface is almost stress free. With increasing time, the higher stress region propagates to the lower surface of the

Fig. 4. Predicted compressive stress-strain curves for a cube of foam: solid curves for standard FEA, dashed curves for the table stress and striker stress in dynamic FEA.


___________________________________________________________________________________

____________________________________________________________________________________


specimen, then a new, yet higher stress region forms near this surface and propagates upwards through the foam. This phenomenon is due to the inertia of the relatively dense foam, the low sound wave velocity, and the relatively low elastic modulus of the foam. The pulses appear as steps on the loading part of the stress strain curve. They have effectively disappeared by the time the foam is at maximum strain. This appears to be the first prediction of such an effect in open-cell PU foam. A initial step of 5 kPa height, with some small oscillations, was observed in the striker stress vs. foam strain graphs derived from impacts on 100 mm thick blocks of PU chip foam (3). In another simulation with a 5 kg striker and a lower impact velocity, there were many more, smaller steps on the stress-strain curve. Hence there are advantages in using a heavier striker, but as the magnitude of the striker acceleration is reduced, the initial impact can be difficult to detect.

A compression/extension cycle was also simulated using the viscoelastic option in Standard FEA, with a maximum strain of 65% reached in 20 ms, approximately the conditions in the dynamic simulation. Figure 4 shows a smoothly increasing stress-strain curve, which is close to the experimental data for the foam (3). The 22% hysteresis in the standard FEA prediction is less than the 35% in the dynamic FEA, due to the constancy of the strain rate in former simulation.

3 HEAD IMPACT ON A LARGE FOAM MAT

Large blocks of PU chip foam, of typical density 72 kg m-3, are used for fall protection in sport applications. An athlete’s head was simulated falling vertically onto the top surface of the PU mat. The headform was modelled as a rigid sphere of radius 87 mm (the mean radius of curvature of the headform at the crown impact site used in experiments) and mass 4.1 kg, while the foam was a cylindrical block of radius 600 mm and thickness 100 or 400 mm. The foam was supported by a rigid flat table, but able to move away from it (Figure 5). An axisymmetric 2-Dimensional rather than a 3-Dimensional problem was considered, since this reduced the computation time by more than an order of magnitude.

A coefficient of friction of 0.75 was used between the foam and both the headform and the table. The undeformed FEA mesh for the foam consisted of horizontal and vertical straight lines, with spacing biased towards the initial impact point, providing greater detail of the strain field in that region. The foam elements used were CAX4, 4-node bilinear axisymmetric quadrilaterals.


___________________________________________________________________________________

____________________________________________________________________________________


Fig. 5. Axisymmetric model of head impact on a mat, with predicted mat shape just as the head rebounds at 102 ms, with principal compressive stress contours (kPa) in the foam.

The predicted headform force (head acceleration multiplied by its mass) versus the central mat deflection graphs in figure 6 show hysteresis that is almost constant when the headform drop distance (hence initial impact velocity) is varied. The values of the energy in, energy out and % energy lost are given in Table 4. For the 400 mm thick mat, the shape of the predicted loading response (Fig. 6b) is approximately the same as for the experimental data in figure 7. The model is more successful than that using a hyperelastic foam material (3). For the 100 mm thick mat, a head impact velocity of 4.43 ms-1 (a drop from 1.0 m) causes the foam to almost bottom out. Figure 5 shows the temporary residual dent in the upper surface of the foam as the headform rebounds, and the small temporary residual stresses in the mat.

However, as the impact velocity (drop height) increases, the experimental graphs show increased hysteresis, and the unloading curves fall below the predicted responses. Hence linear viscoelasticity cannot completely explain the foam mat response. Non-linear viscoelasticity in the foam is the most likely cause of the extra hysteresis. There is also possibly a contribution from air flow through the foam, modelled by Mills and Lyn (4). The hysteresis in the simulations is the same or slightly higher than the 35% for the dynamic FEA simulation of the cube impact.


___________________________________________________________________________________

____________________________________________________________________________________


Fig. 6. Predicted headform force versus central mat deflection graphs for the marked drop heights in metres: a) 100 mm, b) 400 mm thick mats.


___________________________________________________________________________________

____________________________________________________________________________________


Fig. 7. Experimental headform force versus central mat deflection graphs for the marked drop heights in metres: a) 100 mm, b) 400 mm thick mats. The dashed curves are hyperelastic FEA predictions [3].


___________________________________________________________________________________

____________________________________________________________________________________


Table 4. Energy flows and hysteresis predicted for head impacts on foam blocks

Mat thickness m

Drop height

m

Maximum deflection mm

energy input

Ein J

Hysteresis

%

0.1 0.25 63.9 10.0 35

0.1 0.5 77.7 20.1 35

0.1 1.0 88.8 40.3 35

0.4 0.25 77.0 10.0 39

0.4 0.5 101.0 20.1 40

0.4 1.0 131.3 40.2 42

4 MICROMECHANICS OF A FOAM COMPRESSION CYCLE

Initial analysis of a Gibson-Ashby model

When standard (implicit) FEA models of PU foam were converted to dynamic FEA, a number of changes were made. The introduction of viscoelasticity slows the computation and may cause artefacts. Gibson and Ashby’s (14) model (Figure 8) was initially used because the circa 2 minute computation times allowed experimentation with the model parameters. However, the weakly supported edges labelled A are almost free to vibrate in bending, which is not the case in real PU foams. Consequently, such vibrations may affect the dynamic FEA simulations.

Using an elastic model with quadratic-order elements, the predicted stress-strain curve reduces in slope as the edges marked B and V in figure 8a buckle. The unloading curve is identical with the loading curve (Figure 9a). Although its geometry is inappropriate for PU foams, its predicted compressive stress-strain curve is a reasonable first approximation for that of PU foam. The square cross-section edges have a width that is 0.10 of their length, so the relative density is 0.023 if the structure is repeated in the direction into the paper (Figure 8). Relatively large elements were used, with only two elements across each edge. In the dynamic FEA for a cell size D = 1 mm, ABAQUS CAE did not allow meshing with hexahedral elements. Consequently quadratic-order tetrahedral elements (C3D10M) were used for all the simulations, to avoid errors from changes of meshing.


___________________________________________________________________________________

____________________________________________________________________________________


Fig. 8. Gibson and Ashby’s micromechanics model: a) initial shape, b) half model after 0.04 s, c) after 0.4 s.

First, a viscoelastic step was used in ABAQUS Standard to study the effect of viscoelasticity alone. Using Prony coefficient set 1 or 3 and an instantaneous Young’s modulus of E0 = 100 MPa, additional negative curvature was predicted to occur in the loading part of the stress-strain curve, compared with the purely elastic simulation, causing the stress to nearly reach a maximum during loading. In simulations of cyclic loading to various maximum strains, the strain rate was kept constant at 0.5 s-1. The predicted loading curves (Figure 9a) follow the same path, but the unloading curves are separate, as expected. Table 5 shows that the hysteresis is almost proportional to the gi


___________________________________________________________________________________

____________________________________________________________________________________


Fig. 9. Compressive stress-strain curves predicted for the viscoelastic Gibson-Ashby model: a) viscoelastic standard analysis for loading to 3 maximum strains, compared with the elastic solution, b) dynamic FEA prediction of the striker and table stresses for cell diameter D = 1 m.


___________________________________________________________________________________

____________________________________________________________________________________


values of the viscoelastic model, and increases slightly with the maximum strain in the simulation.

Table 5. Hysteresis values for various micromechanics simulations

Test type model gi maximum strain %

Energy input

kJ m-3

hysteresis %

10 0.23 28

30 1.63 35

50 3.30 39

0.10

70 5.03 41

Constant

strain rate

Gibson Ashby

0.05 60 5.23 18

Impact Gibson Ashby

0.05 45.3 3.56 22

0.05 74 2.91 17 Constant strain rate

wet Kelvin

0.10 74 2.91 40

Second, dynamic FEA was used with the same E0 = 100 MPa, and Prony coefficient set 3 of Table 1. In dynamic FEA the elements cannot be quadratic-order, so linear-order elements were used. Initially the cell diameter was 1 m, which allowed a reasonable ∆t = 5 x 10-5 s. The impact of a 4000 kg rigid cube, initially travelling at 2 ms-1, provided an initial strain rate of 1 s-1 and a maximum compressive strain of 45%. The predicted striker-stress curve (Figure 9b) has several sub-maxima associated with bending vibrations of the edge A1 in figure 8, running into a stress plateau when the edges B and V parallel to the compression direction buckle (the phenomenon predicted in the elastic analysis). The table-stress curve also shows maxima during loading, out of phase with the striker-stress maxima. The edge-vibration effects die away by the start of the unloading curve. Compared with figure 9a, it is difficult to see the onset of non-linearity in the loading response.

When the cell size was reduced to 10 mm, both the striker mass and its initial velocity were reduced by a factor of 100, to keep the initial strain rate and the input energy density constant. Mass scaling of the polyurethane kept the ∆t at the previous level. The effect of the bending vibration of edge A1 in the predicted stress-strain curves was larger, making it difficult to interpret. When the cell was scaled down to D = 1 mm, the edge vibration effect remains large. The edge bending vibrations are an artefact of the regular lattice, and the unrealistic model geometry. In a larger model, with irregular cells, the vibrations of a number of edges would be averaged, so the effect would be


___________________________________________________________________________________

____________________________________________________________________________________


minimal. The hysteresis (Table 5) predicted by the dynamic FEA model, for the same gi = 0.05, is slightly higher than for the standard model.

Viscoelastic analysis of the Wet Kelvin model

The evolved wet Kelvin foam has edges that vary in width, along their length. Figure 10a shows one complete cell of the foam; the [111] direction runs from the centre of one hexagonal face to the centre of the opposite hexagonal face. The relative density (Table 6) is typical of PU foams. The mid-edge width is given, as is the diameter of the holes in the hexagonal faces. The edge width parameter S is for the ‘initial’ Kelvin foam structure that has its surface energy minimized by the Surface Evolver program.

Table 6. Parameters of evolved wet Kelvin foam cells (Cell diameter D = 1 unit)

Edge width

S

Edge width at mid length

Relative density R

hexagonal face

hole diameter

0.2 0.1072 0.0276 0.478

The RUC for [111] direction compression is a prism with an equilateral triangle cross-section (Figure 10b), and a three-fold screw symmetry axis along its axis. Zhu et al. (15) showed that the vertices move directly towards the screw axis by equal amounts, maintaining their 120° angular separation when viewed along the screw axis. The angles between the boundary planes at the sides of the prism remain at 120°. Modelling was performed for a complete repeat unit of the helix of edges, having ‘cut’ half-edges at each end. The boundary conditions on the cut lateral cell-edges at the sides of the prism were achieved by using frictionless contact (which cannot break if tension is applied) between the element faces and a discrete-rigid plane. Three such planes have a single degree of freedom; to move (by equal amounts) directly towards the prism axis.

The cross-sectional area A of the prism is 0.5 D2/√3, where D is the cell diameter. Hence the compressive engineering stress on the top surface of the foam can be calculated from the compressive force on the edge that meets that surface. The lateral strain in the foam is 6U1 where U1 is the lattice 1 axis displacement component of one bounding rigid plane. The initial length of the RUC is √(3/4)D, hence the compressive strain can be evaluated. To evaluate the edge ending and torsional stiffness to within 5% of the analytical values, 10-node quadratic tetrahedral elements C3D10M were used,


___________________________________________________________________________________

____________________________________________________________________________________


Fig. 10. Wet Kelvin foam model a) One complete cell, b) end view of RUC for [111] direction compression, containing a helix of edges.

with typically 21 triangular element faces across the cut-edge ends. There were typically 8000 elements in the models.

The large deformation option was used with ABAQUS 6.5 standard. The periodic boundary conditions from the elastic analysis (7) were used in the ‘visco’ step. It is difficult to simulate contact between foam edges, due to the ‘orphan mesh’ used. Therefore the maximum compressive strain in the simulation was chosen to be 74%, before edge contact occurs. The PU material properties were a combination of

a) density 1200 kg m-3

b) elasticity, with short time Young’s modulus E0 = 50 MPa and Poisson’s ratio ν0 = 0.45.

c) viscoelasticity, with a Prony series of shear moduli of data set 1 or 3 in Table 1.

Figure 11 shows the bent and twisted edges at 74% strain, with two edges approaching but not touching. Figure 12 shows the predicted compressive stress-strain curve of the foam, for a strain cycle to 74% compressive strain in 1 s. As with the Gibson-Ashby model, the addition of viscoelasticity increases the negative curvature of the stress-strain graph during loading, and causes the unloading response to fall below the loading response. When the model was reloaded, the second loading curve falls below the initial one, a phenomenon noted with PU foams.


___________________________________________________________________________________

____________________________________________________________________________________


Fig. 11. Predicted shape of Kelvin foam model at 74% compressive strain in the [111] direction.

Fig. 12. Predicted stress-strain curves of Kelvin foam for compression in the [111] direction. Solid graph for gi = 0.1 (set 3 of table 1) dashed graphs for 1.5 cycles for gi = 0.05 (set 1 of table 1)


___________________________________________________________________________________

____________________________________________________________________________________


These predictions for a model relative density R = 0.0276 can be compared with the experimental data in figure 1a for virgin foam with R = 0.0215. Since the stresses (7) should be proportional to R2, the predicted stresses should be 65% higher than the experimental ones for the same strain. The initial stress plateau of the virgin foam of 3.3 kPa, if increased by 65%, is 5.4 kPa. This compares well with the prediction of 5.7 kPa for the Prony set 3 of Table 1. The predicted unloading curve for gi = 0.1 is closer to the experimental data than that for gi = 0.05, in spite of the latter being a better representation of the solid PU viscoelasticity. In neither case is the predicted hysteresis (Table 5) as high as the experimental values (Table 3).

Dynamic FEA of the wet Kelvin model

It was unclear how to implement periodic boundary conditions in dynamic FEA, so these were not used. It is not possible to impose a non-zero displacement of a dummy point, used in standard FEA (7) to create the deformation, nor can the striker mass be easily linked to the forces on further dummy points. However, dynamic FEA could be better for modelling non-uniform strain in the compressed foam; the RUC could be made two or three times as long in the compression direction as that used in the standard FEA, and self contact conditions can be easily applied to the foam structure.

The end boundary conditions affect the model response. ‘Encastre’ support of the cut lower end of the helix of edges is too rigid compared with the conditions in a larger block of foam. However if the cut edge is allowed to rotate freely, this is too flexible. The same applies to the impacted upper end of the helix of edges. Such distinctions would be unimportant if the model size exceeded 10 cell diameters in length, but in the small models that are possible to analyse in a reasonable time, they have a critical effect on the predictions.

When a realistically small cell diameter of D = 1 mm was used, ∆t = 1 x 10-11 s, which made the computation unacceptably slow. Hence the foam cell size was chosen as D = 1 m, causing ∆t = 5.3 x 10-7. Mass scaling was then used to further speed up the calculations; to increase ∆t to 1 x 10-5 s. There is a limit to the mass scaling, when the deformation speed exceeds the wave speed in an element during the analysis, so the analysis ceases. When simulations were attempted, stability and contact problems prevented success. Nevertheless, further attempts will be made to solve these problems.


___________________________________________________________________________________

____________________________________________________________________________________


DISCUSSION

Dynamic FEA, considering the polymer viscoelasticity, confirms that it is reasonable to interpret drop impact tests on PU foam blocks in terms of the foam stress-strain curve (3). The small step at the start of the curve is due to a pressure (or densification) pulse which propagates through the foam; the step should be ignored in extracting the foam response. This makes the hyperelastic model of Table 2 a better fit to the data for remoulded PU foams than previously assumed (3).

The predictions from the viscoelastic analysis of head impacts on large blocks of remoulded PU foams are improvements on the previous hyperelastic analysis, both in terms of the shapes of the force vs. deflection curves and the hysteresis. However non-linear viscoelasticity in the foam probably causes the hysteresis to increase with the impact velocity in experimental data. Air flow also contributes to losses, when large foam blocks are impacted at velocities in excess of 5 m s-1, make the analysis of such systems complex. Viscoelasticity could be added to previous elastic analyses of PU foam responses during the Indentation Force Deflection test (13).

There are limitations in modelling foam micromechanics using FEA. Only a small amount of the foam structure can be considered, because the computation time must not be excessive. This necessitates the use of a small RUC from a regularly-repeating foam structure. The dynamic FEA stress-strain predictions for the Gibson Ashby model were noisy, due to the bending vibration of edges close to the striker surface. In real foams, with random rather than a regular cell structures, the vibrations of individual edges will not be in phase, so will cancel out. The time interval in dynamic FEA is small due both to the small cell size and the fine mesh required to capture the edge geometry.

The use of viscoelasticity in standard FEA was more successful. Simulations with the Gibson Ashby model, which predicts stress-strain non-linearity due to large foam geometry changes, indicated that the foam hysteresis is almost proportional to the polymer damping (measured by the Prony coefficients gi), as expected. It also increases slightly with the maximum compressive strain in the foam. Hysteresis is predicted to be slightly greater in PU foams impacted in compression than in those cyclically compressed in a testing machine; the reduction in the strain rate, towards the maximum strain in the impact test, allows more time for the effective modulus to drop. Since the hysteresis of virgin PU foam is higher than that predicted by the linear viscoelastic models, the viscoelastic response of the foam is probably non-linear.

A small stress drop was predicted during the [111] direction compression of a viscoelastic, wet Kelvin model. This might cause strain localisation in a larger model, in which a highly deformed band of foam forms and propagates while the compressive stress remains constant. In spite of the stress relaxation data for solid PU being best


___________________________________________________________________________________

____________________________________________________________________________________


fitted by the generalised Maxwell model, with parameter set 1 in Table 1, it appears that the hysteresis in a compression cycle on virgin PU foam is best matched by a model with at least double the damping (double the gi values). This model can now predict the main features of PU foam compression responses, via quantitative relationships with the polymer properties and foam density. However, the influences of anisotropic cell shapes, and of variable cell diameters and packing, remain to be incorporated in a more complex model. A micromechanics model for remoulded PU foam is still remote, given the complexity of the microstructure. Such a model would consider the material as a composite, with a statistical distribution of foam density and a distribution of pre-compression of the chips.

REFERENCES

1. Davies O.L. & Mills N.J., The rate dependence of Confor PU foams, Cell. Polym., 18, (1999) 117-136.

2. Mott, P.H., Roland, C.M. and Corsaro R.D., Acoustic and dynamic mechanical properties of a polyurethane rubber, J. Acoust. Soc. Am. 111 (2002) 1782-1790.

3. Lyn G. & Mills N.J., Design of foam crash mats for head impact protection, Sports Engng., 4, (2001) 153-163.

4. Mills N.J. & Lyn G., Modelling of air flow in impacted flexible polyurethane foams, Cell. Polym., 21, (2002) 343-367.

5. Huang J.S. & Gibson L.J., Creep of polymer foams, J. Mater. Sci. 26, (1991) 637-647.

6. Zhu H.X. & Mills N.J., Modelling the creep in open cell foams, J. Mech. Phys. Solids, 47, (1999) 1437-1457.

7. Mills N.J., The high strain mechanical response of wet Kelvin foams open cell foams, Int. J. Solids Struct., accepted. (2006)

8. Surface Evolver (2004) at //www.susqu.edu/facstaff/b/brakke/evolver/

9. McKown S., Ph D thesis, University of Liverpool, (2005)

10. Elliott J.A., Windle A.H. et al., In-situ deformation of an open cell flexible polyurethane foam, J. Mater. Sci., 37 (2002) 1547-1555.

11. ABAQUS 6.5 manuals, Hibbitt, Karlsson and Sorensen Inc. Pawtucket, R.I., USA (2005).

12. Ogden R.W., Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids, Proc. Roy. Soc. Lond. A 328, (1972) 567-583.


___________________________________________________________________________________

____________________________________________________________________________________


13. Mills N.J. & Gilchrist A., Modelling the indentation of low density polymer foams, Cell. Polym. 19, (2000) 389-412.

14. Gibson L.J. & Ashby M, Cellular Solids, 2nd Ed., Cambridge University Press (1998)

15. Zhu H.X., Mills N.J. & Knott J.F., Analysis of the high strain compression of open-cell foams, J. Mech. Phys. Solids, 45, (1997) 1875-1904.

Finite Element Models for Viscoelasticity in Polyurethane Foams

Documents

Transcript of Finite Element Models for Viscoelasticity in Polyurethane Foams