Models for elastomer stress strain properties
As discussed in Section 4.2.1, models predicting the modulus of polyurethanes have been proposed. Based on realistic models of polyurethane structure, several have been shown to reproduce measured properties with good fidelity. Part of the success of these models is that they occur in the low strain-linear response range for the materials such that there are no irreversible plastic deformations that occur during the test. Modeling the whole stress strain curve is a much larger challenge. High strain experiments on polyurethanes enter the domain of nonlinear viscoelasticity and irreversible processes that occur to the soft segment (i.e., disentanglement and bond rupture), hard segment (shear yielding and orientation), and physical processes that may be reversible, but only partially so on the time scale of the measurement (i.e., hysteresis and elastic set). To some extent, polyurethane stress-strain behavior can be anticipated based on composition, phase structure, and reasonable assumptions about how polymers behave.
Factors that Affect Polyurethane Stress-Strain Behavior
For any given polyurethane sample undergoing mechanical testing, there are four scenarios that one should consider when trying to anticipate large strain behavior. These scenarios are shown schematically in Figure 4.13.
Scenario 1: Hard and soft phase are below their respective glass transition temperatures exhibiting properties of a hard glassy polymer. The plastic exhibits linear elastic behavior up until large scale flow is initiated at the yield point, accompanied by a clear necking as the material exhibits Poisson narrowing initiating at some sample nonuniformity. The material elongates plastically up until failure which is a function of strain rate, temperature, defectivity, and details of the material structure. An example of a polyurethane sample tested within this range is shown in Figure 4.14.
FIGURE 4.13 Diagram of the categories of phase-separated polyurethanes that affect stress-strain properties apart from molecular weight.
Figure 4.14 Illustrative stress-strain cure for a polyurethane elastomer characterized by Scenario 1 with both soft and hard segments below their glass transition temperatures. Reprinted with permission from Ref. . © American Chemical Society.
Scenario 2: Hard phase is below its glass transition temperature and soft phase above its glass transition temperature. This is the typical situation for a polyurethane elastomer in application. In this case, the material strain properties will be sensitive to the microphase separation such that if the material is not microphase separated it will elongate like a crosslinked rubber (Fig. 4.15a). If the material has microphase separated but is in a spherical phase, or is a different phase below its percolation threshold, it may elongate like a filled rubber (Fig. 14.15b). If the material is phase separated and the hard phase is above its percolation threshold, the material will behave as a composite (Fig. 14.15c) having a high initial modulus that drops rapidly at elongations above a poorly defined yield type point.
Scenario 3: This is a special case that is rarely observed in practice. Such a case is conceivable in the instance where the soft segment is an aromatic polyester phase and the hard segment is comprised of an aliphatic isocyanate and a branched diol. Depending on their volume fractions, the stress-strain curves may appear as a special case of those observed in Figure 4.15.
Scenario 4: In the case, where the soft segment and the hard segment are both above their respective T, one should expect that the material will extend as a weakly crosslinked elastic network, elongating at relatively low stress and breaking at relatively high extensions and low stress (Fig. 14.16).
Attempts to model stress strain curves have usually followed an "Equivalent Box Model" such as employed by Kolarik [46, 49, 50] and others. This model separates the total stress into the sum of stress contained by the hard segment and a soft segment (Eq. 4.14) that may have dissolved components of hard segment in it (Fig. 4.17).
A representation of the functional dependence of stress with strain that allows a separable and additive calculation would possibly be represented by Equation 4.15. In Equation 4.15, <ph is the volume fraction of hard phase, e is the strain, ey is the yield strain, Eh is the modulus of the hard phase (estimated from the modulus of a conventional thermoplastic below its glass transition temperature), Es is the modulus of the soft segment, and P(K) is a complex function reflecting the stress amplification of the dissolved hard phase within the soft segment as it reinforces the soft segment as a function of the soft segment strain. This phenomenon of strain reinforcement was first observed as a deviation of measurements on polyurethanes from the behavior predicted by the Mooney-Rivlin equation . The Mooney-Rivlin equation has been used with good success to model stress-strain behavior to moderate strain levels at which point the limited or constrained extensibility of chains becomes significant [52-54]
Figure 4.15 Illustrative stress-strain curves for a polyurethane elastomers of Scenario 2. (a) The hard phase is below the microphase separation limit, (b) the hard phase is microphase separated but not co-continuous, and (c) the hard segment is phase separated and above the percolation threshold. Reprinted with permission from Ref. . © American Chemical Society.
Figure 4.16 Illustrative stress-strain curve for a polyurethane elastomer with both the hard and soft segments above their respective glass transitions.
Figure 4.17 The Equivalent Box model for calculating large strain properties of polyurethanes.
As strain is increased, the stress amplification of dissolved hard segments decreases since some of the hard segment clusters yield. At the limit of maximum extension, when all the soft segment chains have fully extended, all or most of the hard segment clusters would have yielded prior to soft segment chains breaking, and the stress amplification of dissolved hard segment is low (assuming the dissolved cluster cohesive energy is less than the bond energy of the soft segment chain) [55, 56].
Calculation of ultimate elongation and toughness (the integral under the stress strain curve) for an unfilled Gaussian network is usually related to the contour length of the chains at which point they are completely stretched, and incremental strain increases will result in chain breakage. For polyurethanes within Scenario 2 (Fig. 4.13), the situation is substantially more complex. Elongation of these systems is known to depend on unquantifiable factors associated with sample history and processing minutiae. Reasonable attempts have been made to model the stress-strain behavior of crosslinked glassy materials like epoxies. Predictions from semi-empirical equations (i.e., the Martin, Roth, and Stiehler equation) and an analogue equation replacing stress and strain variables with normalized failure stress and strain (Eq. 4.16) and experimental validation have been obtained [57, 58].
In Equation 4.16, E is the tensile modulus, e is the strain normalized to gauge length, X is the extension ratio (equal to e + 1), eb is the elongation at break, ve is the crosslink density, and A is an adjustable constant on the order of 0.5. A plot of log ((ob/ve) (273/T)) versus log eb results in a consistent envelope of failure stress and failure strain for a variety of materials when normalized for their molecular weight between crosslinks A graphical representation of Equation 4.16 for elastomers with different molecular weights between crosslinks as a function of elongation at break is shown in Figure 4.18.
If one assumes that the ultimate strength of the elastomer is dictated by the ultimate strength of the soft segment chains, it is expected that as the hard segment volume increases, the molecular weight between (physical) crosslinks will go down and elongation at break will decrease. This is in qualitative agreement with observations. As the hard segment traverses the phase morphology space to a co-continuous structure, the effect of increasing crosslink density on strength and elongation to break becomes more obvious as seen for example in Figures 4.15 and 4.16.