All materials are temperature dependent, but some more than others: metals tend to be crystalline solids and will melt at sufficiently high temperatures; in contrast, crosslinked elastomers are always solids. They can be glassy or rubbery, crystalline or amorphous. When heated to extreme temperatures, they burn rather than melt, producing new substances, usually low molecular weight hydrocarbons (i.e. tarry substances and smoke).
Of course, you do not have to melt or burn a material to see the effects of temperature. In fatigue analysis, we are concerned with stress-strain and crack growth behaviour. These can be temperature dependent for both metals and rubbers. However, while metals have a very high thermal conductivity, rubbers have almost the lowest. Therefore, fatigue analyses involving large temperature gradients are much more common in rubber than in metal.
As shown in Fig.1, while a 100°C temperature gradient in a metal can affect the fatigue tensile strength or the fatigue limit by 10% [1], the same 100°C temperature gradient in rubber can reduce the fatigue life by four orders of magnitude [2]!
Fig. 1. Left – Effects of temperature on carbon steel showing tensile strength (+), yield stress (●), and fatigue limit (○), [1]; Right – Effects of temperature on natural rubber (Δ) and styrene butadiene rubber (●) [2].
The mechanisms underlying the elasticity of metals and rubbers could hardly be more different. Under stress, atoms in a metal’s crystal lattice are displaced from their equilibrium positions, and potential energy is stored in strained interatomic bonds. In rubber, however, the strain energy is not predominantly stored in strained atomic bonds. Rather, elasticity arises because the constituent long-chain molecules are much more likely to be randomly coiled than to be fully extended.
Thus, provided that the molecules are sufficiently agitated by random thermal fluctuations, an entropic spring effect is created, meaning that potential energy can be stored by working to reduce the entropy of the polymer chain network by increasing the end-to-end distance of individual polymer chains [3].
Polymers in general can exhibit both glassy and rubbery behavior, depending on the temperature. The rubbery state – in which entropic elasticity dominates – exists above the glass transition temperature Tg, if the molecular motion rate is sufficiently high. In the rubbery state, very large strains are possible and the rubbery elastic storage modulus E’r determines the stress-strain curve.
Below Tg, however, the lack of thermal molecular mobility prevents molecular reconfiguration, resultng in a glassy stiffness E’g that is several orders of magnitude higher than E’r. Polymers operating below Tg are thus not capable of large elastic strains and instead exhibit inelastic behavior when strains exceed a few percent. Figure 2 shows how the storage and loss moduli vary through the glass transition (left), and how the rate of molecular motion rate depends on temperature (right). The relative rate φ(T)/φ(Tg) of molecular motion as a function of temperature T is described by the WLF equation [4], which has material constants A and B.
Since the fracture mechanical properties of rubber depend on the viscoelastic dissipation in the crack tip process zone, with higher dissipation associated with lower crack growth rates, frequency and temperature effects can be inferred accordingly. Viscoelastic master curves, such as those shown in Fig. 2, can be used as part of the material property rate dependence specification in the Endurica solver.
Fig. 2, Left – Rubber’s elastic and viscous responses depend on temperature relative to the glass transition temperature Tg; Right: The rate of molecular motion depends on temperature relative to the glass transition temperature Tg.
Self-Heating and Thermal Runaway
During a charge cycle, work WL is done on the charge stroke, some of which WU is recovered on the discharge stroke, as shown in Fig. 3. The unrecovered part of the work H remains in the material as heat energy, increasing the temperature.
Fig. 3. Work input WL on the loading stroke is partially recovered as WUon the unloading stroke. A portion H of the energy remains in the material as heat.
The rate of viscoelastic heating of rubber depends on strain amplitude, cycle rate (i.e. frequency) and temperature. The strain amplitude dependence of the viscoelastic storage and loss modulii, G’ and G” respectively, can be specified using the Kraus model [5,6]:
where εa is the strain amplitude, and where G’∞, G’0, εa,c, m, G”∞, G”max, and ΔG”U are material parameters. The viscoelastic heat rate per unit volume can be calculated from:
Due to the low thermal conductivity of rubber, small amounts of viscoelastic self-heating can produce large temperature gradients. Accurately accounting for thermal effects on rubber durability generally requires both structural finite element analysis to calculate stress and strain fields, and a thermal finite element analysis to calculate the temperature field. Endurica fatigue solvers can provide heat rate calculations in a coupled finite element simulation for both transient and steady state thermal analyses.
In cases where the temperature in the rubber exceeds a critical value Tx, an additional heat rate contribution q ̇x occurs due to exothermic chemical reactions. The effect is illustrated in Fig. 4, for a rubber cylinder subjected to a rotating bending load [7]. The thermal runaway starts after about 250 seconds. Both experimental (dashed line) and Endurica-calculated (solid line) simulation results are plotted for the cylinder centreline (blue) and for the cylinder outer surface (green). The thermal runaway event typically results in rapid decomposition of the rubber into hydrocarbon gases (i.e. smoke/burning rubber) and low-molecular weight substances (tar).
Fig. 4. When temperature exceeds a critical value Tx, exothermic chemical reactions can produce a thermal runaway failure. Plot (right) shows Endurica calculated transient temperature history (solid lines) for a rotating bending cylinder (structural finite element model shown on left). For comparison, experimentally measured temperature histories are also shown (dashed lines).
Reversible Temperature Effects
The crack growth properties of rubber reversibly depend on temperature. Higher temperatures tend to reduce the tear strength Tc of rubber and increase the crack growth rate, as shown in Fig. 5 [8]. At lower temperatures, the tear strength is increased and crack growth is retarded. Endurica’s crack growth models can be specified with a temperature dependence via the temperature sensitivity coefficient (see Table 1) or via a table look-up function.
Fig. 6 shows the fatigue life as a function of temperature calculated from the parameters in Table 1 [2]. Over a range of 100°C, natural rubber loses approximately a factor of two in fatigue life, and styrene butadiene rubbers loses four orders of magnitude!
Table 1. Crack growth properties and temperature sensitivity for natural rubber (NR) and styrene butadiene rubber (SBR), estimated from measurements reported in [2].
Fig. 5 – Increasing temperature causes the crack growth rate to increase. Results are shown for natural rubber [8].
Fig. 6. Endurica calculated dependence of fatigue life on temperature for natural rubber (Δ) and for styrene butadiene rubber (●) [2]. Compare to Fig. 1.
Fig. 7 shows the Haigh diagram calculated by Endurica for three different temperatures: 23, 90 and 110°C. For example, at a mean strain of 100% and a strain amplitude of 20%, the fatigue life at 23°C exceeds 106 cycles, but at 110°C the fatigue life is approximately 103 cycles. This effect has been confirmed experimentally in recent work by [9].
Fig. 7. Endurica calculated Haigh diagrams for natural rubber at 23, 90 and 110°C . Increasing temperature tends to reduce strain crystallization, with the result that the mean strain benefit associated with strain crystallization is reduced or even eliminated at high temperatures.
Irreversible Temperature Effects / Ageing
Prolonged exposure to high temperatures can cause permanent changes in the cross link density and mechanical properties of rubber, including stiffness and crack growth properties. The effect depends on the availability of oxygen [10], as shown in Fig. 8.
Fig. 8 – The evolution of rubber’s properties during ageing depends on the availability of oxygen, and on the temperature [10]. Under aerobic conditions, ageing tends to increase stiffness while strain at break decreases. Under anaerobic conditions, ageing tends to decrease stiffness while strain at break decreases.
The rate at which thermochemical ageing of rubber progresses can be specified in Endurica using the Arrhenius law [11] and its activation energy parameter Ea. When following a temperature history θ(t), Endurica integrates the Arrhenius law to determine an equivalent exposure time τ at the reference temperature θ0. R is the real gas constant.
The equivalent exposure time controls the evolution of the stiffness and crack growth properties with thermal history. As shown in Fig. 9, the evolution of the crack growth rate law is specified by a tabular function that gives the stiffness E(τ), tensile strength Tc(τ) and the fatigue limit T0(τ). The material properties are then updated iteratively according to the co-simulation workflow shown in Fig. 10. This allows the effects of thermal history and ageing on fatigue performance to be considered.
Fig. 9. The crack growth rate law evolves as a function of the equivalent exposure time τ. Crack growth property evolution is specified in Endurica by the dependence of the rubber’s tear strength Tc(τ) and its fatigue limit T0(τ) on exposure time.
Fig. 10. Endurica DT’s co-simulation workflow updates the crack length c, exposure time τ, and stiffness E so that stress, strain and temperature fields can be updated during solution.
Conclusion
There are many ways in which metals and rubbers differ in their behaviour, and thermal behaviour is one of the most important.
Rubber more often requires careful attention to thermal effects due to its exceptionally low thermal conductivity, its entropy-elasticity, its visco-elastic properties and tendency to self-heat under cyclic loading, the sensitivity of crack growth properties and strain crystallization to temperature, oxidation, and ageing.
Endurica’s fatigue solvers provide material models and workflows that capture these thermal effects, enabling accurate analysis and “right the first time” engineering.
References
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