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Fig. 14. Example using JHC concrete model
Fig. 15. Typical stress-strain curves for concrete and reinforcing steel: (a) stress-strain curve for concrete (data from USDOD 2008); (b) stress-strain curve for steel (data from USDOD 2008); (c) stress-strain curves for steel reinforcing rebar (data from Malvar 1998)
The illustration in Fig. 15 demonstrates elastic behavior, nonlinear behavior, and damage of the material. During blast load events, the failed portion of the material needs to be removed based on certain criteria. There are six failure criteria besides the failure time in LS-DYNA. These criterions are based on maximum pressure, maximum principal stress, deviatoric stress, maximum principal strain, shear strain, and Tuler-Butcher criterion (impulse criterion) (Hallquist 2006). In this research, the maximum principal strain criterion is considered as failure criteria for materials because structural members usually have ductility and fail because of excessive displacement after yielding. Another competitive criterion for the blast load effects is the Tuler-Butcher criterion (Tuler and Butcher 1968). This criterion is based on an applied impulse on the structures. It works well during high pressure loads and can predict the spalling phenomenon reasonably well when the loading duration is short. However, if the simulation time is long and the structure sustains a relatively low pressure, it overestimates the damage. Hence, this criterion has not been considered in this research where the structural behavior needs to be investigated beyond the blast loading duration.
Under the action of rapidly applied loads, the rate of strain increases. This can affect mechanical properties of structural materials significantly. Figs. 15(a and b) show typical stress-strain curves for concrete and steel, respectively, including the effects of highstrain rates. Malvar (1998) provides a clearer explanation of the properties of steel rebars, as shown in Fig. 15(c). The main characteristics of the strain-strain relationship observed in Figs. 15 can be summarized in the following:
1. The material behavior shown in Fig. 15 is under monotonic loading. Repeated loading is usually not considered for blast loads.
2. The yield stress of steel and the compressive strength of concrete are increased by more than 25% because of highstrainrate effects. The percentage of elongation at failure remains roughly unchanged for both concrete and steel. Fu et al. (1991) has also confirmed this.
3. The ultimate strain is slightly reduced (Malvar 1998) or the strain at maximum stress and rupture remains nearly constant according to TM-1300 (USDOD 2008).
4. The elastic modulus remains the same for steel and increases slightly for concrete due to high-strain rate effects.
Current research on the behavior of materials during high-strain rate loadings has shown that strain effects primarily increase the structural material strength and have very little influence on the failure strain. The strain-rate effect is usually considered a factor of static yield stress [Drysdale and Zak 1985; Livermore Software Technology (LST) 2007]. This strain-effect factor is only a function of strain rate and material intrinsic properties. Hence, tensile failure strain can be assumed to be 0.002 for the concrete cover, 0.005 for the concrete core, 0.23 for reinforcing rebar, and 0.20 for steel stringers based on the behavior of materials during static loading.
The material behavior of elastomers is simulated using a linear viscoelastic material model in LS-DYNA (Mat 6). This model is for rubber material and can consider the rate effects by a convolution integral of the stress tensor. HooFatt and Ouyang (2006) have studied the behavior of elastomers at strain rates lower than 400 s21 and have found that the elastomer stiffness increased when the strain rates were lower than 180 s21 and decreased when the strain rates were between 180 and 280 s21. The strain rate during the blast load is generally between the order of 103 and 105 s21.