Refined constitutive, bridging, and contact laws for including effects of the impact-induced temperature rise in impact responses of composite plates with embedded SMA wires

• Low-velocity impact analysis of a composite plate with SMA wires is accomplished.
• Effects of the impact-induced temperature rise are considered, for the first time.
• A free energy function including the phase transformation latent heat is presented.
• Brinson's model for the SMAs and Hertz contact law are refined.
• A refined bridging law is employed instead of the traditional homogenization rules.

In the present research, effects of the impact-induced temperature rise on the phase transformation of shape memory alloy (SMA) wires embedded in composite plates and consequently, impact responses of the hybrid plates are investigated, for the first time. In the available approaches, effects of the temperature rise have been studied only for single SMA wires under uniaxial tensile loads whereas the stress and martensite volume fraction vary along the SMA wires embedded in a hybrid composite plate under transverse impact. The impact-induced heat of the SMA wires is determined based on proposing a refined free energy function that includes the latent heat of the phase transformation. Furthermore, Brinson's constitutive law of the superelastic SMAs and Hertz contact law are revised based on the refined phase version of the free energy function and a proper bridging law wherein, distributions of the SMA phases are considered to be both localized and time-dependent. Results of the employed bridging law are more accurate than those of the common homogenization techniques wherein, stresses of the wires cannot be easily distinguished from those of the mixture. The resulting governing equations are solved by the finite element method and the refined nonlinear constitutive laws are solved through a return-mapping Newton-Raphson procedure, within each time step. Results reveal that incorporation of the realistic heat generation phenomenon leads to remarkably higher local stresses; a fact that has to be considered in the design stage.