To model rupture dynamics, a friction law must be assumed. Commonly used constitutive laws for modeling friction include slip-weakening laws which are characterized by a drop from static to dynamic frictional stress. Within this framework, the prevailing understanding asserts that the frictional behavior is solely controlled by the fracture energy -- the area beneath the frictional stress versus the cumulative slip curve. In particular, it is claimed that the curve's shape itself has no influence on the system's response. Here we perform fully dynamic rupture simulations to challenge prevailing beliefs by demonstrating that the constitutive law shape exerts an intimate control over rupture profiles. For a consistent fracture energy but varying constitutive law shapes, the velocity profile is different: each abrupt slope transition leads to the localization of a distinct velocity peak. For example, in the case of a bilinear slip-weakening law featuring two different slopes, the rupture exhibits two distinct velocity peaks. This phenomenon arises from the transition between a constant weakening rate to another. However, this distinction does not seem to influence how a rupture responds to a stress barrier or the cumulative radiated energy emitted. These results are derived through two separate numerical schemes (spectral boundary integral and finite element methods) ensuring their independence from the computational approach employed.