rateindependent

Rate-independent refers to systems whose evolution depends on the history of loading rather than the speed at which the load is applied. If a loading path t → ℓ(t) is reparameterized by a nondecreasing map, the resulting state path is the same up to reparameterization. This contrasts with rate-dependent (viscous) models, where changing the loading rate changes the response.

In mathematical and mechanical modeling, rate-independent systems are described by an energy functional E(t, z) defined on a state space Z and a dissipation mechanism D that measures the cost of changing state. A common structural feature is 1-homogeneous dissipation: the cost of a given state change scales linearly with the size of the change. Evolution is often formulated via energetic solutions, which satisfy a global stability condition and an energy balance. Global stability means the current state minimizes energy plus dissipation relative to all admissible states, while the energy balance accounts for work input and the dissipation accumulated over time.

Numerical and analytical approaches frequently use time-incremental minimization schemes. At each time step, one solves for z_k by minimizing E(t_k, z) plus the dissipation distance to the previous state z_{k-1}. In the limit of vanishing time steps, the resulting evolution is rate-independent and may exhibit instantaneous jumps in state when loading reaches a threshold.

Applications and common examples include plasticity, fracture mechanics, damage models, and phase transformations with hysteresis. Rate-independent models capture systems where internal friction or yield phenomena depend on the amount of deformation rather than its rate, making them suitable for processes with abrupt changes and path-dependent behavior.