Because of rubber’s nonlinear behavior, the linear superposition
of unit load cases is not a valid method for converting load history
into strain history. Nor is it an option to run a full FEA of the
load history at each update, given that typical model run-time is
approximately one hour for only a few seconds of real-time loading
history. Instead, we used the Endurica EIE interpolation engine.
Endurica EIE uses a pre-computed, nonlinear map that connects
possible loading states of the rubber tip, as specified by the x, y,
and z force components acting on the tip, to the corresponding
strain states that occur in each element of the FE model. EIE
leverages the assumption that there is a unique, one-to-one
correspondence between points in the 3-channel loading space
and the deformation states of the material. Since rubber is
nonlinearly elastic, this assumption is generally accurate.
Once the strain history for the surface elements of the rubber tip
was known, the next step was to accrue the associated damage.
This was accomplished using the Endurica DT incremental fatigue
solver. The incremental solver is based on the following calculation
(Mars et al 2018), and is made for each possible failure plane in
each element. This formulation uses the same crack growth rate
laws and input variables as are used to initialize the residual life,
but integrates over cycles from time Ni to time Ni+1, rather than
over crack size.
The result of the integration is the change of length ΔCi->i+1,j,k of
a crack for each finite element j and each potential failure plane k
of the model.
Equation 1: Incremental fatigue solver formulation
The accumulated crack lengths for each element are
written to a file during the analysis, so that future additions of load
history may begin at the point at which the prior increment left off.