Fatigue Material

The fatigue material uses a modified rainflow cycle counting algorithm to accumulate damage in a material using Miner’s Rule. Element stress/strain relationships become zero when fatigue life is exhausted.

uniaxialMaterial Fatigue $matTag $tag <-E0 $E0> <-m $m> <-min $min> <-max $max>

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$matTag	integer tag identifying material
$K1	initial stiffness
$K2	secondary stiffness
$δy	yield displacement
$gap	initial gap*

NOTES:

This material is implemented as a compression-only gap material. Delta_y and gap should be input as negative values.

DESCRIPTION:

This material is based on an approximation to the Hertz contact model proposed by Muthukumar (See REFERENCES below). The energy dissipated during impact is:

E = kh * δm^(n+1) * (1-e^2) / (N+1)

where kh is the impact stiffness parameter, with a typical value of EA/L or 25,000 k-in.-3/2; n is typically taken as 3/2 for the exponent associated with the Hertz power rule; e is the coefficient of restitution, with typical values from 0.6-0.8; and δm is the maximum penetration during the pounding event. The effective stiffness, Keff, is:

Keff = kh * sqrt(δm)

The yield displacement is:

δy = a * δm

where a is typically taken as 0.1. The initial stiffness, K1, and secondary stiffness, K2, are then selected such that the Impact model dissipates an amount of energy during a pounding event that is consistent with the associated energy dissipated in the Hertz model.

K1 = Keff + E / (a*δm^2)

K2 = Keff - E / ((1-a)*δm^2)

Response of Impact Material during a pounding event.

Response of Impact Material for displacement cycles of increasing amplitude.

EXAMPLE:

REFERENCES:

Muthukumar, S., and DesRoches, R. (2006). “A Hertz Contact Model with Non-linear Damping for Pounding Simulation.” Earthquake Engineering and Structural Dynamics, 35, 811-828.

Muthukumar, S. (2003). “A Contact Element Approach with Hysteresis Damping for the Analysis and Design of Pounding in Bridges.” PhD Thesis, Georgia Institute of Technology. http://smartech.gatech.edu/

Nielson, B. (2005). “Analytical Fragility Curves for Highway Bridges in Moderate Seismic Zones.” PhD Thesis, Georgia Institute of Technology. http://smartech.gatech.edu/

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Code Developed by: Mathew Dryden, UC Berkeley