This model is very similar to the Kinematic model, but uses a degrading hysteretic loop that accounts for decreasing energy dissipation and unloading stiffness with increasing plastic deformation.
Two measures are used for plastic deformation:
Maximum plastic deformation in each the positive and negative directions
Accumulated plastic deformation, which is the absolute sum of each increment of positive or negative plastic deformation. Plastic deformation is that which does not occur on the two elastic segments of the action-deformation curve
Accumulated plastic deformation can occur under cyclic loading of constant amplitude, and can be used to represent fatigue.
For this model, the following parameters are required:
Separately for positive and negative deformations
Initial energy factor at yield, f0, usually 1.0
Energy factor at moderate deformation, f1
Energy factor at maximum deformation, f2
Moderate deformation level, x1, as a multiple of the yield deformation
Maximum deformation level, x2, as a multiple of the yield deformation
Accumulated deformation weighting factor, a
Stiffness degradation weighting factor, s
Larger-smaller weighting factor, w, usually 0.0
The energy factors represent the area of a degraded hysteresis loop divided by the energy of the non-degraded loop, such as for the kinematic model. For example, an energy factor of 0.3 means that a full cycle of deformation would only dissipate 30% of the energy that the non-degraded material would. The energy factors must satisfy 1.0 > f0 > f1 > f2 > 0.0. The deformation levels must satisfy 1.0 < x1 < x2.
All weighting factors may take any value from 0.0 to 1.0, inclusive. Because the accumulated plastic deformation is constantly increasing, it is recommended that the weighting factor a generally be small or zero.
For each increment of deformation:
The absolute maximum positive and negative plastic deformations that have occurred up to this point in the analysis are determined, dmaxpos and dmaxneg, as well as the accumulated plastic deformation dacc.
A plastic deformation level is calculated as dpos = adacc+(1-a)dmax pos where a is the accumulated weighting factor for positive deformation.
Comparing dpos with the positive deformation levels d1 and d2, obtained by multiplying x1 and x2 with the positive yield deformation, an energy factor fpos can be determined by interpolation. If dpos> d2, then fpos = f2.
Following the same approach, the energy factor for negative deformation, fneg, is computed using the corresponding parameters for negative deformation
The larger of these two energy factors is called fmax, and the smaller is fmin, The final energy factor is computed as f = wfmax + (1+w) fmin.
In the most common case, w = 0 and f = fmin.
Degradation does not occur during monotonic loading. However, upon load reversal, the curve for unloading and reverse loading is modified according to the energy factor computed for the last deformation increment. This is done by squeezing, or flattening, the curve toward the diagonal line that connects the two points of maximum positive and negative deformation.
This squeezing is scaled to achieve the desired decrease in energy dissipation. The scaling can occur in two directions:
Parallel to the elastic unloading line, called elastic degradation
Parallel to the horizontal axis, called stiffness degradation
The amount of scaling in each direction is controlled by the stiffness degradation weighting parameter, s. For s = 0.0, all degradation is elastic type. For s= 1.0, all degradation is stiffness type. For intermediate values, the degradation is apportioned accordingly.
While the deformation and individual energy levels are computed separately for the positive and negative directions, the final energy level is a single parameter that affects the shape of the hysteresis loop in both directions.
Note that if all the energy factors are equal to 1.0, this model degenerates to the kinematic hysteresis model.
The figures show the shape of the hysteresis loop for elastic degradation, stiffness degradation, and a mixture with a stiffness degradation factor of s = 0.5. Each of these three cases dissipates the same amount of energy for a given cycle of loading, and less than the energy dissipated for the equivalent kinematic model.
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