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This command is used to construct a uniaxial material that represents a 'pinched' load-deformation response and exhibits degradation under cyclic loading. Cyclic degradation of strength and stiffness occurs in three ways: unloading stiffness degradation, reloading stiffness degradation, strength degradation.
uniaxialMaterial Pinching4 $matTag $ePf1 $ePd1 $ePf2 $ePd2 $ePf3 $ePd3 $ePf4 $ePd4 <$eNf1 $eNd1 $eNf2 $eNd2 $eNf3 $eNd3 $eNf4 $eNd4> $rDispP $rForceP $uForceP <$rDispN $rForceN $uForceN > $gK1 $gK2 $gK3 $gK4 $gKLim $gD1 $gD2 $gD3 $gD4 $gDLim $gF1 $gF2 $gF3 $gF4 $gFLim $gE $dmgType
$matTag |
unique material object integer tag |
$ePf1 $ePf2 $ePf3 $ePf4 |
floating point values defining force points on the positive response envelope |
$ePd1 $ePd2 $ePd3 $ePd4 |
floating point values defining deformation points on the positive response envelope |
$eNf1 $eNf2 $eNf3 $eNf4 |
floating point values defining force points on the negative response envelope (optional, default: negative of positive envelope values) |
$eNd1 $eNd2 $eNd3 $eNd4 |
floating point values defining deformations points on the negative response envelope (optional, default: negative of positive envelope values) |
$rDispP |
floating point value defining the ratio of the deformation at which reloading occurs to the maximum historic deformation demand |
$rForceP |
floating point value defining the ratio of the force at which reloading begins to force corresponding to the maximum historic deformation demand |
$uForceP |
floating point value defining the ratio of strength developed upon unloading from negative load to the maximum strength developed under monotonic loading |
$rDispN |
floating point value defining the ratio of the deformation at which reloading occurs to the minimum historic deformation demand (optional, default: $rDispP) |
$rForceN |
floating point value defining the ratio of the force at which reloading begins to the force corresponding to the minimum historic deformation demand (optional, default: $rForceP) |
$uForceN |
floating point value defining the ratio of the strength developed upon unloading from a positive load to the minimum strength developed under monotonic loading (optional, default: $rForceP) |
$gK1 $gK2 $gK3 $gK4 $gKLim |
floating point values controlling cyclic degradation model for unloading stiffness degradation |
$gD1 $gD2 $gD3 $gD4 $gDLim |
floating point values controlling cyclic degradation model for reloading stiffness degradation |
$gF1 $gF2 $gF3 $gF4 $gFLim |
floating point values controlling cyclic degradation model for strength degradation |
$gE |
floating point value used to define maximum energy dissipation under cyclic loading. Total energy dissipation capacity is defined as this factor multiplied by the energy dissipated under monotonic loading. |
$dmgType |
string to indicate type of damage (option: "cycle", "energy") |
NOTE:
Figure 1: Definition of Pinching4 Uniaxial Material Model
Damage Models:
Stiffness and strength are assumed to deteriorate due to the imposed "load" history. The same basic equations are used to describe deterioration in strength, unloading stiffness and reloading stiffness:
where ki is the unloading stiffness at time ti, ko is the initial unloading stiffness (for the case of no damage), and
(defined below) is the value of the stiffness damage index at time ti.
where 
is the deformation demand that defines the end of the reload cycle for increasing deformation demand, 
is the maximum historic deformation demand (which would be the deformation demand defining the end of the reload cycle if degradation of reloading stiffness is ignored), and 
(defined below) is the value of reloading stiffness damage index at time ti.
where 
is the current envelope maximum strength at time ti, 
is the initial envelope maximum strength for the case of no damage, and 
(defined below) is the value of strength value index at time ti.
The damage indices, 
, 
and 
, may be defined to be a function of displacement history only ($dmgType = "cycle") or displacement history and energy accumulation ($dmgType = "energy"). For either case, all of the damage indices are computed using the same basic equation.
If the damage indices are assumed to be a function of displacement history and energy accumulation, the unloading stiffness damage index, 
is computed as follows:
where
with Emonotic equal to the energy required to achieve under monotonic loading the deformation that defines failure, defmax and defmin the positive and negative deformations that define failure. The other damage indices, 
and 
, are computed using the same equations with degradation model parameters gK* replaced by gF* and gD*, as is appropriate.
The above expressions were meant for "Energy" type damage. The user specification of "Energy" type damage implements damage due to displacement as well as energy. Other type of damage can be activated: "Cycle" which implements damage due to displacement as well as damage accrued due to load cycle counting. The expressions for the "Cycle" damage are given below.
If the damage indices are assumed to be a function only of the displacement history, the unloading stiffness damage index, 
is computed as follows:
where
with Cycle equal to the number of cycles accrued in the loading history, defmax and defmin the positive and negative deformations that define failure. The other damage indices, 
and 
, are computed using the same equations with degradation model parameters gK* replaced by gF* and gD*, as is appropriate.
EXAMPLE:
main input file:
supporting files:
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