Drucker-Prager model with tension cut-off and isotropic damage - DruckerPragerCut

The Drucker-Prager plasticity model with tension cut-off is a multisurface model, appropriate for cohesive-frictional materials such as concrete loaded both in compression and tension. The plasticity model is formulated for isotropic hardening and enhanced by isotropic damage, which is driven by the cumulative plastic strain. The model can be used only in the small-strain context, with additive split of the strain tensor into the elastic and plastic parts.

The basic equations include the additive decomposition of strain into elastic and plastic parts,

$$\mbox{\boldmath$\varepsilon$} = \mbox{\boldmath$\varepsilon$} _{\rm {e}} + \mbox{\boldmath$\varepsilon$} _{\rm {p}},$$ (32)

the stress strain law

$$\mbox{\boldmath$\sigma$} = (1-\omega)\bar{\mbox{\boldmath$\sigma$}}=(1-\omega)\mbox{\boldmath$D$} :(\mbox{\boldmath$\varepsilon$}-\mbox{\boldmath$\varepsilon$}_{\rm {p}}),$$ (33)

the definition of the yield function in terms of the effective stress,

$$f \left(\bar{\mbox{\boldmath$\sigma$}},\kappa \right) = \alpha I_1 + \sqrt{J_2} - \tau_Y,$$ (34)

the flow rule

$$g \left( \bar{\mbox{\boldmath$\sigma$}} \right) = \alpha_{\psi}I_1 + \sqrt{J_2},$$ (35)

the linear hardening law

$$\tau_Y(\kappa) = \tau_0 + H\kappa,$$ (36)

where $\tau_0$ represents the initial yield stress under pure shear, the damage law

$$\omega(\kappa) = \omega_c(1- ^{-a\kappa}),$$ (37)

where $\omega_c$ is critical damage and $a$ is a positive dimensionless parameter. More detailed descriptioin of some parameters is in Section 1.4.1.

The dilatancy coefficient $\alpha_{\psi}$ controls flow associativeness; if $\alpha_{\psi}=\alpha$, an associate model is recovered, which overestimates the dilatancy of concrete. Hence, the dilatancy coefficient is usually chosen smaller, $\alpha_{\psi}\leq \alpha$, and the non-associated model is formulated.

Table 9: Drucker Prager material with tension cut-off - summary.

Description	Drucker Prager material with tension cut-off
Record Format	DruckerPragerCut num(in) # d(rn) # tAlpha(rn) # E(rn) # n(rn) # tau0(rn) # alpha(rn) # [alphaPsi(rn) #] [H(rn) #] [omega_crit(rn) #] [a(rn) #] [yieldtol(rn) #] [NewtonIter(in) #]
Parameters	- num material model number
	- d material density
	- tAlpha thermal dilatation coefficient
	- E Young modulus
	- n Poisson ratio
	- tau0 initial yield stress in shear $\tau_0$
	- alpha friction coefficient
	- alphaPsi dilatancy coefficient, equals to alpha by default
	- H hardening modulus (can be negative in the case of plastic softening), 0 by default
	- omega_crit critical damage in damage law (37), 0 by default
	- a exponent in damage law (37), 0 by default
	- yieldtol tolerance of the error in the yield criterion, default value 1.e-14
	- newtonIter maximum number of iterations in $\lambda$ search, default value 30
Supported modes	1dMat, 3dMat, PlaneStrain