Isotropic damage model for interfaces

The model provides an interface law which can be used to describe a damageable interface between two materials (e.g. between steel reinforcement and concrete). The law is formulated in terms of the traction vector and the displacement jump vector. The basic response is elastic, with stiffness kn in the normal direction and ks in the tangential direction. Similar to other isotropic damage models, this model assumes that the stiffness degradation is isotropic, i.e., both stiffness moduli decrease proportionally and independently of the loading direction. The damaged stiffnesses are kn $ \times(1-\omega)$ and ks $ \times(1-\omega)$ where $ \omega$ is a scalar damage variable. The damage evolution law is postulated in an explicit form, relating the damage variable $ \omega$ to the largest previously reached equivalent ``strain'' level, $ \kappa$.

The equivalent ``strain'', $ \tilde\varepsilon$, is a scalar measure of the displacement jump vector. The choice of the specific expression for the equivalent strain affects the shape of the elastic domain in the strain space and plays a similar role to the choice of a yield condition in plasticity. Currently, in the present implementation, the equivalent strain is given by

$\displaystyle \tilde\varepsilon = \sqrt{\langle w_n\rangle^2 + \beta w_s^2}

where $ \langle w_n\rangle$ is the positive part of the normal displacement jump (opening of the interface) and $ w_s$ is the norm of the tangential part of displacement jump (sliding of the interface). Parameter $ \beta$ is optional and its default value is 0, in which case damage depends on the opening only (not on the sliding). The dependence of damage $ \omega$ on maximum equivalent strain $ \kappa$ is described by the following damage law which corresponds to exponential softening:

$\displaystyle \omega = \left\{ \begin{array}{ll}
0 & \mbox{ for } \kappa\le \va...
...ilon_0 )}{G_f} \right) & \mbox{ for } \kappa> \varepsilon_0

Here, $ \varepsilon_0=f_t/k_n$ is the value of equivalent strain at the onset of damage. Note that if the interface is subjected to shear traction only (with zero or negative normal traction), the propagation of damage starts when the magnitude of the sliding displacement is $ \vert w_s\vert=\varepsilon_0/\sqrt{\beta}$, i.e., when the magnitude of the shear traction is equal to

$\displaystyle f_s = \frac{k_s \varepsilon_0}{\sqrt{\beta}} = f_t\frac{k_s }{k_n\sqrt{\beta}}

So the ratio between the shear strength and tensile strength of the interface, $ f_s/f_t$, is equal to $ k_s/k_n\sqrt{\beta}$.

The model parameters are summarized in Tab. 28.

Table 28: Isotropic damage model for interface elements - summary.
Description Isotropic damage model for concrete in tension
Record Format isointrfdm01 kn(rn) # ks(rn) # ft(rn) # gf(rn) # [ maxomega(rn) #] talpha(rn) # d(rn) #
Parameters - d material density
- tAlpha thermal dilatation coefficient
- kn elastic stifness in normal direction
- ks elastic stifness in tangential direction
- ft tensile strength
- gf fracture energy
- [maxomega] maximum damage, used for convergence improvement (its value is between 0 and 0.999999 (default), and it affects only the secant stiffness but not the stress)
- [beta] parameter controlling the effect of sliding part of displacement jump on equivalent strain, default value 0
Supported modes 2dInterface, 3dInterface

Borek Patzak