Cap mode

For the cap mode, an ellipsoid interface model is used. The yield condition is assumed as

$\displaystyle f_3($$\displaystyle \mbox{\boldmath$\sigma$}$$\displaystyle , \kappa_3) = C_{nn}\sigma^2+C_{ss}\tau^2 + C_n\sigma-\bar{\sigma}^2(\kappa_3)$ (65)

where $ C_{nn},\ C_{ss}$, and $ C_n$ are material model parameters and $ \bar{\sigma}$ is yield value, originally assumed in the following form of hardening/softening law [17]
$\displaystyle \bar{\sigma}_1(\kappa_3)$ $\displaystyle =$ $\displaystyle \bar{\sigma}_i+(\bar{\sigma}_p-\bar{\sigma}_i)\sqrt{\mbox{$\displ...
$\displaystyle \bar{\sigma}_2(\kappa_3)$ $\displaystyle =$ $\displaystyle \bar{\sigma}_p+(\bar{\sigma}_m-\bar{\sigma}_p)\left(\mbox{$\displ...
...a_3-\kappa_p}{\kappa_m-\kappa_p}$}\right)^2;\;\;\kappa_3\in(\kappa_p, \kappa_m)$ (66)
$\displaystyle \bar{\sigma}_3(\kappa_3)$ $\displaystyle =$ $\displaystyle \bar{\sigma}_r+(\bar{\sigma}_m-\bar{\sigma}_r)\exp\left(m\mbox{$\...
...pa_m}{\bar{\sigma}_m-\bar{\sigma}_r}$}\right);\;\;\kappa_3\in(\kappa_m, \infty)$  

with $ m=2(\bar{\sigma}_m-\bar{\sigma}_p)/(\kappa_m-\kappa_p)$. The hardening/softening law (66) is shown in fig.(5). Note that the curved diagram is a $ C^1$ continuous $ \sigma-\kappa_3$ relation. The energy under the load-displacement diagram can be related to a ``compressive fracture energy''. The original hardening law (66.1) exhibits indefinite slope for $ \kappa_3=0$, which can cause the problems with numerical implementation. This has been overcomed by replacing this hardening law with parabolic equation given by

$\displaystyle \bar{\sigma}_1(\kappa_3) = \bar{\sigma}_i-2*(\bar{\sigma}_i-\bar{\sigma}_p)*$$\displaystyle \mbox{$\displaystyle\frac{\kappa_3}{\kappa_p}$}$$\displaystyle +(\bar{\sigma}_i-\bar{\sigma}_p)$$\displaystyle \mbox{$\displaystyle\frac{\kappa_3}{\kappa_p}$}$ (67)

An associated flow and strain hardening hypothesis are being considered. This yields

$\displaystyle \dot\kappa_3=\lambda_3\sqrt{(2C_{nn}\sigma+C_n)*(2C_{nn}\sigma+C_n) + (2C_{ss}\tau)*(2C_{ss}\tau)}$ (68)

Figure 5: Hardening/softening law for cap mode

The model parameters are summarized in Tab. 17. There is one algorithmic issue, that follows from the model formulation. Since the cap mode hardening/softening is not coupled to hardening/softening of shear and tension modes the it may happen that when the cap and shear modes are activated, the return directions become parallel for both surfaces. This should be avoided by adjusting the input parameters accordingly (one can modify dilatancy angle, for example).

Table 17: Composite model for masonry - summary.
Description Composite plasticity model for masonry
Record Format Masonry02 num(in) # d(rn) # E(rn) # n(rn) # ft0(rn) # gfi(rn) # gfii(rn) # kn(rn) # ks(rn) # c0(rn) # tanfi0(rn) # tanfir(rn) # tanpsi(rn) # si(rn) # sp(rn) # sm(rn) # sr(rn) # kp(rn) # km(rn) # kr(rn) # cnn(rn) # css(rn) # cn(rn) #
Parameters - num material model number
  - d material density
  - E Young modulus
  - n Poisson ratio
  - ft0 tensile strength
  - gfi fracture energy for mode I
  - gfii fracture energy for mode II
  - kn joint elastic property
  - ks joint elastic property
  - c0 initial cohesion
  - tanfi0 initial friction angle
  - tanfir residual friction angle
  - tanpsi dilatancy
  - {si, sp, sm, sr} cap parameters $ \{\bar{\sigma}_i, \bar{\sigma}_p, \bar{\sigma}_m, \bar{\sigma}_r\}$
  - {kp, km,kr} cap parameters $ \{\kappa_p, \kappa_m, \kappa_r\}$
  - cnn,css,cn cap mode parametrs
Supported modes _2dInterface

Borek Patzak