Material model for composite brittle damage

The model is designed for transversely isotropic elastic material defined by five elastic material constants. Axis $1$ represents the principal direction. The relation to orthotropic material reads

$$\nu_{12}=\nu_{13},~\nu_{21}=\nu_{31},~\nu_{23}=\nu_{32},~E_{22}=E_{33}$$ (46)

$$G_{12}=G_{13}=G_{21}=G_{31},G_{23}=G_{32}$$ (47)

$$\frac{\nu_{12}=\nu_{13}}{E_{11}} = \frac{\nu_{21}=\nu_{31}}{E_{22}},~\frac{\nu_{31}=\nu_{21}}{E_{33}} = \frac{\nu_{13}=\nu_{12}}{E_{11}}$$ (48)

Material orientation on finite element can be specified with mlcs optional parameter. If unspecified, material orientation is the same as the global coordinate system. This array contains six numbers, where first three numbers represent directional vector of local $x$-axis, and next three numbers represent directional vector of local $y$-axis with the reference to the global coordinate system. The composite material is extended to 1d and is suitable for trusses. In such particular case, only $xx$ components are considered from material definition.

The linear softening occurs after reaching a critical stress $f_t$ in mode I, see Fig. 7. Orientation of cracks is assumed to be orthogonal and aligned with orientation of material axis [7, pp.236]. The transverse isotropy is generally lost upon fracture, material becomes orthotropic and six damage parameters $d_1, d_2, \ldots d_6$ are introduced.

Figure 7: Implemented stress-strain evolution with damage for 1D case. Tension and compression are separated, but sharing the same damage parameter.

The compliance matrix, in the secant form and including damage parameters, reads

$$\left[ \begin{array}{cccccc} \frac{1-d_1}{E_{11}} & -\frac{\nu_{2... ...ac{1-d_5}{G_{31}} & 0\\ 0& 0&0 &0& 0& \frac{1-d_6}{G_{12}} \end{array} \right]$$ (49)

The evolution of damage in 3D space is based on the evolution of strain in corresponding 1D direction according to 7 and expressed as

$$(1-d_i)E_i = \frac{\sigma _i}{\mbox{\boldmath$\varepsilon$}_i}$$ (50)

$$d_i = 1-\frac{\sigma _i}{E_i\varepsilon _i}$$ (51)

$$\varepsilon _{0,i} - \varepsilon _{E,i} = \frac{2G_{f,i}}{f_{0,i} l_i}$$ (52)

$$\sigma _i = \frac{f_{0,i}(\varepsilon _{0,i}-\varepsilon _i)}{(\varepsilon _{... ...)} = f_{0,i} - \frac{f_{0,i}^2l_i}{2G_{f,i}}(\varepsilon _i-\varepsilon _{E,i})$$ (53)

where $\varepsilon _{0,i}$ is strain at zero stress, $f_{0,i}$ is maximum given stress, $\varepsilon _{E,i}$ is strain at maximum stress, $G_{f,i}$ is fracture energy disregarding the characteristic size of finite element, $l_i$ is the characteristic length associated with element size and interpolation order. The solution is similar to section  and proceeds in total strain and strees formulation. Fig. 8 shows a typical performance for damage in one direction.

Figure 8: Typical loading/unloading material performance for homogenized stress and strain in direction$_{11}$. Note that damage parameter is common for tension and compression.

Table 21: Brittle damage for composites - summary.

Description	Material for composite brittle damage
Record Format	compdammat num(in) # d(rn) # Exx(rn) # EyyEzz(rn) # nuxynuxz(rn) # nuyz(rn) # GxyGxz(rn) # [Tension_f0_Gf(rn) #] [Compres_f0_gf(rn) #]
Parameters	- num material model number
	- d material density
	- Exx Young's modulus for principal direction $xx$
	- EyyEzz Young's modulus in othogonal directions to the principal direction $xx$
	- nuxynuxz Poisson's ratio in $xy$ and $xz$ directions
	- nuyz Poisson's ratio in $yz$ direction
	- GxyGxz shear modulus in $xy$ and $xz$ directions
	- Tension_f0_Gf array with six pairs of positive numbers. Each pair desribes maximum stress in tension and fracture energy for each direction ($xx$, $yy$, $zz$, $yz$, $zx$, $xy$)
	- Compres_f0_gf array with six pairs of numbers. Each pair desribes maximum stress in compression (given as a negative number) and positive fracture energy for each direction ($xx$, $yy$, $zz$, $yz$, $zx$, $xy$)
Supported modes	3dMat, 1dMat