/MAT/LAW122 (MODIFIED_LADEVEZE)

$\left\{\begin{array}{l}{\sigma }_{xx}=C_{11}{\epsilon }_{xx}+C_{12}{\epsilon }_{xy}\\ {\sigma }_{yy}=C_{21}{\epsilon }_{xx}+C_{22}{\epsilon }_{xy}\\ {\sigma }_{xy}=G_{12}{\epsilon }_{xy}\\ {\sigma }_{yz}=\kappa G_{23}{\epsilon }_{yz}\\ {\sigma }_{zx}=\kappa G_{13}{\epsilon }_{zx}\end{array}\right.$ with $C=\frac{1}{1-{\nu }_{12}{\nu }_{21}}\left[\begin{array}{cc}{E}_1& {\nu }_{12}{E}_2\\ {\nu }_{21}{E}_1& E_2\end{array}\right]$

Where, $E_1=\left\{\begin{array}{ccc}{E}_1^T& \text{if}& {\epsilon }_{xx}\ge 0\\ \frac{E_1^C}{1+\gamma E_1^C\left|{\epsilon }_{xx}\right|}& \text{if}& {\epsilon }_{xx}<0\end{array}\right.$ .

This nonlinear evolution of compression Young’s modulus in fiber direction is used to represent the effect of fiber micro buckling and misalignment. $\kappa$ is the shear factor used for shells only and defined in the property.

The same non-linearity of fiber direction Young’s modulus in compression is used.

• For Shells:(2)
  $$f=\sqrt{{\sigma }_{xy}^2+A{\sigma }_{yy}^2}-{\sigma }_Y$$
• For Solids:(3)
  $$f=\sqrt{{\sigma }_{xy}^2+{\sigma }_{yz}^2+{\sigma }_{zy}^2+A\left({\sigma }_{yy}^2+{\sigma }_{zz}^2\right)}-{\sigma }_Y$$

This describes an isotropic hardening following a power law. The hardening modulus $\beta$ is numerically bounded by the value $\max \left(E,2G_{12}\right)$ to avoid stability issues.

• Fiber damage $d_f$ affects the behavior along the fiber direction 1. Under tension loading condition, fiber damage evolves following the equations.(5)
  $$d_f=\left\{\begin{array}{ccc}0& \text{if}& {\epsilon }_f^{eq}\le {\epsilon }_f^{ti}\\ d_f^{tu}\frac{{\epsilon }_f^{eq}-{\epsilon }_f^{ti}}{{\epsilon }_f^{tu}-{\epsilon }_f^{ti}}& \text{if}& {\epsilon }_f^{ti}<{\epsilon }_f^{eq}\le {\epsilon }_f^{tu}\\ 1-\left(1-d_f^{tu}\right)\frac{{\epsilon }_f^{tu}}{{\epsilon }_f^{eq}}& \text{if}& {\epsilon }_f^{eq}>{\epsilon }_f^{tu}\end{array}\right.$$
  Where, ${\epsilon }_f^{ti}$ is the strain at the onset of damage, ${\epsilon }_f^{tu}$ is the ultimate strain, $d_f^{tu}$ is the ultimate damage value and ${\epsilon }_f^{eq}$ is the equivalent fiber strain defined by:

  • For Shells:(6)
    $${\epsilon }_f^{eq}={\epsilon }_{xx}+{\nu }_{21}{\epsilon }_{yy}$$
  • For Solids:(7)
    $${\epsilon }_f^{eq}=\left(1-{\nu }_{23}{\nu }_{32}\right){\epsilon }_{xx}+\left({\nu }_{23}{\nu }_{31}+{\nu }_{21}\right){\epsilon }_{yy}+\left({\nu }_{21}{\nu }_{32}+{\nu }_{31}\right){\epsilon }_{zz}$$
  The compression damage on fiber due to matrix buckling can be activated with the IBUCK flag and is described with a similar equation:(8)

  $$d_f=\left\{\begin{array}{ccc}0& \text{if}& \left|{\epsilon }_f^{eq}\right|\le {\epsilon }_f^{ci}\\ d_f^{cu}\frac{\left|{\epsilon }_f^{eq}\right|-{\epsilon }_f^{ci}}{{\epsilon }_f^{cu}-{\epsilon }_f^{ci}}& \text{if}& {\epsilon }_f^{ci}<\left|{\epsilon }_f^{eq}\right|\le {\epsilon }_f^{cu}\\ 1-\left(1-d_f^{cu}\right)\frac{{\epsilon }_f^{cu}}{\left|{\epsilon }_f^{eq}\right|}& \text{if}& \left|{\epsilon }_f^{eq}\right|>{\epsilon }_f^{cu}\end{array}\right.$$
  This fiber damage then affects the stress computation as:(9)

  $${\sigma }_{xx}^{dam}=\left(1-d_f\right){\sigma }_{xx}$$
  Figure 2 shows the expected behavior in tension/compression along the fiber direction. The dashed line allows to highlight the nonlinear Young’s modulus in compression.

  
  
  Figure 2. Tension/compression test in fiber direction showing fiber damage effect on stress

  Note: Along fiber direction, the behavior is purely elastic and damaging.
• Shear matrix damage $d$ is introduced to represent the debonding between matrix and fibers. Its evolution is influenced by the energy release rate often used in Lemaitre-type damage model. Two elastic energy rates are considered in this model.
  • For Shells:(10)
    $$\begin{array}{l}{Z}_d=\frac{1}{2}\left(\frac{{\sigma }_{12}^2}{G_{12}}+\frac{{\sigma }_{13}^2}{G_{13}}\right)\\ Z_d^{\text{'}}=\frac{1}{2}\left(\frac{{〈{\sigma }_{22}〉}_{+}^2}{E_2}\right)\end{array}$$

    Where, ${〈〉}_{+}$ are the Macauley’s brackets that only considers the positive values of ${\sigma }_{22}$ . However, if compression damage described below is considered (for shells only), these brackets becomes simple parenthesis.

  • For Solids:(11)
    $$\begin{array}{l}{Z}_d=\frac{1}{2}\left(\frac{{\sigma }_{12}^2}{G_{12}}+\frac{{\sigma }_{23}^2}{G_{23}}+\frac{{\sigma }_{13}^2}{G_{13}}\right)\\ Z_d^{\text{'}}=\frac{1}{2}\left(\frac{{〈{\sigma }_{22}〉}_{+}^2}{E_2}+\frac{{〈{\sigma }_{33}〉}_{+}^2}{E_3}\right)\end{array}$$
    This then leads to the following computation:(12)

    $$Y=Su{p}_{t\le \tau }\sqrt{Z_d+b{Z}_d^{\text{'}}}$$

    Where, $b$ is a coupling factor.

  Depending on the value of the flag ISH, the shear matrix damage can evolve with different shapes.

  • ISH = 1: linear shape (Figure 3)
    $d=\left\{\begin{array}{ccc}0& \text{if}& Y(t)\le Y_0\\ \frac{Y(t)-Y_0}{Y_C}& \text{if}& d<d_{MAX},Y(t)<Y_S,\\ 1-(1-d_{MAX})\frac{Y(t-\Delta t)}{Y(t)}& \text{otherwise}& \end{array}\right.Y(t)<Y_R$

    
    
    Figure 3. Shear test showing shear matrix damage effect with a linear shape

  • ISH = 2: exponential shape (Figure 4)
    $d=\left\{\begin{array}{ccc}{d}_{sat1}\left(1-\exp \left(\frac{Y_0-Y\left(t\right)}{Y_C}\right)\right)& \text{if}& Y\left(t\right)>Y_0\\ 0& \text{otherwise}& \end{array}\right.$

    
    
    Figure 4. Shear test showing shear matrix damage effect with an exponential shape

  • ISH = 3: tabulated shape (Figure 5)

    $d=f_{D1}\left(\frac{Y\left(t\right)}{Y_0}\right)$

    Where, $f_{D1}$ is the function identified by IFUNCD1.

    
    
    Figure 5. Shear test showing shear matrix damage effect with a tabulated shape

  The shear matrix damage then affects the stresses computation as:

  • For Shells:(13)
    $$\begin{array}{l}{\sigma }_{xy}^{dam}=\left(1-d\right){\sigma }_{xy}\\ {\sigma }_{yz}^{dam}=\min \left(1-d,1-d\text{'}\right){\sigma }_{yz}\\ {\sigma }_{zx}^{dam}=\min \left(1-d,1-d\text{'}\right){\sigma }_{zx}\end{array}$$
  • For Solids:(14)
    $$\begin{array}{l}{\sigma }_{xy}^{dam}=\left(1-d\right){\sigma }_{xy}\\ {\sigma }_{yz}^{dam}=\left(1-d\right){\sigma }_{yz}\\ {\sigma }_{zx}^{dam}=\left(1-d\right){\sigma }_{zx}\end{array}$$
• Transverse matrix damage $d\text{'}$ allows to represent the microcracking of the matrix. Its evolution is very similar to shear matrix damage, except from the fact that a different elastic energy release rate is used:(15)
  $$Y^{\text{'}}=Su{p}_{t\le \tau }\sqrt{Z_d^{\text{'}}}$$
  Then, like shear matrix damage, three different evolution shapes are available depending on the ITR flag value.

  • ITR = 1: linear shape (Figure 6)
    $d\text{'}=\left\{\begin{array}{ccc}0& \text{if}& Y\text{'}(t)\le Y_0\text{'}\\ \frac{Y\text{'}(t)-Y_0\text{'}}{Y_C\text{'}}& \text{if}& d<d_{MAX},Y\text{'}(t)<Y_S,\\ 1-(1-d_{MAX})\frac{Y\text{'}(t-\Delta t)}{Y\text{'}(t)}& \text{otherwise}& \end{array}\right.Y\text{'}(t)<Y_R$

    
    
    Figure 6. Tensile test in transverse direction showing transverse matrix damage effect with a linear shape

  • ITR = 2: exponential shape (Figure 7)
    $d\text{'}=\left\{\begin{array}{ccc}{d}_{sat2}\left(1-\exp \left(\frac{Y_0\text{'}-Y\text{'}\left(t\right)}{Y_C\text{'}}\right)\right)& \text{if}& Y\text{'}\left(t\right)>Y_0\text{'}\\ 0& \text{otherwise}& \end{array}\right.$

    
    
    Figure 7. Tensile test in transverse direction showing transverse matrix damage effect with an exponential shape

  • ITR = 3: tabulated shape (Figure 8)

    $d\text{'}=f_{D2}\left(\frac{Y\text{'}\left(t\right)}{Y_0\text{'}}\right)$

    Where, $f_{D2}$ is the function identified by IFUNCD2.

    
    
    Figure 8. Tensile test in transverse direction showing transverse matrix damage effect with a tabulated shape

    This damage variable is supposed to occur in tension only. In compression, the microcracks of the matrix are assumed too close to recover the initial undamaged stiffness (Figure 9). However, for shells only, a specific transverse matrix damage evolution in compression can be described the same way using parameters: $Y_{0C}\text{'}$ , $Y_{CC}\text{'}$ , $d_{sat2C}$ or IFUNCD2C.

    
    
    Figure 9. Tension/compression test in transverse direction

    This last damage variable affects the stress computation with:(16)

    $$\begin{array}{l}{\sigma }_{yy}^{dam}=\left\{\begin{array}{ccc}\left(1-d\text{'}\right){\sigma }_{yy}& \text{if}& {\epsilon }_{yy}\ge 0\\ {\sigma }_{yy}& \text{if}& {\epsilon }_{yy}<0\end{array}\right.\\ {\sigma }_{zz}^{dam}=\left\{\begin{array}{ccc}\left(1-d\text{'}\right){\sigma }_{zz}& \text{if}& {\epsilon }_{zz}\ge 0\\ {\sigma }_{zz}& \text{if}& {\epsilon }_{zz}<0\end{array}\right.\end{array}$$

• In fiber direction, the viscosity affects the Young’s modulus through the introduction of a rate factor denoted $F_{11}$ :(17)
  $$E_1^{vis}=E_1\left(1+F_{11}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{11}}\right)\right)$$

  Where, $\dot{\epsilon }$ is the equivalent strain rate and ${\dot{\epsilon }}_{11}$ is the reference strain rate in direction 1.

  The rate factor equation can take different shape depending on the value of the flag LTYPE11:

  • LTYPE11 = 1: power law (18)
    $$F_{11}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{11}}\right)=D_{11}{\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{11}}\right)}^{n_{11}}$$
  • LTYPE11 = 2: linear law(19)
    $$F_{11}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{11}}\right)=D_{11}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{11}}\right)+n_{11}$$
  • LTYPE11 = 3: logarithmic law(20)
    $$F_{11}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{11}}\right)=D_{11}{〈\ln \left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{11}}\right)〉}_{+}+\log n_{11}$$
  • LTYPE11 = 4: tangent hyperbolic law(21)
    $$F_{11}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{11}}\right)=D_{11}\tanh \left(n_{11}{〈\dot{\epsilon }-{\dot{\epsilon }}_{11}〉}_{+}\right)$$
  The fibers failure can also be affected by strain rate through the introduction of the factor, $F_{11R}$ whose evolution will also depend on the flag LTYPE11 using parameters $D_{11R}$ and $n_{11R}$ :(22)

  $$\begin{array}{l}{\epsilon }_f^{ti}{}^{vis}={\epsilon }_f^{ti}\left(1+F_{11R}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{11}}\right)\right)\\ {\epsilon }_f^{tu}{}^{vis}={\epsilon }_f^{tu}\left(1+F_{11R}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{11}}\right)\right)\\ {\epsilon }_f^{ci}{}^{vis}={\epsilon }_f^{ci}\left(1+F_{11R}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{11}}\right)\right)\\ {\epsilon }_f^{cu}{}^{vis}={\epsilon }_f^{cu}\left(1+F_{11R}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{11}}\right)\right)\end{array}$$
  The expected behavior is detailed below in Figure 10.

  
  
  Figure 10. Strain rate effect on fibers direction behavior

• In matrix direction, the shear and transverse behavior are affected by strain rate, as well. The elasticity is then modified with the introduction of the factors $F_{22}$ and $F_{12}$ as follows:(23)
  $$\begin{array}{l}{E}_2^{vis}=E_2\left(1+F_{22}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{12}}\right)\right)\\ E_3^{vis}=E_3\left(1+F_{22}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{12}}\right)\right)\\ G_{12}^{vis}=G_{12}\left(1+F_{12}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{12}}\right)\right)\\ G_{23}^{vis}=G_{23}\left(1+F_{12}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{12}}\right)\right)\\ G_{13}^{vis}=G_{13}\left(1+F_{12}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{12}}\right)\right)\end{array}$$

  You can notice that the two factors are using the same strain rate reference value ${\dot{\epsilon }}_{12}$ and that $E_3$ , $G_{23}$ and $G_{13}$ are only modified for solids. The shape of the factors $F_{22}$ and $F_{12}$ will be fixed by the flag LTYPE12, and will depend respectively on the values of $D_{22}$ , $n_{22}$ and $D_{12}$ , $n_{12}$ .

  The fracture energy is increased with strain rate, as well, using the same factors:(24)

  $$\begin{array}{l}{Y}_0^{vis}=Y_0\left(1+F_{12}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{12}}\right)\right)\\ Y_C^{vis}=Y_C\left(1+F_{12}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{12}}\right)\right)\\ Y_0{\text{'}}^{vis}=Y_0\text{'}\left(1+F_{22}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{12}}\right)\right)\\ Y_C{\text{'}}^{vis}=Y_C\text{'}\left(1+F_{22}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{12}}\right)\right)\\ Y_{0C}{\text{'}}^{vis}=Y_{0C}\text{'}\left(1+F_{22}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{12}}\right)\right)\\ Y_{CC}{\text{'}}^{vis}=Y_{CC}\text{'}\left(1+F_{22}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{12}}\right)\right)\end{array}$$

  Note: The compression damage parameter for transverse directions $Y_{0C}\text{'}$ and $Y_{CC}\text{'}$ are only modified for shells.
• Finally, the last parameter affected by viscous effect is the initial yield stress, with the use of factor $F_{R0}$ .(25)
  $${\sigma }_{Y0}^{vis}={\sigma }_{Y0}\left(1+F_{R0}\left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{R0}}\right)\right)$$

  Similarly, the shape of factor $F_{R0}$ is dictated by the flag LTYPER0 using parameters $D_{R0}$ and $n_{R0}$ .

  The expected behavior for transverse matrix direction (which is similar for tension and shear) is detailed below in Figure 11.

  
  
  Figure 11. Strain rate effect on matrix transverse (or shear) direction behavior