/MAT/LAW72 (HILL_MMC)

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Example (Metal)

Material softening and failure are considered in the material example. Using the MMC parameters C₁, C₂, and C₂ the failure strain in a uniaxial tension test is calculated to be 0.98. In a uniaxial tension test with m=0.5, the material starts to soften at 0.98 until $0.98\times D_c$ is reached.

Figure 1.

#RADIOSS STARTER
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/UNIT/1
unit for mat
                   g                  mm                  ms
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#-  2. MATERIALS:
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/MAT/LAW72/1/1
Metal
#              RHO_I
              0.0028
#                  E                  nu
              200E+3                 0.3
#               Sig0                Eps0                   n                   F                   G
                1276             1.63E-3               0.265                 0.5                 0.5
#                  H                   N                   L                   M
                 0.5                 1.5                   0                   0
#                 C1                  C2                  C3                   m                  Dc
                0.12                 720               1.095                 0.5                 1.1
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
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Comments

1. 3D equivalent Hill stress:(1)
   $$f=\sqrt{F{\left({\sigma }_{\mathit{yy}}-{\sigma }_{\mathit{zz}}\right)}^2+G{\left({\sigma }_{\mathit{zz}}-{\sigma }_{\mathit{xx}}\right)}^2+H{\left({\sigma }_{\mathit{xx}}-{\sigma }_{\mathit{yy}}\right)}^2+\text{}2L{\sigma }_{\mathit{yz}}^2+2M{\sigma }_{\mathit{zx}}^2+2N{\sigma }_{\mathit{xy}}^2}$$
   For shell element, take:(2)

   $$f=\sqrt{F{\sigma }_{\mathit{yy}}^2+G{\sigma }_{\mathit{xx}}^2+H{\left({\sigma }_{\mathit{xx}}-{\sigma }_{\mathit{yy}}\right)}^2+2N{\sigma }_{\mathit{xy}}^2}$$
2. MMC fracture criteria:(3)
   $$D=\underset{0}{\overset{{\epsilon }_p}{\int }}\frac{d{\epsilon }_p}{{\epsilon }_f\left(\theta ,\eta \right)}$$
   • Anisotropic 3D model(4)
     $${\epsilon }_f\left(\theta ,\eta \right)={\left\{\frac{{\sigma }_y^0}{C_2}\left[C_3+\frac{\sqrt{3}}{2-\sqrt{3}}\left(1-C_3\right)\left(\sec \left(\frac{\theta \pi }{6}\right)-1\right)\right]\left[\sqrt{\frac{1+C_1^2}{3}\cos \left(\frac{\theta \pi }{6}\right)+C_1\left(\eta +\frac{1}{3}\right)\sin \left(\frac{\theta \pi }{6}\right)}\right]\right\}}^{-\frac{1}{n}}$$

     While,

     $\theta$ is Lode angle $\theta =1-\frac{2}{\pi }\mathit{ar}\hspace{0.17em}\cos \hspace{0.17em}\zeta$

     with Lode angle parameter $\zeta =\frac{27}{2}\frac{J_3}{{\sigma }_{\mathit{VM}}^3}$

     $J_3$ is the third invariant of the deviatoric stress.

     $\eta$ is stress triaxiality with $\eta =\frac{\frac{1}{3}\left({\sigma }_{\mathit{xx}}+{\sigma }_{\mathit{yy}}+{\sigma }_{\mathit{zz}}\right)}{{\sigma }_{\mathit{VM}}}$

   • 2D Anisotropic Model(5)
     $${\epsilon }_f\left(\theta ,\eta \right)={\left\{\frac{{\sigma }_y^0}{C_2}{f}_3\left[\left(\sqrt{\frac{1+C_1^2}{3}}{f}_1\right)+C_1\left(\eta +\frac{f_2}{3}\right)\right]\right\}}^{-\frac{1}{n}}$$
     with,(6)

     $$f_1=cos\left\{\frac{1}{3}\text{arcsin}\left[-\frac{27}{2}\eta \left({\eta }^2-\frac{1}{3}\right)\right]\right\}$$

     (7)

     $$f_2=sin\left\{\frac{1}{3}\text{arcsin}\left[-\frac{27}{2}\eta \left({\eta }^2-\frac{1}{3}\right)\right]\right\}$$

     (8)

     $$f_3=C_3+\frac{\sqrt{3}}{2-\sqrt{3}}\left(1-C_3\right)\left(\frac{1}{f_1}-1\right)$$
3. Fracture and damage with MMC fracture criteria:
   • When D = 1: fracture initiate
   • By 1 < D < D_c: the yield stress is multiplied by softening function $\beta$ to reduce the deformation resistance.

     ${\sigma }_y=\beta {\sigma }_y^0{\left({\epsilon }_p+{\epsilon }_p^0\right)}^n$ with $\beta ={\left(\frac{D_c-D}{D_c-1}\right)}^{m}0<\beta <1$

   • If D ≥ D_c, the element is deleted.
   • The exponent m is used to describe the softening behavior. It is recommended to use m > 0.

     If 0 < m < 1, then the softening curve is convex.

     If m > 1, then the softening curve is concave. The softening is between ${\epsilon }_f$ and $D_c\cdot {\epsilon }_f$ . Once the plastic strain is reached $D_c\cdot {\epsilon }_f$ (in this case $D>D_c$ ), then the element is deleted.

     
     
     Figure 2.

     
     
     Figure 3.

4. It is possible to display user variables in animation files (with Engine /ANIM/Eltyp/Restype) and in Time history file (with Starter /TH/SHEL and /TH/BRIC):
   • USER1: Plastic strain
   • USER2: Damage value