/MAT/LAW74

Format

Definition

Comments

1. This material law must be used with property set /PROP/TYPE6 (SOL_ORTH), /PROP/TYPE14 (SOLID), /PROP/TYPE20 (TSHELL) or /PROP/TYPE21 (TSH_ORTH).
2. The yield stress is defined by a user function and the yield stress is compared to equivalent stress:(1)
   $${\sigma }_{\mathit{eq}}=\sqrt{F{\left({\sigma }_2-{\sigma }_3\right)}^2+G{\left({\sigma }_3-{\sigma }_1\right)}^2+H{\left({\sigma }_1-{\sigma }_2\right)}^2+2L{{\sigma }_{23}}^2+2M{{\sigma }_{31}}^2+2N{{\sigma }_{12}}^2}$$
   Where, HILL parameters are:(2)

   $$F=\frac{1}{2}\left(\frac{1}{{\sigma }_{22}^2}+\frac{1}{{\sigma }_{33}^2}-\frac{1}{{\sigma }_{11}^2}\right),G=\frac{1}{2}\left(\frac{1}{{\sigma }_{11}^2}+\frac{1}{{\sigma }_{33}^2}-\frac{1}{{\sigma }_{22}^2}\right),H=\frac{1}{2}\left(\frac{1}{{\sigma }_{11}^2}+\frac{1}{{\sigma }_{22}^2}-\frac{1}{{\sigma }_{33}^2}\right)$$

   (3)

   $$L=\frac{1}{2{\sigma }_{23}^2},M=\frac{1}{2{\sigma }_{31}^2},N=\frac{1}{2{\sigma }_{12}^2}$$

   Where, ${\sigma }_{11},{\sigma }_{22},{\sigma }_{33},{\sigma }_{12},{\sigma }_{23}$ and ${\sigma }_{31}$ represent the stress components either in the orthotropic frame if an orthotropic property is used, or in the orthogonalized isoparametric frame.

3. If ${\epsilon }_p$ (plastic strain) reaches ${\epsilon }_p^{max}$ , in one integration point, the solid element is deleted.
4. If largest principal strain ${\epsilon }_1>{\epsilon }_t$ , stress is reduced using the following relation:(4)
   $$\sigma =\sigma \left(\frac{{\epsilon }_m-{\epsilon }_1}{{\epsilon }_m-{\epsilon }_t}\right)$$
5. If ${\epsilon }_1>{\epsilon }_m$ , the stress is reduced to 0 (but the element is not deleted).
6. The table for yield stress definition can be 2-dimensional or 3-dimensional.
   • If the table is 2-dimensional, its parameters are assumed to represent respectively plastic strain and strain rate $\left({\epsilon }^p,\dot{\epsilon }\right)$ .

     Then if ${\epsilon }_{m-1}^p\le {\epsilon }^p\le {\epsilon }_m^p$ and ${\dot{\epsilon }}_{n-1}\le \dot{\epsilon }\le {\dot{\epsilon }}_n$ yield is linearly interpolated between the four values of the table corresponding to $\left({\epsilon }_i^p,{\dot{\epsilon }}_j\right),i=m-1,m;j=n-1,n$ .

   • If the table is 3-dimensional, its parameters are assumed to represent respectively plastic strain, strain rate, and temperature $\left({\epsilon }^p,\dot{\epsilon },T\right)$ .

     Then if ${\epsilon }_{m-1}^p\le {\epsilon }^p\le {\epsilon }_m^p$ and ${\dot{\epsilon }}_{n-1}\le \dot{\epsilon }\le {\dot{\epsilon }}_n$ and $T_{q-1}\le T\le T_q$ yield is linearly interpolated between the eight values of the table corresponding to $\left({\epsilon }_i^p,{\dot{\epsilon }}_j,T_k\right),i=m-1,m;j=n-1,n;k=q-1,q$ .

     If $\left({\epsilon }^p,\dot{\epsilon }\right)$ or $\left({\epsilon }^p,\dot{\epsilon },T\right)$ falls out of the range of the table, yield stress is obtained by linear extrapolation. Thus, it is necessary to input into the table the static curves corresponding to zero strain rate (entry $\dot{\epsilon }=0$ should belong to the table definition).

     Values of the table are yield stress values.

7. If yield stress also depends on temperature, the table is 3-dimensional:
   If the /HEAT/MAT option is not associated to the material identifier, adiabatic conditions are assumed and temperature is computed as:(5)

   $${\rm T}=T_i+\frac{E_{\mathit{int}}}{\rho C_p\left(\mathit{Volume}\right)}$$
   Where,

   E_int

   Internal energy computed by $\rho$ ,

   Volume

   Current density and volume

   C_p

   Heat capacity per mass unit

   Otherwise, the finite element formulation for heat transfer must be asked for (I_form =1 in option /HEAT/MAT); initial temperature and specific heat input in the option /HEAT/MAT will then be used.

8. The evolution of Young's modulus:
   • If fct_ID_E > 0, the curve defines a scale factor for Young's modulus evolution with equivalent plastic strain, which means the Young's modulus is scaled by the function $\mathrm{f}\left({\overline{\epsilon }}_p\right)$ :(6)
     $$E\left(t\right)=E\cdot \text{f}\left({\overline{\epsilon }}_p\right)$$

     The initial value of the scale factor should be equal to 1 and it decreases.

   • If fct_ID_E = 0, the Young's modulus is calculated as:(7)
     $$E\left(t\right)=E-\left(E-E_{\mathit{inf}}\right)\left[1-\exp \left(-C_E{\overline{\epsilon }}_p\right)\right]$$

     Where, E and E_inf are respectively, the initial and asymptotic value of Young's modulus, ${\overline{\epsilon }}_p$ accumulated equivalent plastic strain.

     Note: If fct_ID_E = 0 and C_E = 0, Young's modulus E is kept constant.