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  1. Material
  2. Material3D
  3. Hoffman

NonlinearHoffman

PreviousNonlinearHillNextTimberPD

Last updated 1 year ago

Orthotropic Hoffman Material

References

Theory

The NonlinearHoffman defines an orthotropic material using Hoffman yield criterion and associative plasticity.

The yield surface is defined as

with σ=[σ11 σ22 σ33 σ12 σ23 σ13]T\sigma=[\sigma_{11}~\sigma_{22}~\sigma_{33}~\sigma_{12}~\sigma_{23}~\sigma_{13}]^\mathrm{T}σ=[σ11​ σ22​ σ33​ σ12​ σ23​ σ13​]T is the stress, C1C_1C1​ to C9C_9C9​ are material constants. K(ϵˉp)K(\bar\epsilon_p)K(ϵˉp​) is the isotropic hardening function.

The constants are defined as follows.

C1=12(1σ11tσ11c+1σ22tσ22c−1σ33tσ33c),C2=12(1σ22tσ22c+1σ33tσ33c−1σ11tσ11c),C3=12(1σ33tσ33c+1σ11tσ11c−1σ22tσ22c),C4=1σ120σ120,C5=1σ230σ230,C6=1σ130σ130,C7=σ11c−σ11tσ11tσ11c,C8=σ22c−σ22tσ22tσ22c,C9=σ33c−σ33tσ33tσ33c.\begin{align*} C_1&=\dfrac{1}{2}(\dfrac{1}{\sigma_{11}^t\sigma_{11}^c}+\dfrac{1}{\sigma_{22}^t\sigma_ {22}^c}-\dfrac{1}{\sigma_{33}^t\sigma_{33}^c}),\\[4mm] C_2&=\dfrac{1}{2}(\dfrac{1}{\sigma_{22}^t\sigma_{22}^c}+\dfrac{1}{\sigma_{33}^t\sigma_{33}^c}-\dfrac{1}{\sigma_ {11}^t\sigma_{11}^c}),\\[4mm] C_3&=\dfrac{1}{2}(\dfrac{1}{\sigma_{33}^t\sigma_{33}^c}+\dfrac{1}{\sigma_{11}^t\sigma_{11}^c}-\dfrac{1}{\sigma_ {22}^t\sigma_{22}^c}),\\[4mm] C_4&=\dfrac{1}{\sigma_{12}^0\sigma_{12}^0},\quad{}C_5=\dfrac{1}{\sigma_{23}^0\sigma_{23}^0},\quad{}C_6=\dfrac{1}{\sigma_ {13}^0\sigma_{13}^0},\\[4mm] C_7&=\dfrac{\sigma_{11}^c-\sigma_{11}^t}{\sigma_{11}^t\sigma_{11}^c},\quad{}C_8=\dfrac{\sigma_{22}^c-\sigma_ {22}^t}{\sigma_{22}^t\sigma_{22}^c},\quad{}C_9=\dfrac{\sigma_{33}^c-\sigma_{33}^t}{\sigma_{33}^t\sigma_{33}^c}. \end{align*}C1​C2​C3​C4​C7​​=21​(σ11t​σ11c​1​+σ22t​σ22c​1​−σ33t​σ33c​1​),=21​(σ22t​σ22c​1​+σ33t​σ33c​1​−σ11t​σ11c​1​),=21​(σ33t​σ33c​1​+σ11t​σ11c​1​−σ22t​σ22c​1​),=σ120​σ120​1​,C5​=σ230​σ230​1​,C6​=σ130​σ130​1​,=σ11t​σ11c​σ11c​−σ11t​​,C8​=σ22t​σ22c​σ22c​−σ22t​​,C9​=σ33t​σ33c​σ33c​−σ33t​​.​

The Hoffman function allows different yield stresses for tension and compression. To recover the original Hill yield function, simply set σiit=σiic\sigma_{ii}^t=\sigma_{ii}^cσiit​=σiic​ for i=1, 2, 3i=1,~2,~3i=1, 2, 3.

The hardening function K(εˉp)K(\bar\varepsilon_p)K(εˉp​) can be user defined. It shall be noted that K(0)=1K(0)=1K(0)=1. The following method shall be implemented.

virtual double compute_k(double) const = 0;
virtual double compute_dk(double) const = 0;

History Layout

location
parameter

initial_history(0)

equivalent plastic strain

initial_history(1:7)

plastic strain

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