ABAQUS材料子程序学习（线性各向同性硬化塑性）

  • 前言
    • 塑性力学增量形式实现
    • umat子程序
    • 参数
    • 计算结果

前言

记录自己学习abaqus软件umat子程序的t过程，本文主要参考了《非线性本构关系在ABAQUS中的实现》第四章和技术邻的视频课程“非线性各向同性强化弹塑性umat子程序教程”

塑性力学增量形式实现

计算过程中，体应变和体应力是弹性关系 Δ σ v = K ⋅ Δ ε v (1) {Δσ_v}={K}\cdot{Δε_v}\tag{1} Δσv​=K⋅Δεv​(1)

K K K为体积模量， K = E 3 ( 1 − ν ) K=\cfrac{E}{3(1-ν)} K=3(1−ν)E​ , Δ ε v = Δ ε 11 + Δ ε 22 + Δ ε 33 Δε_v=Δε_{11}+Δε_{22}+Δε_{33} Δεv​=Δε11​+Δε22​+Δε33​

所以在下面的讨论中只考虑偏应力张量和偏应力张量

试应力： σ t r ′ ( t ) = σ ′ ( t ) + C ′ : Δ ε ′ (2) \bm{σ^{tr'}(t)=σ'(t)+\mathbb{C'}:Δε'}\tag{2} σtr′(t)=σ′(t)+C′:Δε′(2)

viogt表记：
{ σ 11 t r ′ σ 22 t r ′ σ 33 t r ′ σ 12 t r ′ σ 23 t r ′ σ 13 t r ′ } = { σ ′ 11 ( t ) σ ′ 22 ( t ) σ ′ 33 ( t ) σ ′ 12 ( t ) σ ′ 23 ( t ) σ ′ 13 ( t ) } + [ 2 G 0 0 0 0 0 0 2 G 0 0 0 0 0 0 2 G 0 0 0 0 0 0 G 0 0 0 0 0 0 G 0 0 0 0 0 0 G ] ⋅ { Δ ε ′ 11 ( t ) Δ ε ′ 22 ( t ) Δ ε ′ 33 ( t ) Δ γ 12 ( t ) Δ γ 23 ( t ) Δ γ 13 ( t ) } \left \{ \begin{matrix} σ_{11}^{tr'}\\ σ_{22}^{tr'}\\ σ_{33}^{tr'}\\ σ_{12}^{tr'}\\ σ_{23}^{tr'}\\ σ_{13}^{tr'}\end{matrix} \right \} =\left \{ \begin{matrix} {σ'}_{11}(t)\\ {σ'}_{22}(t)\\{σ'}_{33}(t)\\{σ'}_{12}(t)\\{σ'}_{23}(t)\\{σ'}_{13}(t)\end{matrix} \right \} +\left [\begin{matrix} 2G&0&0&0&0&0 \\ 0&2G&0&0&0&0 \\ 0&0& 2G&0&0&0 \\ 0&0&0& G&0&0 \\ 0&0&0&0& G&0 \\ 0&0&0&0&0 &G \end{matrix} \right ] \cdot \left \{ \begin{matrix} {Δε'}_{11}(t)\\ {Δε'}_{22}(t)\\{Δε'}_{33}(t)\\{Δγ}_{12}(t)\\{Δγ}_{23}(t)\\{Δγ}_{13}(t)\end{matrix} \right \} ⎩⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎧​σ11tr′​σ22tr′​σ33tr′​σ12tr′​σ23tr′​σ13tr′​​⎭⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎫​=⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧​σ′11​(t)σ′22​(t)σ′33​(t)σ′12​(t)σ′23​(t)σ′13​(t)​⎭⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎫​+⎣⎢⎢⎢⎢⎢⎢⎡​2G00000​02G0000​002G000​000G00​0000G0​00000G​⎦⎥⎥⎥⎥⎥⎥⎤​⋅⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧​Δε′11​(t)Δε′22​(t)Δε′33​(t)Δγ12​(t)Δγ23​(t)Δγ13​(t)​⎭⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎫​
静态分析umat用的工程应变： { Δ ε ′ 11 ( t ) Δ ε ′ 22 ( t ) Δ ε ′ 33 ( t ) Δ γ 12 ( t ) Δ γ 23 ( t ) Δ γ 13 ( t ) } = { Δ ε ′ 11 ( t ) Δ ε ′ 22 ( t ) Δ ε ′ 33 ( t ) 2 Δ ε ′ 12 ( t ) 2 Δ ε ′ 23 ( t ) 2 Δ ε ′ 13 ( t ) } \left \{ \begin{matrix} {Δε'}_{11}(t)\\ {Δε'}_{22}(t)\\{Δε'}_{33}(t)\\{Δγ}_{12}(t)\\{Δγ}_{23}(t)\\{Δγ}_{13}(t)\end{matrix} \right \} = \left \{ \begin{matrix} {Δε'}_{11}(t)\\ {Δε'}_{22}(t)\\{Δε'}_{33}(t)\\{2Δε'}_{12}(t)\\{2Δε'}_{23}(t)\\{2Δε'}_{13}(t)\end{matrix} \right \} ⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧​Δε′11​(t)Δε′22​(t)Δε′33​(t)Δγ12​(t)Δγ23​(t)Δγ13​(t)​⎭⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎫​=⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧​Δε′11​(t)Δε′22​(t)Δε′33​(t)2Δε′12​(t)2Δε′23​(t)2Δε′13​(t)​⎭⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎫​

Mises等效应力：

主应力形式：
σ e = { 1 2 [ ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 ] } 1 2 {σ_e}=\left \{ \frac{1}{2} \left [ \left (\sigma_1- \sigma_2 \right )^2+ \left (\sigma_2- \sigma_3 \right )^2+\left (\sigma_3- \sigma_1 \right )^2 \right ] \right \}^{\frac{1}{2}} σe​={21​[(σ1​−σ2​)2+(σ2​−σ3​)2+(σ3​−σ1​)2]}21​

偏应力形式:
σ e = [ 3 2 ( σ 11 ′ 2 + σ 22 ′ 2 + σ 33 ′ 2 + 2 σ 12 2 + 2 σ 23 2 + 2 σ 13 2 ) ] 1 2 (3) {σ_e}= \left [{\frac{3}{2}} \left ({\sigma'_{11}}^2+ {\sigma'_{22}}^2+{\sigma'_{33}}^2+2{\sigma_{12}}^2+2{\sigma_{23}}^2+2{\sigma_{13}}^2\right )\right ]^{\frac{1}{2}}\tag{3} σe​=[23​(σ11′​2+σ22′​2+σ33′​2+2σ12​2+2σ23​2+2σ13​2)]21​(3)

其中: σ 11 ′ = σ 11 − σ v \sigma'_{11}=\sigma_{11}-\sigma_{v} σ11′​=σ11​−σv​ ， σ 22 ′ = σ 22 − σ v \sigma'_{22}=\sigma_{22}-\sigma_{v} σ22′​=σ22​−σv​， σ 33 ′ = σ 33 − σ v \sigma'_{33}=\sigma_{33}-\sigma_{v} σ33′​=σ33​−σv​； σ v = 1 3 [ σ 11 + σ 22 + σ 33 ] \sigma_{v}=\frac{1}{3} \left [\sigma_{11}+\sigma_{22}+\sigma_{33} \right] σv​=31​[σ11​+σ22​+σ33​]

Mises等效塑性应变增量：
Δ ε ˉ p = 2 3 [ ( Δ ε 1 p − Δ ε 2 p ) 2 + ( Δ ε 2 p − Δ ε 3 p ) 2 + ( Δ ε 3 p − Δ ε 1 p ) 2 ] 1 2 Δ\bar{ε}^p=\frac{\sqrt{2}}{3} \left[ \left(Δε_{1}^p-Δε_{2}^p \right)^2 +\left(Δε_{2}^p-Δε_{3}^p \right)^2 +\left(Δε_{3}^p-Δε_{1}^p \right)^2\right]^{\frac{1}{2}} Δεˉp=32 ​​[(Δε1p​−Δε2p​)2+(Δε2p​−Δε3p​)2+(Δε3p​−Δε1p​)2]21​

线性硬化塑性 σ y = σ y 0 + h ε ˉ p (4) {σ_y}={σ_{y0}}+h\bar{ε}^p\tag{4} σy​=σy0​+hεˉp(4)
初始屈服加上等效塑性应变乘以硬化系数。
等效塑性变形作为状态变量存在STATEV(NSTATV)中，提取 ε ˉ p \bar{ε}^p εˉp 并计算当前的屈服应力 σ y {σ_y} σy​

(2)代入(3)，计算试应力的等效Mises应力 σ M i s e s t r {σ}_{Mises}^{tr} σMisestr​

判断 σ M i s e s t r {σ}_{Mises}^{tr} σMisestr​ 与 σ y {σ_y} σy​ 关系:

若 σ M i s e s t r ＜ σ y {σ}_{Mises}^{tr}＜{σ_y} σMisestr​＜σy​ ： σ ( t + Δ t ) = σ t r ( t ) \bm{σ(t+Δt)}=\bm{σ^{tr}(t)} σ(t+Δt)=σtr(t)

一致切线刚度矩阵，为弹性刚度矩阵
D D S D D E ( i , j ) = [ 2 G + λ λ λ 0 0 0 λ 2 G + λ λ 0 0 0 λ λ 2 G + λ 0 0 0 0 0 0 G 0 0 0 0 0 0 G 0 0 0 0 0 0 G ] (5) \bm{DDSDDE}(i,j) =\left [\begin{matrix} 2G+λ&λ&λ&0&0&0 \\ λ&2G+λ&λ&0&0&0 \\ λ&λ& 2G+λ&0&0&0 \\ 0&0&0& G&0&0 \\ 0&0&0&0& G&0 \\ 0&0&0&0&0 &G \end{matrix} \right ]\tag{5} DDSDDE(i,j)=⎣⎢⎢⎢⎢⎢⎢⎡​2G+λλλ000​λ2G+λλ000​λλ2G+λ000​000G00​0000G0​00000G​⎦⎥⎥⎥⎥⎥⎥⎤​(5)

若 σ M i s e s t r ≥ σ y {σ}_{Mises}^{tr}≥{σ_y} σMisestr​≥σy​ ，计算新增的 Δ ε ˉ p Δ\bar{ε}^p Δεˉp

注：单轴拉伸下Mises等效应力 σ M i s e s = [ 1 2 ( σ 11 2 + σ 11 2 ) ] 1 2 = σ 11 {σ}_{Mises}=\left[ \frac{1}{2} \left( {σ}_{11}^2+ {σ}_{11}^2\right) \right]^{\frac{1}{2}}={σ}_{11} σMises​=[21​(σ112​+σ112​)]21​=σ11​
等效应变 ε M i s e s = 2 3 [ 2 ( ε 11 − ( − μ ε 11 ) ) 2 ] 1 2 = 2 3 ( 1 + μ ) ε 11 {ε}_{Mises}=\frac{\sqrt{2}}{3} \left[ 2\left( {ε}_{11}-(-μ{ε}_{11}) \right)^2 \right]^{\frac{1}{2}}={\frac{2}{3}}(1+μ){ε}_{11} εMises​=32 ​​[2(ε11​−(−με11​))2]21​=32​(1+μ)ε11​
单轴拉伸： σ 11 = E ⋅ ε 11 {σ}_{11}=E·{ε}_{11} σ11​=E⋅ε11​ → σ M i s e s = 3 E 2 ( 1 + μ ) ⋅ ε M i s e s = 3 G ⋅ ε M i s e s {σ}_{Mises}=\frac{3E}{2(1+μ)}·{ε}_{Mises}=3G·{ε}_{Mises} σMises​=2(1+μ)3E​⋅εMises​=3G⋅εMises​

计算新增的 Δ ε ˉ p Δ\bar{ε}^p Δεˉp, h h h为塑性硬化系数
σ M i s e s t r − 3 G Δ ε ˉ p = σ y 0 + h ⋅ ( ε ˉ p + Δ ε ˉ p ) {σ}_{Mises}^{tr}-3GΔ\bar{ε}^p=σ_{y0}+h·(\bar{ε}^p+Δ\bar{ε}^p) σMisestr​−3GΔεˉp=σy0​+h⋅(εˉp+Δεˉp)
σ M i s e s t r − h ⋅ ε ˉ p − σ y 0 = ( h + 3 G ) Δ ε ˉ p {σ}_{Mises}^{tr}- h·\bar{ε}^p-σ_{y0}=(h+3G)Δ\bar{ε}^p σMisestr​−h⋅εˉp−σy0​=(h+3G)Δεˉp
Δ ε ˉ p = σ M i s e s t r − h ⋅ ε ˉ p − σ y 0 ( h + 3 G ) Δ\bar{ε}^p=\frac{{σ}_{Mises}^{tr}- h·\bar{ε}^p-σ_{y0}}{(h+3G)} Δεˉp=(h+3G)σMisestr​−h⋅εˉp−σy0​​
非线性硬化时 h h h 是随等效塑性应变 ε ˉ p \bar{ε}^p εˉp变化的，通常利用Newton迭代法求解等效塑性应变增量 Δ ε ˉ p Δ\bar{ε}^p Δεˉp

根据正交流动法则：
Δ ε p = Δ ε ˉ p 3 2 σ ′ σ M i s e s = Δ ε ˉ p 3 2 σ t r ′ σ M i s e s t r \bm{Δε^p}=Δ\bar{ε}^p\frac{3}{2}\frac{\bm{σ'}}{σ_{Mises}}=Δ\bar{ε}^p\frac{3}{2}\frac{\bm{σ^{tr'}}}{σ_{Mises}^{tr}} Δεp=Δεˉp23​σMises​σ′​=Δεˉp23​σMisestr​σtr′​
偏应力张量与试应力的偏应力张量方向相同，如下图

屈服后一致切线刚度矩阵：
应力增量=弹性刚度矩阵×弹性应变增量
d σ = D e ⋅ d ε e \bm{dσ=D_e·dε^e} dσ=De​⋅dεe
弹性应变=总应变-塑性应变增量
d σ = D e ⋅ ( d ε − d ε p ) (6) \bm{dσ=D_e·(dε-dε^p)}\tag{6} dσ=De​⋅(dε−dεp)(6)
将 d ε p \bm{dε^p} dεp 用 d ε \bm{dε} dε 表示出来，再代回式(6)就可求出屈服后的DDSDDE(NTENS,NTENS)
一致性条件
{ f ( σ , ε p ˉ ) = 0 f ( σ + d σ , ε p ˉ + d ε p ˉ ) = 0 ⟹ ∂ f ∂ σ ⋅ d σ + ∂ f ∂ ε p ˉ d ε p ˉ = 0 (7) \begin{cases} f(\bm{σ},\bar{ε^p})=0 \\ f(\bm{σ}+d\bm{σ},\bar{ε^p}+d\bar{ε^p})=0 \end{cases} \implies \frac{\partial f}{\partial \bm{σ}}\cdot \bm{dσ}+\frac{\partial f}{\partial \bar{ε^p}} d\bar{ε^p}=0\tag{7} {f(σ,εpˉ)=0f(σ+dσ,εpˉ+dεpˉ)=0​⟹∂σ∂f​⋅dσ+∂εpˉ∂f​dεpˉ=0(7)
屈服函数： f ( σ , ε p ˉ ) = σ M i s e s − h ⋅ ε p ˉ f(\bm{σ},\bar{ε^p})=σ_{Mises}-h·\bar{ε^p} f(σ,εpˉ)=σMises​−h⋅εpˉ
∂ f ∂ σ = ∂ σ M i s e s ∂ σ ∂ f ∂ ε p ˉ = − h (8) \frac{\partial f}{\partial \bm{σ}} =\frac{\partial σ_{Mises}}{\partial \bm{σ}}\qquad \frac{\partial f}{\partial \bar{ε^p}} =-h\tag{8} ∂σ∂f​=∂σ∂σMises​​∂εpˉ∂f​=−h(8)
为了方便计算改写为矩阵形式：

用Voigt表记应力， σ \bmσ σ为列向量 [ σ 11 , σ 22 , σ 33 , σ 12 , σ 23 , σ 13 ] T [σ_{11},σ_{22},σ_{33},σ_{12},σ_{23},σ_{13}]^T [σ11​,σ22​,σ33​,σ12​,σ23​,σ13​]T

因此， ∂ f ∂ σ = ∂ σ M i s e s ∂ σ = 3 2 σ M i s e s [ σ 11 ′ , σ 22 ′ , σ 33 ′ , 2 σ 12 ′ , 2 σ 23 ′ , 2 σ 13 ′ ] \frac{\partial f}{\partial \bm{σ}}=\frac{\partial σ_{Mises}}{\partial \bm{σ}}=\frac{3}{2σ_{Mises}}\left[σ'_{11},σ'_{22},σ'_{33}, 2σ'_{12}, 2σ'_{23}, 2σ'_{13}\right] ∂σ∂f​=∂σ∂σMises​​=2σMises​3​[σ11′​,σ22′​,σ33′​,2σ12′​,2σ23′​,2σ13′​]为行向量

正交条件: d ε p = d ε p ˉ [ ∂ σ M i s e s ∂ σ ] T (9) \bm{dε^p}=d\bar{ε^p} \left[ \frac{\partial σ_{Mises} }{\partial \bmσ}\right]^T\tag{9} dεp=dεpˉ[∂σ∂σMises​​]T(9)
(8)(9)代入(7)：
d ε p ˉ = ∂ σ M i s e s ∂ σ ⋅ d σ h = 1 h ∂ σ M i s e s ∂ σ ⋅ D e ⋅ d ε e = 1 h ∂ σ M i s e s ∂ σ ⋅ D e ⋅ ( d ε − d ε p ) = 1 h ∂ σ M i s e s ∂ σ D e ( d ε − d ε p ˉ [ ∂ σ M i s e s ∂ σ ] T ) \begin{gathered} d\bar{ε^p}=\frac{\partial σ_{Mises}}{\partial \bmσ} \cdot\frac {\bm{dσ}}{h}=\frac{1}{h}\frac{\partial σ_{Mises}}{\partial \bm{σ}} \cdot\bm{D_e}\cdot\bm{dε^e}\\ \quad\\ = \frac{1}{h}\frac{\partial σ_{Mises}}{\partial \bm{σ}}\cdot\bm{D_e}\cdot(\bm{dε-dε^p})\\ \quad\\ =\frac{1}{h}\frac{\partial σ_{Mises}}{\partial \bm{σ}} \bm{D_e}({\bm{dε}-d\bar{ε^p}\left[ \frac{\partial σ_{Mises}} {\partial \bmσ}\right]^T}) \end{gathered} dεpˉ=∂σ∂σMises​​⋅hdσ​=h1​∂σ∂σMises​​⋅De​⋅dεe=h1​∂σ∂σMises​​⋅De​⋅(dε−dεp)=h1​∂σ∂σMises​​De​(dε−dεpˉ[∂σ∂σMises​​]T)​
{ h + ∂ σ M i s e s ∂ σ ⋅ D e ⋅ [ ∂ σ M i s e s ∂ σ ] T } d ε p ˉ = ∂ σ M i s e s ∂ σ ⋅ D e ⋅ d ε d ε p ˉ = ∂ σ M i s e s ∂ σ ⋅ D e ⋅ d ε h + ∂ σ M i s e s ∂ σ ⋅ D e ⋅ [ ∂ σ M i s e s ∂ σ ] T d ε p = [ ∂ σ M i s e s ∂ σ ] T ⊗ ∂ σ M i s e s ∂ σ ⋅ D e ⋅ d ε h + ∂ σ M i s e s ∂ σ ⋅ D e ⋅ [ ∂ σ M i s e s ∂ σ ] T \left\{h+\frac{\partial σ_{Mises}}{\partial \bmσ}\cdot \bm{D_e}\cdot\left[\frac{\partial σ_{Mises}}{\partial \bmσ} \right]^T\right\}d\bar{ε^p}=\frac{\partial σ_{Mises}}{\partial \bmσ} \cdot\bm{D_e}\cdot\bm{dε}\\ \quad\\ d\bar{ε^p}=\frac{\frac{\partial σ_{Mises}}{\partial \bmσ} \cdot\bm{D_e}\cdot\bm{dε}}{ h+\frac{\partial σ_{Mises}}{\partial \bmσ}\cdot \bm{D_e}\cdot\left[\frac{\partial σ_{Mises}}{\partial \bmσ} \right]^T}\\ \quad\\ d\bm{ε^p}= \frac{\left[\frac{\partial σ_{Mises}}{\partial \bmσ} \right]^T\otimes\frac{\partial σ_{Mises}}{\partial \bmσ} \cdot\bm{D_e}\cdot\bm{dε}}{ h+\frac{\partial σ_{Mises}}{\partial \bmσ}\cdot \bm{D_e}\cdot\left[\frac{\partial σ_{Mises}}{\partial \bmσ} \right]^T} {h+∂σ∂σMises​​⋅De​⋅[∂σ∂σMises​​]T}dεpˉ=∂σ∂σMises​​⋅De​⋅dεdεpˉ=h+∂σ∂σMises​​⋅De​⋅[∂σ∂σMises​​]T∂σ∂σMises​​⋅De​⋅dε​dεp=h+∂σ∂σMises​​⋅De​⋅[∂σ∂σMises​​]T[∂σ∂σMises​​]T⊗∂σ∂σMises​​⋅De​⋅dε​
代入(6):
d σ = D e ⋅ ( d ε − d ε p ) = { D e − [ ∂ σ M i s e s ∂ σ ] T ⊗ ∂ σ M i s e s ∂ σ ⋅ D e h + ∂ σ M i s e s ∂ σ ⋅ D e ⋅ [ ∂ σ M i s e s ∂ σ ] T } ⋅ d ε \bm{dσ=D_e·(dε-dε^p)}=\left\{ \bm{D_e}-\frac{\left[\frac{\partial σ_{Mises}}{\partial \bmσ} \right]^T\otimes\frac{\partial σ_{Mises}}{\partial \bmσ}\cdot\bm{D_e}}{ h+\frac{\partial σ_{Mises}}{\partial \bmσ}\cdot \bm{D_e}\cdot\left[\frac{\partial σ_{Mises}}{\partial \bmσ} \right]^T}\right\} \cdot\bm{dε} dσ=De​⋅(dε−dεp)={De​−h+∂σ∂σMises​​⋅De​⋅[∂σ∂σMises​​]T[∂σ∂σMises​​]T⊗∂σ∂σMises​​⋅De​​}⋅dε
d σ = D e q ⋅ d ε (11) \bm{dσ}=\bm{D_{eq}}\cdot\bm{d\varepsilon}\tag{11} dσ=Deq​⋅dε(11)
D e ⋅ [ ∂ σ M i s e s ∂ σ ] T = 3 G 2 σ M i s e s [ 2 G + λ λ λ 0 0 0 λ 2 G + λ λ 0 0 0 λ λ 2 G + λ 0 0 0 0 0 0 G 0 0 0 0 0 0 G 0 0 0 0 0 0 G ] ⋅ { σ 11 ′ σ 22 ′ σ 33 ′ 2 σ 12 ′ 2 σ 23 ′ 2 σ 13 ′ } = 3 G σ M i s e s { σ 11 ′ σ 22 ′ σ 33 ′ σ 12 ′ σ 23 ′ σ 13 ′ } \bm{D_e}\cdot\left[\frac{\partial σ_{Mises}}{\partial \bm{σ}}\right]^T= \frac{3G}{2σ_{Mises}}\left [\begin{matrix} 2G+λ&λ&λ&0&0&0 \\ λ&2G+λ&λ&0&0&0 \\ λ&λ& 2G+λ&0&0&0 \\ 0&0&0& G&0&0 \\ 0&0&0&0& G&0 \\ 0&0&0&0&0 &G \end{matrix} \right]\cdot\left\{\begin{matrix} σ'_{11}\\σ'_{22}\\σ'_{33}\\2σ'_{12}\\ 2σ'_{23}\\2σ'_{13}\end{matrix} \right\} =\frac{3G}{σ_{Mises}}\left\{\begin{matrix} σ'_{11}\\σ'_{22}\\σ'_{33}\\σ'_{12}\\ σ'_{23}\\σ'_{13}\end{matrix} \right\} De​⋅[∂σ∂σMises​​]T=2σMises​3G​⎣⎢⎢⎢⎢⎢⎢⎡​2G+λλλ000​λ2G+λλ000​λλ2G+λ000​000G00​0000G0​00000G​⎦⎥⎥⎥⎥⎥⎥⎤​⋅⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧​σ11′​σ22′​σ33′​2σ12′​2σ23′​2σ13′​​⎭⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎫​=σMises​3G​⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧​σ11′​σ22′​σ33′​σ12′​σ23′​σ13′​​⎭⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎫​
[ ∂ σ M i s e s ∂ σ ] ⋅ D e ⋅ [ ∂ σ M i s e s ∂ σ ] T = 9 G 2 σ M i s e s 2 { σ 11 ′ , σ 22 ′ , σ 33 ′ , 2 σ 12 ′ , 2 σ 23 ′ , 2 σ 13 ′ } { σ 11 ′ σ 22 ′ σ 33 ′ σ 12 ′ σ 23 ′ σ 13 ′ } = 9 G 2 σ M i s e s 2 2 3 [ 3 2 ( σ 11 ′ 2 + σ 22 ′ 2 + σ 33 ′ 2 + 2 σ 12 ′ 2 + 2 σ 23 ′ 2 + 2 σ 13 ′ 2 ) ] = 3 G \left[\frac{\partial σ_{Mises}}{\partial \bm{σ}}\right]\cdot\bm{D_e}\cdot\left[\frac{\partial σ_{Mises}}{\partial \bm{σ}}\right]^T =\frac{9G^2}{σ_{Mises}^2}\left\{ σ'_{11},σ'_{22},σ'_{33},2σ'_{12},2 σ'_{23},2σ'_{13} \right\}\left\{\begin{matrix} σ'_{11}\\σ'_{22}\\σ'_{33}\\σ'_{12}\\ σ'_{23}\\σ'_{13}\end{matrix} \right\}\\ \quad =\frac{9G}{2σ_{Mises}^2}\frac{2}{3}\left[\frac{3}{2}\left( {σ'_{11}}^2+{σ'_{22}}^2+{σ'_{33}}^2+2{σ'_{12}}^2+2{σ'_{23}}^2+2{σ'_{13}}^2 \right)\right]=3G [∂σ∂σMises​​]⋅De​⋅[∂σ∂σMises​​]T=σMises2​9G2​{σ11′​,σ22′​,σ33′​,2σ12′​,2σ23′​,2σ13′​}⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧​σ11′​σ22′​σ33′​σ12′​σ23′​σ13′​​⎭⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎫​=2σMises2​9G​32​[23​(σ11′​2+σ22′​2+σ33′​2+2σ12′​2+2σ23′​2+2σ13′​2)]=3G
因此：
[ ∂ σ M i s e s ∂ σ ] ⋅ D e ⋅ [ ∂ σ M i s e s ∂ σ ] T = 3 G (12) \left[\frac{\partial σ_{Mises}}{\partial \bm{σ}}\right]\cdot\bm{D_e}\cdot\left[\frac{\partial σ_{Mises}}{\partial \bm{σ}}\right]^T =3G\tag{12} [∂σ∂σMises​​]⋅De​⋅[∂σ∂σMises​​]T=3G(12)
[ ∂ σ M i s e s ∂ σ ] T ⊗ ∂ σ M i s e s ∂ σ = 9 4 σ M i s e s 2 { σ 11 ′ ⋅ σ 11 ′ σ 11 ′ ⋅ σ 22 ′ σ 11 ′ ⋅ σ 33 ′ σ 11 ′ ⋅ σ 12 ′ σ 11 ′ ⋅ σ 23 ′ σ 11 ′ ⋅ σ 13 ′ σ 22 ′ ⋅ σ 11 ′ σ 22 ′ ⋅ σ 22 ′ σ 22 ′ ⋅ σ 33 ′ σ 22 ′ ⋅ σ 12 ′ σ 22 ′ ⋅ σ 23 ′ σ 22 ′ ⋅ σ 13 ′ σ 33 ′ ⋅ σ 11 ′ σ 33 ′ ⋅ σ 22 ′ σ 33 ′ ⋅ σ 33 ′ σ 33 ′ ⋅ σ 12 ′ σ 33 ′ ⋅ σ 23 ′ σ 33 ′ ⋅ σ 13 ′ σ 12 ′ ⋅ σ 11 ′ σ 12 ′ ⋅ σ 22 ′ σ 12 ′ ⋅ σ 33 ′ σ 12 ′ ⋅ σ 12 ′ σ 12 ′ ⋅ σ 23 ′ σ 12 ′ ⋅ σ 13 ′ σ 23 ′ ⋅ σ 11 ′ σ 23 ′ ⋅ σ 22 ′ σ 23 ′ ⋅ σ 33 ′ σ 23 ′ ⋅ σ 12 ′ σ 23 ′ ⋅ σ 23 ′ σ 23 ′ ⋅ σ 13 ′ σ 13 ′ ⋅ σ 11 ′ σ 13 ′ ⋅ σ 22 ′ σ 13 ′ ⋅ σ 33 ′ σ 13 ′ ⋅ σ 12 ′ σ 13 ′ ⋅ σ 23 ′ σ 13 ′ ⋅ σ 13 ′ } (13) \begin{array}{l} \large{\left[\frac{\partial σ_{Mises}}{\partial \bmσ} \right]^T\otimes\frac{\partial σ_{Mises}}{\partial \bmσ}=}\\ \quad\\ \frac{9}{4σ_{Mises}^2}\left\{\begin{matrix} σ'_{11}\cdotσ'_{11}&σ'_{11}\cdotσ'_{22}&σ'_{11}\cdotσ'_{33}&σ'_{11}\cdotσ'_{12}&σ'_{11}\cdotσ'_{23}&σ'_{11}\cdotσ'_{13}\\ σ'_{22}\cdotσ'_{11}&σ'_{22}\cdotσ'_{22}&σ'_{22}\cdotσ'_{33}&σ'_{22}\cdotσ'_{12}&σ'_{22}\cdotσ'_{23}&σ'_{22}\cdotσ'_{13}\\ σ'_{33}\cdotσ'_{11}&σ'_{33}\cdotσ'_{22}&σ'_{33}\cdotσ'_{33}&σ'_{33}\cdotσ'_{12}&σ'_{33}\cdotσ'_{23}&σ'_{33}\cdotσ'_{13}\\ σ'_{12}\cdotσ'_{11}&σ'_{12}\cdotσ'_{22}&σ'_{12}\cdotσ'_{33}&σ'_{12}\cdotσ'_{12}&σ'_{12}\cdotσ'_{23}&σ'_{12}\cdotσ'_{13}\\ σ'_{23}\cdotσ'_{11}&σ'_{23}\cdotσ'_{22}&σ'_{23}\cdotσ'_{33}&σ'_{23}\cdotσ'_{12}&σ'_{23}\cdotσ'_{23}&σ'_{23}\cdotσ'_{13}\\ σ'_{13}\cdotσ'_{11}&σ'_{13}\cdotσ'_{22}&σ'_{13}\cdotσ'_{33}&σ'_{13}\cdotσ'_{12}&σ'_{13}\cdotσ'_{23}&σ'_{13}\cdotσ'_{13} \end{matrix} \right\}\tag{13} \end{array} [∂σ∂σMises​​]T⊗∂σ∂σMises​​=4σMises2​9​⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧​σ11′​⋅σ11′​σ22′​⋅σ11′​σ33′​⋅σ11′​σ12′​⋅σ11′​σ23′​⋅σ11′​σ13′​⋅σ11′​​σ11′​⋅σ22′​σ22′​⋅σ22′​σ33′​⋅σ22′​σ12′​⋅σ22′​σ23′​⋅σ22′​σ13′​⋅σ22′​​σ11′​⋅σ33′​σ22′​⋅σ33′​σ33′​⋅σ33′​σ12′​⋅σ33′​σ23′​⋅σ33′​σ13′​⋅σ33′​​σ11′​⋅σ12′​σ22′​⋅σ12′​σ33′​⋅σ12′​σ12′​⋅σ12′​σ23′​⋅σ12′​σ13′​⋅σ12′​​σ11′​⋅σ23′​σ22′​⋅σ23′​σ33′​⋅σ23′​σ12′​⋅σ23′​σ23′​⋅σ23′​σ13′​⋅σ23′​​σ11′​⋅σ13′​σ22′​⋅σ13′​σ33′​⋅σ13′​σ12′​⋅σ13′​σ23′​⋅σ13′​σ13′​⋅σ13′​​⎭⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎫​​(13)
(12)(13)代入(11):
D e q = { D e − [ ∂ σ M i s e s ∂ σ ] T ⊗ ∂ σ M i s e s ∂ σ ⋅ D e h + ∂ σ M i s e s ∂ σ ⋅ D e ⋅ [ ∂ σ M i s e s ∂ σ ] T } \bm{D_{eq}}=\left\{ \bm{D_{e}}-\frac{\left[\frac{\partial σ_{Mises}}{\partial \bmσ} \right]^T\otimes\frac{\partial σ_{Mises}}{\partial \bmσ}\cdot\bm{D_{e}}}{ h+\frac{\partial σ_{Mises}}{\partial \bmσ}\cdot \bm{D_e}\cdot\left[\frac{\partial σ_{Mises}}{\partial \bmσ} \right]^T}\right\} Deq​={De​−h+∂σ∂σMises​​⋅De​⋅[∂σ∂σMises​​]T[∂σ∂σMises​​]T⊗∂σ∂σMises​​⋅De​​}
[ ∂ σ M i s e s ∂ σ ] T ⊗ ∂ σ M i s e s ∂ σ ⋅ D e h + ∂ σ M i s e s ∂ σ ⋅ D e ⋅ [ ∂ σ M i s e s ∂ σ ] T = 9 4 σ M i s e s 2 ( h + 3 G ) [ σ 11 ′ ⋅ σ 11 ′ σ 11 ′ ⋅ σ 22 ′ σ 11 ′ ⋅ σ 33 ′ σ 11 ′ ⋅ σ 12 ′ σ 11 ′ ⋅ σ 23 ′ σ 11 ′ ⋅ σ 13 ′ σ 22 ′ ⋅ σ 11 ′ σ 22 ′ ⋅ σ 22 ′ σ 22 ′ ⋅ σ 33 ′ σ 22 ′ ⋅ σ 12 ′ σ 22 ′ ⋅ σ 23 ′ σ 22 ′ ⋅ σ 13 ′ σ 33 ′ ⋅ σ 11 ′ σ 33 ′ ⋅ σ 22 ′ σ 33 ′ ⋅ σ 33 ′ σ 33 ′ ⋅ σ 12 ′ σ 33 ′ ⋅ σ 23 ′ σ 33 ′ ⋅ σ 13 ′ σ 12 ′ ⋅ σ 11 ′ σ 12 ′ ⋅ σ 22 ′ σ 12 ′ ⋅ σ 33 ′ σ 12 ′ ⋅ σ 12 ′ σ 12 ′ ⋅ σ 23 ′ σ 12 ′ ⋅ σ 13 ′ σ 23 ′ ⋅ σ 11 ′ σ 23 ′ ⋅ σ 22 ′ σ 23 ′ ⋅ σ 33 ′ σ 23 ′ ⋅ σ 12 ′ σ 23 ′ ⋅ σ 23 ′ σ 23 ′ ⋅ σ 13 ′ σ 13 ′ ⋅ σ 11 ′ σ 13 ′ ⋅ σ 22 ′ σ 13 ′ ⋅ σ 33 ′ σ 13 ′ ⋅ σ 12 ′ σ 13 ′ ⋅ σ 23 ′ σ 13 ′ ⋅ σ 13 ′ ] ⋅ [ 2 G + λ λ λ 0 0 0 λ 2 G + λ λ 0 0 0 λ λ 2 G + λ 0 0 0 0 0 0 G 0 0 0 0 0 0 G 0 0 0 0 0 0 G ] = 9 G 4 σ M i s e s 2 ( h + 3 G ) [ 2 σ 11 ′ ⋅ σ 11 ′ 2 σ 11 ′ ⋅ σ 22 ′ 2 σ 11 ′ ⋅ σ 33 ′ σ 11 ′ ⋅ σ 12 ′ σ 11 ′ ⋅ σ 23 ′ σ 11 ′ ⋅ σ 13 ′ 2 σ 22 ′ ⋅ σ 11 ′ 2 σ 22 ′ ⋅ σ 22 ′ 2 σ 22 ′ ⋅ σ 33 ′ σ 22 ′ ⋅ σ 12 ′ σ 22 ′ ⋅ σ 23 ′ σ 22 ′ ⋅ σ 13 ′ 2 σ 33 ′ ⋅ σ 11 ′ 2 σ 33 ′ ⋅ σ 22 ′ 2 σ 33 ′ ⋅ σ 33 ′ σ 33 ′ ⋅ σ 12 ′ σ 33 ′ ⋅ σ 23 ′ σ 33 ′ ⋅ σ 13 ′ 2 σ 12 ′ ⋅ σ 11 ′ 2 σ 12 ′ ⋅ σ 22 ′ 2 σ 12 ′ ⋅ σ 33 ′ σ 12 ′ ⋅ σ 12 ′ σ 12 ′ ⋅ σ 23 ′ σ 12 ′ ⋅ σ 13 ′ 2 σ 23 ′ ⋅ σ 11 ′ 2 σ 23 ′ ⋅ σ 22 ′ 2 σ 23 ′ ⋅ σ 33 ′ σ 23 ′ ⋅ σ 12 ′ σ 23 ′ ⋅ σ 23 ′ σ 23 ′ ⋅ σ 13 ′ 2 σ 13 ′ ⋅ σ 11 ′ 2 σ 13 ′ ⋅ σ 22 ′ 2 σ 13 ′ ⋅ σ 33 ′ σ 13 ′ ⋅ σ 12 ′ σ 13 ′ ⋅ σ 23 ′ σ 13 ′ ⋅ σ 13 ′ ] \begin{array}{l} \large {\frac{\left[\frac{\partial σ_{Mises}}{\partial \bmσ} \right]^T\otimes\frac{\partial σ_{Mises}}{\partial \bmσ}\cdot\bm{D_{e}}}{ h+\frac{\partial σ_{Mises}}{\partial \bmσ}\cdot \bm{D_e}\cdot\left[\frac{\partial σ_{Mises}}{\partial \bmσ} \right]^T}}=\frac{9}{4σ_{Mises}^2(h+3G)}\\ \quad\\ \small{ \left[\begin{matrix} σ'_{11}\cdotσ'_{11}&σ'_{11}\cdotσ'_{22}&σ'_{11}\cdotσ'_{33}&σ'_{11}\cdotσ'_{12}&σ'_{11}\cdotσ'_{23}&σ'_{11}\cdotσ'_{13}\\ σ'_{22}\cdotσ'_{11}&σ'_{22}\cdotσ'_{22}&σ'_{22}\cdotσ'_{33}&σ'_{22}\cdotσ'_{12}&σ'_{22}\cdotσ'_{23}&σ'_{22}\cdotσ'_{13}\\ σ'_{33}\cdotσ'_{11}&σ'_{33}\cdotσ'_{22}&σ'_{33}\cdotσ'_{33}&σ'_{33}\cdotσ'_{12}&σ'_{33}\cdotσ'_{23}&σ'_{33}\cdotσ'_{13}\\ σ'_{12}\cdotσ'_{11}&σ'_{12}\cdotσ'_{22}&σ'_{12}\cdotσ'_{33}&σ'_{12}\cdotσ'_{12}&σ'_{12}\cdotσ'_{23}&σ'_{12}\cdotσ'_{13}\\ σ'_{23}\cdotσ'_{11}&σ'_{23}\cdotσ'_{22}&σ'_{23}\cdotσ'_{33}&σ'_{23}\cdotσ'_{12}&σ'_{23}\cdotσ'_{23}&σ'_{23}\cdotσ'_{13}\\ σ'_{13}\cdotσ'_{11}&σ'_{13}\cdotσ'_{22}&σ'_{13}\cdotσ'_{33}&σ'_{13}\cdotσ'_{12}&σ'_{13}\cdotσ'_{23}&σ'_{13}\cdotσ'_{13} \end{matrix} \right]\cdot\left [\begin{matrix} 2G+λ&λ&λ&0&0&0 \\ λ&2G+λ&λ&0&0&0 \\ λ&λ& 2G+λ&0&0&0 \\ 0&0&0& G&0&0 \\ 0&0&0&0& G&0 \\ 0&0&0&0&0 &G \end{matrix} \right]}\\ \quad\\ =\frac{9G}{4σ_{Mises}^2(h+3G)} \small{ \left[\begin{matrix} 2σ'_{11}\cdotσ'_{11}&2σ'_{11}\cdotσ'_{22}&2σ'_{11}\cdotσ'_{33}&σ'_{11}\cdotσ'_{12}&σ'_{11}\cdotσ'_{23}&σ'_{11}\cdotσ'_{13}\\ 2σ'_{22}\cdotσ'_{11}&2σ'_{22}\cdotσ'_{22}&2σ'_{22}\cdotσ'_{33}&σ'_{22}\cdotσ'_{12}&σ'_{22}\cdotσ'_{23}&σ'_{22}\cdotσ'_{13}\\ 2σ'_{33}\cdotσ'_{11}&2σ'_{33}\cdotσ'_{22}&2σ'_{33}\cdotσ'_{33}&σ'_{33}\cdotσ'_{12}&σ'_{33}\cdotσ'_{23}&σ'_{33}\cdotσ'_{13}\\ 2σ'_{12}\cdotσ'_{11}&2σ'_{12}\cdotσ'_{22}&2σ'_{12}\cdotσ'_{33}&σ'_{12}\cdotσ'_{12}&σ'_{12}\cdotσ'_{23}&σ'_{12}\cdotσ'_{13}\\ 2σ'_{23}\cdotσ'_{11}&2σ'_{23}\cdotσ'_{22}&2σ'_{23}\cdotσ'_{33}&σ'_{23}\cdotσ'_{12}&σ'_{23}\cdotσ'_{23}&σ'_{23}\cdotσ'_{13}\\ 2σ'_{13}\cdotσ'_{11}&2σ'_{13}\cdotσ'_{22}&2σ'_{13}\cdotσ'_{33}&σ'_{13}\cdotσ'_{12}&σ'_{13}\cdotσ'_{23}&σ'_{13}\cdotσ'_{13} \end{matrix} \right]} \end{array} h+∂σ∂σMises​​⋅De​⋅[∂σ∂σMises​​]T[∂σ∂σMises​​]T⊗∂σ∂σMises​​⋅De​​=4σMises2​(h+3G)9​⎣⎢⎢⎢⎢⎡​σ11′​⋅σ11′​σ22′​⋅σ11′​σ33′​⋅σ11′​σ12′​⋅σ11′​σ23′​⋅σ11′​σ13′​⋅σ11′​​σ11′​⋅σ22′​σ22′​⋅σ22′​σ33′​⋅σ22′​σ12′​⋅σ22′​σ23′​⋅σ22′​σ13′​⋅σ22′​​σ11′​⋅σ33′​σ22′​⋅σ33′​σ33′​⋅σ33′​σ12′​⋅σ33′​σ23′​⋅σ33′​σ13′​⋅σ33′​​σ11′​⋅σ12′​σ22′​⋅σ12′​σ33′​⋅σ12′​σ12′​⋅σ12′​σ23′​⋅σ12′​σ13′​⋅σ12′​​σ11′​⋅σ23′​σ22′​⋅σ23′​σ33′​⋅σ23′​σ12′​⋅σ23′​σ23′​⋅σ23′​σ13′​⋅σ23′​​σ11′​⋅σ13′​σ22′​⋅σ13′​σ33′​⋅σ13′​σ12′​⋅σ13′​σ23′​⋅σ13′​σ13′​⋅σ13′​​⎦⎥⎥⎥⎥⎤​⋅⎣⎢⎢⎢⎢⎡​2G+λλλ000​λ2G+λλ000​λλ2G+λ000​000G00​0000G0​00000G​⎦⎥⎥⎥⎥⎤​=4σMises2​(h+3G)9G​⎣⎢⎢⎢⎢⎡​2σ11′​⋅σ11′​2σ22′​⋅σ11′​2σ33′​⋅σ11′​2σ12′​⋅σ11′​2σ23′​⋅σ11′​2σ13′​⋅σ11′​​2σ11′​⋅σ22′​2σ22′​⋅σ22′​2σ33′​⋅σ22′​2σ12′​⋅σ22′​2σ23′​⋅σ22′​2σ13′​⋅σ22′​​2σ11′​⋅σ33′​2σ22′​⋅σ33′​2σ33′​⋅σ33′​2σ12′​⋅σ33′​2σ23′​⋅σ33′​2σ13′​⋅σ33′​​σ11′​⋅σ12′​σ22′​⋅σ12′​σ33′​⋅σ12′​σ12′​⋅σ12′​σ23′​⋅σ12′​σ13′​⋅σ12′​​σ11′​⋅σ23′​σ22′​⋅σ23′​σ33′​⋅σ23′​σ12′​⋅σ23′​σ23′​⋅σ23′​σ13′​⋅σ23′​​σ11′​⋅σ13′​σ22′​⋅σ13′​σ33′​⋅σ13′​σ12′​⋅σ13′​σ23′​⋅σ13′​σ13′​⋅σ13′​​⎦⎥⎥⎥⎥⎤​​
屈服后的DDSDDE(NTENS,NTENS)
D D S D D E ( N T E N S , N T E N S ) = D e q DDSDDE(NTENS,NTENS)=\bm{D_{eq}} DDSDDE(NTENS,NTENS)=Deq​

umat子程序

      SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD,1 RPL,DDSDDT,DRPLDE,DRPLDT,2 STRAN,DSTRAN,TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,CMNAME,3 NDI,NSHR,NTENS,NSTATV,PROPS,NPROPS,COORDS,DROT,PNEWDT,4 CELENT,DFGRD0,DFGRD1,NOEL,NPT,LAYER,KSPT,JSTEP,KINC)
CINCLUDE 'ABA_PARAM.INC'
CCHARACTER*80 CMNAMEDIMENSION STRESS(NTENS),STATEV(NSTATV),1 DDSDDE(NTENS,NTENS),DDSDDT(NTENS),DRPLDE(NTENS),2 STRAN(NTENS),DSTRAN(NTENS),TIME(2),PREDEF(1),DPRED(1),3 PROPS(NPROPS),COORDS(3),DROT(3,3),DFGRD0(3,3),DFGRD1(3,3),4 JSTEP(4),5 STRESS_TR(NTENS),DSTRAN_P(NTENS),DSTRAN_E(NTENS),STRESS_D(NTENS),6 S_S(NTENS,NTENS)EE=PROPS(1)  !modulusEMU=PROPS(2)   !Possion's ritioEH=PROPS(3)   !线性硬化模量SGM_Y=PROPS(4)LMD=EE*EMU/((1.0+EMU)*(1.0-2.0*EMU))EG=EE/(2.0*(1.0+EMU))DDSDDE=0.0DO I=1,NDIDDSDDE(I,I)=2*EGENDDODO I=NDI+1,NTENSDDSDDE(I,I)=EGENDDODO I=1,NDIDO J=1,NDIDDSDDE(I,J)=DDSDDE(I,J)+LMDENDDOENDDODO I=1,NTENSSTRESS_TR(I)=STRESS(I)ENDDODO I=1,NTENSDO J=1,NTENSSTRESS_TR(I)=STRESS_TR(I)+DDSDDE(I,J)*DSTRAN(J)ENDDOENDDO
C     计算MISES等效应力STRESS_TR_V=0.0DO I=1,NDISTRESS_TR_V=STRESS_TR_V+STRESS_TR(I)ENDDOSTRESS_TR_V=STRESS_TR_V/3.0STRESS_TR_MISES=0.0DO I=1,NDISTRESS_TR_MISES=STRESS_TR_MISES+(STRESS_TR(I)-STRESS_TR_V)**2.0ENDDODO I=NDI+1,NTENSSTRESS_TR_MISES=STRESS_TR_MISES2.0*+STRESS_TR(I)**2。0ENDDOSTRESS_TR_MISES=SQRT(STRESS_TR_MISES*1.5)P=STATEV(1)
C     判断是否屈服IF(STRESS_TR_MISES .LE. (SGM_Y+EH*P)) THENDO I=1,NTENSSTRESS(I)=STRESS_TR(I)ENDDOELSEDP=(STRESS_TR_MISES-(SGM_Y+EH*P))/(EH+3.0*EG)  STATEV(1)=P+ABS(DP)DO I=1,NDIDSTRAN_P(I)=1.5*DP*(STRESS_TR(I)-STRESS_TR_V)/STRESS_TR_MISESENDDODO I=1+NDI,NTENSDSTRAN_P(I)=1.5*DP*(STRESS_TR(I))/STRESS_TR_MISESENDDODO I=1,NTENSDSTRAN_E(I)=DSTRAN(I)-DSTRAN_P(I)ENDDODO I=1,NTENSDO J=1,NTENSSTRESS(I)=STRESS(I)+DDSDDE(I,J)*DSTRAN_E(J)  !应力更新ENDDOENDDOSTRESS_V=0.0DO I=1,NDISTRESS_V=STRESS_V+STRESS(I)ENDDOSTRESS_V=STRESS_V/3.0STRESS_MISES=0.0DO I=1,NDISTRESS_D(I)=STRESS(I)-STRESS_VSTRESS_MISES=STRESS_MISES+STRESS_D(I)**2.0ENDDODO I=1+NDI,NTENSSTRESS_D(I)=STRESS(I)STRESS_MISES=STRESS_MISES+2.0*STRESS_D(I)**2.0ENDDOSTRESS_MISES=SQRT(1.5*STRESS_MISES)DO I=1,NTENSDO J=1,NDIS_S(I,J)=18.0*EG*STRESS_D(I)*STRESS_D(J)/(4.0*(EH+3.0*EG)*STRESS_MISES**2.0)ENDDODO J=1+NDI,NTENSS_S(I,J)=9.0*EG*STRESS_D(I)*STRESS_D(J)/(4.0*(EH+3.0*EG)*STRESS_MISES**2.0)ENDDOENDDODO I=1,NTENSDO J=1,NTENSDDSDDE(I,J)=DDSDDE(I,J)-S_S(I,J)ENDDOENDDOENDIFRETURNEND

参数

名称 参数 单位
杨氏模量 PROPS(1)=200000 MPa
泊松比 PROPS(2)=0.3
塑性硬化系数 PROPS(3)=10000 MPa
初始屈服强度 PROPS(4)=1000 MPa

step-1，增加位移载荷 U2=0.2

Time Amp-1
0 0
0.25 1
0.5 0
0.75 -1
1 0

计算结果

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