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 力学学报  2016, Vol. 48 Issue (3): 660-674  DOI: 10.6052/0459-1879-15-333 0

### 引用本文 [复制中英文]

[复制中文]
Meng Lingkai, Zhou Changdong, Guo Kunpeng, Zhang Xiaoyang. A NEW FORMULATION OF CONSTITUTIVE MODEL FOR HYPERELASTIC-CYCLIC PLASTICITY[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(3): 660-674. DOI: 10.6052/0459-1879-15-333.
[复制英文]

### 文章历史

2015-09-05 收稿
2016-01-19 录用
2016-01-28 网络版发表.

0 引言

1 变形运动学 1.1 基本理论

 $F=\frac{\partial x}{\partial X}F=\text{det}F>0$ (1)

 $F=RU=VR$ (2)

 \begin{align} & \text{L= D + W (4a)} \\ & D=\frac{L+{{L}^{\text{T}}}}{2},W=\frac{L-{{L}^{\text{T}}}}{2}(4b) \\ \end{align} (3)

 ${{U}^{2}}={{F}^{\text{T}}}F=C$ (5)
 ${{V}^{2}}=F{{F}^{\text{T}}}=B$ (6)

1.2 变形梯度乘法分解定理

 图1 初始或参考构C0 ，中间或卸载构型ˉC 与当前或即时构型C Fig.1 The initial configuration C0, intermediate configuration ˉC and current configuration C

 ${{F}^{\text{p}}}=\frac{\partial \bar{X}}{\partial X},{{J}^{\text{p}}}=\det {{F}^{\text{p}}}=1$ (7)
 $d\bar{X}={{F}^{\text{p}}}\text{d}X$ (8)

 ${{F}^{\text{e}}}=\frac{\partial x}{\partial \bar{X}},{{J}^{\text{e}}}=\det {{F}^{\text{e}}}>0$ (9)
 $\text{d}x={{F}^{\text{e}}}d\bar{X}$ (10)

 $\text{d}x=F\text{ d}X$ (11)

 $\text{d}x={{F}^{\text{e}}}{{F}^{\text{p}}}\text{ }\backslash \text{ d}X$ (12)

 $F={{F}^{\text{e}}}{{F}^{\text{p}}}\text{ }$ (13)

 图2 卸载状态A Fig.2 The unloading state A

 $L={{L}^{\text{e}}}+{{F}^{\text{e}}}{{L}^{p}}{{({{F}^{\text{e}}})}^{-1}}$ (14)

 ${{F}^{\text{e}}}={{R}^{\text{e}}}{{U}^{\text{e}}}={{V}^{\text{e}}}{{R}^{\text{e}}}$ (15)
 ${{F}^{\text{p}}}={{R}^{\text{p}}}{{U}^{\text{p}}}={{V}^{\text{p}}}{{R}^{\text{p}}}$ (16)

 ${{({{U}^{\text{e}}})}^{2}}={{({{F}^{\text{e}}})}^{\text{T}}}{{F}^{\text{e}}}={{C}^{\text{e}}}$ (17)
 ${{({{V}^{\text{e}}})}^{2}}={{F}^{\text{e}}}{{({{F}^{\text{e}}})}^{\text{T}}}={{B}^{\text{e}}}$ (18)
 ${{({{U}^{\text{p}}})}^{2}}={{({{F}^{\text{p}}})}^{\text{T}}}{{F}^{\text{p}}}={{C}^{\text{p}}}$ (19)
 ${{({{V}^{\text{p}}})}^{2}}={{F}^{\text{p}}}{{({{F}^{\text{p}}})}^{\text{T}}}={{B}^{\text{p}}}$ (20)

2 变形梯度分解理论的拓展

 ${{F}^{\text{p}}}={{F}^{\text{en}}}{{F}^{\text{dis}}},\text{ }\det {{F}^{\text{en}}}=\det {{F}^{\text{dis}}}=1\text{ }$ (21)

 图3 Lion 塑性变形梯度分解理论示意图 Fig.3 The schematic diagram of the Lion deformation decomposition theory
 图4 Lion 变形梯度分解理论流变学模型 Fig.4 The rheological model of the Lion deformation decomposition theory

 \begin{align} & {{F}^{\text{p}}}=F_{1}^{\text{en}}F_{1}^{\text{dis}}=F_{2}^{\text{en}}F_{2}^{\text{dis}}=\cdots =F_{M}^{\text{en}}F_{M}^{\text{dis}} \\ & \det F_{i}^{\text{en}}=\det F_{i}^{\text{dis}}=1,i=1,2,\cdots ,M \\ \end{align} (22)

 ${{L}^{\text{p}}}=L_{i}^{\text{en}}+F_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}L_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{dis}}{{(F_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}})}^{-1}}\text{ }$ (23)

 图5 拓展的Lion 变形分解理论示意图 Fig.5 The schematic diagram of the generalized Lion deformation decomposition theory
 图6 拓展的Lion 变形分解理论流变学模型 Fig.6 The rheological model of the generalized Lion deformation decomposition theory

 $F_{i}^{\text{en}}=R_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}U_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}=V_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}R_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}\text{ }$ (24)

 $B_{i}^{\text{en}}=F_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}{{(F_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}})}^{\text{T}}}={{(V_{i}^{\text{en}})}^{2}}\text{ }$ (25)
 $C_{i}^{\text{en}}={{(F_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}})}^{\text{T}}}F_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}={{(U_{i}^{\text{en}})}^{2}}\text{ }$ (26)

 $\psi =\text{ }{{\psi }^{\text{e}}}({{B}^{\text{e}}})+\text{ }\psi _{1}^{\text{en}}(B_{1}^{\text{en}})+\psi _{2}^{\text{en}}(B_{2}^{\text{en}})+\cdots +\text{ }\psi _{M}^{\text{en}}(B_{M}^{\text{en}})\text{ }$ (27)
$\psi$ 本文只考虑各向同性材料，因此，式(26)中$\psi^{\rm e}({ C}^{\rm e})$及$\psi _i^{\rm en}({ B}_i^{\rm en})$分别为其自变量${ C}^{\rm e}$和 ${ B}_i^{\rm en}$的各向同性标量函数，仿照Lion[17]以及文献[21]的定义，将第$i$个背应力分量${ S}_i^{\rm en}$定义为
 $S_{i}^{\text{en}}=\text{sy}{{\text{m}}_{0}}(2\frac{\partial \psi _{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}(B_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}})}{\partial B_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}}B_{i}^{\text{en}})$ (28)

 ${{S}^{\text{en}}}=\sum\limits_{i=1}^{M}{S_{i}^{\text{en}}}\text{ }$ (29)

 $\tau :L\tau :L=\tau :{{L}^{\text{e}}}+[{{({{F}^{\text{e}}})}^{\text{T}}}\tau {{({{F}^{\text{e}}})}^{\text{-T}}}-{{S}^{\text{en}}}]:{{L}^{p}}+{{S}^{\text{en}}}:{{L}^{\text{p}}}$ (30)

 \begin{align} & \tau :L=\tau :{{L}^{e}}+[{{({{F}^{e}})}^{T}}\tau {{({{F}^{e}})}^{-T}}-{{S}^{en}}]:{{L}^{p}}+ \\ & \sum\limits_{i=1}^{M}{S_{i}^{en}}:L_{i}^{en}+\sum\limits_{i=1}^{M}{[{{(F_{i}^{en})}^{T}}S_{i}^{en}{{(F_{i}^{en})}^{-T}}]}:L_{i}^{dis} \\ \end{align} (31)

 \begin{align} & (\tau -2\frac{\partial {{\psi }^{\text{e}}}}{\partial {{B}^{\text{e}}}}{{B}^{\text{e}}}):{{D}^{\text{e}}}+({{M}^{\text{e}}}-{{S}^{\text{en}}}):{{L}^{\text{p}}}+ \\ & \sum\limits_{i=1}^{M}{M_{i}^{\text{en}}}:L_{i}^{\text{dis}}\ge 0 \\ \end{align} (71)

 $(\tau -2\frac{\partial {{\psi }^{\text{e}}}}{\partial {{B}^{\text{e}}}}{{B}^{\text{e}}}):D\text{ }\ge 0$ (72)

 $\tau =2\frac{\partial {{\psi }^{\text{e}}}}{\partial {{B}^{\text{e}}}}{{B}^{\text{e}}}$ (73)

 $(M_{0}^{\text{e}}-{{S}^{\text{en}}}):{{D}^{\text{p}}}+\sum\limits_{i=1}^{M}{M_{i}^{\text{en}}}:D_{i}^{\text{dis}}\text{ }\ge 0$ (74)

4.2 流动法则与随动硬化法则

 $(M_{0}^{\text{e}}-{{S}^{\text{en}}}):{{D}^{\text{p}}}\ge 0$ (75)
 $M_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}:D_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{dis}}\ge 0,i=1,2,\cdots ,M$ (76)

 ${{D}^{\text{p}}}=\sqrt{\frac{3}{2}}{{\dot{\kappa }}^{\text{p}}}\frac{M_{0}^{\text{e}}-{{S}^{\text{en}}}}{\left\| M_{0}^{\text{e}}-{{S}^{\text{en}}} \right\|}$ (77a)

 $\left\| M_{0}^{\text{e}}-{{S}^{\text{en}}} \right\|=\sqrt{(M_{0}^{\text{e}}-{{S}^{\text{en}}}):(M_{0}^{\text{e}}-{{S}^{\text{en}}})}$ (77b)

 $\psi _{i}^{\text{en}}(B_{i}^{\text{en}})=\frac{1}{3}{{r}_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}{{\gamma }_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}{{\left\| h_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}} \right\|}^{2}}$ (88)

 $S_{i}^{\text{en}}=\frac{2}{3}{{r}_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}{{\gamma }_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}h_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}\text{ }$ (89)

 ${{(\overset{.}{\mathop{\overline{h_{i}^{\text{en}}}}}\,)}^{\log }}=D_{i}^{\text{en}}\text{ }$ (90)

 ${{(\overset{.}{\mathop{\overline{h_{i}^{\text{en}}}}}\,)}^{\log }}=\overset{.}{\mathop{\overline{h_{i}^{\text{en}}}}}\,-\Omega _{i}^{\log }h_{i}^{\text{en}}+h_{i}^{\text{en}}\Omega _{i}^{\log }$ (91)

 $\Omega _{i}^{\log }=W_{i}^{\text{en}}+N_{i}^{\log }$ (92)

 ${{W}^{p}}=W_{i}^{\text{en}},i=1,2,\cdots ,M\text{ }$ (93)

 $\Omega _{i}^{\log }={{W}^{\text{p}}}+N_{i}^{\log }$ (94)
${ N}_i^{\log }$的表达式如下
 N_{i}^{\log }=\left\{ \begin{align} & \mathbf{0},{{b}_{1}}={{b}_{2}}={{b}_{3}} \\ & \nu \left[B_{{\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}D_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}} \right],{{b}_{1}}e{{b}_{2}}={{b}_{3}} \\ & {{\nu }_{1}}\left[B_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}D_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}} \right]+{{\nu }_{2}}\left[{{(B_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}})}^{2}}D_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}} \right]+ \\ & {{\nu }_{3}}\left[{{(B_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}})}^{2}}D_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}B_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}}}{i}}}^{\text{en}} \right],{{b}_{1}}\ne {{b}_{2}}\ne {{b}_{3}}\ne {{b}_{1}} \\ \end{align} \right. (95)

 ${{(\overset{.}{\mathop{\overline{S_{i}^{\text{en}}}}}\,)}^{\log }}=\frac{2}{3}{{r}_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}{{\gamma }_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}{{(\dot{h}_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}})}^{\log }}=\frac{2}{3}{{r}_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}{{\gamma }_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}D_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}$ (96)

 \begin{align} & {{(\overset{.}{\mathop{\overline{S_{i}^{\text{en}}}}}\,)}^{\log }}=\frac{2}{3}{{r}_{\underset{-}{\mathop{i}}\,}}{{\gamma }_{\underset{-}{\mathop{i}}\,}}{{D}^{\text{p}}}-\left[{{\mu }_{i}}{{{\dot{\kappa }}}^{\text{p}}}+\left\langle {{{\dot{\lambda }}}_{i}} \right\rangle \cdot \right. \\ & \left. H\left( \frac{3}{2}S_{\underset{-}{\mathop{i}}\,}^{\text{en}}:S_{\underset{-}{\mathop{i}}\,}^{\text{en}}-r_{i}^{2} \right) \right]{{\gamma }_{\underset{-}{\mathop{i}}\,}}S_{\underset{-}{\mathop{i}}\,}^{\text{en}} \\ \end{align} (97)

(1) A-F随动硬化法则 若假定背应力分支总数$M =1$，则拓展的Lion塑性变形分解定理退化为原有的Lion变形梯度分解定理，此时假定变形率张量${ D}^{\rm dis}$的演化方程为
 ${{D}^{\text{dis}}}=\frac{3}{2}{{\dot{\kappa }}^{\text{p}}}\frac{{{M}^{\text{en}}}}{r}$ (98)

 ${{(\overset{.}{\mathop{\overline{{{S}^{\text{en}}}}}}\,)}^{\log }}=\frac{2}{3}r\gamma {{D}^{\text{p}}}-{{\dot{\kappa }}^{\text{p}}}\gamma {{S}^{\text{en}}}$ (99)

(2) Chaboche随动硬化法则 若单纯从形式上讲，Chaboche随动硬化法则与A-F随动硬化法则并无本质上的区别，只是Chaboche模型中背应力分支总数取为$M =3$. 即
 $D_{i}^{\text{dis}}=\frac{3}{2}{{\dot{\kappa }}^{\text{p}}}\frac{M_{i}^{\text{en}}}{{{r}_{i}}},i=1,2,3$ (100)

 ${{(\overset{.}{\mathop{\overline{S_{i}^{\text{en}}}}}\,)}^{\log }}=\frac{2}{3}{{r}_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}{{\gamma }_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}{{D}^{\text{p}}}-{{\dot{\kappa }}^{\text{p}}}{{\gamma }_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}S_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}},i=1,2,3$ (101)

(3) O-W随动硬化法则-初始模型 若假定变形率张量${ D}_i^{\rm dis}$的演化方程为
 \begin{align} & D_{i}^{\text{dis}}=\frac{3}{2}{{{\dot{\kappa }}}^{\text{p}}}\cdot \left\langle \frac{{{D}^{\text{p}}}}{\left\| {{D}^{\text{p}}} \right\|}:\frac{S_{i}^{\text{en}}}{\left\| S_{i}^{\text{en}} \right\|} \right\rangle \cdot \\ & H\left( \frac{3}{2}M_{{\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}:{{(M_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}}}{i}}}^{\text{en}})}^{\text{T}}}-r_{i}^{2} \right)\frac{M_{i}^{\text{en}}}{{{r}_{i}}} \\ \end{align} (102)

 \begin{align} & {{(\overset{.}{\mathop{\overline{S_{i}^{\text{en}}}}}\,)}^{\log }}=\frac{2}{3}{{r}_{{\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}$}}{i}}}}{{\gamma }_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}{{D}^{\text{p}}}-{{{\dot{\kappa }}}^{\text{p}}}\cdot \left\langle \frac{{{D}^{\text{p}}}}{\left\| {{D}^{\text{p}}} \right\|}:\frac{S_{i}^{\text{en}}}{\left\| S_{i}^{\text{en}} \right\|} \right\rangle \cdot \\ & H\left( \frac{3}{2}S_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}:S_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}-r_{i}^{2} \right){{\gamma }_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}S_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}}}{i}}}^{\text{en}} \\ \end{align} (103)

(4) O-W随动硬化法则-修正模型 为了更好地模拟材料的循环塑性，文献[13] 对模型做了修正，引入了非线性项，而反映到大变形理论中，即将式(102)修改为
 \begin{align} & D_{i}^{\text{dis}}=\frac{3}{2}{{{\dot{\kappa }}}^{\text{p}}}\cdot \left\langle \frac{{{D}^{\text{p}}}}{\left\| {{D}^{\text{p}}} \right\|}:\frac{S_{i}^{\text{en}}}{\left\| S_{i}^{\text{en}} \right\|} \right\rangle \cdot \\ & {{(\frac{\sqrt{\frac{3}{2}M_{{\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}:{{(M_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}}}{i}}}^{\text{en}})}^{\text{T}}}}}{{{r}_{i}}})}^{{{m}_{i}}}}\frac{M_{i}^{\text{en}}}{{{r}_{i}}} \\ \end{align} (104)

 \begin{align} & {{(\overset{.}{\mathop{\overline{S_{i}^{\text{en}}}}}\,)}^{\log }}=\frac{2}{3}{{r}_{{\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}$}}{i}}}}{{\gamma }_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}{{D}^{\text{p}}}- \\ & {{{\dot{\kappa }}}^{\text{p}}}\cdot \left\langle \frac{{{D}^{\text{p}}}}{\left\| {{D}^{\text{p}}} \right\|}:\frac{S_{i}^{\text{en}}}{\left\| S_{i}^{\text{en}} \right\|} \right\rangle {{(\frac{\sqrt{\frac{3}{2}S_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}:S_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{en}}}}{{{r}_{i}}})}^{{{m}_{i}}}}{{\gamma }_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}}S_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}}}{i}}}^{\text{en}} \\ \end{align} (105)

4.3 屈服面方程

 $f\left( {{M}^{\text{e}}},{{S}^{\text{en}}},{{\sigma }_{\text{y}}} \right)=0$ (106)

 \begin{align} & f\left( {{M}^{\text{e}}},{{S}^{\text{en}}},{{\sigma }_{\text{y}}} \right)=\quad \\ & \sqrt{\frac{3}{2}\left( \text{sy}{{\text{m}}_{0}}\left( {{M}^{\text{e}}} \right)-{{S}^{\text{en}}} \right):\left( \text{sy}{{\text{m}}_{0}}\left( {{M}^{\text{e}}} \right)-{{S}^{\text{en}}} \right)}-{{\sigma }_{\text{y}}}=0 \\ \end{align} (107)

 ${{\sigma }_{\text{y}}}\left( {{\kappa }^{\text{p}}} \right)={{\sigma }_{{{\text{y}}_{0}}}}+\left( {{\sigma }_{{{\text{y}}_{\text{s}}}}}-{{\sigma }_{{{\text{y}}_{0}}}} \right)\left( 1-{{\text{e}}^{-{{C}_{\text{iso}}}{{\kappa }^{\text{p}}}}} \right)$ (108)

 ${{\dot{\kappa }}^{\text{p}}}={{\left[\frac{2}{3}{{D}^{\text{p}}}:{{D}^{\text{p}}} \right]}^{1/2}}$ (109a)
 ${{\kappa }^{\text{p}}}=\int{{{\left[\frac{2}{3}{{D}^{\text{p}}}:{{D}^{\text{p}}} \right]}^{1/2}}dt}$ (109b)

 \begin{align} & {{{\dot{\kappa }}}^{\text{p}}}\ge 0f\left( {{M}^{\text{e}}},{{S}^{\text{en}}},{{\sigma }_{\text{y}}} \right)\le 0,\\ & {{{\dot{\kappa }}}^{\text{p}}}f\left( {{M}^{\text{e}}},{{S}^{\text{en}}},{{\sigma }_{\text{y}}} \right)=0 \\ \end{align} (110)

4.4 塑性旋率 由前文叙述可知，两类塑性变形梯度$F^{\rm p}$和$F_i^{\rm dis}$ 分别确定了卸载构型和随动硬化构型，而为了将小变形理论中的随动硬化率在形式上推广到大变形理论中，本文假定随动硬化构型旋率为0，即$W_i^{\rm dis} = 0$(式(81)，而 $D_i^{\rm dis}$则由本文的随动硬化法则(式 (83)，式(98)，式(100)，式(102)，式(104))确定，故变形梯度 $F_i^{\rm dis}$ 的演化过程由此确定为
 $\dot{F}_{i}^{\text{dis}}=D_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{dis}}F_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i}}}^{\text{dis}},i=1,2,\cdots ,M$ (111)

4.5 本构理论总结

5 结论

K-O随动硬化模型与O-W随动硬化模型的共性在于Heaviside阶跃函数的引入，这种阶跃性质使得背应力分量始终在相应的临界面内演化，故在数值积分过程中背应力的积分需要利用径向返回映射法，同时考虑到对数共旋率以及多种构型的引入，本文本构理论的数值化过程会相对复杂一些.

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A NEW FORMULATION OF CONSTITUTIVE MODEL FOR HYPERELASTIC-CYCLIC PLASTICITY
Meng Lingkai, Zhou Changdong, Guo Kunpeng, Zhang Xiaoyang
School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China
Abstract: Until now, a number of classical hyperelastic-finite plasticity constitutive models have been proposed. However, most of them are based on the classical Armstrong-Frederick kinematic hardening rule in consideration of the complexity brought by the introduction of the intermediate configuration in the hyperelasticity theory. Hence, based on the existed constitutive theories, the methodology of Lion decomposition theory was extended utilizing the notion of the multi-mechanism process and clearly put forward the conception of the multi-intermediate configuration. Furthermore, the classical concept of the objectivity in the continuum mechanics for better application to the hyperelasticity theory was generalized and then a new hyperelastic-finite plastic constitutive model was proposed. The new constitutive model not only meets the thermal dynamic laws but also can incorporate several classical kinematic hardening rules which were usually adopted in cyclic plasticity of infinitesimal deformation theory (e.g. the A-F model, Chaboche model, O-W model and the K-O model, etc.). Therefore, this model corresponding to finite deformation problems contains two typical characteristics adopted by infinitesimal deformation theory: the additive decomposition property and step mutation feature of the backstress on the critical surface. Thus, the present model can be treated as parallel to the corresponding form in the small deformation case. Finally, the situation accounting for Karim-Ohno kinematic hardening rule is under specific consideration and compared with the hypoelasticity constitutive model.
Key words: constitutive model    objectivity    decomposition theory of deformation gradient    kinematic hardening rule    cyclic plasticity    structural analysis