﻿ 材料构型力学及其在复杂缺陷系统中的应用
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 力学学报  2015, Vol. 47 Issue (2): 197-214  DOI: 10.6052/0459-1879-14-240 0

### 引用本文 [复制中英文]

[复制中文]
Li Qun. MATERIAL CONFIGURATIONAL MECHANICS WITHAPPLICATION TO COMPLEX DEFECTS[J]. Chinese Journal of Ship Research, 2015, 47(2): 197-214. DOI: 10.6052/0459-1879-14-240.
[复制英文]

### 文章历史

2014-08-18收稿
2014-09-29录用
2014-11-28网络版发表

1 材料构型力学的基本概念

 $\Lambda ({x_i},{u_{k,j}}) = - W({x_i},{u_{k,j}})$ (1)

1.1 $J_{k}$积分及其构型应力

 $\nabla (\Lambda ) = - {(W)_{,i}} = - {\left( {\frac{{\partial W}}{{\partial {x_i}}}} \right)_{{\rm{expl}}}} - {\sigma _{kj}}{u_{k,ji}}$ (2)

 ${b_{ji}} = W{\delta _{ji}} - {\sigma _{jk}}{u_{k,i}}$ (3)

 ${R_i} = - {\left( {\frac{{\partial W}}{{\partial {x_i}}}} \right)_{{\rm{expl}}}}$ (4)

 ${b_{ji,j}} + {R_i} = 0$ (5)

 $J = {J_1} = \oint_\Gamma {{b_{j1}}{n_j}} ds = \oint_\Gamma {(W{n_1} - {\sigma _{jk}}{u_{k,1}}{n_j}} )ds$ (6)
 ${J_2} = \oint_\Gamma {{b_{j2}}{n_j}} ds = \oint_\Gamma {(W{n_2} - {\sigma _{jk}}{u_{k,2}}{n_j}} )ds$ (7)
 图 1 Jk, M, L 积分路径 Fig.1 Jk, M, L integrals

1.2 M积分及其构型应力

 $\begin{array}{l} \nabla \cdot (\Lambda x) = - {(W{x_i})_{,i}} = - {x_{i,i}}W - \\ {\left( {\frac{{\partial W}}{{\partial {x_i}}}} \right)_{{\rm{expl}}}}{x_i} - \frac{{\partial W}}{{\partial {u_{k,j}}}}{u_{k,ji}}{x_i} \end{array}$ (8)

 ${M_j} = W{x_i}{\delta _{ij}} - {\sigma _{jk}}{u_{k,i}}{x_i} + \frac{{2 - m}}{2}{\sigma _{ji}}{u_i}$ (9)

 $R = - {\left( {\frac{{\partial W}}{{\partial {x_i}}}} \right)_{{\rm{expl}}}}{x_i}$ (10)

 ${M_{j,j}} + R = 0$ (11)

 $M = \oint_\Gamma {\left( {W{x_i}{n_i} - {\sigma _{jk}}{u_{k,i}}{x_i}{n_j}} \right)} d\Gamma$ (12)

1.3 L积分及其构型应力

 $\begin{array}{l} \nabla \times (\Lambda x) = - {e_{mij}}{(W{x_j})_{,i}} = \\ [{(\frac{{\partial W}}{{\partial {x_i}}})_{{\rm{expl}}}}{x_j} + \frac{{\partial W}}{{\partial {u_{k,l}}}}{u_{k,li}}{x_j}] \end{array}$ (13)

 ${L_{ml}} = {e_{mij}}(W{x_j}{\delta _{il}} + {\sigma _{il}}{u_j} - {\sigma _{kl}}{u_{k,i}}{x_j})$ (14)

 ${R_m} = - {e_{mij}}\left[{{{(\frac{{\partial W}}{{\partial {x_i}}})}_{{\rm{expl}}}}{x_j} + \left( {{\sigma _{ik}}{u_{j,k}} - {\sigma _{kj}}{u_{k,i}}} \right)} \right]$ (15)

 ${L_{ml,l}} + {R_m} = 0$ (16)

 $L = {L_3} = \oint_\Gamma {{e_{3ij}}(W{x_j}{n_i} + {\sigma _{il}}{u_j}{n_l} - {\sigma _{kl}}{u_{k,i}}{x_j}{n_l}){\rm{d}}\Gamma }$ (17)

2 基于构型力学的材料屈服、断裂、破坏新准则 2.1 基于材料构型力学(材料构型力学)的屈服准则

 $\left. {\begin{array}{*{20}{l}} {{I_1} = {b_{ii}}}\\ {{I_2} = \frac{1}{2}({b_{ii}}{b_{jj}} - {b_{ij}}{b_{ji}})}\\ {{I_3} = {\rm{det}}({b_{ij}})} \end{array}} \right\}$ (18)

 $\left| {\begin{array}{*{20}{c}} {{b_{11}} - {b_N}}&{{b_{12}}}&{{b_{13}}}\\ {{b_{21}}}&{{b_{22}} - {b_N}}&{{b_{23}}}\\ {{b_{31}}}&{{b_{32}}}&{{b_{33}} - {b_N}} \end{array}} \right| = 0$ (19)

 $b_N^3 - {I_1}b_N^2 - {I_2}{b_N} - {I_3} = 0$ (20)

 ${b_{{\rm{I}},{\rm{II}}}} = \frac{{ - {b_{12}} + {b_{21}}}}{2} \pm \sqrt {{{(\frac{{ - {b_{12}} - {b_{21}}}}{2})}^2} + b_{11}^2}$ (21)

 $\left. {\begin{array}{*{20}{l}} {\left| {{b_{\rm{I}}} - {b_{{\rm{II}}}}} \right| = {b_{\rm{s}}},\;{b_{\rm{I}}} \ge {b_{{\rm{III}}}} = 0 \ge {b_{{\rm{II}}}}}\\ {\left| {{b_{\rm{I}}} - {b_{{\rm{III}}}}} \right| = {b_{\rm{s}}},{b_{\rm{I}}} \ge {b_{{\rm{II}}}} \ge {b_{{\rm{III}}}} = 0}\\ {\left| {{b_{{\rm{III}}}} - {b_{{\rm{II}}}}} \right| = {b_{\rm{s}}},{b_{{\rm{III}}}} = 0 \ge {b_{\rm{I}}} \ge {b_{{\rm{II}}}}} \end{array}} \right\}$ (22)

 ${b_{\rm{v}}} = \sqrt {\frac{1}{2}[{{({b_{\rm{I}}} - {b_{{\rm{II}}}})}^2} + {{({b_{{\rm{II}}}} - {b_{{\rm{III}}}})}^2} + {{({b_{{\rm{III}}}} - {b_{\rm{I}}})}^2}]} = {b_{\rm{s}}}$ (23)

 ${b_{\rm{v}}} = b_{12}^2 + {b_{12}}{b_{21}} + b_{21}^2 + 3b_{11}^2$ (24)

 图 2 含斜裂纹无限大板的单轴拉伸 Fig.2 An inclined crack subjected to tension

 图 3 不同屈服准则预测Ⅰ-Ⅱ复合型裂纹的裂尖塑性屈服区 Fig.3 Predictions of the elastic-plastic boundaries by the MCM yield criterion

2.2 基于材料构型力学的断裂准则

(1)裂纹初始起裂扩展的方向定义为：裂纹尖端至材料构型力学屈服准则所预测的弹-塑性边界最小距离的方向.

(2)裂纹起裂扩展定义为: 当弹-塑性边界包围的塑性区总势能改变量达到某个门槛值时,裂纹开始失稳扩展.

 ${\left. {\frac{{\partial {r_{{\rm{MCM}}}}}}{{\partial \theta }}} \right|_{\theta = {\theta _0}}} = 0,\;\;{\left. {\frac{{{\partial ^2}{r_{{\rm{MCM}}}}}}{{\partial {\theta ^2}}}} \right|_{\theta = {\theta _0}}} > 0$ (25)

 $P = \int_{{\Gamma _{\rm{A}}}} {{b_{\rm{v}}}{\rm{d}}{\Gamma _{\rm{A}}}} = {P_{\rm{C}}}$ (26)

 图 4 材料构型力学断裂准则和其他断裂准则对Ⅰ-Ⅱ复合型裂纹起裂角 θ0 的预测对比 Fig.4 Prediction of initial fracture angle θ0 versus inclination angle β of cracks by the MCM fracture criterion

 图 5 材料构型力学断裂准则和其他断裂准则对Ⅰ-Ⅱ复合型裂纹断裂载荷的预测对比 Fig.5 Prediction of critical (KⅠ/KⅠC, KⅡ/KⅠC)-diagram by the MCM fracture criterion

2.3 基于M积分描述材料复杂缺陷最终失效的破坏准则

 $M \ge {M_{\rm{C}}}$ (27)

 $\varepsilon /{\varepsilon _0} = \sigma /{\sigma _0} + \rho {(\sigma /{\sigma _0})^n}$ (28)
 图 6 弹塑性材料内含复杂缺陷问题 Fig.6 Multi-defects in an elastic-plastic material

 图 8 不同多裂纹缺陷构型的临界 M 积分 Fig.8 The critical values of the M-integral corresponding to the critical external load at failure for variable microvoids patterns

 $\Pi = \frac{{ME}}{{{\sigma ^2}{A_{\rm{D}}}}}$ (29)

 $\Pi \ge {\Pi _{\rm{C}}}$ (30)

 图 7 用于有限元模拟的复杂缺陷构型,该缺陷范围内随机分布一定 数量的微裂纹(a)9条裂纹;(b)16条裂纹;(c)25条裂纹;(d)36条裂 纹;(e)49条裂纹;(f)64条裂纹;(g)81条裂纹;(h)100条裂纹; (i)121条裂纹;(j)144条裂纹;(k)169条裂纹 Fig.7 Schematic of multiple microcracks randomly distributed in the local damage zone (a)9 cracks;(b)16 cracks;(c)25 cracks;(d)36 cracks;(e)49 cracks;(f)64 cracks;(g)81 cracks;(h)100 cracks; (i)121 cracks;(j)144 cracks;(k)169 cracks.
3 材料构型力学基本量的相关实验测量

 图 9 材料构型力学基本量的实验测量 Fig.9 Experimental measurement within material configurational mechanics

 图 10 测量得到并经过平滑后的面内位移场 Fig.10 The measured values of the displacements after smooth technique

 $\left. {\begin{array}{*{20}{l}} {{\sigma _{xx}} = E({\varepsilon _{xx}} + \upsilon {\varepsilon _{yy}})/(1 - {\upsilon ^2})}\\ {{\sigma _{yy}} = E({\varepsilon _{yy}} + \upsilon {\varepsilon _{xx}})/(1 - {\upsilon ^2})}\\ {{\tau _{xy}} = E{\gamma _{xy}}/(2 + 2\upsilon )} \end{array}} \right\}$ (31)

 $\begin{array}{l} {\varepsilon _{ij}}/{\varepsilon _0} = (1 + \upsilon ){\sigma _{ij}}/{\sigma _0} - {\delta _{ij}}\upsilon {\sigma _{kk}}/{\sigma _0} + \\ \qquad \frac{3}{2}\alpha {(\bar \sigma /{\sigma _0})^{n - 1}}{S_{ij}}/{\sigma _0} \end{array}$ (32)

 图 11 材料构型应力实验测量数据 Fig.11 The evaluated material configurational stresses

4 材料构型力学在纳米损伤力学中的应用

 图 12 无限大平面含多个纳米尺寸圆形夹杂 Fig.12 An infinite plane containing multiple nano-sized circular inclusions

 图 13 不同纳米孔排列模式下薄膜材料的应力分布 Fig.13 Distribution of the von Mises stress in nanoporous membrane

 图 14 不同纳米孔洞尺寸、排列构型对$\Pi$参数得影响 Fig.14 Variable tendencies of $\Pi$ parameter against the size of nanopores for three patterns of nanoporous membranes

5 材料构型力学在铁电材料断裂力学中的应用

 $\left. {\begin{array}{*{20}{l}} {\varepsilon _{ij}^{\rm{T}} = \varepsilon _{ij}^{\rm{L}} + \varepsilon _{ij}^{\rm{R}}}\\ {D_i^{\rm{T}} = D_i^{\rm{L}} + P_i^{\rm{R}}} \end{array}} \right\}$ (33)

 $dh = \underbrace {{\sigma _{ij}}{\rm{ d}}\varepsilon _{ij}^{\rm{L}} - D_i^{\rm{L}}{\rm{ d}}{E_i}}_{{\rm{d}}{h_0}} + \underbrace {{\sigma _{ij}}{\rm{ d}}\varepsilon _{ij}^{\rm{R}} - P_i^{\rm{R}}{\rm{d}}{E_i}}_{d{h_{\rm{R}}}}$ (34)

 $\left. {\begin{array}{*{20}{l}} {{\sigma _{ij}} = \frac{{\partial {h_0}}}{{\partial \varepsilon _{kl}^{\rm{L}}}} = {c_{ijkl}}\left( {{\varepsilon _{kl}} - \varepsilon _{kl}^{\rm{R}}} \right) - {e_{kij}}{E_k}}\\ {D_i^{\rm{L}} = {D_i} - P_i^{\rm{R}} = - \frac{{\partial {h_0}}}{{\partial {E_i}}} = {e_{ikl}}\left( {{\varepsilon _{kl}} - \varepsilon _{kl}^{\rm{R}}} \right) + {\kappa _{ij}}{E_j}} \end{array}} \right\}$ (35)

 $\left. {\begin{array}{*{20}{l}} {{\sigma _{ij,j}} + {b_i} = 0}\\ {D_{i,i}^{\rm{L}} + P_{i,i}^{\rm{R}} - {\omega _V} = 0} \end{array}} \right\}$ (36)

 $\sum\nolimits_{_{kj}} { = {h_0}{\delta _{kj}}} - {\sigma _{ij}}{u_{i,k}} - {D_j}{\phi _k}$ (37)

 $\sum\nolimits_{_{kj,j}} { + {g_k}} = 0$ (38)

 $\begin{array}{l} {g_k} = {\sigma _{ij}}\varepsilon _{ij,k}^{\rm{R}} + P_j^{\rm{R}}{\phi _{,kj}} - {b_i}{u_{i,k}} + {\omega _V}{\phi _{,k}} + {\sigma _{ij}}\varepsilon _{ij,k}^{\rm{L}} + \\ \qquad D_j^{\rm{L}}{\phi _{,kj}} - {h_{0,k}} = {\sigma _{ij}}\varepsilon _{ij,k}^{\rm{R}} + P_j^{\rm{R}}{\phi _{,kj}} - \\ \qquad {b_i}{u_{i,k}} + {\omega _V}{\phi _{,k}} - {\left( {\frac{{\partial {h_0}}}{{\partial {x_k}}}} \right)_{\exp }} \end{array}$ (39)

 ${g_k} = {\sigma _{ij}}\varepsilon _{ij,k}^{\rm{R}} + P_j^{\rm{R}}{\phi _{jk}}$ (40)

 ${J_k} = \int_\Gamma {\left( {{h_0}{\delta _{kj}} - {\sigma _{ij}}{u_{i,k}} - {D_j}{\phi _{,k}}} \right){n_j}ds}$ (41)

 图 15 铁电材料非线性断裂中全局和局部 J 积分 Fig.15 The global and local J integral within the nonlinear ferroelectric fracture mechanics

 $\begin{array}{l} {{\tilde J}_k} = \mathop {\lim }\limits_{\varepsilon \to 0} \int_{{\Gamma _\varepsilon }} {{{\sum\nolimits_{_{kj}} n }_j}ds} = \int_\Gamma {\sum\nolimits_{_{kj}} {{n_j}} ds} - \int_V {\sum\nolimits_{_{kj,j}} d V} = \\ \qquad \int_\Gamma {\left( {{h_0}{\delta _{kj}} - {\sigma _{ij}}{u_{i,k}} - {D_j}{\phi _k}} \right){n_j}ds} + \int_V {{g_k}dV} = \\ \qquad {J_k} + \int_V {{g_k}dV} \end{array}$ (42)

 图 16 铁电多晶材料的构型力计算模型 Fig.16 Computational model of material configurational forces in ferroelectric polycrystals
6 总结与展望

(1) 完善了材料构型力学理论,建立了构型力和复杂缺陷力学之间的桥梁,基于材料构型力和守恒积分分别提出材料屈服准则、断裂准则、复杂缺陷材料破坏准则,此类准则可用于各种形式的复杂缺陷材料损伤与结构完整性评估.

(2) 提出了一种基于DIC方法测量材料构型力学中基本物理量的实验方法. 此方法简便、有效、可行,填补材料构型力学发展所遇到的实验工作空白.

(3) 首次将构型力概念应用于纳米损伤力学,认清纳米表面效应对纳米孔洞扩展、聚合的影响; 基于材料微结构变化构架上的构型力理论,可成功描述纳米级别的损伤演化、尺度效应.

(4) 构型力及守恒积分在铁电材料裂纹问题中的应用,为多晶铁电材料断裂行为研究提供一种新思路,可以为智能结构材料的破坏问题提供理论支撑.

(5) 材料构型力学具有独立的理论框架体系,因而其基本的平衡方程、几何方程、本构方程可独立存在,对于某个力学问题,材料构型力学的基本解、动态问题等方面的研究工作仍需要深入研究. 材料构型力学在材料构型演化中的应用,如裂纹扩展、相变、位错滑移、物质质量迁移、甚至有限元网格划分的优化处理等方面具有广阔的应用空间.

(6) 材料构型力学的工程应用前景. 构型力学的优势在于处理材料缺陷演化问题. 利用构型力学研究材料的缺陷演化,国内外学者已经做了大量的研究工作. 但其直接的工程应用,可能还不像应力强度因子$K$、J积分、能量释放率、米泽斯准则等被工程界普遍接受. 但本文的研究工作表明,构型力学完全可以取代以往的破坏准则,对材料损伤与断裂进行预测和评估. 例如: (1) J积分,实际上就是构型力学的概念之一.(2)利用等效构型应力概念,可以取代米泽斯等效应力概念,对材料屈服进行预测. (3)利用基于M积分的$\Pi$概念,可以对多缺陷材料的失效进行预测. 可以预见,施加必要的实验验证以及工程推广,构型力学的工程应用前景广阔.

 $\nabla (\Lambda ) = - {\left( {\frac{{\partial W}}{{\partial {x_i}}}} \right)_{{\rm{expl}}}} - {\sigma _{kj}}{u_{k,ji}}$ (A1)

 ${b_{ji}} = W{\delta _{ji}} - {\sigma _{jk}}{u_{k,i}}$ (A2)

 ${R_i} = - {\left( {\frac{{\partial W}}{{\partial {x_i}}}} \right)_{{\rm{expl}}}}$ (A3)

 ${b_{ji,j}} + {R_i} = 0$ (A4)

 $\left. {\begin{array}{*{20}{l}} {J = {J_1} = \oint\limits_C {{b_{j1}}{n_j}} {\rm{d}}s = \oint\limits_C {(w{n_1} - {\sigma _{jk}}{u_{k,1}}{n_j}} ){\rm{d}}s}\\ {{J_2} = \oint\limits_C {{b_{j2}}{n_j}} {\rm{d}}s = \oint\limits_C {(w{n_2} - {\sigma _{jk}}{u_{k,2}}{n_j}} ){\rm{d}}s} \end{array}} \right\}$ (A5)

 $\nabla \cdot (\Lambda x) = - mW - {\left( {\frac{{\partial W}}{{\partial {x_i}}}} \right)_{{\rm{expl}}}}{x_i} - \frac{{\partial W}}{{\partial {u_{k,j}}}}{u_{k,ji}}{x_i}$ (A6)

 ${M_j} = W{x_i}{\delta _{ij}} - {\sigma _{jk}}{u_{k,i}}{x_i} + \frac{{2 - m}}{2}{\sigma _{ji}}{u_i}$ (A7)

 $R = - {\left( {\frac{{\partial W}}{{\partial {x_i}}}} \right)_{{\rm{expl}}}}{x_i}$ (A8)

 ${M_{j,j}} + R = 0$ (A9)

 $M = \oint_\Gamma {\left( {W{x_i}{n_i} - {\sigma _{jk}}{u_{k,i}}{x_i}{n_j} + \frac{{2 - m}}{2}{\sigma _{ji}}{u_i}{n_j}} \right)} {\rm{ d}}\Gamma$ (A10)

 $\nabla \times (\Lambda x) = - {e_{mij}}[{(\frac{{\partial W}}{{\partial {x_i}}})_{{\rm{expl}}}}{x_j} + \frac{{\partial W}}{{\partial {u_{k,l}}}}{u_{k,li}}{x_j}]$ (A11)

 ${L_{ml}} = {e_{mij}}(W{x_j}{\delta _{il}} + {\sigma _{il}}{u_j} - {\sigma _{kl}}{u_{k,i}}{x_j})$ (A12)

 ${R_m} = - {e_{mij}}[{(\frac{{\partial W}}{{\partial {x_i}}})_{{\rm{expl}}}}{x_j} + \left( {{\sigma _{ik}}{u_{j,k}} - {\sigma _{kj}}{u_{k,i}}} \right)]$ (A13)

 ${L_{ml,l}} + {R_m} = 0$ (A14)

 $L = {L_3} = \oint_\Gamma {{e_{3ij}}(W{x_j}{n_i} + {\sigma _{il}}{u_j}{n_l} - {\sigma _{kl}}{u_{k,i}}{x_j}{n_l})d\Gamma }$ (A15)