﻿ 考虑夹杂相互作用的复合陶瓷夹杂界面的断裂分析
«上一篇
 文章快速检索 高级检索

 力学学报  2016, Vol. 48 Issue (1): 154-162  DOI: 10.6052/0459-1879-14-399 0

### 引用本文 [复制中英文]

[复制中文]
Fu Yunwei, Zhang Long, Ni Xinhuay, Liu Xiequan, Yu Jinfeng, Chen Cheng. INTERFACE CRACKING ANALYSIS WITH INCLUSIONS INTERACTION IN COMPOSITE CERAMIC[J]. Chinese Journal of Ship Research, 2016, 48(1): 154-162. DOI: 10.6052/0459-1879-14-399.
[复制英文]

### 文章历史

2014-12-10 收稿
2015-10-21 录用
2015-11-25 网络版发表

1. 军械工程学院火炮工程系, 石家庄 050003;
2. 军械工程学院车辆与电气工程系, 石家庄 050003;
3. 军械工程学院基础部, 石家庄 050003;
4. 中国人民解放军78638部队, 什邡 618400

1 相互作用直推估下的等效加载应力场

 图 1 含随机三相胞元的四相模型 Fig. 1 The four phase model with random three-phase-cell

 ${\pmb \sigma}_0 = ({\pmb I}-{\pmb \varOmega}_{i} {\pmb H})^{- 1}{\pmb \sigma}_\infty$ (1)

 ${\pmb \sigma}_i = \left( {{\pmb I}+{\pmb \varOmega}_i {\pmb H}_i } \right)^{ - 1}{\pmb \sigma}_0$ (2)

 ${\pmb H }= \Big({\pmb I} - \sum {\pmb \varOmega}_{i} {\pmb H}_{i}^{\rm d} \Big )^{ - 1}{\pmb H}^{\rm d}$ (3)

 ${\pmb H}_{i}^{\rm d} = \sum f_{i} ({\pmb H}_{i}^{ - 1}+ {\pmb \varOmega}_{i} )^{ - 1}$ (4)

 ${\pmb H}^{\rm d} = \sum {\pmb H}_i^{\rm d}$ (5)

 ${\pmb C} = ( {\pmb S}_0 +{\pmb H})^{-1}$ (6)

 ${\pmb \sigma}_1^{\rm t} = - {\pmb \varOmega}_1 \left( {{\pmb I} + {\pmb \varOmega}_1{\pmb H}_1 } \right)^{ - 1}{\pmb \varepsilon }_{10}^{\rm t} + \left( {{\pmb I} + {\pmb \varOmega}_1 {\pmb H}_1 } \right)^{ - 1}{\pmb \sigma}_0$ (7)

 ${\pmb \sigma}_0 ={\pmb \varOmega} \left( {{\pmb I} - {\pmb \varOmega} {\pmb H}}\right)^{-1}{\pmb \varepsilon }$ (8)

 ${\pmb \varepsilon} = \left[{\pmb I } + \left( f_1 {\pmb \omega}_1 + f_2 {\pmb \omega}_2 \right) \left( {{\pmb I} - {\pmb \varOmega} {\pmb H}} \right)^{ - 1} \right]^{ - 1} \cdot \left( {f_1 {\pmb \omega}_1 {\pmb \varepsilon}_{10}^{\rm t} + f_2 {\pmb \omega}_2 {\pmb \varepsilon}_{20}^{\rm t} } \right)$ (9)

 ${\pmb \varepsilon} = \left[{{\pmb I} + f_1 {\pmb \omega}_1 \left( {{\pmb I} -{\pmb \varOmega} {\pmb H}} \right)^{ - 1}} \right]^{-1}f_1 {\pmb \omega}_1{\pmb \varepsilon}_{10}^{\rm t}$ (10)

 ${\pmb \sigma }'_\infty = {\pmb \omega }_1^{ - 1} {\pmb \sigma}_{i} = ({\pmb I} - {\pmb \varOmega}_i{\pmb H})^{-1 }{\pmb \sigma}_\infty$ (11)

 ${\pmb \sigma}_\infty ^{\rm t} = {\pmb \omega}_1^{ - 1} {\pmb \sigma}_1^{\rm t}$ (12)

 ${\pmb \sigma}_{\rm eff}^\infty = {\pmb \sigma }'_\infty + {\pmb \sigma}_\infty ^{\rm t}$ (13)

2 界面裂纹开裂分析

 $K = K_{\rm I} - {\rm i}K_{\rm II} = \dfrac{1}{2}\sqrt {\dfrac{\pi}{a\sin \beta }} {\rm e}^{ - \lambda \beta }{\rm e}^{ -{\rm i}(\psi + \tfrac{\beta }{2})} \cdot \\ \qquad \left( { - {\rm i}a {\rm e}^{{\rm i}\beta }P_1 + P_0 +\dfrac{{\rm i}P_{ - 1} }{a{\rm e}^{ - {\rm i}\beta }} - \dfrac{P_{ - 2}}{a^2{\rm e}^{2{\rm i}\beta }}} \right)$ (14)

 $P_1 = P_{\rm 1r} + {\rm i}P_{\rm 1m} \\ P_{\rm 1r} = \dfrac{m(1 + \alpha )}{ 1 + m }\dfrac{(N + T)(1 + m) - (N - T)F\cos 2\phi }{\alpha (1 + 2m) - {\rm e}^{ - 2\lambda \beta }(\cos \beta +2\lambda \sin \beta )} \\ P_{\rm 1m} = - \dfrac{N - T}{1 + m}\dfrac{m(1 + \alpha )F\sin2\phi }{\alpha +{\rm e}^{- 2\lambda \beta }(\cos \beta + 2\lambda \sin \beta )} \\ F =(\cos \beta - 2\lambda \sin \beta )^2 + 2\lambda \sin 2\beta -\\ \qquad \lambda ^2(1 -\cos 2\beta ) - \dfrac{1 + 3\cos 2\beta }{4} \\ P_0 = {\rm i}aP_1 (\cos \beta + 2\lambda\sin \beta ) \\ P_{ - 2} = {\rm i}ma^3{\rm e}^{2\lambda \beta }\dfrac{1 + \alpha }{1 +m}(N - T){\rm e}^{2{\rm i}\phi } \\ P_{ - 1} = \dfrac{{\rm i}P_{ - 2} }{a}(\cos \beta -2\lambda \sin \beta ) \\ \alpha = \dfrac{E_0 + E_1 \kappa _0 }{E_1 + E_0 \kappa _1 } ,\ \ \lambda = \dfrac{ - \ln \alpha }{2\pi } \\ m = \dfrac{E_0 (1 + \kappa _1 )}{E_1 (1+ \kappa _0 )} ,\ \ \psi = \lambda \ln (2a\sin \beta ) \\ \kappa _0 = \dfrac{3 - v_0}{1 + v_0 } ,\ \ \kappa _1 = \dfrac{3 - v_1 }{1 + v_1 } \ \ \hbox{(平面应力)} \\ \kappa_0 = 3 - 4v_0 ,\ \ \kappa _1 = 3 - 4v_1 \ \ \hbox{ (平面应变)}$

 图 2 远场加载下的界面弧形裂纹示意图[18] Fig. 2 Arc crack along the interface subjected to far field loading[18]

 $K = N\sqrt { \pi a\sin \beta } P$ (15)

 $P = \dfrac{(1 + \alpha )m(1 + 2{\rm i}\lambda ){\rm e}^{ - \lambda \beta }{\rm e}^{ - {\rm i}(\psi + \tfrac{\beta }{2})}}{\alpha (1 + 2m) - {\rm e}^{ - 2\lambda \beta}(\cos \beta + 2\lambda \sin \beta )}$ (16)

 $K_{\rm I} = N\sqrt {\pi a\sin \beta } Q$ (17)

 $Q = \dfrac{1}{2}\left [(D-1)\cos \dfrac{\beta }{2} +{\rm i} (D+1)\sin \dfrac{\beta }{2} \right]$ (18)

 $K = K_\infty + K_{\rm t}$ (19)

3 界面裂纹开裂的极限外载应力

 ${\pmb \sigma}_{\rm r} = {\pmb \sigma}'_\infty + {\pmb \sigma} _\infty ^{\rm t} = {\pmb L}{\pmb \sigma}_\infty +{\pmb \sigma}_\infty ^{\rm t}$ (20)

 ${\pmb \sigma}_\infty = (\sigma _\infty ,0,0)^{\rm T}$ (21)

 $\left.\begin{array}{l} \sigma _{\rm r11} = L_{11} \sigma _\infty + \sigma _{\infty 11}^{\rm t} \\ \sigma _{\rm r22} = L_{21} \sigma _\infty + \sigma _{\infty 22}^{\rm t} \end{array} \right \}$ (22)

 $\left. \begin{array}{l} {\sigma _{\rm{R}}} = {\sigma _{{\rm{r22}}}} = {L_{21}}{\sigma _\infty } + \sigma _{\infty 22}^{\rm{t}}\\ {\sigma _{\rm{T}}} = ({L_{11}} - {L_{21}}){\sigma _\infty } + (\sigma _{\infty 11}^{\rm{t}} - \sigma _{\infty 22}^{\rm{t}}) \end{array} \right\}$ (23)

 $K = \sigma _{\rm R} P\sqrt { \pi a\sin \beta } + \sigma _{\rm T}Q\sqrt {\pi a\sin \beta }$ (24)

 $G = \dfrac{(b_0 + b_1 )K\bar {K}}{16\cosh ^2 (\pi \lambda) }$ (25)

 $G_{\rm c} = \dfrac{(b_0 + b_1 ) \pi a\sin \beta }{16\cosh ^2 \pi \lambda } \cdot \\ \qquad \left[{\sigma _{\rm R}^2 P\bar {P} + \sigma _{\rm R} \sigma_{\rm T} (P\bar {Q} + \bar {P}Q) + \sigma _{\rm T}^2 Q\bar {Q}} \right]$ (26)

 $g_1 \sigma _\infty ^2 + g_2 \sigma _\infty + g_3 = 0$ (27)

 $g_1 = P\bar {P}L_{21}^2 + (P\bar {Q} + \bar {P}Q)L_{21} (L_{11} - L_{21} ) + \\ \qquad Q\bar {Q}(L_{11} - L_{21} )^2 \\ g_2 = (P\bar {Q} + \bar {P}Q)[(L_{11} - L_{21} )\sigma _{\infty 22}^{\rm t} +L_{21} (\sigma _{\infty 11}^{\rm t} - \sigma _{\infty 22}^{\rm t} )] + \\ \qquad 2P\bar {P}L_{21} + 2Q\bar{Q}(L_{11} - L_{21} )(\sigma _{\infty 11}^{\rm t} - \sigma _{\infty 22}^{\rm t} ) \\ g_3 = P\bar {P}(\sigma _{\infty 22}^{\rm t} )^2 + (P\bar {Q} + \bar {P}Q)\sigma_{\infty 11}^{\rm t} (\sigma _{\infty 11}^{\rm t} - \sigma _{\infty 22}^{\rm t} ) + \\ \qquad (\sigma_{\infty 11}^{\rm t} - \sigma _{\infty 22}^{\rm t} )^2 - \dfrac{16[\cosh (\pi \lambda )]^2G_{\rm c} }{(b_0+ b_1 ) \pi a\sin \beta }$

 $\sigma _{\rm c} = \dfrac{\sqrt {g_2^2 - 4g_1 g_3 } - g_2 }{2g_1 }$ (28)

4 数值结果及讨论

 图 3 有限元材料模型和裂纹张开情况 Fig. 3 FEM model of material and its opening interface crack

 图 4 不同刚度夹杂中平均应力和等效外载应力 Fig. 4 Average stress in inclusion and effective applied loading with different inclusion stiffness

 图 5 等效外载应力随界面裂纹含量增加而增大 Fig. 5 Effective applied loading increased with interface crack volume fraction

 图 6 简单拉伸下应力强度因子和应变能释放率随脱粘角的 变化规律 Fig. 6 Variation of stress intensity factor (SIF) and strain energy release rate with crack angle β under simple tension

 图 7 不同刚度夹杂在简单拉伸下应力强度因子和应变能释放率变化规律 Fig. 7 Variation of stress intensity factor (SIF) and strain energy release rate under simple tension of different inclusion stiffness

 图 8 残余应力下应力强度因子和应变能释放率随 脱粘角的变化规律 Fig. 8 Variation of stress intensity factor (SIF) and strain energy release rate with crack angle β under residual stress

 图 9 不同刚度夹杂材料的界面裂纹开裂极限应力 Fig. 9 Critical applied stress for interface cracking in different inclusion stiffness

 图 10 界面裂纹开裂极限应力随夹杂直径的变化规律 Fig. 10 Critical applied stress for interface cracking variation with inclusion diameter

5 结 论

 1 Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc Royal Soc, 1957, A241: 376-396 2 Hill R. A self-consistent mechanics of composite materials. J Mech Phys Solids, 1965, 13: 213-222 3 Budiansky B. On the elastic modules of some heterogeneous materials. J Mech Phys Solids, 1965, 13: 223-227 4 Mori T, Tanaka K. Average stress in matrix and average energy of materials with misfitting inclusion. Act Metal, 1973, 21: 571-574 5 Zheng QS, Du DX. An explicit and universally applicable estimate for the e ective properties of multiphase composites which accounts for inclusion distribution. J Mech Phys Solids, 2001, 49(11): 2765-2788 6 Du DX, Zheng QS. A further exploitation on the e ective selfconsistent scheme and IDD estimate for the e ective properties of multiphase composites which accounts for inclusion distribution. Acta Mech, 2001, 157(1-4): 61 7 Kang H, Milton GW. Solutions to the Pólya-Szegö conjecture and the weak Eshelby conjecture. Archive for Rational Mechanics and Analysis, 2008, 188(1): 93-116 8 Liu LP. Solutions to the Eshelby conjectures. Proceedings of the Royal Society of London A, 2008, 464: 573-594 9 Zou WN, He QC, Huang MJ, et al. Eshelby's problem of nonelliptical inclusions. Journal of the Mechanics and Physics of Solids,2010, 58: 346-372 10 Huang MJ, Wu P, Guan GY, et al. Explicit expressions of the Eshelby tensor for an arbitrary 3D weakly non-spherical inclusion. Acta Mech, 2011, 217: 17-38 11 Xu BX,Wang MZ. Special properties of Eshelby tensor for a regular polygonal inclusion. Acta Mech Sinica, 2005, 21: 267-271 12 Shodja HM. Shokrolahi-Zadeh B. Ellipsoidal domains: piecewise nonuniform and impotent eigenstrain fields. J Elasticity, 2007, 86:1-18 13 Jasiuk I, Tsuchida E, Mura T. The sliding inclusion under shear. Int J Solids Struct, 1987, 23(10): 1373-1385 14 Mura T, Jasiuk I, Tsuchida B. The stress field of a sliding inclusion. Int J Solids Struct, 1985, 21(12): 1165-1179 15 Xu BX, Mueller R, Wang MZ. The Eshelby property of sliding inclusions. Arch Appl Mech, 2011, 81: 19-35 16 Bonfoh N, Hounkpati V, Sabar H. New micromechanical approach of the coated inclusion problem: Exact solution and applications. Computational Materials Science, 2012, 62: 175-183 17 Duan HL, Wang J, Huang ZP, et al. Eshelby formalism for nanoinhomogeneities. Mathematical, Physical and Engineering Sciences,2005, 461(2062): 3335-3353 18 Prasad PBN, Simha KRY. Interface crack around circular inclusion: SIF, kinking, debonding energetics. Engineering Fracture Mechanics,2003, 70: 285-307 19 Toya M. A crack along the interface of a circular inclusion embedded in an infinite solid. J Mech Phys Solids, 1974, 22: 325-348 20 Chao R, Laws N. Closure of an arc crack in an isotropic homogeneous material due to uniaxial loading. Q Jl Mech Appl Math, 1992,45: 629-640 21 Chen YZ, Chen RS. Interaction between curved crack and elastic inclusion in an infinite plate. Archive of Applied Mechanics, 1997, 67:566-575 22 Chen JK,Wang GT, Yu ZZ, et al. Critical particle size for interfacial debonding in polymer/nanoparticle composites. Composites Science and Technology, 2010, 70: 861-872 23 Moorthy S, Ghosh S. A Voronoi cell finite element model for particle cracking in elastic-plastic composite materials. Comput Methods Appl Mech Engrg, 1998, 151: 377-400 24 Caballero A, López CM, Carol I. 3D meso-structural analysis of concrete specimens under uniaxial tension. Comput. Methods Appl Mech Engrg, 2006, (195): 7182-7195 25 Wang D, Zhao J, Zhou Yonghui, et al. Extended finite element modeling of crack propagation in ceramic tool materials by considering the microstructural features. Computational Materials Science,2013, 77: 236-244 26 England AH. An arc crack around a circular elastic inclusion. J Appl Mech, 1966, 32: 637-640 27 郭荣鑫, Lormand G, 李俊昌. 夹杂物问题应力场的数值计算. 昆明理工大学学报(理工版), 2004, 29(3): 51-55 (Guo Rongxin, Lormand G, Li Junchang. Numerical calculation of the stress field for the inclusion problem. Journal of Kunming University of Science and Technology(Science and Technology), 2004, 29(3): 51-55 (in Chinese)) 28 Shodja HM, Rad IZ, Soheilifard R. Interacting cracks and ellipsoidal inhomogeneities by the equivalent inclusion method. Journal of the Mechanics and Physics of Solids, 2003, 51: 945-960 29 Li ZH, Yang LH. The application of the Eshelby equivalent inclusion method for unifying modulus and transformation toughening. International Journal of Solids and Structures, 2002, 39: 5225-5240 30 Othmani Y, Delannay L, Doghri I. Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law. International Journal of Solids and Structures, 2011, 48: 3326-3335 31 Lee HK, Pyo SH. Multi-level modeling of e ective elastic behavior and progressive weakened interface in particulate composites. Composites Science and Technology, 2008, 68: 387-397 32 Hong T. Sti ness properties of particulate composites containing debonded particles. International Journal of Solids and Structures,2010, 47: 2191-2200 33 Mura T. Micromechanics of Defects in Solids. 2nd edn. Martinus Nijho Publishers, 1987
INTERFACE CRACKING ANALYSIS WITH INCLUSIONS INTERACTION IN COMPOSITE CERAMIC
Fu Yunwei1, Zhang Long1, Ni Xinhuay2, Liu Xiequan3, Yu Jinfeng3, Chen Cheng4
1. Department of Artillery Engineering, Ordnance Engineering College, Shijiazhuang 050003, China;
2. Department of Vehicle and Electric Engineering, Ordnance Engineering College, Shijiazhuang 050003, China;
3. Department of Basic Course, Ordnance Engineering College, Shijiazhuang 050003, China;
4. Unit No.78638 of PLA, Shifang 618400, China
Abstract: Micro strain and stress field and the average field is significantly influenced by the interaction of inclusions of high volume fraction, and the composite mechanical properties changes a lot by the introduced inclusions. To reduce the error of using the theoretical model with single inclusion in real multi-inclusion-material property prediction, an inclusion interface cracking model is established based on the interaction direct derivative (IDD) estimate. Average e ective stress field is obtained under applied loading and residual stress based on the IDD estimate, then the stress intensity factor (SIF) of interface arc-crack around circular inclusion under the stress boundary condition is calculated. The critical applied stress is evaluated according to the e ective stress field and the interface arc-crack SIF, and the influence of inclusion elastic modulus, volume fraction, residual stress, and inclusion size are analysed. The result indicates that the method is e ective. Residual stress has remarkable influence to interface cracking; the critical applied stress is not monotonic and complicated with the inclusion sti ness variation when the residual stress is large, while the critical applied stress is increasing with the soft inclusion decreasing the sti ness and the sti inclusion increasing the sti ness when the residual stress is small; large size inclusion decreases the critical applied stress and detrimental to composite strength.
Key words: circular inclusion    interface crack    interaction    the interaction direct derivative estimate    crack stress