Gao, Y. and Oterkus, S., 2019. Ordinary state-based peridynamic modelling for fully coupled thermoelastic problems.Continuum Mechanics and Thermodynamics,31(4), pp.907-937.

地球物理局 地震波动力学实验室 热弹性大组 译“热弹性大组是地球物理局的排头兵、急先锋，所有的组都是为了这个组服务的，
而这个组也必须与时俱进，不断追求卓越，才能不辜负全局上下的期待......”声明：
# 译文优先满足本人研究需求
# 欢迎批评指正，禁止转载

目 录

石中居士：【文献选译】完全耦合热弹性问题的普通态基近场动力学模拟​zhuanlan.zhihu.com

1 引 言

由于近年来航空航天和机械工业的发展，机械和热冲击载荷已成为典型且重要的载荷类型。例如，飞机的燃气涡轮发动机机壳在极短的时间内会经历高达1700°C的温度上升^[1]。在这种载荷条件下的分析中，热机耦合(thermomechanical coupling)效应经常起着至关重要的作用，因此，在热场和结构场中都应考虑它们^[2]。不仅温度对变形的影响不可忽略，而且变形对温度场的影响也是不可忽略的。因此，在处理此类问题时，必须进行完全耦合热弹性分析。

线性耦合热弹性的基本理论已广为人知，并已发展了许多年。Biot^[3]首先在热传导方程中引入了耦合项，以求解热弹性耦合问题。后来，Herrmann^[4] 将Biot的原理推广到三维各向异性体。Jabbari等^[5][6]给出了柱坐标系和球坐标系中经典耦合热弹性的精确方程。尽管解析方法为一些简单的问题提供了一些解析解，但是许多复杂的问题尚未通过解析方法完全求解^[7]。因此，诸如有限元法(FEM)和边界元法(BEM)之类的数值方法已被广泛应用，以获得近似解^[8]。例如，Cannarozzi和Ubertini^[9]使用混合变分方法，对线性耦合准静态热弹性问题进行了有限元分析。位移和温度作为他们研究中的主要变量。另一方面，应力和热通量作为双重变量，它们也直接参与其分析。Tehrani和Eslami^[10]使用BEM研究了耦合系数和弛豫时间对热波和弹性波运动的影响。当断裂涉及完全耦合分析时，裂纹尖端周围的温度分布成为主要问题。裂纹附近的高能量浓度会产生大量的热能，并且导致不可忽略的温度升高。Atkinson和Craster^[11]推导了裂纹扩展过程中裂纹尖端附近区域的一些简单且渐近的温度分布。Weichert和Schönert^[12]研究了具有非常小的塑性区和高裂纹速度的脆性材料中裂纹尖端附近的温度。裂纹尖端被模拟为热源，因此，可以预测温度分布。Bhalla等^[13]进行的实验研究估计了裂纹尖端附近的温度分布。在他们的实验中观察到温度升高。Miehe等^[14]提出了热弹性力学中脆性断裂的连续相场(phase-field)模型。提供了考虑裂纹扩展和耗散产热的数值模拟试验，并讨论了其相应的温度场。

当热弹性问题涉及不连续性时，基于经典力学理论的上述数值模拟方法将预测到无限应力和能量密度。即使对于线性弹性断裂力学(LEFM)和位错动力学，也需要补充本构方程来确定位错运动。相反，近场动力学(PD)^{[15][16][17][18][19][20]}是一种非局部理论，其中包括失效(failure)，作为材料响应的一部分。近场动力学的公式是积分-微分形式，而不是经典连续介质力学中的空间微分方程。结果，近场动力学方程在裂纹或不连续接合处仍然有效^[21]。因此，近场动力学理论特别适用于具有不连续性的问题。因此，本文采用这种理论。在近场动力学理论领域，许多研究人员已经对裂纹的成核和扩展进行了研究^{[22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]}，但大多数人只研究机械场。在热场中，Oterkus等^[38]用近场动力学理论推导了热扩散的公式，并利用它来捕获燃料颗粒破裂^[39]。Bobaru和Duangpanya ^[40][41]在键基近场动力学理论中研究了有/没有连续性的物体中的热传导。关于热机械学(thermomechanics)，Oterkus等^[42]、Oterkus^[43]、Madenci和Oterkus^[44]提出了完全耦合键基近场动力学理论。此外，他们成功地将他们的模型应用于预测裂纹扩展^[45][46]。然而，键基近场动力学限于固定的泊松比，即，对于二维问题是

，对于三维问题是

^[47]。考虑到此约束，消除上述限制的广义近场动力学模型对于热机械问题是必需的。

本文旨在建立一种近场动力学模型，以克服键基近场动力学对材料特性的限制，以求解完全耦合的热弹性问题。因此，采用普通态基近场动力学理论，并在完全耦合的热机械学领域中推导其公式。然后提供在变形场和热场中的耦合方程的有量纲和无量纲形式，并随后进行验证。研究了在各种加载条件下各向同性薄板和砌块的温度变化和位移。通过与ANSYS和BEM解的比较，进行了验证。惊人的一致性表明，所提出的近场动力学模型具有以完全耦合的方式准确预测机械和热响应的能力。在本文的最后部分，预测了三点弯曲试验Kalthoff问题的裂纹扩展方式。另外，对于具有预先存在的裂纹的板施加了压力负荷。给出了裂纹的扩展路径和温度分布，并将裂纹模式与纯机械分析的预测结果进行了比较。因此，估计并讨论了耦合项对裂纹扩展的影响。

封面：^[48]

参考

1. ^Spitsberg, I., Steibel, J.: Thermal and environmental barrier coatings for SiC/Sic CMCs in aircraft engine applications. Int. J. Appl. Ceram. Technol. 1(4), 291–301 (2004)
2. ^Nali, P ., Carrera, E., Calvi, A.: Advanced fully coupled thermo-mechanical plate elements for multilayered structures sub- jected to mechanical and thermal loading. Int. J. Numer. Meth. Eng. 85(7), 896–919 (2011)
3. ^Biot, M.A.: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27(3), 240–253 (1956)
4. ^Herrmann, G.: On variational principles in thermoelasticity and heat conduction. Q. Appl. Math. 21(2), 151–155 (1963)
5. ^Jabbari, M., Dehbani, H., Eslami, M.R.: An exact solution for classic coupled thermoelasticity in spherical coordinates. J. Press. V essel Technol. 132(3), 031201–031211 (2010)
6. ^Jabbari, M., Dehbani, H., Eslami, M.R.: An exact solution for classic coupled thermoelasticity in cylindrical coordinates. J. Press. V essel Technol. 133(5), 051204–051210 (2011)
7. ^Dillon, O., Tauchert, T.: The experimental technique for observing the temperatures due to the coupled thermoelastic effect. Int. J. Solids Struct. 2(3), 385–391 (1966)
8. ^Abali, B.E.: Computational Reality: Solving Nonlinear and Coupled Problems in Continuum Mechanics, vol. 55. Springer, New Y ork (2016)
9. ^Eugster, S.R., Dell’Isola, F.: Exegesis of Sect. III.B from Fundamentals of the Mechanics of Continua by E. Hellinger. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. 98(1), 69–105 (2017)
10. ^Tehrani, P .H., Eslami, M.R.: Boundary element analysis of coupled thermoelasticity with relaxation times in finite domain. AIAA J. 38(3), 534–541 (2000)
11. ^Atkinson, C., Craster, R.V .: Fracture in fully coupled dynamic thermoelasticity. J. Mech. Phys. Solids 40(7), 1415–1432 (1992)
12. ^Weichert, R., Schönert, K.: Heat generation at the tip of a moving crack. J. Mech. Phys. Solids 26(3), 151–161 (1978)
13. ^Bhalla, K.S., Zehnder, A.T., Han, X.: Thermomechanics of slow stable crack growth: closing the loop between experiments and computational modeling. Eng. Fract. Mech. 70(17), 2439–2458 (2003)
14. ^Miehe, C., Schanzel, L.M., Ulmer, H.: Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagationin thermo-elasticsolids.Comput. Method Appl. Mech.294, 449–485 (2015)
15. ^Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48(1), 175–209 (2000)
16. ^Dell’Isola, F., Ugo, A., Luca, P .: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math. Mech. Solids. 20(8), 887–928 (2015)
17. ^Eugster, S.R., Dell’Isola, F.: Exegesis of the introduction and Sect. I from Fundamentals of the Mechanics of Continua by E. Hellinger. ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik. 97(4), 477–506 (2017)
18. ^Eugster, S.R., Dell’Isola, F.: Exegesis of Sect. II and III.A from Fundamentals of the Mechanics of Continua by E. Hellinger. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. (2017). https://doi.org/10.1002/zamm.201600293
19. ^Eugster, S.R., Dell’Isola, F.: Exegesis of Sect. III.B from Fundamentals of the Mechanics of Continua by E. Hellinger. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. 98(1), 69–105 (2017)
20. ^Dell’Isola, F., Alessandro, D.C., Ivan, G.: Higher-gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math. Mech. Solids. (2016). https://doi.org/10.1177/1081286515616034
21. ^Demmie, P.N., Silling, S.A.: An approach to modeling extreme loading of structures using peridynamics. J. Mech. Mater. Struct. 2(10), 1921–1945 (2007)
22. ^Silling, S., Weckner, O., Askari, E., Bobaru, F.: Crack nucleation in a peridynamic solid. Int. J. Fract. 162(1–2), 219–227 (2010)
23. ^Ha, Y .D., Bobaru, F.: Characteristics of dynamic brittle fracture captured with peridynamics. Eng. Fract. Mech. 78(6), 1156–1168 (2011)
24. ^Huang, D., Lu, G.D., Qiao, P .Z.: An improved peridynamic approach for quasi-static elastic deformation and brittle fracture analysis. Int. J. Mech. Sci. 94–95, 111–122 (2015)
25. ^Hu, W.K., Ha, Y .D., Bobaru, F.: Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput. Method Appl. Mech. 217, 247–261 (2012)
26. ^Bobaru, F., Hu, W.K.: The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials. Int. J. Fract. 176(2), 215–222 (2012)
27. ^Agwai, A., Guven, I., Madenci, E.: Predicting crack propagation with peridynamics: a comparative study. Int. J. Fract.171(1), 65–78 (2011)
28. ^V azic, B., Wang, H., Diyaroglu, C., Oterkus, S., Oterkus, E.: Dynamic propagation of a macrocrack interacting with parallel small cracks. AIMS Mater. Sci. 4(1), 118–136 (2017)
29. ^Wang, H., Oterkus, E., Celik, S., Toros, S.: Thermomechanical analysis of porous solid oxide fuel cell by using peridynamics. AIMS Energy 5(4), 585–600 (2017)
30. ^Oterkus, S., Madenci, E., Oterkus, E., Hwang, Y ., Bae, J., Han, S.: Hygro-thermo-mechanical analysis and failure prediction in electronic packages by using peridynamics. In: 2014 IEEE 64th Electronic Components and Technology Conference (ECTC), pp. 973–982. IEEE (2014)
31. ^Amani, J., Oterkus, E., Areias, P ., Zi, G., Nguyen-Thoi, T., Rabczuk, T.: A non-ordinary state-based peridynamics formulation for thermoplastic fracture. Int. J. Impact Eng. 87, 83–94 (2016)
32. ^Diyaroglu, C., Oterkus, E., Madenci, E., Rabczuk, T., Siddiq, A.: Peridynamic modeling of composite laminates under explosive loading. Compos. Struct. 144, 14–23 (2016)
33. ^Diyaroglu, C., Oterkus, E., Oterkus, S., Madenci, E.: Peridynamics for bending of beams and plates with transverse shear deformation. Int. J. Solids Struct. 69, 152–168 (2015)
34. ^Oterkus, E., Madenci, E.: Peridynamics for failure prediction in composites. In: 53rd AIAA/ASME/ASCE/AHS/ASC Struc- tures, Structural Dynamics and Materials Conference 20th AIAA/ASME/AHS Adaptive Structures Conference 14th AIAA. p. 1692 (2012)
35. ^Oterkus, S., Madenci, E.: Peridynamics for antiplane shear and torsional deformations. J. Mech. Mater. Struct 10, 167–193 (2015)
36. ^Oterkus, S., Fox, J., Madenci, E.: Simulation of electro-migration through peridynamics. In: 2013 IEEE 63rd Electronic Components and Technology Conference (ECTC). pp. 1488–1493. IEEE (2013)
37. ^Oterkus, S., Madenci, E., Oterkus, E.: Fully coupled poroelastic peridynamic formulation for fluid-filledfractures.Eng.Geol. 225, 19–28 (2017)
38. ^Oterkus, S., Madenci, E., Agwai, A.: Peridynamic thermal diffusion. J. Comput. Phys. 265, 71–96 (2014)
39. ^Oterkus, S., Madenci, E.: Peridynamic modeling of fuel pellet cracking. Eng. Fract. Mech. 176, 23–37 (2017)
40. ^Bobaru, F., Duangpanya, M.: A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities. J. Comput. Phys. 231(7), 2764–2785 (2012)
41. ^Bobaru, F., Duangpanya, M.: The peridynamic formulation for transient heat conduction. Int. J. Heat Mass Transf.53(19–20), 4047–4059 (2010)
42. ^Oterkus, S., Madenci, E., Agwai, A.: Fully coupled peridynamic thermomechanics. J. Mech. Phys. Solids 64, 1–23 (2014)
43. ^Oterkus, S.: Peridynamics for the Solution of Multiphysics Problems. The University of Arizona, Tucson (2015)
44. ^Madenci, E., Oterkus, S.: Peridynamics for coupled field equations. In: Bobaru, F., Foster, J.T., Geubelle, P.H., Silling, S.A. (eds.) Handbook of Peridynamic Modeling, pp. 489–531. Fl, Boca Raton (2016)
45. ^Oterkus, S., Madenci, E.: Fully coupled thermomechanical analysis of fiber reinforced composites using peridynamics. In: 55th AIAA/ASMe/ASCE/AHS/SC Structures, Structural Dynamics, and Materials Conference-SciTech Forum and Exposi- tion, Maryland (2014)
46. ^Oterkus,S.,Madenci,E.:Crackgrowthpredictioninfully-coupledthermalanddeformationfieldsusingperidynamictheory. In: 54th AIAA Structures, Structural Dynamics and Materials Conference, Boston, Massachusetts (2013)
47. ^Silling, S.A., Askari, E.: A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83(17), 1526–1535 (2005)
48. ^https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.thoughtco.com%2Fwhat-is-fire-made-of-607313&psig=AOvVaw0AXVwSqy9a6tPenkFAMpo_&ust=1598005785925000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCPjCk8_JqesCFQAAAAAdAAAAABAF