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**Nodally integrated thermomechanical RKPM. II: Generalized thermoelasticity and hyperbolic finite-strain thermoplasticity.**
*(English)*
Zbl 1478.74085

Summary: In this two-part paper, a stable and efficient nodally-integrated reproducing kernel particle method (RKPM) approach for solving the governing equations of generalized thermomechanical theories is developed. Part I [the authors, ibid. 68, No. 4, 795–820 (2021, Zbl 1478.74084)] investigated quadrature in the weak form using classical thermoelasticity as a model problem, and a stabilized and corrected nodal integration was proposed. In this sequel, these methods are developed for generalized thermoelasticity and generalized finite-strain plasticity theories of the hyperbolic type, which are more amenable to explicit time integration than the classical theories. Generalized thermomechanical models yield finite propagation of temperature, with a so-called second sound speed. Since this speed is not well characterized for common engineering materials and environments, equating the elastic wave speed with the second sound speed is investigated to obtain results close to classical thermoelasticity, which also yields a uniform critical time step. Implementation of the proposed nodally integrated RKPM for explicit analysis of finite-strain thermoplasticity is also described in detail. Several benchmark problems are solved to demonstrate the effectiveness of the proposed approach for thermomechanical analysis.

### MSC:

74S99 | Numerical and other methods in solid mechanics |

74F05 | Thermal effects in solid mechanics |

74B05 | Classical linear elasticity |

74C15 | Large-strain, rate-independent theories of plasticity (including nonlinear plasticity) |

74J99 | Waves in solid mechanics |

### Keywords:

meshfree reproduced kernel particle method; generalized thermoelasticity; finite-strain thermoplasticity; nodal integration; elastic wave; second sound speed### Citations:

Zbl 1478.74084
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\textit{M. Hillman} and \textit{K.-C. Lin}, Comput. Mech. 68, No. 4, 821--844 (2021; Zbl 1478.74085)

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### References:

[1] | Abouelregal, AE; Zenkour, AM, The effect of fractional thermoelasticity on a two-dimensional problem of a mode I crack in a rotating fiber-reinforced thermoelastic medium, Chin Phys B, 22, 10, 108102 (2013) |

[2] | Adam, L.; Ponthot, J-P, Thermomechanical modeling of metals at finite strains: first and mixed order finite elements, Int J Solids Struct, 42, 21-22, 5615-5655 (2005) · Zbl 1113.74418 |

[3] | Armero, F.; Simo, JC, A priori stability estimates and unconditionally stable product formula algorithms for nonlinear coupled thermoplasticity, Int J Plast, 9, 6, 749-782 (1993) · Zbl 0791.73026 |

[4] | Baek J, Chen J, Zhou G, Arnett K, Hillman M, Hegemier G, Hardesty S (in press) A semi-Lagrangian RKPM with node-based shock algorithm for explosive welding simulation. Comput Mech · Zbl 1467.74097 |

[5] | Bagri, A.; Taheri, H.; Eslami, MR; Fariborz, S., Generalized coupled thermoelasticity of a layer, J Therm Stresses, 29, 4, 359-370 (2006) |

[6] | Bathe, K-J, Finite element procedures (2006), Berlin: Klaus-Jurgen Bathe, Berlin · Zbl 1326.65002 |

[7] | Beni, YT; Movahhedy, MR, Consistent arbitrary Lagrangian Eulerian formulation for large deformation thermo-mechanical analysis, Mater Des, 31, 8, 3690-3702 (2010) |

[8] | Camacho, GT; Ortiz, M., Adaptive Lagrangian modelling of ballistic penetration of metallic targets, Comput Methods Appl Mech Eng, 142, 3-4, 269-301 (1997) · Zbl 0892.73056 |

[9] | Cattaneo, C., On a form of the heat equation eliminating the paradox of an instantaneous propagation, Account Render, 247, 431-433 (1958) · Zbl 1339.35135 |

[10] | Chen, J.; Dargush, GF, Boundary element method for dynamic poroelastic and thermoelastic analyses, Int J Solids Struct, 32, 15, 2257-2278 (1995) · Zbl 0869.73075 |

[11] | Chen J-S, Hillman M, Chi S-W (2016) Meshfree methods: progress made after 20 years. J Eng Mech |

[12] | Chen, J-S; Hillman, M.; Rüter, M., An arbitrary order variationally consistent integration for Galerkin meshfree methods, Int J Numer Methods Eng, 95, 5, 387-418 (2013) · Zbl 1352.65481 |

[13] | Chen, J-S; Liu, WK; Hillman, M.; Chi, SW; Lian, Y.; Bessa, MA; Stein, E.; de Borst, R.; Hughes, TJR, Reproducing Kernel approximation and discretization, Encyclopedia of computational mechanics (2017), Chichester: Wiley, Chichester |

[14] | Chen, J-S; Pan, C.; Roque, C.; Wang, H-P, A lagrangian reproducing kernel particle method for metal forming analysis, Comput Mech, 22, 3, 289-307 (1998) · Zbl 0928.74115 |

[15] | Chen, J-S; Pan, C.; Wu, C-T; Liu, WK, Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures, Comput Methods Appl Mech Eng, 139, 1-4, 195-227 (1996) · Zbl 0918.73330 |

[16] | Chen, J-S; Wu, C-T; Yoon, S.; You, Y., A stabilized conforming nodal integration for Galerkin mesh-free methods, Int J Numer Methods Eng, 50, 2, 435-466 (2001) · Zbl 1011.74081 |

[17] | Chen J-S, Wu Y (2007) Stability in Lagrangian and semi-Lagrangian reproducing kernel discretizations using nodal integration in nonlinear solid mechanics. In: Advances in meshfree techniques. Springer, pp 55-76 · Zbl 1323.74104 |

[18] | Chen, J-S; Yoon, S.; Wu, C-T, Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods, Int J Numer Methods Eng, 53, 12, 2587-2615 (2002) · Zbl 1098.74732 |

[19] | Chen, J-S; Zhang, X.; Belytschko, T., An implicit gradient model by a reproducing kernel strain regularization in strain localization problems, Comput Methods Appl Mech Eng, 193, 27-29, 2827-2844 (2004) · Zbl 1067.74564 |

[20] | Chester, M., Second sound in solids, Phys Rev, 131, 5, 2013 (1963) |

[21] | Danilouskaya, V., Thermal stresses in elastic half space due to sudden heating of its boundary, Pelageya Yakovlevna Kochina, 14, 316-321 (1950) |

[22] | Eringen, AC, Mechanics of continua (1980), Huntington: Robert E. Krieger Publishing Co., Huntington · Zbl 0181.53802 |

[23] | Fan, Z.; Li, B., Meshfree simulations for additive manufacturing process of metals, Integr Mater Manuf Innov, 8, 2, 144-153 (2019) |

[24] | Green, AE; Lindsay, KA, Thermoelasticity, J Elast, 2, 1, 1-7 (1972) · Zbl 0775.73063 |

[25] | Guan, P-C; Chen, J-S; Wu, Y.; Teng, H.; Gaidos, J.; Hofstetter, K.; Alsaleh, M., Semi-Lagrangian reproducing kernel formulation and application to modeling earth moving operations, Mech Mater, 41, 6, 670-683 (2009) |

[26] | Guan, P-C; Chi, S-W; Chen, J-S; Slawson, T.; Roth, MJ, Semi-Lagrangian reproducing kernel particle method for fragment-impact problems, Int J Impact Eng, 38, 12, 1033-1047 (2011) |

[27] | Hillman, M.; Chen, J-S, An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics, Int J Numer Methods Eng, 107, 603-630 (2016) · Zbl 1352.74119 |

[28] | Hillman, M.; Chen, J-S; Chi, S-W, Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems, Comput Part Mech, 1, 3, 245-256 (2014) |

[29] | Hosseini, SM; Sladek, J.; Sladek, V., Meshless local Petrov-Galerkin method for coupled thermoelasticity analysis of a functionally graded thick hollow cylinder, Eng Anal Bound Elem, 35, 6, 827-835 (2011) · Zbl 1259.74084 |

[30] | Hosseini-Tehrani, P.; Eslami, MR; Azari, S., Analysis of thermoelastic crack problems using Green-Lindsay Theory, J Therm Stresses, 29, 4, 317-330 (2006) |

[31] | Hughes, TJ, The finite element method: linear static and dynamic finite element analysis (2012), Mineola: Dover Publications Inc, Mineola |

[32] | Hughes, TJR; Winget, J., Finite rotation effects in numerical integration of rate constitutive equations arising in large deformation analysis, Int J Numer Methods Eng, 15, 12, 1862-1867 (1980) · Zbl 0463.73081 |

[33] | Kouchakzadeh, MA; Entezari, A., Analytical solution of classic coupled thermoelasticity problem in a rotating disk, J Therm Stresses, 38, 11, 1267-1289 (2015) |

[34] | Li, B.; Habbal, F.; Ortiz, M., Optimal transportation meshfree approximation schemes for fluid and plastic flows, Int J Numer Methods Eng, 83, 12, 1541-1579 (2010) · Zbl 1202.74200 |

[35] | Li, S.; Liu, WK, Synchronized reproducing kernel interpolant via multiple wavelet expansion, Comput Mech, 21, 28-47 (1998) · Zbl 0912.76057 |

[36] | Li, S.; Liu, WK, Meshfree and particle methods and their applications, Appl Mech Rev, 55, 1, 1-34 (2002) |

[37] | Lindgren, L-E, Numerical modelling of welding, Comput Methods Appl Mech Eng, 195, 48, 6710-6736 (2006) · Zbl 1120.74822 |

[38] | Liu, WK; Jun, S.; Zhang, YF, Reproducing kernel particle methods, Int J Numer Methods Fluids, 20, 8-9, 1081-1106 (1995) · Zbl 0881.76072 |

[39] | Lord, HW; Shulman, Y., A generalized dynamical theory of thermoelasticity, J Mech Phys Solids, 15, 5, 299-309 (1967) · Zbl 0156.22702 |

[40] | Mallik, SH; Kanoria, M., A unified generalized thermoelasticity formulation: application to penny-shaped crack analysis, J Therm Stresses, 32, 9, 943-965 (2009) |

[41] | Marusich, TD; Ortiz, M., Modelling and simulation of high-speed machining, Int J Numer Methods Eng, 38, 21, 3675-3694 (1995) · Zbl 0835.73077 |

[42] | Norris, DM Jr; Moran, B.; Scudder, JK; Quinones, DF, A computer simulation of the tension test, J Mech Phys Solids, 26, 1, 1-19 (1978) |

[43] | Pan, X.; Wu, CT; Hu, W.; Wu, Y., A momentum-consistent stabilization algorithm for Lagrangian particle methods in the thermo-mechanical friction drilling analysis, Comput Mech, 64, 3, 625-644 (2019) · Zbl 1468.74089 |

[44] | Prevost, J-H; Tao, D., Finite element analysis of dynamic coupled thermoelasticity problems with relaxation times, J Appl Mech, 50, 4, 817-822 (1983) · Zbl 0527.73010 |

[45] | Puso, MA; Zywicz, E.; Chen, JS, A new stabilized nodal integration approach, Lect Notes Comput Sci Eng, 57, 207-217 (2007) · Zbl 1111.74048 |

[46] | Seitz, A.; Wall, WA; Popp, A., A computational approach for thermo-elasto-plastic frictional contact based on a monolithic formulation using non-smooth nonlinear complementarity functions, Adv Model Simul Eng Sci, 5, 1, 5 (2018) |

[47] | Sherief, HH; El-Maghraby, NM, A mode-I crack problem for an infinite space in generalized thermoelasticity, J Therm Stresses, 28, 5, 465-484 (2005) |

[48] | Sherief, HH; El-Maghraby, NM; Allam, AA, Stochastic thermal shock problem in generalized thermoelasticity, Appl Math Model, 37, 3, 762-775 (2013) · Zbl 1351.74020 |

[49] | Simkins, DC; Li, S., Meshfree simulations of thermo-mechanical ductile fracture, Comput Mech, 38, 3, 235-249 (2006) · Zbl 1162.74052 |

[50] | Simo, JC; Miehe, C., Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation, Comput Methods Appl Mech Eng, 98, 1, 41-104 (1992) · Zbl 0764.73088 |

[51] | Tehrani, PH; Eslami, MR, Boundary element analysis of coupled thermoelasticity with relaxation times in finite domain, AIAA J, 38, 3, 534-541 (2000) |

[52] | Vernotte, P., Les paradoxes de la theorie continue de l’equation de la chaleur, Comptes rendus, 246, 3154-3155 (1958) · Zbl 1341.35086 |

[53] | Wang, H.; Liao, H.; Fan, Z.; Fan, J.; Stainier, L.; Li, X.; Li, B., The Hot Optimal Transportation Meshfree (HOTM) method for materials under extreme dynamic thermomechanical conditions, Comput Methods Appl Mech Eng, 364, 112958 (2020) · Zbl 1442.74252 |

[54] | Wu, CT; Wu, Y.; Lyu, D.; Pan, X.; Hu, W., The momentum-consistent smoothed particle Galerkin (MC-SPG) method for simulating the extreme thread forming in the flow drill screw-driving process, Comput Part Mech, 7, 2, 177-191 (2020) |

[55] | Wu, J.; Wang, D., An accuracy analysis of Galerkin meshfree methods accounting for numerical integration, Comput Methods Appl Mech Eng, 375, 113631 (2021) · Zbl 07340457 |

[56] | Yang, Q.; Stainier, L.; Ortiz, M., A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids, J Mech Phys Solids, 54, 2, 401-424 (2006) · Zbl 1120.74367 |

[57] | Yousefi, H.; Kani, AT; Kani, IM, Multiscale RBF-based central high resolution schemes for simulation of generalized thermoelasticity problems, Front Struct Civil Eng, 13, 2, 429-455 (2019) |

[58] | Zamani, A.; Hetnarski, RB; Eslami, MR, Second sound in a cracked layer based on Lord-Shulman theory, J Therm Stresses, 34, 3, 181-200 (2011) |

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