Abstract: This paper demonstrates a novel formulation of structural analysis. A novel stress-based formulation of structural analysis for material nonlinear problems was proposed in earlier work. In this paper, this methodology is further extended for 3D finite element analysis. The approach avoids use of elastic moduli as the material input in the analysis procedure. It utilizes the whole stress-strain curve of the material. It can be shown that this analysis procedure solved the nonlinear or plasticity problem with relative ease. This paper solves a uniaxial bar, in which the results are compared with the solutions of Green-Lagrange strain and Piola-Kirchhoff stresses. The uniaxial bar is also solved by a regression model in the �scikit-learn� module in Python. The second problem solved is of a beam in pure bending for which the energy release rate is measured. For the beam in pure bending, the bending moment carrying capacity of the beam section is evaluated by this methodology as the crack propagates through the depth of the beam. It can be shown that the methodology is very simple, accurate, and clear in its physical steps.

一种新的基于应力的有限元分析公式

[1]Amiri F, Millán D, Shen Y, et al., 2014a. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 69:102-109.

[2]Amiri F, Anitescu C, Arroyo M, et al., 2014b. XLME interpolants, a seamless bridge between XFEM and enriched meshless methods. Computational Mechanics, 53(1):45-57.

[3]Anitescu C, Atroshchenko E, Alajlan N, et al., 2019. Artificial neural network methods for the solution of second order boundary value problems. Computers, Materials & Continua, 59(1):345-359.

[4]Areias P, Rabczuk T, 2013. Finite strain fracture of plates and shells with configurational forces and edge rotations. International Journal for Numerical Methods in Engineering, 94(12):1099-1122.

[5]Areias P, Rabczuk T, 2017. Steiner-point free edge cutting of tetrahedral meshes with applications in fracture. Finite Elements in Analysis and Design, 132:27-41.

[6]Areias P, Rabczuk T, Dias-da-Costa D, 2013. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 110:113-137.

[7]Areias P, Rabczuk T, Camanho PP, 2014. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 72:50-63.

[8]Areias P, Rabczuk T, Msekh MA, 2016. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 312:322-350.

[9]Areias P, Reinoso J, Camanho PP, et al., 2018. Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation. Engineering Fracture Mechanics, 189:339-360.

[10]Bathe K, 2014. Finite Element Procedures, 2nd Edition. Massachusetts Institute of Technology, Cambridge, USA.

[11]Berg J, Nyström K, 2018. A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing, 317:28-41.

[12]Budarapu PR, Gracie R, Bordas SPA, et al., 2014a. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 53(6):1129-1148.

[13]Budarapu PR, Gracie R, Yang SW, et al., 2014b. Efficient coarse graining in multiscale modeling of fracture. Theoretical and Applied Fracture Mechanics, 69:126-143.

[14]Chau-Dinh T, Zi G, Lee PS, et al., 2012. Phantom-node method for shell models with arbitrary cracks. Computers & Structures, 92-93:242-256.

[15]Einstein A, 1916. The foundation of the general theory of relativity. Annalen der Physik, 354:769-822.

[16]ES6, 2018. Mini Tensile Test Specimens. TecQuipment Ltd., Long Eaton, Nottingham, UK.

[17]Evans PH, Marathe MS, 1968. Microcracking and stress-strain curves for concrete in tension. Matériaux et Construction, 1:61-64.

[18]Gaur H, 2019. A new stress based approach for nonlinear finite element analysis. Journal of Applied and Computational Mechanics, 5:563-576.

[19]Gaur H, Srivastav A, 2020. A novel formulation of material nonlinear analysis in structural mechanics. Defence Technology, 17(1):36-49.

[20]Gauss CF, 1867. Werke. Cambridge Library Collection– Mathematics. Cambridge University Press, Cambridge, UK.

[21]Ghorashi SS, Valizadeh N, Mohammadi S, et al., 2015. T-spline based XIGA for fracture analysis of orthotropic media. Computers & Structures, 147:138-146.

[22]Griffith AA, 1921. VI. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London Series A, 221(582-593):163-198.

[23]Guo HW, Zhuang XY, Rabczuk T, 2019. A deep collocation method for the bending analysis of Kirchhoff plate. Computers, Materials & Continua, 59(2):433-456.

[24]Hillerborg A, Modéer M, Petersson PE, 1976. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research, 6(6):773-781.

[25]Kumar P, 2014. Elements of Fracture Mechanics. Tata McGraw-Hill Education, New Delhi, India.

[26]Lagaris IE, Likas A, Fotiadis DI, 1998. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 9(5):987-1000.

[27]McCulloch WS, Pitts W, 1990. A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biology, 52(1-2):99-115.

[28]Pedregosa F, Varoquaux G, Gramfort A, et al., 2011. Scikit-learn: machine learning in Python. The Journal of Machine Learning Research, 12(85):2825-2830.

[29]Rabczuk T, Belytschko T, 2004. Cracking particles: a simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 61(13):2316-2343.

[30]Rabczuk T, Belytschko T, 2007. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 196(29-30):2777-2799.

[31]Rabczuk T, Areias PMA, Belytschko T, 2007. A meshfree thin shell method for non-linear dynamic fracture. International Journal for Numerical Methods in Engineering, 72(5):524-548.

[32]Rabczuk T, Zi G, Bordas S, et al., 2008. A geometrically non-linear three-dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 75(16):4740-4758.

[33]Rabczuk T, Zi G, Bordas S, et al., 2010a. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 199(37-40):2437-2455.

[34]Rabczuk T, Bordas S, Zi G, 2010b. On three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 88(23-24):1391-1411.

[35]Rabczuk T, Song JH, Zhuang X, et al., 2019. Extended Finite Element and Meshfree Methods. Elsevier, Amsterdam, the Netherlands.

[36]Raissi M, Karniadakis EG, 2018. Hidden physics models: machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357:125-141.

[37]Samaniego E, Anitescu C, Goswami S, et al., 2020. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 362:112790.

[38]Sirignano J, Spiliopoulos K, 2018. DGM: a deep learning algorithm for solving partial differential equations. Journal of Computational Physics, 375:1339-1364.

[39]Talebi H, Silani M, Bordas SPA, et al., 2014. A computational library for multiscale modeling of material failure. Computational Mechanics, 53(5):1047-1071.

[40]Talebi H, Silani M, Rabczuk T, 2015. Concurrent multiscale modeling of three dimensional crack and dislocation propagation. Advances in Engineering Software, 80: 82-92.

[41]Timoshenko SP, Goodier JN, 1970. Theory of Elasticity. 3rd Edition. Mcgraw Hill, New York, USA.

[42]van Rossum G, Drake FL, 2009. Python 3 Reference Manual. CreateSpace, Scotts Valley, CA, USA.

[43]Vu-Bac N, Duong TX, Lahmer T, et al., 2018. A NURBS-based inverse analysis for reconstruction of nonlinear deformations of thin shell structures. Computer Methods in Applied Mechanics and Engineering, 331:427-455.

[44]Weinan E, Yu B, 2018. The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics, 6:1-12. https://arxiv.org/abs/1710.00211

[45]Zhang YM, Gao ZR, Li YY, et al., 2020. On the crack opening and energy dissipation in a continuum based disconnected crack model. Finite Elements in Analysis and Design, 170:103333.