An efficient finite element method for computing the response of a strain-limiting elastic solid containing a V-notch and inclusions
G. Shylaja
Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India
Search for more papers by this authorCorresponding Author
V. Kesavulu Naidu
Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India
Correspondence
V. Kesavulu Naidu, Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, 560035, India.
Email: [email protected]
S. M. Mallikarjunaiah, Department of Mathematics, and Statistics, Texas A&M University-Corpus Christi, Corpus Christi, TX 78412, USA. Email: [email protected]
Search for more papers by this authorB. Venkatesh
Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India
Search for more papers by this authorCorresponding Author
S. M. Mallikarjunaiah
Department of Mathematics & Statistics, Texas A&M University-Corpus Christi, Corpus Christi, Texas, USA
Correspondence
V. Kesavulu Naidu, Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, 560035, India.
Email: [email protected]
S. M. Mallikarjunaiah, Department of Mathematics, and Statistics, Texas A&M University-Corpus Christi, Corpus Christi, TX 78412, USA. Email: [email protected]
Search for more papers by this authorG. Shylaja
Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India
Search for more papers by this authorCorresponding Author
V. Kesavulu Naidu
Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India
Correspondence
V. Kesavulu Naidu, Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, 560035, India.
Email: [email protected]
S. M. Mallikarjunaiah, Department of Mathematics, and Statistics, Texas A&M University-Corpus Christi, Corpus Christi, TX 78412, USA. Email: [email protected]
Search for more papers by this authorB. Venkatesh
Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India
Search for more papers by this authorCorresponding Author
S. M. Mallikarjunaiah
Department of Mathematics & Statistics, Texas A&M University-Corpus Christi, Corpus Christi, Texas, USA
Correspondence
V. Kesavulu Naidu, Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, 560035, India.
Email: [email protected]
S. M. Mallikarjunaiah, Department of Mathematics, and Statistics, Texas A&M University-Corpus Christi, Corpus Christi, TX 78412, USA. Email: [email protected]
Search for more papers by this authorAbstract
The precise triangulation of the domain assumes a critical role in calculating numerical approximations of differential operators utilizing a collocation method. A well-executed triangulation contributes significantly to the reduction of discretization errors. Conventional collocation techniques typically represent the smooth curved domain by triangulating a mesh, wherein boundary points are approximated using polygons. However, this methodology frequently introduces geometrical errors that adversely impact the accuracy of the numerical approximation. To mitigate such geometrical inaccuracies, isoparametric, subparametric, and iso-geometric methods have been proposed, facilitating the approximation of curved surfaces or line segments. This paper proposes an efficient finite element method tailored to approximate the elliptic boundary value problem (BVP) solution that governs the response of an elastic solid containing a V-notch and inclusions. The algebraically nonlinear constitutive equation and the balance of linear momentum are reduced to a second-order quasi-linear elliptic partial differential equation. Our approach encompasses the representation of complex curved boundaries through a smooth, distinctive point transformation. The principal objective is to utilize higher-order shape functions to accurately compute the entries within the finite element matrices and vectors and obtain a precise approximate solution to the BVP. A Picard-type linearization addresses the nonlinearities inherent in the governing differential equation. Numerical results derived from the test cases demonstrate a significant enhancement in accuracy.
REFERENCES
- 1Kesavulu Naidu, V., Nagaraja, K.V.: Advantages of cubic arcs for approximating curved boundaries by subparametric transformations for some higher order triangular elements. Appl. Math. Comput. 219(12), 6893–6910 (2013)
10.1016/j.amc.2013.01.004 Google Scholar
- 2Sasikala, J., Kesavulu Naidu, V., Venkatesh, B., Mallikarjunaiah, S.M.: On an efficient octic order sub-parametric finite element method on curved domains. Comput. Math. Appl. 143, 249–268 (2023)
10.1016/j.camwa.2023.05.006 Google Scholar
- 3Vasilyeva, M., Mallikarjunaiah, S.M., Palaniappan, D.: Multiscale model reduction technique for fluid flows with heterogeneous porous inclusions. J. Comput. Appl. Math. 424, 114976 (2023)
10.1016/j.cam.2022.114976 Google Scholar
- 4Gou, K., Mallikarjunaiah, S.M.: Computational modeling of circular crack-tip fields under tensile loading in a strain-limiting elastic solid. Commun. Nonlinear Sci. Numer. Simul. 121, 107217 (2023)
10.1016/j.cnsns.2023.107217 Google Scholar
- 5Gou, K., Mallikarjunaiah, S.M.: Finite element study of v-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body. Math Mech. Solids 28(10), 2155–2176 (2023)
10.1177/10812865221152257 Google Scholar
- 6Vasilyeva, M., Mallikarjunaiah, S.M.: Generalized multiscale finite element treatment of a heterogeneous nonlinear strain-limiting elastic model. Multiscale Model. Simul. 22(1), 334–368 (2024)
10.1137/22M1514179 Google Scholar
- 7Ferguson, L.A., Muddamallappa, M., Walton, J.R.: Numerical simulation of mode-iii fracture incorporating interfacial mechanics. Int. J. Fract. 192, 47–56 (2015)
10.1007/s10704-014-9984-y Google Scholar
- 8Ergatoudis, I., Irons, B.M., Zienkiewicz, O.C.: Curved, isoparametric,“quadrilateral” elements for finite element analysis. Int. J. Solids Struct. 4(1), 31–42 (1968)
10.1016/0020-7683(68)90031-0 Google Scholar
- 9McLeod, R.J.Y., Mitchell, A.R.: The use of parabolic arcs in matching curved boundaries in the finite element method. IMA J. Appl. Math. 16(2), 239–246 (1975)
10.1093/imamat/16.2.239 Google Scholar
- 10Mitchell, A.R.: Advantages of cubics for approximating element boundaries. Comput. Math. Appl. 5(4), 321–327 (1979)
10.1016/0898-1221(79)90091-9 Google Scholar
- 11Zienkiewicz, O.C., Taylor, R.L.: The finite element method for solid and structural mechanics. Elsevier (2005)
- 12Ciarlet, P.G., Raviart, P.-A.: Interpolation theory over curved elements, with applications to finite element methods. Comput. Meth. Appl. Mech. Eng. 1(2), 217–249 (1972)
10.1016/0045-7825(72)90006-0 Google Scholar
- 13Scott, L.R.: Finite element techniques for curved boundaries. PhD thesis, Massachusetts Institute of Technology (1973).
- 14Rathod, H.T., Nagaraja, K.V., Naidu, V.K., Venkatesudu, B.: The use of parabolic arcs in matching curved boundaries by point transformations for some higher order triangular elements. Finite Elem. Anal. Des. 44(15), 920–932 (2008)
10.1016/j.finel.2008.07.001 Google Scholar
- 15Nagaraja, K.V., Naidu, V.K., Rathod, H.T.: The use of parabolic arc in matching curved boundary by point transformations for sextic order triangular element. Int. J. Math. Anal. 4, 357–374 (2010)
- 16Kesavulu Naidu, V., Nagaraja, K.V.: The use of parabolic arc in matching curved boundary by point transformations for septic order triangular element and its applications. Adv. Stud. Contemp. Math. 20(3), 437–456 (2010)
- 17Rajagopal, K.R.: The elasticity of elasticity. Z. Angew. Math. Phys. 58(2), 309–317 (2007)
- 18Rajagopal, K.R.: On implicit constitutive theories. Appl. Math. 48(4), 279–319 (2003)
10.1023/A:1026062615145 Google Scholar
- 19Rajagopal, K.R.: Non-linear elastic bodies exhibiting limiting small strain. Math. Mech. Solids 16(1), 122–139 (2011)
10.1177/1081286509357272 Google Scholar
- 20Rajagopal, K.R.: On the nonlinear elastic response of bodies in the small strain range. Acta Mech. 225(6), 1545–1553 (2014)
- 21Rajagopal, K.R., Srinivasa, A.R.: On a class of non-dissipative materials that are not hyperelastic. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 465, pp. 493–500. The Royal Society, 2009.
- 22Rajagopal, K.R., Walton, J.R.: Modeling fracture in the context of a strain-limiting theory of elasticity: A single anti-plane shear crack. Int. J. Fract. 169(1), 39–48 (2011)
10.1007/s10704-010-9581-7 Google Scholar
- 23Gou, K., Mallikarjuna, M., Rajagopal, K.R., Walton, J.R.: Modeling fracture in the context of a strain-limiting theory of elasticity: A single plane-strain crack. Int. J. Eng. Sci. 88, 73–82 (2015)
10.1016/j.ijengsci.2014.04.018 Google Scholar
- 24Mallikarjunaiah, S.M.: On Two Theories for Brittle Fracture: Modeling and Direct Numerical Simulations. PhD thesis, Texas A&M University (2015)
- 25Bulíček, M., Málek, J., Süli, E.: Analysis and approximation of a strain-limiting nonlinear elastic model. Math. Mech. Solids 20(1), 92–118 (2015)
10.1177/1081286514543601 Google Scholar
- 26Bulíček, M., Málek, J., Rajagopal, K.R., Walton, J.R.: Existence of solutions for the anti-plane stress for a new class of strain-limiting elastic bodies. Calc. Var. Partial Differ. Equ. 54(2), 2115–2147 (2015)
10.1007/s00526-015-0859-5 Google Scholar
- 27Bulíček, M., Málek, J., Rajagopal, K.R., Süli, E.: On elastic solids with limiting small strain: modeling and analysis. EMS Surv. Math. Sci. 1(2), 283–332 (2014)
10.4171/emss/7 Google Scholar
- 28Yoon, H.C., Vasudeva, K.K., Mallikarjunaiah, S.M.: Finite element model for a coupled thermo-mechanical system in nonlinear strain-limiting thermoelastic body. Commun. Nonlinear Sci. Numer. Simul. 108, 106262 (2022)
10.1016/j.cnsns.2022.106262 Google Scholar
- 29Bonito, A., Girault, V., Süli, E.: Finite element approximation of a strain-limiting elastic model. IMA J. Numer. Anal. 40(1), 29–86 (2020)
10.1093/imanum/dry065 Google Scholar
- 30Bogdanov, V.L., Dovzhyk, M.V., Nazarenko, V.M.: Fracture of highly elastic and composite materials in compression along near-surface crack. Mech. Compos. Mater. 59(2), 411–418 (2023)
10.1007/s11029-023-10105-x Google Scholar
- 31Selivanov, M., Bogdanov, V., Altenbach, H.: Solving some problems of crack mechanics for a normal edge crack in orthotropic solid within the cohesive zone model approach. Mech. Compos. Mater. 59(2), 335–362 (2023)
10.1007/s11029-023-10099-6 Google Scholar
- 32Trivedi, N., Das, S., Craciun, E.-M.: The mathematical study of an edge crack in two different specified models under time-harmonic wave disturbance. Mech. Compos. Mater. 58(1), 1–14 (2022)
10.1007/s11029-022-10007-4 Google Scholar
- 33Tanwar, A., Das, S., Craciun, E.-M., Altenbach, H.: Interaction among interfacial offset cracks in composite materials under the anti-plane shear loading. Z. Angew. Math. Mech. 103(11), e202300081 (2023)
10.1002/zamm.202300081 Google Scholar
- 34Singh, A., Das, S., Altenbach, H., Craciun, E.-M.: Semi-infinite moving crack in an orthotropic strip sandwiched between two identical half planes. Z. Angew. Math. Mech. 100(2), e201900202 (2020)
- 35Rajagopal, K.R.: Conspectus of concepts of elasticity. Math. Mech. Solids 16(5), 536–562 (2011)
- 36Rajagopal, K.R., Srinivasa, A.R.: On the response of non-dissipative solids. Proc. Royal Soc. A: Math. Phys. Eng. Sci. 463(2078), 357–367 (2007)
- 37Mallikarjunaiah, S.M., Walton, J.R.: On the direct numerical simulation of plane-strain fracture in a class of strain-limiting anisotropic elastic bodies. Int. J. Fract. 192(2), 217–232 (2015)
- 38Mai, T., Walton, J.R.: On strong ellipticity for implicit and strain-limiting theories of elasticity. Math. Mech. Solids 20(2), 121–139 (2015)
10.1177/1081286514544254 Google Scholar
- 39Mai, T., Walton, J.R.: On monotonicity for strain-limiting theories of elasticity. J. Elasticity 120(1), 39–65 (2015)
10.1007/s10659-014-9503-4 Google Scholar
- 40Rajagopal, K.R., Srinivasa, A.R.: On thermomechanical restrictions of continua. Proc. R. Soc. Lond. A: Math. Phys. Sci. 460(2042), 631–651 (2004)
- 41Hao, Y.L., Li, S.J., Sun, S.Y., Zheng, C.Y., Hu, Q.M., Yang, R.: Super-elastic titanium alloy with unstable plastic deformation. Appl. Phys. Lett. 87(9), 091906 (2005)
10.1063/1.2037192 Google Scholar
- 42Saito, T., Furuta, T., Hwang, J.-H., Kuramoto, S., Nishino, K., Suzuki, N., Chen, R., Yamada, A., Ito, K., Seno, Y., Nonaka, T., Ikehata, H., Nagasako, N., Iwamoto, C., Ikuhara, Y., Sakuma, T.: Multifunctional alloys obtained via a dislocation-free plastic deformation mechanism. Science 300(5618), 464–467 (2003)
- 43Kulvait, V., Málek, J., Rajagopal, K.R.: The state of stress and strain adjacent to notches in a new class of nonlinear elastic bodies. J. Elasticity 135(1), 375–397 (2019)
10.1007/s10659-019-09724-0 Google Scholar
- 44Tian, Y., Yu, Z., Ong, C.Y.A., Cui, W.: Nonlinear elastic behavior induced by nano-scale phase in matrix of -type ti–25nb–3zr–2sn–3mo titanium alloy. Mater. Lett. 145, 283–286 (2015)
- 45Devendiran, V.K., Sandeep, R.K., Kannan, K., Rajagopal, K.R.: A thermodynamically consistent constitutive equation for describing the response exhibited by several alloys and the study of a meaningful physical problem. Int. J. Solids Struct. 108, 1–10 (2017)
- 46Muliana, A., Rajagopal, K.R., Tscharnuter, D., Schrittesser, B., Saccomandi, G.: Determining material properties of natural rubber using fewer material moduli in virtue of a novel constitutive approach for elastic bodies. Rubber Chem. Technol. 91(2), 375–389 (2018)
- 47Kowalczyk-Gajewska, K., Pieczyska, E.A., Golasiński, K., Maj, M., Kuramoto, S., Furuta, T.: A finite strain elastic-viscoplastic model of gum metal. Int. J. Plast. 119, 85–101 (2019)
- 48Bustamante, R., Montero, S., Ortiz-Bernardin, A.: A novel nonlinear constitutive model for rock: Numerical assessment and benchmarking. Appl. Eng. Sci. 3, 100012 (2020)
- 49Bustamante, R., Rajagopal, K.R.: A new type of constitutive equation for nonlinear elastic bodies. Fitting with experimental data for rubber-like materials. Proc. R. Soc. A 477(2252), 20210330 (2021)
10.1098/rspa.2021.0330 Google Scholar
- 50Ogden, R.W.: Large deformation isotropic elasticity–on the correlation of theory and experiment for incompressible rubberlike solids. Proc. R. Soc. London A 326(1567), 565–584 (1972)
- 51Kulvait, V., Málek, J., Rajagopal, K.R.: Modeling gum metal and other newly developed titanium alloys within a new class of constitutive relations for elastic bodies. Arch. Mech. 69(3), 223–241 (2017)
- 52Kulvait, V., Malek, J., Rajagopal, K.R.: Anti-plane stress state of a plate with a v-notch for a new class of elastic solids. Int. J. Fract. 179(1), 59–73 (2013)
10.1007/s10704-012-9772-5 Google Scholar
- 53Yoon, H.C., Mallikarjunaiah, S.M.: A finite-element discretization of some boundary value problems for nonlinear strain-limiting elastic bodies. Math. Mech. Solids 27(2), 281–307 (2022)
10.1177/10812865211020789 Google Scholar
- 54Shylaja, G., Venkatesh, B., Naidu, V.K., Murali, K.: Improved finite element triangular meshing for symmetric geometries using matlab. Materials Today: Proc. 46, 4375–4380 (2021)
10.1016/j.matpr.2020.09.665 Google Scholar
- 55Shylaja, G., Venkatesh, B., Naidu, V.K., Murali, K.: Two-dimensional non-uniform mesh generation for finite element models using matlab. Mater. Today. 46, 3037–3043 (2021).
- 56Murali, K., Naidu, V.K., Venkatesh, B.: Darcy–brinkman–forchheimer flow over irregular domain using finite elements method. IOP Conf. Ser. Mater. Sci. Eng. 577(1), 012158 (2019)
- 57Shylaja, G., Venkatesh, B., Naidu, V.K.: Finite element method to solve poisson's equation using curved quadratic triangular elements. IOP Conf. Ser. Mater. Sci. Eng. 577(1), 012165 (2019)
10.1088/1757-899X/577/1/012165 Google Scholar
- 58Murali, K., Naidu, V.K., Venkatesh, B.: Solution of darcy-brinkman flow over an irregular domain by finite element method. J. Phys. Conf. Ser. 1172(1), 012091 (2019)
- 59Murali, K., Naidu, V.K., Venkatesh, B.: Solution of darcy–brinkman–forchheimer equation for irregular flow channel by finite elements approach. J. Phys. Conf. Ser. 1172(1), 012033 (2019)
- 60Yoon, H.C., Lee, S., Mallikarjunaiah, S.M.: Quasi-static anti-plane shear crack propagation in nonlinear strain-limiting elastic solids using phase-field approach. Int. J. Fract. 227(2), 153–172 (2021)
10.1007/s10704-020-00501-y Google Scholar
- 61Lee, S., Yoon, H.C., Mallikarjunaiah, S.M.: Finite element simulation of quasi-static tensile fracture in nonlinear strain-limiting solids with the phase-field approach. J. Comput. Appl. Math. 399, 113715 (2022)
10.1016/j.cam.2021.113715 Google Scholar
- 62Yoon, H.C., Mallikarjunaiah, S.M., Bhatta, D.: Finite element solution of crack-tip fields for an elastic porous solid with density-dependent material moduli and preferential stiffness. Adv. Mech. Eng. 16(2) 1–23 (2024)
10.1177/16878132241231792 Google Scholar
- 63Mallikarjunaiah, S.M., Bhatta, D.: A finite element model for hydro-thermal convective flow in a porous medium: Effects of hydraulic resistivity and thermal diffusivity. arXiv preprint arXiv:2402.15917 (2024)
- 64Manohar, R., Mallikarjunaiah, S.M.: An -adaptive discontinuous galerkin discretization of a static anti-plane shear crack model. arXiv preprint arXiv:2411.00021 (2024)
