MIXED FORMULATION OF FINITE ELEMENT METHOD WITHIN STRAIN GRADIENT ELASTICITY

Authors

DOI:

https://doi.org/10.15407/dopovidi2025.03.061

Keywords:

gradient theory of elasticity, stress and strain gradients, variational equations, finite element method, mixed approximation, convergence

Abstract

One of the generalized continuum theories related to the microstructure scale is the Toupin-Mindlin gradient theory of elasticity, which allows one to take into account structural inhomogeneities and material damage at the micro level. The peculiarity of solving variational equations in gradient theory is the consideration of the first partial derivatives of the components of the small strain tensor. A necessary condition for the convergence of a solution based on the finite element method is the property of the approximation functions that ensures the continuity of displacements and their first derivatives at the boundary between elements. This study considers an alternative approach, according to which the solution of boundary value problems in gradient elasticity theory is based on the use of a variational formulation of displacements, deformations, stresses, and their gradients. This formulation greatly simplifies the choice of approximation functions, since there is no need to use finite elements that ensure the continuity of the first derivatives of displacements between elements. To analyze the correctness of the mixed approximation, the variational equations of the mixed method are transformed into an equivalent form with respect to displacements, strains, and their gradients. Based on the obtained a priori estimates, conditions that ensure the convergence of approximate solutions based on the mixed approximation are determined.

Downloads

Download data is not yet available.

References

Toupin, R. A. (1962). Elastic materials with couple-stresses. Arch. Rational Mech. Anal., 11, pp. 385-414. https://doi.org/10.1007/BF00253945

Mindlin, R. D. (1964). Micro-structure in linear elasticity. Arch. Ration. Mech. Anal., 16, pp. 51-78. https://doi. org/10.1007/BF00248490

Mindlin, R. D. & Eshel, N. N. (1968). On first strain-gradient theories in linear elasticity. Int. J. Solids Struct., 4, No. 1, pp. 109-124. https://doi.org/10.1016/0020-7683(68)90036-X

Aifantis, E. C. (1992). On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci., 30, No. 10, pp. 1279-1299. https://doi.org/10.1016/0020-7225(92)90141-3

Altan, B. C. & Aifantis, E. C. (1997). On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mater., 8, No. 3, pp. 231-282. https://doi.org/10.1515/JMBM.1997.8.3.231

Askes, H. & Aifantis, E. C. (2011). Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct., 48, No. 13, pp. 1962-1990. https://doi.org/10.1016/j.ijsolstr.2011.03.006

Ru, C. Q. & Aifantis, E. C. (1993). A simple approach to solve boundary-value problems in gradient elasticity. Acta. Mech., 101, pp. 59-68. https://doi.org/10.1007/BF01175597

Grentzelou, C. G. & Georgiadis, H. G. (2005). Uniqueness for plane crack problems in dipolar gradient elasticity and in couple-stress elasticity. Int. J. Solids Struct., 42, No. 24-25, pp. 6226-6244. https://doi.org/10.1016/j. ijsolstr.2005.02.045

Grentzelou, C. G. & Georgiadis, H. G. (2008). Balance laws and energy release rates for cracks in dipolar gradient elasticity. Int. J. Solids Struct., 45, No. 2, pp. 551-567. https://doi.org/10.1016/j.ijsolstr.2007.08.007

Gourgiotis, P. A. & Georgiadis, H. G. (2009). Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity. J. Mech. Phys. Solids., 57, No. 11, pp. 1898-1920. https://doi.org/10.1016/j. jmps.2009.07.005

Lazar, M. & Maugin, G. A. (2005). Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity. Int. J. Eng. Sci., 43, No. 13-14, pp. 1157-1184. https://doi.org/10.1016/j. ijengsci.2005.01.006

Nazarenko, L., Glüge, R. & Altenbach, H. (2022). Uniqueness theorem in coupled strain gradient elasticity with mixed boundary conditions. Continuum Mech. Thermodyn., 34, pp. 93-106. https://doi.org/10.1007/s00161- 021-01048-6

Bleustein, J. L. (1967). A note on the boundary conditions of Toupin’s strain-gradient theory. Int. J. Solids Struct., 3, No. 6, pp. 1053-1057. https://doi.org/10.1016/0020-7683(67)90029-7

Deng, G. & Dargush, G. F. (2021). Mixed variational principle and finite element formulation for couple stress elastostatics. Int. J. Mech. Sci., 202-203, 106497. https://doi.org/10.1016/j.ijmecsci.2021.106497

Gao, X.-L. & Park, S. K. (2007). Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. Int. J. Solids Struct., 44, No. 22-23, pp. 7486-7499. https://doi.org/10.1016/j.ijsolstr.2007.04.022

Amanatidou, E. & Aravas, N. (2002). Mixed finite element formulations of strain-gradient elasticity problems. Comput. Methods Appl. Mech. Eng., 191, No. 15-16, pp. 1723-1751. https://doi.org/10.1016/S0045- 7825(01)00353-X

Markolefas, S., Papathanasiou, T. K. & Georgantzinos, S. K. (2019). p-Extension of C0 continuous mixed finite

elements for plane strain gradient elasticity. Arch. Mech., 71, No. 6, pp. 567-593. https://doi.org/10.24423/ aom.3219

Akarapu, S. & Zbib, H. M. (2006). Numerical analysis of plane cracks in strain-gradient elastic materials. Int. J. Fract., 141, pp. 403-430. https://doi.org/10.1007/s10704-006-9004-y

Skalka, P., Navrátil, P. & Kotoul, M. (2016). Novel approach to FE solution of crack problems in the Laplacian- based gradient elasticity. Mech. Mater., 95, pp. 28-48. https://doi.org/10.1016/j.mechmat.2015.12.007

Lurie, S. & Solyaev, Y. (2019). Anti-plane inclusion problem in the second gradient electroelasticity theory. Int. J. Eng. Sci., 144, 103129. https://doi.org/10.1016/j.ijengsci.2019.103129

Chirkov, A.Yu., Nazarenko, L. & Altenbach, H. (2024). Plane crack problems within strain gradient elasticity and mixed finite element implementation. Comput. Mech., 74, pp. 703-721. https://doi.org/10.1007/s00466- 024-02451-x

Chirkov, O.Yu., Nazarenko, L. & Altenbach, H. (2024). Mixed formulation of finite element method within Toupin—Mindlin gradient elasticity theory. Strength Mater., 56, pp. 223-233. https://doi.org/10.1007/s11223- 024-00642-8

Ciarlet, P. G. (1978). The finite element method for elliptic problems. Studies in mathematics and its applications (Vol. 4). Amsterdam: North-Holland, 1978.

Zienkiewicz, O. C. & Taylor, R. L. (2000). The finite element method. Vol. 2: Solid mechanics. Oxford: Butterworth-Heinemann.

Chirkov, A. Yu., Nazarenko, L. & Altenbach, H. (2025). Mixed FEM implementation of three-point bending of the beam with an edge crack within strain gradient elasticity theory. Continuum Mech. Thermodyn., 37, 1. https://doi.org/10.1007/s00161-024-01333-0

Chirkov, O. Yu. (2024). Computational analysis of fracture mechanics model problems based on the Toupin— Mindlin gradient elasticity theory equations. Strength Mater., 56, No. 5, pp. 907-916. https://doi.org/10.1007/ s11223-024-00702-z

Nazarenko, L., Chirkov, A. Yu., Altenbach, H. (2025). Axisymmetric problem within strain-gradient elasticity and mixed FEM solution for cylinders with cracks under tension. Math. Mech. Solids. https://doi. org/10.1177/10812865251321800

Chirkov, O.Yu. (2024). Material microstructure scale effect on brittle fracture resistance of WWER-1000 reac- tor pressure vessel steel. Strength Mater., 56, No. 6, pp. 1103-1107. https://doi.org/10.1007/s11223-025-00720-5

Published

29.06.2025

How to Cite

Chirkov, O. (2025). MIXED FORMULATION OF FINITE ELEMENT METHOD WITHIN STRAIN GRADIENT ELASTICITY. Reports of the National Academy of Sciences of Ukraine, (3), 61–70. https://doi.org/10.15407/dopovidi2025.03.061