zbMATH — the first resource for mathematics

A control volume-based finite element method for plane micropolar elasticity. (English) Zbl 1195.74206
Summary: This paper describes the development of a numerical procedure for predicting deformations and stresses in a loaded two-dimensional membrane exhibiting micropolar or Cosserat constitutive behaviour. The procedure employs a conventional finite element (FE) mesh together with a dual mesh of interconnected control volumes, each of which must satisfy equilibrium. A series of patch tests covering a variety of simple strain states are used to validate the procedure that is then employed to predict the stress concentration in a membrane containing a small hole. The predictions provided by the procedure are compared with those given previously by FEs.

74S05 Finite element methods applied to problems in solid mechanics
74A35 Polar materials
74K15 Membranes
Full Text: DOI
[1] Aifantis, On the microstructural origin of certain inelastic models, Journal of Engineering Materials and Technology 106 pp 326– (1984)
[2] Eringen, Linear theory of micropolar elasticity, Journal of Mathematical Mechanics 15 pp 909– (1966) · Zbl 0145.21302
[3] Shu, Finite elements for materials with strain gradient effects, International Journal for Numerical Methods in Engineering 44 pp 373– (1999) · Zbl 0943.74072
[4] Zervos, A finite element displacement formulation for gradient elastoplasticity, International Journal for Numerical Methods in Engineering 50 pp 1369– (2001) · Zbl 1047.74073
[5] Soh, Finite element formulations of strain gradient theory of microstructures and the C0-1 patch test, International Journal for Numerical Methods in Engineering 61 pp 433– (2004) · Zbl 1075.74678
[6] Steinmann, An improved FE expansion for microploar localization, Communications in Numerical Methods in Engineering 10 pp 1005– (1994) · Zbl 0813.73070
[7] Nakamura, Finite element method for orthotropic micropolar elasticity, International Journal of Engineering Science 22 pp 319– (1984) · Zbl 0536.73007
[8] Providas, Finite element method in plane Cosserat elasticity, Computers and Structures 80 pp 2059– (2002)
[9] Li, Finite element method for linear micropolar elasticity and numerical study of some scale effects phenomena in MEMS, International Journal of Mechanical Sciences 46 pp 1571– (2004) · Zbl 1098.74054
[10] Diegele, Linear micropolar elastic crack tip fields under mixed mode loading conditions, International Journal of Fracture 129 pp 309– (2004) · Zbl 1187.74171
[11] Baliga, A new finite element formulation for convection diffusion problems, Numerical Heat Transfer 3 pp 393– (1980)
[12] Patankar, Numerical Fluid Flow and Heat Transfer (1980)
[13] Malan, Modelling coupled heat and mass transfer in drying non hygroscopic capillary particulate materials, Communications in Numerical Methods in Engineering 19 pp 669– (2003) · Zbl 1032.80009
[14] Bailey, A finite volume procedure to solve elastic solid mechanics problems in three dimensions on an unstructured mesh, International Journal for Numerical Methods in Engineering 38 pp 1757– (1995) · Zbl 0822.73079
[15] Greenshields, A fluid structure model for fast brittle fracture in plastic pipes, Journal of Fluids and Structures 14 pp 221– (2000)
[16] Greenshields, A unified formulation for continuum mechanics applied to fluid-structure interaction in flexible tubes, International Journal for Numerical Methods in Engineering 64 pp 1575– (2005) · Zbl 1122.74379
[17] Slone, Dynamic fluid-structure interaction using finite volume unstructured mesh procedures, Computers and Structures 80 pp 371– (2002)
[18] Ivankovic, Application of the finite volume method to the analysis of dynamic fracture problems, International Journal of Fracture 66 pp 357– (1994)
[19] Taylor, Solution of the elastic visco-plastic constitutive equations: a finite volume approach, Applied Mathematical Modelling 19 pp 747– (1995) · Zbl 0852.73078
[20] Fallah, Comparison of finite element and finite volume methods application in geometrically nonlinear stress analysis, Applied Mathematical Modelling 24 pp 439– (2000) · Zbl 0991.74068
[21] Bijelonja, A finite volume method for large strain analysis of incompressible hyperelastic materials, International Journal for Numerical Methods in Engineering 64 pp 1594– (2005) · Zbl 1122.74532
[22] Wenke, A finite volume method for solid mechanics incorporating rotational degrees of freedom, Computers and Structures 81 pp 321– (2003)
[23] Allman, A compatible triangular element including vertex rotations for plane elasticity analysis, Computers and Structures 19 pp 1– (1984) · Zbl 0548.73049
[24] Nakamura, Finite element analysis of Saint Venant end effects in micropolar elastic solids, Engineering Computations 12 pp 571– (1995) · Zbl 0843.73015
[25] Cook, Concepts and Applications of Finite Element Analysis (1989) · Zbl 0696.73039
[26] Kaloni, Stress concentration effects in micropolar elasticity, Zeitschrift für Angewandte Mathematik und Physik 18 pp 136– (1967)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.