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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.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74A35 Polar materials
74K15 Membranes
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[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)
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