A large strain finite volume method for orthotropic bodies with general material orientations

Abstract

This paper describes a finite volume method for orthotropic bodies with general principal material directions undergoing large strains and large rotations.The governing and constitutive relations are presented and the employed updated Lagrangian mathematical model is outlined. In order to maintain equivalence with large strain total Lagrangian methods, the constitutive stiffness tensor is updated transforming the principal material directions to the deformed configuration. Discretisation is performed using the cell-centred finite volume method for unstructured convex polyhedral meshes. The current methodology is successfully verified by numerically examining two separate test cases: a circular hole in an orthotropic plate subjected to a traction and a rotating orthotropic plate containing a hole subjected to a pressure. The 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 variable material orientation and parallel processing.

A Lagrangian cell‐centred finite volume method for metal forming simulation

“…The implicit diffusion term (first term on the right‐hand side of Equation () is discretised using central differencing with over‐relaxed non‐orthogonal correction : where F is the number of internal faces in cell P , , k u f = n u f − Δ u f and n u f is the unit normal of the face. The first term on the right‐hand side of the equation is treated implicitly, whereas the second term on the right‐hand side, representing a correction for the face non‐orthogonality, is treated explicitly; further details are given in .…”

“…The final discretised form of the linear momentum equation for each control volume P can be arranged in the form of M linear algebraic equations, for example see : where F is the number of internal faces of control volume P , a P is the central coefficient, a N are the neighbour coefficients representing interactions with neighbour cell‐centred unknowns and b P is the source vector contribution.…”

“…Due to the prominence of the FV method in the field of computational fluid dynamics, it is natural to expect the application of the FV method in Eulerian form to metal forming; a number of authors have applied FV‐based flow formulations to simulate plastic deformation processes, for example see . When examining the literature related to the application of Lagrangian FV methods to solid mechanics, a large range of practical problems have been successfully analysed [, ]; nonetheless, there has been relatively little research into Lagrangian FV approaches for finite strain elastoplastic analysis. Maneeratana and Ivanković developed such a procedure, where a number of separate mathematical models were examined including those based on hypoelastoplastic relations with objective Jaumann and Truesdell stress rates; Bijeonja later adopted a similar approach.…”

Development of a Hip Joint Model for Finite Volume Simulations

“…Nonetheless, the finite volume (FV) method, being attractively simple, yet strongly conservative in nature, has become a viable alternative in many solid mechanics applications [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40]70,71]. Accordingly, the current work employs a cell-centered FV structural contact procedure to numerically examine the hip joint, implemented in open-source Cþþ based software OpenFOAM (Open Source Field Operation and Manipulation, version 1.6-ext) [41][42][43], where the implemented physics and numerical algorithms may be viewed and are open to academic scrutiny, which is not possible with black box commercial codes.…”

Convergence of a Cell-Centered Finite Volume Discretization for Linear Elasticity

“…Most work where node-centered [6] and cell-centered [26] finite volume methods are introduced for elasticity contain only numerical validation. This includes also recent work on cellcentered methods [23,10]. When additional variables are introduced, convergence of finite volume methods has been established [20].…”