Computational solution to the torsion of rectangular cross-section prismatic bars made of functionally graded materials - Cabrera, Ryan Mendez (College of Engineering, University of the Philippines Diliman, 2014)
The main goal of this study is in developing a solution to the torsion boundary value problem (BVP) of linearly elastic prismatic bars with rectangular cross-section.Solutions to such BVP are well- established in the literature but only for those materials which are conventional (e.g.,metals).For those made of materials called functionally graded materials (FGMs),solutions are rarely available.The torsion BVP for FGMs is fairly difficult because of it governing partial differential equation (PDE),which is a consequence of the inhomogenity of FGMs,plus the inherent difficulty of the associated Neumann boundary conditions(BCs).The most recent solutions,i.e.,and exact and layerwise,are provided in the literature but only for one-dimensional layered type FGMs.Aside from the exact solution being able to solve on.FGMs which vary exponentially,the layerwise solution (which can, hyphotherically,solve other forms of material variation besides exponential, e.g. power-law ) suffers some serious lack of mathematical rigor because it does not directly solve the torsion BVP.Instead,it only approximates the bar's cross-section to be composed of thin isotropic layers,thus imitating FGMs. Therefore, this study was focused on developing a computational solution which is based on standard, well- established numerical methods.For practically, the finite difference method (FDM) was chosen not only for its simplicity in implementation but also for its geometric suitability to rectangular cross-section bars.A two-stage verification procedure was the carried out to determine the FDMs accuracy and convergence.The first stage was done to benchmark the FDM solution for conventional torsion BVP;the relative errors obtained for the rigidity D and torsion stress Tmax were all lower than 1.00%.A second stage was then done FGM bars whose materials modulus function,gzx,obeys the exponential and power-law forms prescribed in the literature.The percents errors obtained for the rigidity factor K(a quantity related to D)were all lower than 5.00%.However,for Tmaxm using a the FGM sample used in XU et al.,the relative error obtained was about 15.00%,which is only fairly acceptable.Nevertheless, the overall distribution of T(x,y,)by the FDM solution closely resembles to that by the exact solution since they share common characteristics such as the same location for Tmax,which is at the midpoint of(-b<y<b).In addition,the magnitude of this maximum stress is larger, from both methods,than that in a conventional bar of the same aspect ratio.Overall,the FDM was found to be an effective method in solving the torsion BVP for FGMs.