A mathematical modelling approach on the characteristic and mechanical parameters of porous cobalt as a scaffold material for optimal bone growth rate

The purpose of this research is to develop a general predictive mathematical model of the deformation behaviours for various symmetric geometrical tubes under lateral compression between two flat rigid plates. The mathematical model has been proposed based on rigid, perfectly plastic model and the energy balance method. The mathematical models are divided into two cases i.e. 'Case 1' and 'Case 2' based on the geometrical shapes of the tubes. ‘Case 1’ is for shapes with number of sides 6, 10, 14 and so on such as hexagonal, decagonal and tetra-decagonal tubes. Whereas, ‘Case 2’ is for shapes with number of sides 4, 8, 12 and so on such as square, octagonal and dodecagonal tubes. The prediction or assumption used in this mathematical model was that the tubes would deform in phase by phase during plastic deformation. In order to achieve this purpose, the deformation behaviour and the energy-absorption performance of various geometrical tube shapes need to be determined. The geometrical tubes shapes which were studied include square, hexagonal, octagonal, decagonal, dodecagonal and tetra-decagonal tubes. For that, experimental tests and finite element analysis (FEA) simulation were conducted to determine the collapse behaviour of these various symmetrical geometric tubes. First, the quasi-static lateral compression test was conducted on square and cylindrical tubes experimentally and by FEA simulation method by using INSTRON Universal Testing Machine and ABAQUS software respectively. Both results were compared to validate the FEA simulation results. Then, the validated FEA simulation method was performed for these various symmetrical geometric tubes to determine their deformation behaviour and energy-absorption performance and then to validate the newly mathematical model. The comparison between the experiment and FEA simulation had shown good agreement. The simulation study showed that square and symmetric hexagonal tubes deformed with 1 phase of plastic deformation, symmetric octagonal and decagonal tubes deformed with 2 phases of plastic deformation, symmetric dodecagonal and tetra-decagonal tubes deformed with 3 phases of plastic deformation. It was determined that, the general mathematical model had succeeded to predict the deformation behaviour of various symmetric geometrical shapes for both cases but discrepancy occurred for certain specimens due to sudden high peak at the last phase and small angle difference for neighbouring sides. The energy – absorption performance analyses for different types of symmetric geometrical tubes had shown that symmetric hexagonal tube produced the best energy-absorption with high total energy absorption, low yield stress and long stroke without any sudden jump force.

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