Thin films that are functionally gradient improve the mechanical properties of film-substrate layered materials. Mechanical properties of such materials are found by using indentation tests. In this study, finite element models are developed to simulate the indentation test. The models are based on an axisymmetric half space of a specimen subjected to spherical indentation. The film layer through the thickness is modeled to have either homogeneous material properties or nonhomogeneous material properties that vary linearly. Maximum indenter displacement, and maximum normal and shear stresses at the interface are compared between the homogeneous model and the nonhomogeneous model for pragmatic contact length to film thickness ratios of 0.2 to 0.4, and film to substrate moduli ratios of 1 to 200 to 1. Additionally, a coefficient is derived from regression of the stress data produced by these models and compared to that used to define the pressure field in the axisymmetric Hertzian contact model. The results of this study suggest that a displacement boundary condition to an indenter produces the same results as a pressure distribution boundary condition. The critical normal stresses that occur between modeling a film as a nonhomogeneous and as a homogeneous material vary from 19% for a modulus ratio of 2.5:1 to as high as 66% for a modulus ratio of 200:1 indicating that the modeling techniques produced very different maximum normal stresses. The difference in the maximum shear stress between the nonhomogeneous and the homogeneous models varied from 19% for a 2.5:1 modulus ratio to 57% for the 200:1 modulus ratio but reached values as low as 6% for the 50:1 modulus ratio. The maximum contact depth between the nonhomogeneous and the homogeneous models varied from 14% for the 2.5:1 case to as much as 75% in the 200:1 case. The results from the reapplication of the pressure field derived from the regression coefficients and the R2 values from these regression models indicate the correctness of the regression model used as well as its ability to replicate the normal stresses in the contact area and maximum indenter displacements in a FEA model for both the homogeneous and the nonhomogeneous models for modulus ratios ranging from 2.5:1 to 200:1. The agreement between the regression based coefficients and the force based coefficients suggests the validity for the use of the theoretical axisymmetric Hertzian contact model for defining the pressure field in the contact area and displacements for both the homogeneous case and the nonhomogeneous case for the considered film to substrate moduli ratios and contact length to film thickness ratios.

University of Macau / Faculty of Science and Technology / Department of Civil and Environmental Engineering

Functionally graded materials (FGMs) are inhomogeneous materials in which the material properties vary with respect to space. Research has been done by scientific community in developing techniques like photothermal radiometry (PTR) to measure the thermal conductivity and volumetric heat capacity of FGMs. One of the problems involved in the technique is to solve the inverse problem, i.e., estimating the thermal properties after the frequency scan has been obtained. The present work involves finding the unknown thermal conductivity and volumetric heat capacity of the FGMs by using finite volume method. By taking the flux entering the sample as periodic and solving the discretized 1-D thermal wave field equation at a frequency domain, one can obtain the complex temperatures at the surface of the sample for each frequency. These complex temperatures when solved for a range of frequencies gives the phase vs frequency scan which can then be compared to original frequency scan obtained from the PTR experiment by using a residual function. Brute force and gradient descent optimization methods have been implemented to estimate the unknown thermal conductivity and volumetric specific heat of the FGMs through minimization of the residual function. In general, the spatial composition profile of the FGMs can be approximated by using a smooth curve. Three functional forms namely Arctangent curve, Hermite curve, and Bezier curve are used in approximating the thermal conductivity and volumetric heat capacity distributions in the FGMs. The use of Hermite and Bezier curves gives the flexibility to control the slope of the curve i.e. the thermal property distribution along the thickness of the sample. Two-layered samples with constant thermal properties and three layered samples in which one of the layer has varying thermal properties with respect to thickness are considered. The program is written in Fortran and several test runs are performed. Results obtained are close to the original thermal property values with some deviation based on the stopping criteria used in the gradient descent algorithm. Calculating the gradients at each iteration takes considerable amount of time and if these gradient values are already available, the problem can be solved at a faster rate. One of the methods is extending automatic differentiation to complex numbers and calculating the gradient values ahead; this is left for future work.