An improved relation for the effective elastic compliance of a film/substrate system during indentation by a flat cylindrical punch

Nanomechanical and nanotribological properties of bioactive titanium surfaces prepared by alkali treatment

“…The effective elastic modulus of the film, subtracting the substrate influence, can be calculated by means of the analytical method proposed by Xu and Pharr (2006), which extends the Gao et al proposal (1992) to a wide range of situations. The method demands specific information about film and substrate: film thickness, substrate elastic modulus and the respective Poisson's ratios (ν).…”

“…The film elastic moduli, without the substrate effects, were calculated by the analytical method proposed by Xu and Pharr (2006).…”

Nanoindentation of Al2O3/Al2TiO5 composites: Small-scale mechanical properties of Al2TiO5 as reinforcement phase

“…In fact most of the models neglect the influence of the lateral stiffness and model the response of the system by a series of springs in series. The models used in this investigation are the ones proposed by Bec et al, 36 Song and Pharr [37][38][39] (which is based on the one proposed by Gao et al 40 ) and Hay and Crawford. 41 This last model is the only one that takes into account lateral stiffness.…”

Determining the elastic modulus and hardness of an ultra-thin film on a substrate using nanoindentation

“…A number of studies with several different approaches to modeling the substrate effect have been reported [7][8][9][10][11][12][13]. King used numerical techniques to model the elastic indentation of a layered half space with flat-ended punches of various cross-sections [8].…”

“…Gao's approximation becomes increasingly inaccurate as the elastic mismatch between the film and the substrate increases [11]. Xu and Pharr later showed that the accuracy of Gao's expression for the effective indentation modulus could be improved by a slight modification of the formula [13].…”