Table 2.

Mathematical functions used to describe the profile of the optimal elastic modulus gradients.

function	continuity/discontinuity	equation	determination of constants
linear	a continuous linear elastic modulus profile across the whole thickness (C-LIN)	E(t) = At + B	using the values of the elastic modulus at the bottom and top of each model
	a discontinuous linear elastic modulus profile across the thickness of individual layers (DC-LIN)		using the values of the elastic modulus at the bottom and top of each individual layer
exponential	a continuous exponential elastic modulus profile across the whole thickness (C-EXP)	E(t) = CeDt	using the values of the elastic modulus at the bottom and top of each model
	a discontinuous exponential elastic modulus profile across the thickness of individual layers (DC-EXP)		using the values of the elastic modulus at the bottom and top of each individual layer
logistic	a continuous logistic elastic modulus profile across the whole thickness (C-LG)	$E(t)=F+{\displaystyle \frac{G-F}{{(1+H{\text{e}}^{-I(t-J)})}^{1/k}}}$	using the values of the elastic modulus at the bottom and top of each individual layer
constant	a discontinuous constant elastic modulus profile across the thickness of individual layers (DC-CST)	E(t) = L	using the mean values of the elastic modulus of individual layers