The variation in Young's modulus of as much as 50%, introduced over a distance of 2mm, follows the law [2] E(z)= Esurface +E0z k where Esurface=254GPa, E0=85.325GPa.mm-k and k=0.497, 0mm< z <2mm. The authors demonstrated that this gradient in the elastic modulus led to optimal materials properties in the contact-damage behavior. Since we want to numerically analyze the effect of stochasticity on the predicted optimal elastic gradient, a series of "random" microstructures that have the same average surface gradient, but with variable placement of the second phase, is generated. An example is given in Figure 1 (in white alumina, in black glass). Figure 1: example of a random microstructure. For each discrete depth z (corresponding to a layer), the Young modulus of the so represented composite can be calculated using the rule of mixtures relation E graded(z)= Vglass(z)Eglass + (1-Vglass)Ealumina where Vglass is the volume fraction of glass at each z. The profiles of elastic modulus were so calculated for 50 different random generated microstructures and reported in Figure 2. This picture reports the average values for each z and also average values +/- standard deviation (dashed lines). Thus a range of variability in the elastic modulus profile for real (i.e. discrete) microstructures is individuated. Figure 2:Young modulus vs depth z. Average values for 50 different microstructures The average elastic modulus profile approximately coincides with that of a continuous (i.e. homogenized) material. The continuous material is achieved in this analysis generating a microstructure with several different layers, each one with a single value of elastic modulus. In the limit of an infinite number of
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