Dr Kevin M. Knowles (Department of Materials, University of Cambridge, UK)

Fourth-rank tensor algebra of cubic materials
When Oct 09, 2017
from 02:00 PM to 03:00 PM
Where LR8
Contact Name
Contact Phone 01865-283446
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Dr Kevin M. Knowles

Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, UK


In many situations when dealing with the elastic properties of cubic crystalline materials it is convenient to make the assumption that the material is isotropic, for example because the material under consideration is polycrystalline. However, there are clearly circumstances where the variation of elastic properties as a function of orientation in a single crystal is actually quite important, e.g., for silicon, widely used in single crystal form in microelectromechanical systems.


While the formulae for how the Young’s modulus, E, of a single crystal varies as a function of orientation and crystal system are well documented, formulae for other elastic properties have received much less attention, even for cubic materials. The recent surge in interest in auxetic materials has prompted renewed research into the conditions which enable cubic materials to have negative Poisson’s ratios, v; this has also drawn attention to the relative complexity of the general formula for the Poisson’s ratio v (n,m) of a cubic material as a function of the direction n of the uniaxial loading and the direction m of the transverse strain and the choices of n and m which give global maxima and minima of v (n,m).


In this seminar I will consider in detail a number of elastic moduli for cubic materials where the assumption of isotropy when dealing with polycrystalline cubic materials enables formulae to be readily derived which are functions of isotropic values of E and v alone, but where recognition of anisotropy makes the mathematics significantly more complicated. These are: (i) the variation of shear modulus as a function of orientation for a cubic single crystal, (ii) the biaxial moduli of a cubic single crystal substrate subjected to an equi-biaxial elastic strain as a function of substrate orientation, (iii) the plane strain Young’s modulus of a cubic single crystal as a function of orientation and (iv) the ‘effective modulus’ in cubic single crystals subjected to full transverse constraint. Of particular interest are global maxima and minima of these elastic moduli, just as for v (n,m).