Nanoindentation of Poly (Methyl Methacrylate)
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Nanoindentation of Poly (Methyl Methacrylate) Michael J. Adams, David M. Gorman and Simon A. Johnson Unilever Research Port Sunlight, Bebington, Wirral, CH63 3JW, U.K. ABSTRACT For the indentation of an elastic-plastic homogeneous half-space with a pyramidal indenter, the load theoretically increases with the square of the total penetration depth. Experimental data are presented in the current paper that demonstrate the validity of this relationship for an organic polymer and a Berkovich indenter, provided that appropriate account is taken of the tip defect and viscoplasticity. It is also shown that a simple analytical solution exists for the ratio of the contact depth to the total penetration depth. These findings assist in identifying procedures for obtaining the rate-dependent mechanical properties of thin polymer coatings or polymeric materials with depth-dependent mechanical properties. INTRODUCTION Loubet et al. [1] showed that the load increases with the square of the total penetration depth for the indentation of both perfectly-elastic and rigid-plastic homogeneous half-spaces with conical and pyramidal indenters. They also established that this relationship is valid for the indentation of an elastic perfectly-plastic half-space with a Vickers pyramid; this involved the derivation of an expression for the loading curve in terms of material and geometric parameters and carrying out indentation measurements on steel. Malzbender et al. [2,3] obtained a similar result by manipulating the standard nanoindentation equations. Moreover, they found an excellent agreement for the analytical expression with finite element calculations based on relatively large values of the elastic modulus and hardness. Hainsworth et al. [4] empirically extended the approach to other metals and sapphire and also indicated how deviations from the quadratic behaviour could be used to investigate materials with depth or scale-dependent mechanical properties. The current paper examines whether this useful data reduction procedure is also applicable to glassy polymers which typically exhibit elastic-viscoplastic behaviour. THEORY The hardness, H, of a solid corresponds to the mean contact pressure and therefore H = W /A,
(1)
where W is the applied normal load and A is the projected contact area. For a conical or pyramidal indenter A is given by A = b hc2 ,
(2)
where hc is the contact depth, and b is a geometric constant that has a value equal to 24.56 for a perfect Berkovich indenter. The general procedure for nanoindentation experiments involves monitoring the total penetration depth, ht , as a function of W. In order to calculate the hardness, it is necessary to calculate hc from ht. Oliver and Pharr [5] showed that Q7.10.1
hc = ht − κ W , S
(3)
where S is the contact stiffness and κ = (2π - 4)/π = 0.73 for a conical or 0.75 (exactly) for a Berkovich indenter. Thus, the hardness of an elastic-plastic material can be calculated using measurements of S, either from unloading data, or from an appropriate jitter modulation technique [
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