Quasi-static and Oscillatory Indentation in Linear Viscoelastic Solids
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1049-AA06-05
Quasi-static and Oscillatory Indentation in Linear Viscoelastic Solids Yang-Tse Cheng1 and Che-Min Cheng2 1 General Motors R&D Center, Warren, MI, 48090 2 Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100080, China, People's Republic of ABSTRACT Instrumented indentation is often used in the study of small-scale mechanical behavior of “soft” matters that exhibit viscoelastic behavior. A number of techniques have been used to obtain the viscoelastic properties from quasi-static or oscillatory indentations. This paper summarizes our recent findings from modeling indentation in linear viscoelastic solids. These results may help improve methods of measuring viscoelastic properties using instrumented indentation techniques. INTRODUCTION Instrumented indentation [1-13] can be performed in either quasi-static or oscillatory mode for measuring mechanical properties of “soft” matters, such as polymers, composites, and biomaterials, that are often viscoelastic. In the load- or displacement-controlled quasi-static mode, the load-displacement curves are recorded. One of the widely used methods, due to Oliver and Pharr [2], is to obtain the elastic modulus from the initial unloading slope, S = (dF / dh) m , at the maximum indenter displacement, hm , dF 4G 2E S= a= A, (1) h = hm = dh 1 −ν π (1 − ν 2 ) where G is the shear modulus, ν is Poisson’s ratio, E = 2G (1 + ν ) is Young’s modulus, a is the contact radius, and A = πa 2 is the contact area. The contact radius, a , can be obtained from the contact depth, hc , and indenter geometry. Oliver and Pharr [2] proposed an equation for hc : Fm , (2) hc = hm − ξ (dF / dh) m where Fm is the load at hm . The numerical value of ξ is (2 /π )(π − 2) = 0.727 and 3 / 4 for a conical and paraboloid of revolution, respectively. Although Eqs. (1) and (2) were derived from solutions to elastic contact problems, they have been used for indentation in elastic-plastic solids and viscoelastic solids. One of our motivations was to evaluate whether Eqs. (1) and (2) could be used for indentation in linear viscoelastic solids and another was to improve the existing methods [14-18]. In the oscillatory mode, a sinusoidal force is typically superimposed on a quasi-static load on the indenter [1,3,4,6,9,10,13]. The indentation displacement response and the out-of phase angle between the applied harmonic force and the assumed harmonic displacement may be recorded at a given excitation frequency or multiple frequencies. Several authors [1,3,4,6,9,10,13] have proposed analysis procedures for determining the complex Young’s
modulus, E * (ω ) = E ' (ω ) + iE '' (ω ) , where E ' (ω ) is the storage modulus and E '' (ω ) is the loss modulus, from oscillatory indentations using the following equations: π S π Cω E′ E ′′ = = , (3) and 2 2 1 −ν 2 A 1 −ν 2 A where ν is Poisson’s ratio, S is contact stiffness, C is damping coefficient, and A is contact area between the indenter and the sample. For an ideal indenter with infinite system stiffness and zero mass, the contact stiffness and damp
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