Obtaining shear relaxation modulus and creep compliance of linear viscoelastic materials from instrumented indentation u

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Using Laplace transform, we solve the inverse problem of obtaining the shear relaxation modulus and creep compliance of linear viscoelastic solids from indentation by axisymmetric indenters of power-law profiles. We identify several simple, though nontrivial, loading paths for carrying out indentation measurements such that the inverse problem has analytical solutions. We show that the shear relaxation modulus and creep compliance may be readily obtained using the newly derived analytical expressions together with proposed indentation loading paths. Zt


Instrumented indentation is a powerful tool for probing small-scale mechanical behaviors of “soft” materials, such as polymers, composites, and biomaterials.1–4 Because many soft materials deform viscoelastically, it is important to develop robust analysis methods for obtaining linear viscoelastic properties from indentation measurements. Extending the early work by such pioneers as Lee,5 Radok,6 Lee and Radok,7 Hunter,8 Graham,9,10 and Ting,11,12 a number of authors have recently proposed methods for determining viscoelastic properties from indentation measurements.13–38 This work, together with our previous publications on related topics,39–44 provides several complementary methods for obtaining linear viscoelastic properties from instrumented indentation measurements. This work also helps improve the understanding of indentation in linear viscoelastic solids. II. ANALYSIS

sij ðtÞ ¼ 2

Gðt  tÞ

@dij ðtÞ dt ; @t


K ðt  tÞ

@dij ðtÞ dt ; @t




sii ðtÞ ¼ 3 0

where G(t) is the relaxation modulus in shear and K(t) is the relaxation modulus in dilatation. The time dependent Young’s modulus and Poisson’s ratio are then given by E(t) = [9K(t)G(t)]/[3K(t) þ G(t)] and n(t) ¼ [E(t)/2G(t)]-1, respectively. When G(t), K(t), and n(t) are time-independent, Eq. (1) reduces to the ones for elastic solids. The corresponding problem of frictionless indentation in linear elastic solids has been solved previously, for example by Sneddon,46 for the contact radius, a, and indenter displacement, h, relationship: Z1 0 f ðxÞ h ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi dx ; ð2Þ 1  x2 0

We consider a rigid, smooth, frictionless, axisymmetric indenter of power-law profile, f(r) = arn, where a is a constant, indenting a semi-infinite linear viscoelastic solid that can be described by the following constitutive relationships45 between deviatoric stress and strain, sij and dij, and between dilatational stress and strain, sii and eii,


Address all correspondence to this author. e-mail: [email protected] This author was an editor of this journal during the review and decision stage. For the JMR policy on review and publication of manuscripts authored by editors, please refer to http://www.mrs. org/jmr_policy DOI: 10.1557/JMR.2009.0365 J. Mater. Res., Vol. 24, No. 10, Oct 2009

and for the indentation force, F, and displacement relationship: Z1 2 0 4Ga x f ðxÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi dx ; F¼ ð3Þ 1n 1  x2 0

where x = r/a. For the power-law indenter profile, f(t) ¼ arn, we obtain, usin

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