Modeling indentation in linear viscoelastic solids
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Modeling indentation in linear viscoelastic solids Yang-Tse Cheng1 and Che-Min Cheng2 Materials and Processes Laboratory, General Motors Research and Development Center, Warren, Michigan, USA 2 Institute of Mechanics, Chinese Academy of Sciences, Beijing, China 1
ABSTRACT Using analytical and finite element modeling, we study conical indentation in linear viscoelastic solids and examine the relationship between initial unloading slope, contact depth, and viscoelastic properties. We will then discuss whether the Oliver-Pharr method for determining contact depth, originally proposed for indentation in elastic and elastic-plastic solids, is applicable to indentation in viscoelastic solids. 1. INTRODUCTION Instrumented indentation is playing an increasing role in the study of small-scale mechanical behavior of “soft” matters, such as polymers, composites, biomaterials, and food products. Many of these materials exhibit viscoelastic behavior, especially at elevated temperatures. Modeling of indentation into viscoelastic solids thus forms the basis for analyzing indentation experiments in these materials. Theoretical studies of contacting linear viscoelastic bodies became active since the mid 1950s by the work of Lee [1], Radok [2], Lee and Radok [3], Hunter [4], Gramham [5, 6], Yang [7], and Ting [8, 9]. In recent years, a number of authors have extended the early work to the analysis of indentation measurements [10-15]. In this paper, we examine, through analytical and finite element modeling, the relationship between initial unloading slope, contact depth, and viscoelastic properties. We will then discuss whether the commonly used Oliver-Pharr method [16, 17] is applicable to indentation in viscoelastic solids. 2. ANALYTICAL RESULTS We consider a rigid, smooth, and frictionless conical indenter with half-angle θ indenting a viscoelastic solid that can be described by the constitutive relationships between deviatoric stress and strain, sij and d ij , and between dilatational stress and strain, σ ii and ε ii , as: t
s ij (t ) = 2∫ G (t − τ ) 0
∂d ij (τ ) ∂τ
dτ
, (1) ∂ε ii (τ ) σ ii (t ) = 3∫ K (t − τ ) dτ ∂τ 0 where G (t ) is the stress relaxation modulus in shear and K (t ) is the hydrostatic stress relaxation modulus. The time dependent Young’s modulus and Poisson’s ratio are then given by E (t ) = [9 K (t )G (t )] /[3K (t ) + G (t )] and ν (t ) = [E (t ) / 2G (t )] − 1 , respectively. In this paper, we further assume that Poisson’s ratio is time independent, which is possible if K (t ) and G (t ) have the same time dependence. Under these assumptions, the relationship between load, F (t ) , and displacement, h(t ) , are given by [18]: t
dh(τ ) 8 tan θ F (t ) = G (t − τ )h(τ ) dτ . ∫ π (1 − ν ) 0 dτ t
(2)
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The load-displacement relationship can therefore be obtained if the viscoelastic properties of materials, G (t ) and ν , are known. Conversely, the viscoelastic properties may be obtained from measured F (t ) vs. h(t ) relations by solving an integral equation. Eq. (2) reduces to the wellkno
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