Center for Composite Materials - University of Delaware

Research Summary

A Semi-Analytical Method for Investigating Conical Indentation of a Viscoelastic Sphere

Authors: J. K. Phadikar, T. A. Bogetti and A. M. Karlsson


• Indentation: tool for measuring mechanical properties of small scale structures
• Mechanical characterization of viscoelastic spheres are of technological interest
• A semi-analytical method is proposed to determine the force-displacement relationship for conical indentation of a viscoelastic sphere
• The efficiency of the method is investigated


• Conventional testing methods such as tensile test are difficult to employ for to determine the mechanical properties of small scale structures such as thin films.
• An useful test for such structures is indentation testing (Cheng and Cheng, 2004)
• In an indentation test, an indenter is pushed into the substrate and then removed as shown in Figure 1.
• The force, P, and the displacement, h, are continuously measured during the loading and the unloading periods. A force-displacement relationship is obtained as shown in Figure 2.
• From the force-displacement relationship, the material properties of the substrate is determined


• Viscoelastic spheres are encountered in various engineering fields
• Examples include:
-Micron-sized metal coated polymer particles used in the manufacturing of anisotropic conductive adhesives (Kristiansen et al., 2001)
-Polymer latex particles for controlling the mechanical properties of latex films used in synthetic latex materials (Misawa et al., 1991)
-Living cells (Dao et al., 2003)
• In the present work, conical indentation of a viscoelastic sphere resting on a flat rigid plate (shown in Figure 3) is considered.
• A semi-analytical method is proposed to determine the force-displacement relationship by knowing the material properties and geometrical parameters.


• Lee and Radok (1960) proposed the method of functional equations to determine the viscoelastic solution of a problem from the corresponding elastic solution.
• In this method, the elastic solution of the problem is converted into the viscoelastic solution in Laplace domain and then the solution is converted back to the time domain.
• For the present problem, the elastic solution is obtained using finite element simulations in ABAQUS.
• The indenter force is dependent on the displacement and the material properties as follows:

• Using Buckingham’s PI theorem (Buckingham, 1914) and dimensional analysis, the above equation can be simplified to the following equations:

• By finite element simulations, the function f2 was determined to be:

• The method of functional equation can be applied to Eq. (2) to obtain the corresponding viscoelastic solution of the problem as:

• The viscoelastic constitutive relationships for the sphere were assumed to be as shown in Figure 4a and 4b. Further, the loading function was assumed to be as shown in Figure 4c.

Figure 4: Assumed constitutive behavior of the viscoelastic material and the loading function: a) standard three-element solid model for deviatoric behavior, b) spring element for spherical (volumetric) behavior and c) triangular loading

• For the selected constitutive relationships and loading function, the displacement was determined to be (using Eq. 4):


• To investigate the accuracy of the proposed method, a viscoelastic material with the material properties such as G1 = 234.6 MPa, G2 = 25.78 MPa, η = 257.78 and K = 687.62 MPa, is considered and the force-displacement relationship for an indenter with half-angle α = 70.3° is determined using the proposed method.
• The force-displacement relationship is compared with that obtained directly using ABAQUS.
• Figure 5 depicts such comparisons for selected time periods. It can be seen that the proposed method can compute the force-displacement relationship quite accurately.
• The computational cost involved in obtaining the force-displacement relationships by the proposed method is much less compared to that by using ABAQUS directly.
• For example, for the case T = 30 s, it took approximately 5 CPU hours to obtain the force-displacement relationship using finite element simulation, whereas it took less than 1 CPU second for the semi-analytical method (excluding the finite element analysis for the elastic problem). Computations were performed in a DELL Precision T7400 workstation with two Intel(R) Xeon(R) X5472 @3GHz processors.


• Instrumented indentation is a useful technique for measuring the mechanical properties of small scale structures.
• Viscoelastic spheres are encountered in various engineering practices.
• A semi-analytical method has been proposed to determine the force-displacement relationship for conical indentation of a viscoelastic sphere resting on a flat rigid plate.
• The method can capture the response quite accurately and the computational cost required is much less compared to when direct finite element analysis is used.


• Cheng YT, Cheng CM. Scaling, dimensional analysis, and indentation measurements. Mater. Sci. Engg. R: Reports 2004;44:91-149.
• Kristiansen H, Shen Y, Liu J. Characterisation of mechanical properties of metal-coated polymer spheres for anisotropic conductive adhesive. IEEE 2001:344-348.
• Misawa H, Koshioka M, Sasaki K, Kitamura N, Masuhara H, Three dimensional optical trapping and laser ablation of a single polymer latex particle in water. J. Appl. Phys. 1991;70:3829-3836.
• Dao M, Lim C, Suresh S, Mechanics of the human red blood cell deformed by optical tweezers. J. Mech. Phys. Solids 2003;51:2259-2280.
• Lee EH, Radok, JRM. The contact problem for viscoelastic bodies. J. Appl. Mech. 1960; 27:438–444.
• Buckingham E. On physically similar systems; illustrations of the use of dimensional equations. Phys. Rev. 1914;4:345-376.

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