Viscoelasticity in simple indentation-cycle experiments: a computational study

Instrumented indentation has become an indispensable tool for quantitative analysis of the mechanical properties of soft polymers and biological samples at different length scales. These types of samples are known for their prominent viscoelastic behavior, and attempts to calculate such properties from the indentation data are constantly made. The simplest indentation experiment presents a cycle of approach (deepening into the sample) and retraction of the indenter, with the output of the force and indentation depth as functions of time and a force versus indentation dependency (force curve). The linear viscoelastic theory based on the elastic–viscoelastic correspondence principle might predict the shape of force curves based on the experimental conditions and underlying relaxation function of the sample. Here, we conducted a computational analysis based on this theory and studied how the force curves were affected by the indenter geometry, type of indentation (triangular or sinusoidal ramp), and the relaxation functions. The relaxation functions of both traditional and fractional viscoelastic models were considered. The curves obtained from the analytical solutions, numerical algorithm and finite element simulations matched each other well. Common trends for the curve-related parameters (apparent Young’s modulus, normalized hysteresis area, and curve exponent) were revealed. Importantly, the apparent Young’s modulus, obtained by fitting the approach curve to the elastic model, demonstrated a direct relation to the relaxation function for all the tested cases. The study will help researchers to verify which model is more appropriate for the sample description without extensive calculations from the basic curve parameters and their dependency on the indentation rate.

For the acquisition of analytical solutions, the equations were solved symbolically with SageMath [1] and Wolfram Alpha integral calculator [2]. We will reproduce the relaxation functions here and provide analytical solutions for the Ting's equations describing a common indentation experiment with the triangular or sinusoidal displacement. The solutions for the approach (tip deepens into the sample, contact area increasing) and retraction (contact area decreasing) curves will be provided separately. The solutions for the approach curve (Lee-Radok's solution) could also be extended for the case of the retraction curves, we did it for some situations where the complete Ting's solution was not obtained. The Lee-Radok's and Ting's solution match for the cylindrical probe since the contact area is constant or has a zero value.
where  () D t is the Dirac delta function. The common analytical solution of the Ting's equation for all the geometries is: (A4) As expected for a viscous material, the force drops to zero then the cantilever goes up (retracts).
The Kelvin-Voight element, a combination of a spring and a dashpot in parallel, has the following relaxation function: ( ) . We will split the solutions for the approach Fap and retraction Fretr curves. 1 () tt function can be found from Eq. (3), which leads to the condition: which for the triangular ramp leads to: and for the sinusoidal ramp: (A7) The common solution for all the geometries for the triangular (ramp) and sinusoidal (sin) indentation histories are, respectively: solution for the approach curve was also obtained before in [3].
For the Maxwell element, the relaxation function is: . The 1 () tt function for the triangular displacement is: The solution for 1 () tt in the case of the sinusoidal load could not be isolated, but it can be numerically found from the following relation: For the approach curve, the solutions for the triangular displacement and different probe geometries are: where erfi() is the imaginary Gauss error function. For the sinusoidal displacement, there is no closed-form analytical solution for the case of the spherical probe: where W is the Lambert W function. For the sinusoidal displacement, the 1 () tt function can be numerically found from the following relation: The solutions for the approach curves are: For the retraction curves, the analytical solutions for the spherical and conical probes are too long and complex, so here only the solution for the cylindrical probe is presented: The solutions for the sinusoidal displacement were found except for the case of sphereretraction, but are not presented here due to complexity.
appr sphere ramp geom appr cone ramp geom The solutions for the retraction curves for the spherical and conical geometries are too long and not presented here. For the sinusoidal ramp, only the following solutions were obtained for the approach curves:   The parameters of the SLS model: The parameters of the PLR model: 1) 1 E  =1000 Pa, =0.2; 2) 0 E =1000 Pa,  =0.4. The indentation speed for the triangular ramp was 50 nm/s, the frequency of the sinusoidal ramp was 0.25 Hz (total time was 2 s for both cases), the amplitude was 50 nm. There is no analytical Ting's solution for the SLS model-spheretriangular ramp, PLR model-sphere-cone-sinusoidal ramp cases, the Lee-Radok's solution is presented where available. Figure S3. A force curve obtained with AFM indentation on a cell (NIH 3T3 fibroblast) at a high indentation rate. The force curve was obtained at the indentation rate of 660 Hz that corresponds to the indentation time of 0.0015 s, 140 nm diameter spherical (parabolical) probe, sinusoidal displacement. Due to a strong dissipation (the NHA=0.81), the Hertzian fit does not follow the curve closely, and the curve exponent (0.73) is twice lower than the Hertzian one (1.5). The experimental data are taken from [6].   ). The experimental data are taken from [7].