Broadband nanoindentation of glassy polymers: Part II. Viscoplasticity

Abstract

The relationship between hardness and flow stress in glassy polymers is examined. Materials studied include poly(methylmethacrylate), polystyrene, and polycarbonate. Properties are strongly rate dependent, so broadband nanoindentation creep (BNC) is used to measure hardness across a broad range of indentation strain rates (10⁻⁴ to 10 s⁻¹). Molybdenum (Mo) is also studied to serve as a “control” whose rate-dependent hardness properties have been measured previously and whose flow stress, unlike the polymers, is pressure insensitive. The BNC hardness data are converted to uniaxial flow stress using two methods based on the usual Tabor–Marsh–Johnson correlation. With both methods the resulting BNC-derived uniaxial flow stress data agree closely with literature compression uniaxial flow stress data for all materials. For the polymers, the BNC hardness data depend on initial rate of loading, indicating that the measured properties are path dependent. Path dependence is not detected in Mo.

Appendix

We obtain the continuous area-time curves in Fig. 4 by fitting continuous curves to the discrete area-time measurements also shown in the figure. Rather than relying on arbitrary functions like polynomials, power laws, or decaying exponentials to generate continuous area-time curves, we employ another approach that utilizes the nanoindentation load-depth-time traces. The method does a better job of capturing the trend of the data over a wide range of strain rate and has been validated and improved using finite element analysis.34 With this method it is assumed that ${A^{1/2}} \propto h_p^{{\xi _p}}$ at constant load. The power law exponent ζ_p is used as a fitting parameter to fit the continuous area-time curve to the discrete experimental data. During the creep hold, h_p is estimated using

$${h_p} \cong h_p^0 + \frac{{{h_t} - h_p^0 - \frac{P}{{{E_{{\text{eff}}}}{{\left( {{A^0}} \right)}^{1/2}}}}\left( {1 - \left( {1 - {\zeta _p}} \right)\frac{{P - {P^0}}}{{2{P^0}}}} \right)}}{{1 - \frac{{P{\zeta _p}}}{{h_p^0{E_{{\text{eff}}}}{{\left( {{A^0}} \right)}^{1/2}}}}}}$$

((A1))

where P is the instantaneous load (ideally, P = P⁰ although in a real experiment P varies a small amount). The superscripts “0” refer to the values of the parameters at the end of the creep hold and $h_p^0 = h_t^0 - {P^0}{\left( {{A^0}} \right)^{1/2}}$. E_eff is given by

(A2)

where E_s and E_d are Young’s moduli and v_s and v_d are Poisson’s ratios of specimen and indenter, respectively, and β ≈ 1.23.36E* is the usual modulus encountered in Hertz contact problems. Equation (A1) is obtained from a theoretical analysis. However, for polymers Eeff lacks uniqueness because of viscoelasticity, so the polymers do not strictly conform to the theory. In practice, this discrepancy does not present a problem because ζ_p is calculated using a representative value of Eeff and in the end the continuous area-time curve is insensitive to the choice of E_eff. Figure A1 shows ln (A⁰)^1/2 versus ln $h_p^0$ from experiments with different creep hold times at fixed load. The slopes of the curves correspond to ζ_p. Rather than forming straight lines, the data must be approximated as parabolas

FIG. A1

Data used to calculate ?_p for PMMA, PS, and PC. For each material, data include indents with creep hold times that vary from 0.01 to100 s and load times that vary from 0.01 to 100 s. The 0.01-s load time data were fit to Eq. (A3), and these fits were differentiated to calculate ?_p. The coefficients for the fit and the values of E_eff used to calculate h_p are given in the embedded table.

$${\text{ln(}}{A^{\text{0}}}{{\text{)}}^{{\text{1/2}}}}\,{\text{ = }}\,{X_{\text{0}}}\,{\text{ + }}\,{X_{\text{1}}}\ln h_p^{\text{0}}{\text{ + }}{X_{\text{2}}}{{\text{(}}h_p^{\text{0}}{\text{)}}^{\text{2}}},$$

((A3))

in which case ζ_p = X₁ + 2X₂ ln $h_p^0$. Previously34 we had not been able to reliably detect that ζ_p varies with depth. However, our measurement methods are now better, and the finite element analyses neglected viscoelastic deformation, so it should come as no surprise that ζ_p is not a constant for the polymers.

Because ζ_p depends on hp, it is not straightforward to calculate h_p using Eq. (A1). Therefore, to account for variation in ζ_p in Eq. (A1) we divide the raw depth-load-time data into 100 equal segments on a log(time) scale so that for each segment, ζ_p is approximately constant. We then analyze each segment separately starting with the final segment of the creep hold, where A = A⁰ and ${h_p} = h_p^0$ are known, and work backward to the beginning of the creep hold.