<p>The effects of hydrostatic pressure and indentation strain on nanoindentation elastic modulus (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({E}_{\text{s}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mtext>s</mtext> </msub> </math></EquationSource> </InlineEquation>) and hardness (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>H</mi> </math></EquationSource> </InlineEquation>) were investigated in the five polymers poly(methyl methacrylate) (PMMA), polycarbonate (PC), polystyrene (PS), poly(ether ether ketone) (PEEK), and low density polyethylene (LDPE) using four different pyramidal probes with equivalent cone angle (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\theta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>) = 47.7° (cube corner), 35.0°, 19.7° (Berkovich), or 10.0°. Both <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\({E}_{\text{s}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mtext>s</mtext> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(H\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>H</mi> </math></EquationSource> </InlineEquation> generally increased with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\theta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>, with the strongest effects in the glassy polymers. Consistent with a hydrostatic pressure effect, the measured <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\({E}_{\text{s}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mtext>s</mtext> </msub> </math></EquationSource> </InlineEquation> were 10–50% higher than conventional elastic moduli measured using bulk specimens. To remove this effect, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\({E}_{\text{s}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mtext>s</mtext> </msub> </math></EquationSource> </InlineEquation> versus <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(H\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>H</mi> </math></EquationSource> </InlineEquation> was extrapolated to <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(H=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> assuming a linear relationship. These extrapolated values were within 15% of the conventional values for all the materials except for LDPE, which showed anomalous behavior likely associated with its rubbery state. Hardness-derived flow stress <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(({\sigma}_{\text{H}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mtext>H</mtext> </msub> </mrow> </math></EquationSource> </InlineEquation>)–representative strain (<InlineEquation ID="IEq12"> <EquationSource Format="TEX">\({\epsilon}_{\text{r}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϵ</mi> <mtext>r</mtext> </msub> </math></EquationSource> </InlineEquation>) curves were also calculated and compared to literature uniaxial compression stress–strain curves. <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\({\sigma}_{\text{H}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mtext>H</mtext> </msub> </math></EquationSource> </InlineEquation> compared best to the literature for data for a Berkovich probe at an <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\({\epsilon}_{\text{r}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϵ</mi> <mtext>r</mtext> </msub> </math></EquationSource> </InlineEquation> = 0.07. The empirical Tabor–Marsh–Johnson correlation used to calculate <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\({\sigma}_{\text{H}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mtext>H</mtext> </msub> </math></EquationSource> </InlineEquation> does not work as well for these polymers at either higher or lower <InlineEquation ID="IEq16"> <EquationSource Format="TEX">\({\epsilon}_{\text{r}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϵ</mi> <mtext>r</mtext> </msub> </math></EquationSource> </InlineEquation>.</p>

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Pressure-Corrected Elastic Modulus and Plasticity in Polymer Nanoindentation

  • Joseph E. Jakes,
  • Douglas D. Stauffer,
  • Donald S. Stone

摘要

The effects of hydrostatic pressure and indentation strain on nanoindentation elastic modulus ( \({E}_{\text{s}}\) E s ) and hardness ( \(H\) H ) were investigated in the five polymers poly(methyl methacrylate) (PMMA), polycarbonate (PC), polystyrene (PS), poly(ether ether ketone) (PEEK), and low density polyethylene (LDPE) using four different pyramidal probes with equivalent cone angle ( \(\theta \) θ ) = 47.7° (cube corner), 35.0°, 19.7° (Berkovich), or 10.0°. Both \({E}_{\text{s}}\) E s and \(H\) H generally increased with \(\theta \) θ , with the strongest effects in the glassy polymers. Consistent with a hydrostatic pressure effect, the measured \({E}_{\text{s}}\) E s were 10–50% higher than conventional elastic moduli measured using bulk specimens. To remove this effect, \({E}_{\text{s}}\) E s versus \(H\) H was extrapolated to \(H=0\) H = 0 assuming a linear relationship. These extrapolated values were within 15% of the conventional values for all the materials except for LDPE, which showed anomalous behavior likely associated with its rubbery state. Hardness-derived flow stress \(({\sigma}_{\text{H}}\) ( σ H )–representative strain ( \({\epsilon}_{\text{r}}\) ϵ r ) curves were also calculated and compared to literature uniaxial compression stress–strain curves. \({\sigma}_{\text{H}}\) σ H compared best to the literature for data for a Berkovich probe at an \({\epsilon}_{\text{r}}\) ϵ r = 0.07. The empirical Tabor–Marsh–Johnson correlation used to calculate \({\sigma}_{\text{H}}\) σ H does not work as well for these polymers at either higher or lower \({\epsilon}_{\text{r}}\) ϵ r .