Introduction to Mechanical Properties of Materials
AimsBefore you startIntroductionTensile test: force and extensionStress, strain and Poisson's ratioElastic deformation: Hooke's law and stiffnessViscoelasticityPlastic deformation: strength and ductilitySlip: resolved shear stress and Schmid factor, Taylor factorHardness and work hardeningCreepFracture: toughnessDuctile-brittle transition temperatureSummaryQuestionsGoing furtherTLP creditsTLP contentsShow all contentViewing and downloading resourcesAbout the TLPsTerms of useFeedbackCredits Print this page
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The Maxwell model equation
We have the Maxwell relation,
\[k\varepsilon\dot = \frac{1}{{{t_{\rm{R}}}}}\sigma + \dot \sigma \]Impose oscillatory stress and strain (we use complex function to construct oscillatory functions, but only the real part is meaningful),
\[ \sigma = {\sigma _0}{\rm{exp}}\left( {i\omega t} \right) \] \[ \varepsilon = {\varepsilon_0}{\rm{exp}}\left( {i\left( {\omega t - \phi } \right)} \right) \]The Maxwell relation becomes,
\[ k\left( {i\omega } \right){_0}{\rm{exp}}(i\left( {\omega t - \phi )} \right) = \frac{1}{{{t_{\rm{R}}}}}{\sigma _0}\exp \left( {i\omega t} \right) + \left( {i\omega } \right){\sigma _0}\exp \left( {i\omega t} \right) \] \[ k\left( {i\omega } \right){_0}{\rm{exp}}(i\left( {\omega t - \phi )} \right) = {\sigma _0}\exp \left( {i\omega t} \right)\left( {\frac{1}{{{t_{\rm{R}}}}} + i\omega } \right) \]The complex Young’s modulus is defined as
\[ E = \frac{\sigma }{\varepsilon}\] \[ \quad = \frac{{{\sigma _0}\exp \left( {i\omega t} \right)}}{{{\varepsilon_0}{\rm{exp}}\left( {i\left( {\omega t - \phi } \right)} \right)}} \] \[ \quad = \frac{{k\left( {i\omega } \right)}}{{\frac{1}{{{t_{\rm{R}}}}} + i\omega }} \] \[ \quad = \frac{{ki\omega {t_{\rm{R}}}}}{{1 + i\omega {t_{\rm{R}}}}} \] \[ \quad = \frac{{ki\omega {t_{\rm{R}}}}}{{1 + i\omega {t_{\rm{R}}}}}\frac{{1 - i\omega {t_{\rm{R}}}}}{{1 - i\omega {t_{\rm{R}}}}} \] \[ \quad = \frac{{ki\omega {t_{\rm{R}}} + k{\omega ^2}t_{\rm{R}}^2}}{{1 + {\omega ^2}t_{\rm{R}}^2}} \] \[ \quad = \frac{{k{\omega ^2}t_{\rm{R}}^2}}{{1 + {\omega ^2}t_{\rm{R}}^2}} + i\frac{{k\omega {t_{\rm{R}}}}}{{1 + {\omega ^2}t_{\rm{R}}^2}} \]Only the real part has physical meaning, so the Young’s modulus is
\[ E = \frac{{k{\omega ^2}t_{\rm{R}}^2}}{{1 + {\omega ^2}t_{\rm{R}}^2}} \]The ratio between real and complex part indicates the phase difference,
\[ \phi = \arctan \left( {\frac{{\frac{{k\omega {t_{\rm{R}}}}}{{1 + {\omega ^2}t_{\rm{R}}^2}}}}{{\frac{{k{\omega ^2}t_{\rm{R}}^2}}{{1 + {\omega ^2}t_{\rm{R}}^2}}}}} \right) = {\rm{arctan}}\left( {\frac{1}{{\omega {t_{\rm{R}}}}}} \right) \]