During the heating stage of the firing of a ceramic material, the mass \documentclass[12pt]{minimal}
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\begin{document}$$m$$\end{document}, length \documentclass[12pt]{minimal}
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\begin{document}$$l$$\end{document}, and diameter \documentclass[12pt]{minimal}
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\begin{document}$$d$$\end{document} of the sample alter their values depending on the temperature \documentclass[12pt]{minimal}
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\begin{document}$$t$$\end{document}. Young’s modulus \documentclass[12pt]{minimal}
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\begin{document}$$E(f,m,l,d)$$\end{document} measured by a sonic resonance method is also a function of the resonance frequency \documentclass[12pt]{minimal}
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\begin{document}$$f$$\end{document}. Therefore, three thermal analyses (TGA, TDA, modulated force TMA) must be performed to obtain correct values of Young’s modulus. The calculation of Young’s modulus can be simplified if TGA and/or TDA are omitted. This necessarily leads to partly incorrect results. If TGA is not performed, we have \documentclass[12pt]{minimal}
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\begin{document}$$E[f(t),m_0 ,l(t),d(t)]$$\end{document} and the relative difference \documentclass[12pt]{minimal}
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\begin{document}$$(\{E[f(t),m(t),l(t),d(t)]-E[f(t),m_0 ,l(t),d(t)]\}/E[f(t),m(t),l(t),d(t)])$$\end{document} reaches 7 % for \documentclass[12pt]{minimal}
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\begin{document}$$t> 650\,^\circ \text{ C}$$\end{document} and less than 2 % for \documentclass[12pt]{minimal}
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\begin{document}$$t< 500\,^\circ \text{ C}$$\end{document}. If TDA is not performed, we have \documentclass[12pt]{minimal}
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\begin{document}$$E[f(t),m(t),l_0 ,d_0 ]$$\end{document} and the relative difference (\documentclass[12pt]{minimal}
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\begin{document}$$\{E[f(t),m(t),l(t),d(t)]-E[f(t),m(t),l_0 ,d_0 ]\}/E[f(t),m(t),l(t),d(t)])$$\end{document} is less than 0.6 % for \documentclass[12pt]{minimal}
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\begin{document}$$t < 1000\,^\circ \text{ C}$$\end{document}. For the simplest case, we have \documentclass[12pt]{minimal}
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\begin{document}$$E[f(t),m_0 ,l_0 ,d_0 ]$$\end{document} and the relative difference (\documentclass[12pt]{minimal}
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\begin{document}$$\{E[f(t),m(t),l(t),d(t)]-E[f(t),m_0 ,l_0 ,d_0 ]\}/E[f(t),m(t),l(t),d(t)])$$\end{document} is 7.5 % for \documentclass[12pt]{minimal}
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\begin{document}$$t > 600\,^\circ \text{ C}$$\end{document} and less than 2 % for \documentclass[12pt]{minimal}
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\begin{document}$$t<500\,^\circ \text{ C}$$\end{document}.
