Abstract
As shown in Chapters 3 and 4, the nonlinearity in dynamics is caused by changes in the direction of external forces due to large movements and to the nonlinearity of internal forces, caused by non-time-dependent phenomena studied in Chapter 5 and time-dependent phenomena that will be presented in this chapter.
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Notes
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The constitutive law decomposed into its volumetric and deviatoric parts is written as,
$$ {\upsigma}_{ij}={\uplambda}^{\mathrm{Lame}}\;{\upvarepsilon}_{kk}^e\;{\updelta}_{ij}+2\;{\upmu}^{\mathrm{Lame}}\left({\upvarepsilon}_{ij}^e-{\scriptscriptstyle \frac{1}{3}}{\upvarepsilon}_{kk}^e\;{\updelta}_{ij}\right)={\uplambda}^{\mathrm{Lame}}\;{\upvarepsilon}_{kk}^e\;{\updelta}_{ij}+2\;{\upmu}^{\mathrm{Lame}}{e}_{ij}^e $$where Lame’s constants are λLame = E/3(1 − 2 ν) and μLame = G = E/2(1 + ν), E and ν are the Young modulus and the Poisson coefficient respectively. Since the von Mises flow is dominantly deviatoric (\( {\dot{\upvarepsilon}}_{kk}^{v\mathrm{p}}=0\Rightarrow {\dot{\upvarepsilon}}_{ij}^{v\mathrm{p}}\equiv {\dot{e}}_{ij}^{v\mathrm{p}}=\left[\left\langle \Phi \right\rangle /\upxi \right]\;\mathbf{s} \) ), the previous constitutive law is particularized in the following as σ ij  = λLameε kk δ ij  + 2G(e ij  − e vp ij ) = σ vol ij  − s ij
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Oller, S. (2014). Time-dependent Models. In: Nonlinear Dynamics of Structures. Lecture Notes on Numerical Methods in Engineering and Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-05194-9_6
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DOI: https://doi.org/10.1007/978-3-319-05194-9_6
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