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On the coupling of plastic waves (differential constitutive laws). (English) Zbl 0202.57803

In this paper, the author discusses the problem of coupling of plastic waves with the help of the problem of a shearing impact of a body assuming axial symmetry and using only differential constitutive laws. It is supposed that the laws of variation of the components are \(\sigma_{r\theta}\) and \(\sigma_{rz}\) of the stress are prescribed along the cylindrical hole of radius \(r=R\). The axial component \(w\) and the circumferential component \(v\) of the displacement are different from zero. The considerations are similar for all dynamic problems involving one spatial coordinate and two stress components. The partially coupled constitutive laws for the dominant components of strain and stress \(\varepsilon_{r\theta}\), \(\varepsilon_{rz}\), \(\sigma_{r\theta}\) and \(\sigma_{rz}\) are assumed to
\[ \frac{\partial \varepsilon_{r\theta}}{\partial t} = f_1 \frac{\partial \sigma_{r\theta}}{\partial t} + h_1,\quad \frac{\partial \varepsilon_{rz}}{\partial t} = f_2 \frac{\partial \sigma_{rz}}{\partial t} + h_2 \]
where \(f_1\), \(f_2\), \(h_1\) and \(h_2\) are generally functions of all unknown quantities \(\varepsilon_{r\theta}\), \(\varepsilon_{rz}\), \(\sigma_{r\theta}\), \(\sigma_{rz}\) and possibly of \(r\) and \(t\). They are called partially coupled constitutive laws because only the non-instantaneous properties are coupled. In general, there are two partially coupled elastic-plastic shearing waves with different velocities, which may coincide in some special cases. The velocity of elastic shear waves is in a certain way a limit velocity for the velocities of propagation of elastic-plastic shear waves.
Some special cases are considered corresponding to different values of \(f\) and \(h\). Both waves may be uncoupled simple waves in some cases. When both the non-instantaneous and instantaneous response properties are coupled, the constitutive laws assumed are
\[ \frac{\partial \varepsilon_{r\theta}}{\partial t} = \varphi_1 \frac{\partial \sigma_{r\theta}}{\partial t} + \psi_1 \frac{\partial \varepsilon_{rz}}{\partial t} + \chi_1,\quad \frac{\partial \varepsilon_{rz}}{\partial t} = \varphi_2 \frac{\partial \sigma_{rz}}{\partial t} + \psi_2 \frac{\partial \varepsilon_{r\theta}}{\partial t} + \chi_2 \]
where \(\varphi\), \(\psi\) and \(\chi\) are functions of \(\varepsilon_{r\theta}\), \(\varepsilon_{rz}\), \(\sigma_{r\theta}\) and \(\sigma_{rz}\) (in the second equation \(\psi_1\) appears instead of \(\psi_2\) by misprint). In this case also there are two shear waves with different velocities but they are simultaneously shearing waves with respect to \(\sigma_{r\theta}\) and \(\sigma_{rz}\). Therefore these waves are called coupled plastic waves.
A few cases are examined in which some de-coupling occurs. However, it is found that the only constitutive law which always separates the two types of waves is the elastic one.
The mathematical technique used is mainly the theory of characteristics applied to systems of quasilinear partial differential equations. The systems are totally hyperbolic, mixed hyperbolic-parabolic or fully parabolic depending on the functions appearing in the constitutive relations. The equations can be solved by numerical method of integration using an electronic computer but this has not been attempted here.
Finally, comparison is made with a number of constitutive laws which are used in the theory of plasticity. It is shown that the different specific dynamic properties described by them can be obtained from the general constitutive laws discussed in this paper. The paper leaves scope for further work and extensions.
Reviewer: S. K. Mishra

MSC:

74Cxx Plastic materials, materials of stress-rate and internal-variable type
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