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UDC 531.51
MECHANICS
V. A. ZHELNOROVICH
VARIATIONAL PRINCIPLE AND EQUATIONS OF STATE FOR CONTINUOUS MEDIA*
(Presented by Academician L. I. Sedov, 4 VII 1968)
Below, models of material continuous media are considered within the framework of the special theory of relativity on the basis of a variational principle of the form \((^{1-3})\)
\[ \delta \int_{V_4} \Lambda\, d\tau + \delta W^* + \delta W = 0 . \tag{1} \]
In the variational principle (1), as admissible extremals we shall consider piecewise smooth functions that may have a finite discontinuity on certain three-dimensional surfaces. The variation is applied to the defining parameters of the medium \(\mu^A(\xi^p)\), which may be components of tensors or spinors \((^{4-6})\), and to the law of motion of the medium \(\delta x^i = \tilde{x}^i(\xi^p)-x^i(\xi^p)\) (\(\xi^p\) are the variables of the accompanying coordinate system for the medium; the functions \(x^i(\xi^p)\) determine the law of motion of the points of the medium). The total variation of the defining parameters \(\delta_1\mu^A\) is given by the equality \(\delta_1\mu^A=\partial\mu^A+\delta x^i \partial_i\mu^A\), where \(\partial_i=\partial/\partial x^i\), \(\partial\mu^A=\mu^A(x^p)-\tilde{\mu}^A(x^p)\) is the variation of the form of the functions \(\mu^A(x^p)\) at fixed \(x^p\). Variations of the defining parameters in the volume \(V_4\) are regarded as independent arbitrary functions belonging to the same class of functions as the admissible extremals. The relation between the variations of the defining parameters on the sides \((\pm)\) of the discontinuity surface \(S\) is established by the choice of the class of functions for the admissible extremals.
1. Equations of state for continuous media. Consider the case when \(\Lambda\) is defined as a function of \(m\) derivatives with respect to the coordinates of the parameters \(\mu^A\) and of the quantities \(x^i_p=\partial x^i/\partial \xi^p\). We choose the functional \(\delta W^*\) in the form
\[ \begin{aligned} \delta W^* \int_{V_4} \left(P_i^* \delta x^i + M_A^* \delta \mu^A\right)d\tau &- \int_{\Sigma+S_\pm} \left( \sum_{\nu=0}^{m} P_i^{*kj_1\ldots j_\nu}\,\partial_{t_1\ldots j_\nu}\delta x^i + \sum_{\nu=0}^{m-1} M_A^{*kj_1\ldots j_\nu}\,\partial_{j_1\ldots j_\nu}\delta\mu^A \right)n_k\,d\sigma, \end{aligned} \tag{2} \]
where \(\Sigma\) is the surface bounding the volume \(V_4\); \(S\) is the part of the discontinuity surface of the defining parameters lying in \(V_4\). The sign \(S_\pm\) indicates that the integration is carried out over the two sides of the surface \(S\). The tensors \(P^*, M^*\) are assumed to be prescribed functions of the parameters of the medium. \(n_k\) are the components of the unit (imaginary-unit) vector of the outward normal to the surfaces \(\Sigma, S_+, S_-\); \(d\sigma\) is an element of the surfaces \(\Sigma, S\),
\[ \partial_{j_1\ldots j_\nu}=\partial_{j_1}\ldots \partial_{j_\nu}. \]
* The main results of the present work were presented at the Third All-Union Congress on Theoretical and Applied Mechanics, January 1968.
Using the usual methods (7), from (1) we determine the functional \(\delta W\)
\[ \delta W=\int_{\Sigma+S_{\pm}}\left( \sum_{\nu=0}^{m} P_i^{k j_1\ldots j_\nu}\partial_{j_1\ldots j_\nu}\delta x^i + \sum_{\nu=0}^{m-1} M_A^{k j_1\ldots j_\nu}\partial_{j_1\ldots j_\nu}\delta\mu^A \right)n_k\,d\sigma . \tag{3} \]
Here the abbreviated notation has been introduced
\[ \frac{\delta\Lambda}{\delta\partial_{j_1\ldots j_\nu}\mu^A} = \sum_{\gamma=0}^{m-\nu} (-1)^\gamma \partial_{i_1\ldots i_\gamma} \frac{\partial\Lambda}{\partial_{i_1\ldots i_\gamma j_1\ldots j_\nu}\mu^A}, \]
\[ M_A^{k j_1\ldots j_\nu} = {}^{*}M_A^{k j_1\ldots j_\nu} - \frac{\partial\Lambda}{\delta\partial_{k j_1\ldots j_\nu}\mu^A}, \]
\[ P_i^k = {}^{*}P_i^k + \sum_{\gamma=0}^{m-1} \left( \frac{\delta\Lambda}{\delta\partial_{k j_1\ldots j_\gamma}\mu^A} \partial_{j_1\ldots j_\gamma i}\mu^A + \frac{\delta\Lambda}{\delta\partial_{k j_1\ldots j_\gamma}x_p^s} \partial_{j_1\ldots j_\gamma i}x_p^s \right) -\Lambda\delta_i^k, \]
\[ \begin{aligned} P_i^{k j_1\ldots j_\nu} &= {}^{*}P_i^{k j_1\ldots j_\nu} - \sum_{\gamma=\nu-1}^{m-1} C_\gamma^{\nu-1} \frac{\delta\Lambda}{\delta\partial_{k j_1\ldots j_{\nu-1} s_\nu\ldots s_\gamma x_p^i}} \partial_{s_\nu\ldots s_\gamma}x_p^{j_\nu} \\ &\quad+ \sum_{\gamma=\nu}^{m-1} C_\gamma^\nu \left( \frac{\delta\Lambda}{\delta\partial_{k j_1\ldots j_\gamma}\mu^A} \partial_{j_{\nu+1}\ldots j_\gamma i}\mu^A + \frac{\delta\Lambda}{\delta\partial_{k j_1\ldots j_\gamma}x_p^s} \partial_{j_{\nu+1}\ldots j_\gamma i}x_p^s \right), \qquad 0<\nu<m, \\ P_i^{k j_1\ldots j_m} &= {}^{*}P_i^{k j_1\ldots j_m} - \frac{\delta\Lambda}{\delta\partial_{k j_1\ldots j_{m-1}x_p^i}} x_p^{j_m}. \end{aligned} \tag{4} \]
\(C_\gamma^\nu\) is the number of combinations of \(\gamma\) taken \(\nu\) at a time. It is important to note that the tensors \(P,M\), defined by formulas (4) in a certain concrete form, are in fact determined by the variational principle nonuniquely, since the quantities \(\delta x^i\), \(\delta\mu^A\) and their derivatives are dependent on the surfaces \(\Sigma,S\). It is easy to see that, without changing the variational equality (1), one can redefine the equations of state (4) by adding to the right-hand side of equality (3) some functional \(\delta\overline{W}\) that is identically zero or zero by virtue of the Euler equations. In particular, the functional \(\delta\overline{W}\), identically equal to zero, may be defined by a formula of the form \(\int \partial_i\partial_j\Omega^{ij}\,d\tau\), where \(\Omega^{ij}=-\Omega^{ji}\) are arbitrary piecewise smooth functions, linear with respect to the variations \(\delta x^i,\delta\mu^A\) and their gradients. It is easy to see that the concrete determination of the tensors \(P,M\) depends on the adopted technique of variation. It is clear that, owing to the “subjective” arbitrariness in the definition of the tensors \(P,M\), it is not appropriate to attach to them a physical meaning.*
We shall assume that the piecewise smooth surfaces \(\Sigma,S\) admit a parametric representation \(x^i=x^i(u^\alpha)\), and let \(G^{\alpha\beta}\) be the contravariant components of the first metric tensor of the surfaces \(\Sigma,S\). Using the obvious formula
\[ \frac{\partial^\nu}{\partial n^\nu}\partial_j\chi^A = n^{j_1\ldots j_\nu}\xi_j^\beta\,{}^{*}\nabla_\beta\partial_{j_1\ldots j_\nu}\chi^A + (n^p n_p)n_j \frac{\partial^{\nu+1}}{\partial n^{\nu+1}}\chi^A, \qquad n^{j_1\ldots j_\nu}=n^{j_1}\cdots n^{j_\nu}, \]
where \(\partial^\nu/\partial n^\nu=n^{j_1\ldots j_\nu}\partial_{j_1\ldots j_\nu}\); \(\xi_j^\beta=G^{\alpha\beta}g_{ij}\partial x^i/\partial u^\alpha\); \({}^{*}\nabla_\alpha\) is the symbol of covariant differentiation on the surface, one can obtain the recurrence relation
\[ \partial_{j_1\ldots j_{\nu-1}j_\nu}\chi^A = \xi_{j_1\ldots j_{\nu-1}j_\nu}^{i_1\ldots i_{\nu-1}\alpha} \,{}^{*}\nabla_\alpha\partial_{i_1\ldots i_{\nu-1}}\chi^A + (-1)^\nu n_{j_1\ldots j_{\nu-1}j_\nu} \frac{\partial^\nu}{\partial n^\nu}\chi^A, \]
\[ \begin{aligned} \xi_{j_1\ldots j_{\nu-1}j_\nu}^{i_1\ldots i_{\nu-1}\alpha} &= \Bigg[ \delta_{j_1}^{k}\delta_{j_2}^{i_1}\cdots\delta_{j_\nu}^{i_{\nu-1}} + \sum_{\gamma=2}^{\nu-1} (n^p n_p)^{\gamma-1} n^{i_1\ldots i_{\gamma-1}} n_{j_1\ldots j_{\gamma-1}} \delta_{j_\gamma}^{k} \delta_{j_{\gamma+1}}^{i_\gamma}\cdots\delta_{j_\nu}^{i_{\nu-1}} \\ &\quad+ (n^p n_p)^{\nu-1} n^{i_1\ldots i_{\nu-1}} n_{j_1\ldots j_{\nu-1}} \delta_{j_\nu}^{k} \Bigg]\xi_k^\alpha . \end{aligned} \tag{5} \]
* In this connection we note that in the known theories the determination of the tensors of energy–momentum, angular momentum, currents, and other observed quantities is associated with the tensors \(P,M\). In particular, the moment theory of elasticity is based on the physical interpretation of the tensors \(P\).
Taking (5) into account, we obtain
\[ \begin{aligned} M_A^{j_1\ldots j_\nu}\partial_{j_1\ldots j_\nu}\chi^A &=(n^p n_p)^\nu M_A^{j_1\ldots j_\nu} n_{j_1\ldots j_\nu}\frac{\partial^\nu}{\partial n^\nu}\chi^A +\chi^A\overset{*}{\nabla}_{\alpha}H_{A(\nu)}^\alpha \\ &\quad -\overset{*}{\nabla}_{\alpha}\bigl(H_{A(\nu)}^\alpha\chi^A\bigr) +\sum_{\gamma=1}^{\nu-1}(n^p n_p)^\gamma n_{j_1\ldots j_\gamma} \overset{*}{\nabla}_{\alpha}H_{A(\nu)}^{j_1\ldots j_\gamma\alpha} \frac{\partial^\gamma}{\partial n^\gamma}\chi^A \\ &\quad -\sum_{\gamma=1}^{\nu-1} \overset{*}{\nabla}_{\alpha}\bigl(H_{A(\nu)}^{j_1\ldots j_\gamma\alpha} \partial_{j_1\ldots j_\gamma}\chi^A\bigr). \end{aligned} \tag{6} \]
Here the notation is
\[ \begin{aligned} H_{A(\nu)}^{j_1\ldots j_\gamma\alpha} &=(-1)^{\nu-\gamma}\xi_{r_1\ldots r_\gamma r_{\gamma+1}}^{j_1\ldots j_\gamma\alpha} \overset{*}{\nabla}_{\beta} \Bigl\{\xi_{l_1\ldots l_{\gamma+1}l_{\gamma+2}}^{r_1\ldots r_{\gamma+1}\beta} \overset{*}{\nabla}_{\varepsilon} \Bigl[\xi_{q_1\ldots q_{\gamma+2}q_{\gamma+3}}^{l_1\ldots l_{\gamma+2}\varepsilon} \overset{*}{\nabla}_{\rho} \\ &\qquad\qquad\times \bigl[\cdots \xi_{i_1\ldots i_{\nu-2}i_{\nu-1}}^{p_1\ldots p_{\nu-2}\mu} \overset{*}{\nabla}_{\sigma} \bigl(\xi_{m_1\ldots m_{\nu-1}m_\nu}^{i_1\ldots i_{\nu-1}\sigma} M_A^{m_1\ldots m_{\nu-1}m_\nu}\bigr)\bigr] \Bigr]\Bigr\}. \end{aligned} \tag{7} \]
From the definition it is clear that the tensors \(H\) satisfy the recurrence relation
\[ H_{A(\nu)}^{i_1\ldots i_{\gamma-1}\alpha} = -\xi_{j_1\ldots j_{\gamma-1}j_\gamma}^{i_1\ldots i_{\gamma-1}\alpha} \overset{*}{\nabla}_{\beta}H_{A(\nu)}^{j_1\ldots j_\gamma\beta}. \tag{8} \]
Replacing the integrands of the functional \(\delta W\) by formula (6), we obtain the expression of this functional in the following canonical form
\[ \delta W = \int_{\Sigma+S_\pm} \left( \sum_{\nu=0}^{m}\mathcal{P}_{i(\nu)}\frac{\partial^\nu\delta x^i}{\partial n^\nu} + \sum_{\nu=0}^{m-1}\mathcal{M}_{A(\nu)} \frac{\partial^\nu\delta\mu^A}{\partial n^\nu} \right)d\sigma - \]
\[ - \int_{\Sigma+S_\pm} \overset{*}{\nabla}_{\alpha} \left( \sum_{\nu=0}^{m-1}\mathcal{P}_{i}^{j_1\ldots j_\nu\alpha} \partial_{j_1\ldots j_\nu}\delta x^i + \sum_{\nu=0}^{m-2}\mathcal{M}_{A}^{j_1\ldots j_\nu} \partial_{j_1\ldots j_\nu}\delta\mu^A \right)d\sigma, \tag{9} \]
where
\[ \mathcal{M}_A^\alpha=\sum_{\gamma=1}^{m}H_{A(\gamma)}^\alpha,\qquad \mathcal{M}_A^{j_1\ldots j_\nu\alpha} = \sum_{\gamma=1}^{m-\nu}H_{A(\gamma+\nu)}^{j_1\ldots j_\nu\alpha}, \]
\[ \mathcal{M}_{A(\nu)} = (n^p n_p)^\nu \left( M_A^{k j_1\ldots j_\nu}n_k + \overset{*}{\nabla}_{\alpha}\mathcal{M}_A^{j_1\ldots j_\nu\alpha} \right)n_{j_1\ldots j_\nu}. \tag{10} \]
The tensors \(\mathcal{P}\) in formula (9) are obtained from formulas (10) by replacing in them the symbols \(M,\mathcal{M}\) by the symbols \(P,\mathcal{P}\). In the case where the integrands in the second integral in (9) are continuous on the surfaces \(\Sigma+S_\pm\), the second integral in (9) vanishes.
From formulas (7), (8) it follows that the tensors \(\mathcal{M}\) satisfy the recurrence relations
\[ \mathcal{M}_A^{j_1\ldots j_{m-1}\alpha} = -\xi_{j_1\ldots j_{m-1}j_m}^{i_1\ldots i_{m-1}\alpha} M_A^{k j_1\ldots j_m}n_k, \]
\[ \mathcal{M}_A^{i_1\ldots i_{\nu-1}\alpha} = -\xi_{j_1\ldots j_{\nu-1}j_\nu}^{i_1\ldots i_{\nu-1}\alpha} \left( M_A^{k j_1\ldots j_\nu}n_k + \overset{*}{\nabla}_{\beta}\mathcal{M}_A^{j_1\ldots j_\nu\beta} \right). \tag{11} \]
Formulas (11) are considerably simpler than formulas (10), and for the computation of the tensors \(\mathcal{P},\mathcal{M}\) it is more convenient to use the recurrence formulas (11) instead of the direct formulas (10).
Since on the surfaces \(\Sigma,S\) the quantities \(\delta x^i,\delta\mu^A\) and the normal derivatives of these quantities are independent, it is obvious that the components \(\mathcal{P}_{i(\nu)},\mathcal{M}_{A(\nu)}\) are determined by the variational principle in a unique way.
2. Conditions on discontinuity surfaces
If the surface \(S\) is sufficiently smooth and the parameters of the medium are continuous on both sides of the surface \(S\), then the variational conditions on the discontinuity surface may be written in the form
\[ \int_{S_\pm} \left( \sum_{\nu=0}^{m}\mathcal{P}_{i(\nu)}\frac{\partial^\nu\delta x^i}{\partial n^\nu} + \sum_{\nu=0}^{m-1}\mathcal{M}_{A(\nu)} \frac{\partial^\nu\delta\mu^A}{\partial n^\nu} \right)d\sigma =0. \tag{12} \]
Consider the case in which the first \(r\) derivatives \(\partial_{1\ldots r}\mu^A\) of the parameters \(\mu^A\) and the first \(s\) derivatives \(\partial_{j_1\ldots j_{s-1}}x^i_{j_s}\) from the law of motion are continuous on the surface \(S\), while derivatives of the following orders undergo an arbitrary finite discontinuity on \(S\). For definiteness, set \(0<r<s\). From the assumption on the class of functions under consideration it follows that the quantities \(\partial^\nu\delta x^i/\partial n^\nu\), \(\partial^\mu\delta\mu^A/\partial n^\mu\) for \(\nu\le s\), \(\mu\le r\) are continuous, while for \(\nu>s\), \(\mu>r\) they are independent on the surface \(S\). Using the arbitrariness of \(S\) and the indicated properties of the variations of the parameters on \(S\), from (12) we obtain the system of differential conditions on the surface of discontinuity
\[ [\mathcal{P}_{i(\nu)}]\big|_S=0\;(\nu\le s),\qquad \mathcal{P}_{i(\nu)}\big|_S=0\quad(\nu>s), \]
\[ [\mathcal{M}_{A(\nu)}]\big|_S=0\;(\nu\le r),\qquad \mathcal{M}_{A(\nu)}\big|_S=0\quad(\nu>r) \tag{13} \]
(the square brackets \([\,]\big|_S\) denote the difference of the limiting values on \(S\) of the quantities enclosed in the brackets, and it is assumed that \(n_i^+=n_i^-\)).
The functional \(\delta W\), defined by formula (3), was obtained under the assumption that the surface \(S\) is not varied in the accompanying coordinate system. Assuming that, under variation of the surface of discontinuity \(\xi^i=\xi^i(u^\alpha)\), the points \(u^\alpha\) of the surface receive an increment of coordinates \(d\xi^i=\bar{\xi}^i(u^\alpha)-\xi^i(u^\alpha)\), and using (3), (13), we compute the value of the functional \(\delta W\) for zero variations of the defining parameters at the points \(u^\alpha\) of the varied surface \(S\) and on the surface \(\Sigma\):
\[ \delta W=-\int_{S_\pm} \left( \Lambda \hat n_p+ \mathcal{P}_{i(s)}\hat n^{k}n^{j_1\ldots j_{s-1}}x^l_p \partial_{j_1\ldots j_{s-1}}x^i_k+ \right. \]
\[ \left. +\mathcal{M}_{A(r)}n^{j_1\ldots j_r}x^l_s \partial_{j_1\ldots j_r}\mu^A \right)d\xi^p\,d\sigma, \tag{14} \]
where \(\hat n_p=x^i_p n_p\). Taking into account the kinematic conditions for the functions \(\partial_{j_1\ldots j_r}\mu^A\), \(\partial_{j_1\ldots j_{s-1}}x^i_k\), from (14) we obtain an additional condition due to the variation of the surface of discontinuity,
\[ \left[ \Lambda+\mathcal{P}_{i(s)}\hat n^k\partial^s x^i_k/\partial n^s +\mathcal{M}_{A(r)}\partial^{r+1}\mu^A/\partial n^{r+1} \right]_S=0. \tag{15} \]
We separately consider the case in which the quantities \(\partial_i\mu^A\), \(x^i_p\) undergo a discontinuity on \(S\) (the case \(m=s=0\)). In this case (14) passes into the equality
\[ \delta W=-\int_{S_\pm} \left( \mathcal{P}_{i(0)}x^i_p+\mathcal{M}_{A(0)}x^i_p\partial_i\mu^A+\Lambda \hat n_p \right)d\xi^p\,d\sigma=0. \tag{16} \]
Taking into account the kinematic conditions \([\hat n_k/\sqrt{-\hat g}]_S=0\) \((\hat g=\det \hat g_{pq},\ \hat g_{pq}=x^i_p x^j_q g_{ij})\), from (16) we obtain*
\[ \left[ \sqrt{-\hat g}\left(\Lambda+\mathcal{P}_{i(0)}n^i+\mathcal{M}_{A(0)}\partial\mu^A/\partial n\right) \right]_S^+=0. \tag{17} \]
The author expresses sincere gratitude to L. I. Sedov for discussion of the present work.
Moscow State University
named after M. V. Lomonosov
Received
1 VII 1968
REFERENCES
- L. I. Sedov, UMN, 20, issue 5 (1965).
- L. I. Sedov, DAN, 164, No. 3 (1965).
- L. I. Sedov, IUTAM, Symposia, Vienna, 1966.
- V. A. Zhelnorovich, DAN, 169, No. 2 (1966).
- V. A. Zhelnorovich, PMM, 30, issue 6 (1966).
- V. A. Zhelnorovich, in the collection Thermomechanics, dedicated to the anniversary of Academician L. I. Sedov, 1968.
- V. A. Zhelnorovich, DAN, 176, No. 2 (1967).
- M. V. Lurie, PMM, 30, issue 4 (1966).
* Conditions on the surface of discontinuity within the framework of Newtonian mechanics for the case in which \(\Lambda\) depends on \(x^i_p\), \(\partial_k x^i_p\), were considered in work (8). The conditions obtained in that work may be considered satisfactory only in certain particular examples not specified in (8).