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UDC 539.30
THEORY OF ELASTICITY
L. Ya. AINOLA
VARIATIONAL PRINCIPLE OF DYNAMICS IN THE LINEAR THEORY OF ELASTICITY
(Presented by Academician Yu. N. Rabotnov on 18 III 1966)
In all works devoted to integral variational principles of dynamics, a problem of the theory of elasticity is treated as a problem with boundary conditions (see, for example, \((^{1—7})\)). In the present note a variational principle is given that is equivalent to the problem of the theory of elasticity as a mixed problem.
The solution of problems of the dynamics of the linear theory of elasticity leads to a system of differential equations of hyperbolic type
\[ \nabla_i \sigma^{ik} + X^k = \rho \,\partial^2 u^k / \partial t^2 \quad (i, k = 1, 2, 3); \tag{1} \]
\[ \sigma^{ik} = E^{ikjl}\varepsilon_{jl}, \qquad \varepsilon_{ik} = {}^1/_2(\nabla_i u_k + \nabla_k u_i), \tag{2} \]
where \(\sigma^{ik}\), \(\varepsilon_{ik}\) are the components of the stress and strain tensors; \(X^k\), \(u_k\) are the components of the vectors of body forces and displacements; \(\rho\) is the density of the elastic body; \(E^{ikjl}\) are the components of the elasticity tensor; \(\nabla_i\) is the sign of covariant differentiation.
The required solution must satisfy the boundary conditions
\[ \sigma^{ik} n_i = P^k \tag{3} \]
on \(S_1\), a part of the surface \(S\) of the elastic body, and
\[ u_i = U_i \tag{4} \]
on \(S_2\), the remaining part of the surface \(S\), as well as the initial conditions, which without loss of generality in the formulation of the problem may be taken in the form
\[ u_i(x^l, 0) = 0, \qquad \partial u_i(x^l, 0) / \partial t = 0. \tag{5} \]
In conditions (3), (4), \(n_i\) are the components of the unit normal vector to the surface \(S_1\); \(P^k\), \(U_i\) are the components of prescribed vectors of external forces and displacements.
We give the variational formulation of the mathematical problem (1)—(5).
Consider the functional
\[ \begin{aligned} I ={}& \int_0^T \int_V \Bigl\{ \rho \frac{\partial u^i}{\partial t}\frac{\partial u_i^*}{\partial t} - E^{ikjl}\varepsilon_{ik}\varepsilon_{jl}^* + \sigma^{ik}\bigl[\varepsilon_{ik}^* - {}^1/_2(\nabla_i u_k^* + \nabla_k u_i^*)\bigr] \\ &\qquad\qquad + \sigma_*^{ik}\bigl[\varepsilon_{ik} - {}^1/_2(\nabla_i u_k + \nabla_k u_i)\bigr] + X_*^i u_i + X^i u_i^* \Bigr\}\, dV\, dt \\ &+ \int_0^T \int_{S_1} (P_*^i u_i + P^i u_i^*)\, dS\, dt \\ &+ \int_0^T \int_{S_2} \bigl[(u_k^* - U_k^*)\sigma^{ik} n_i + (u_k - U_k)\sigma_*^{ik} n_i\bigr]\, dS\, dt, \tag{6} \end{aligned} \]
where the independent functional arguments are \(u_i\), \(\sigma^{ik}\), \(\varepsilon_{ik}\), \(u_i^*\), \(\sigma_*^{ik}\), \(\varepsilon_{ik}^*\). The functions \(X^i\), \(P^i\), \(U_i\), \(X_*^i\), \(P_*^i\), \(U_i^*\) are assumed to be given,
We form the first variation of functional (6)
\[ \begin{aligned} \delta I &= \int_{0}^{T}\!\!\int_{V} \Bigg\{ \left(\nabla_i \sigma^{ik}+X^k-\frac{\partial^2 u^k}{\partial t^2}\right)\delta u_k^{*} + \left(\nabla_i \sigma_*^{ik}+X_*^k-\frac{\partial^2 u_*^k}{\partial t^2}\right)\delta u_k \\ &\quad +\left[\varepsilon_{ik}-\frac{1}{2}\left(\nabla_i u_k+\nabla_k u_i\right)\right]\delta\sigma_*^{ik} + \left[\varepsilon_{ik}^{*}-\frac{1}{2}\left(\nabla_i u_k^{*}+\nabla_k u_i^{*}\right)\right]\delta\sigma^{ik} \\ &\quad +\left(\sigma^{ik}-E^{ikjl}\varepsilon_{jl}\right)\delta\varepsilon_{ik}^{*} + \left(\sigma_*^{ik}-E^{ikjl}\varepsilon_{jl}^{*}\right)\delta\varepsilon_{ik} \Bigg\}\,dV\,dt \\ &\quad -\int_{0}^{T}\!\!\int_{S_1} \left[(\sigma^{ik}n_k-P^i)\delta u_i^{*} + (\sigma_*^{ik}n_k-P_*^i)\delta u_i\right]\,dS\,dt \\ &\quad +\int_{0}^{T}\!\!\int_{S_2} \left[(u_k-U_k)n_i\delta\sigma_*^{ik} + (u_k^{*}-U_k^{*})n_i\delta\sigma^{ik}\right]\,dS\,dt \\ &\quad +\int_{V}\rho \left[ \left(\frac{\partial u^i}{\partial t}\delta u_i^{*} + \frac{\partial u_*^i}{\partial t}\delta u_i\right)_{t=T} - \left(\frac{\partial u^i}{\partial t}\delta u_i^{*} + \frac{\partial u_*^i}{\partial t}\delta u_i\right)_{t=0} \right]\,dV . \end{aligned} \tag{7} \]
This variation is equal to zero, \(\delta I=0\), if the functions \(u_i,\ \sigma^{ik},\ \varepsilon_{ik}\) satisfy equations (1), (2), the boundary conditions (3), (4), and the initial conditions (5); the system of functions \(u_i^{*},\ \sigma_*^{ik},\ \varepsilon_{ik}^{*},\ X_*^i,\ P_*^i,\ U_i^{*}\) satisfies the same equations (1), (2), the boundary conditions (3), (4), and the conditions
\[ u_i^{*}(x^l,T)=0,\qquad \partial u_i^{*}(x^l,T)/\partial t=0. \tag{8} \]
Introduce a new variable—the time \(\tau=T-t\)—and denote
\[ u_i^{*}(x^l,T-\tau)=v_i(x^l,\tau), \]
\[ \sigma_*^{ik}(x^l,T-\tau)=s^{ik}(x^l,\tau),\qquad \varepsilon_{ik}^{*}(x^l,T-\tau)=e_{ik}(x^l,\tau). \tag{9} \]
In the new variables the last equations and conditions for \(u_i^{*},\ \sigma_*^{ik},\ \varepsilon_{ik}^{*}\) are transformed into the form
\[ \nabla_i s^{ik}+X_*^k(x^l,T-\tau)=\rho\,\partial^2 v^k/\partial \tau^2, \tag{10} \]
\[ s^{ik}=E^{ikjl}e_{jl},\qquad e_{ik}=\frac{1}{2}\left(\nabla_i v_k+\nabla_k v_i\right), \tag{11} \]
\[ s^{ik}n_i=P_*^k(x^l,T-\tau),\qquad v_i=U_i^{*}(x^l,T-\tau) \tag{12} \]
respectively on the surfaces \(S_1,\ S_2\), and
\[ v_i(x^l,0)=0,\qquad \partial v_i(x^l,0)/\partial \tau=0. \tag{13} \]
Problem (10)—(13) completely coincides with problem (1)—(5), if one takes \(\tau=t\) and defines the functions \(v_i,\ s^{ik},\ e_{ik},\ X_*^i,\ P_*^i,\ U_i^{*}\) in the following way:
\[ u_i=v_i,\qquad s^{ik}=\sigma^{ik},\qquad e_{ik}=\varepsilon_{ik},\qquad X_*^k(x^l,T-t)=X^k(x^l,t), \]
\[ P_*^i(x^l,T-t)=P^i(x^l,t),\qquad U_i^{*}(x^l,T-t)=U_i(x^l,t). \tag{14} \]
From relations (9), (14) it follows that
\[ u_i^{*}=u_i(x^l,T-t),\qquad \sigma_*^{ik}=\sigma^{ik}(x^l,T-t),\qquad \varepsilon_{ik}^{*}=\varepsilon_{ik}(x^l,T-t), \]
\[ X_*^i=X^i(x^l,T-t),\qquad P_*^i=P^i(x^l,T-t),\qquad U_i^{*}=U_i(x^l,T-t). \tag{15} \]
Introducing these functions into functional (6), using the notation
\[ F*G=\int_{0}^{T} F(x^l,\tau)G(x^l,T-\tau)\,d\tau \tag{16} \]
and taking into account that
\[ F*G=G*F, \tag{17} \]
we obtain the variational principle of the dynamics of the theory of elasticity.
An elastic body, for given initial displacements and velocities (5), moves in the time interval \((0,T)\) in such a way that the integral
\[ \begin{aligned} &\int_V \left\{ -\frac{1}{2}\rho\,\frac{\partial u^i}{\partial t} * \frac{\partial u_i}{\partial t} -\frac{1}{2}E^{ikjl}\varepsilon_{ik} * \varepsilon_{jl} +\right.\\ &\left. \qquad\qquad +\sigma^{ik} * \left[\varepsilon_{ik}-\frac{1}{2}\left(\nabla_i u_k+\nabla_k u_i\right)\right] +X^i * u_i \right\}\,dV +\\ &\qquad +\int_{S_1} p^i * u_i\,dS +\int_{S_2} (u_k-U_k) * \sigma^{ik} n_i\,dS \end{aligned} \tag{18} \]
has a stationary value.
As special cases of this variational principle one can obtain variational principles in which the varied quantities are the displacements and stresses, or only the displacements. For example, in the latter case one must assume that conditions (2), (4) are satisfied in advance. Then the variational principle takes the form:
An elastic body, for given initial and boundary conditions (4), (5), moves in the time interval \((0,T)\) in such a way that the integral
\[ \int_V \left( -\frac{1}{2}\rho\,\frac{\partial u^i}{\partial t} * \frac{\partial u_i}{\partial t} -\frac{1}{2}E^{ikjl}\varepsilon_{ik} * \varepsilon_{jl} +X^i * u_i \right)\,dV +\int_{S_1} p^i * u_i\,dS \tag{19} \]
has a stationary value.
Institute of Cybernetics
Academy of Sciences of the Estonian SSR
Received
11 III 1966
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