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THEORY OF ELASTICITY
B. M. NAIMARK
SOME NONLINEAR BOUNDARY-VALUE PROBLEMS IN THE THEORY OF A MAXWELL BODY
(Presented by Academician A. A. Dorodnitsyn, 16 II 1961)
Some physical problems concerning the motion of an elastic body in which stresses relax lead to the determination of the displacement vector of the body \(\mathbf u(x_1,x_2,x_3,t)\), with components \(u_1,u_2,u_3\), and of the stress tensor \(\sigma_{x_i x_j}(x_1,x_2,x_3,t)\), \(i,j=1,2,3\), which satisfy the following system of equations:
\[ \vec{\sigma} = N_1 \vec{\varepsilon} - N_2 \int_0^t \exp\left[-\int_\tau^t \frac{ds}{T}\right] \frac{2\mu}{3T}\,\vec{\varepsilon}\,d\tau, \]
\[ \mu \Delta \mathbf u+(\lambda+\mu)\operatorname{grad}\operatorname{div}\mathbf u+\rho \mathbf F = \vec{\Phi}_t(\mathbf u), \tag{1} \]
where \(\vec{\varepsilon}\) is the column with components
\(\partial u_1/\partial x_1,\ \partial u_2/\partial x_2,\ \partial u_3/\partial x_3,\ \partial u_1/\partial x_2+\partial u_2/\partial x_1,\)
\(\partial u_1/\partial x_3+\partial u_3/\partial x_1,\ \partial u_2/\partial x_3+\partial u_3/\partial x_2\);
\(\vec{\sigma}\) is the column with components
\(\sigma_{x_1x_1},\sigma_{x_2x_2},\sigma_{x_3x_3},\sigma_{x_1x_2},\sigma_{x_1x_3},\sigma_{x_2x_3}\);
\(\lambda,\mu\) are positive constants (Lamé constants);
\(\rho(x_1,x_2,x_3)\) is a positive function (density);
\(\mathbf F(x_1,x_2,x_3,t)\) is the vector with components \(F_1,F_2,F_3\) (the body-force vector);
\(T(x_1,x_2,x_3,t,\vec{\sigma})\) is a positive function (the relaxation time);
and \(N_1\) and \(N_2\) are the following matrices of order 6:
\[ N_1= \begin{pmatrix} \lambda+2\mu & \lambda & \lambda & 0 & 0 & 0\\ \lambda & \lambda+2\mu & \lambda & 0 & 0 & 0\\ \lambda & \lambda & \lambda+2\mu & 0 & 0 & 0\\ 0 & 0 & 0 & \mu & 0 & 0\\ 0 & 0 & 0 & 0 & \mu & 0\\ 0 & 0 & 0 & 0 & 0 & \mu \end{pmatrix}, \qquad N_2= \begin{pmatrix} 2 & -1 & -1 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0\\ -1 & -1 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 3/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 3/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 3/2 \end{pmatrix}. \tag{2} \]
The vector \(\vec{\Phi}_t(\mathbf u)\) has the following three components \(\Phi_{it}(\mathbf u)\):
\[ \Phi_{it}(\mathbf u) = \frac{\partial}{\partial x_i} \int_0^t \exp\left[-\int_\tau^t \frac{ds}{T}\right] \frac{2\mu}{3T} \left( 2\frac{\partial u_i}{\partial x_i} - \frac{\partial u_j}{\partial x_j} - \frac{\partial u_k}{\partial x_k} \right)d\tau + \tag{3} \]
\[ + \frac{\partial}{\partial x_j} \int_0^t \exp\left[-\int_\tau^t \frac{ds}{T}\right] \frac{\mu}{T} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)d\tau + \frac{\partial}{\partial x_k} \int_0^t \exp\left[-\int_\tau^t \frac{ds}{T}\right] \frac{\mu}{T} \left( \frac{\partial u_i}{\partial x_k} + \frac{\partial u_k}{\partial x_i} \right)d\tau, \]
where the indices \(i,j,k\) take the values \(1,2,3\) and are obtained from \(1,2,3\) by cyclic permutation.
In what follows we shall assume that the point \(x_1,x_2,x_3\) belongs to a bounded domain \(D\) of three-dimensional space and that the boundary \(\Gamma\) of the domain \(D\) is a surface whose curvature is continuous. In addition, we shall suppose that \(\Gamma=\Gamma_1+\Gamma_2\), and that the boundary separating \(\Gamma_1\) and \(\Gamma_2\) is a smooth curve.
We shall consider the following three boundary-value problems.
- The first boundary-value problem. Find the vector \(\mathbf u(x_1,x_2,x_3,t)\) and the stress tensor \(\sigma_{x_i x_j}(x_1,x_2,x_3,t)\) satisfying equations (1) and the boundary condition
\[ \mathbf u(s)=\vec\varphi(s,t), \qquad s\in \Gamma, \]
where \(\vec\varphi(s,t)\) is a given vector.
- Find the vector \(\mathbf u(x_1,x_2,x_3,t)\) and the stress tensor \(\sigma_{x_i x_j}(x_1,x_2,x_3,t)\), satisfying equations (1) and the boundary conditions
\[ \left. \sigma_{x_i x_1}\cos n x_1+\sigma_{x_i x_2}\cos n x_2+\sigma_{x_i x_3}\cos n x_3 \right|_{\Gamma} = X_i(s,t), \qquad s\in\Gamma,\quad i=1,2,3, \]
where \(\cos n x_1,\ \cos n x_2,\ \cos n x_3\) are the direction cosines of the outward normal to the boundary \(\Gamma\); \(X_i(s,t)\) is a given vector (the vector of external forces).
- Find the vector \(\mathbf u(x_1,x_2,x_3,t)\) and the tensor \(\sigma_{x_i x_j}(x_1,x_2,x_3,t)\), satisfying equations (1) and, on \(\Gamma_1\), boundary conditions 1, while on \(\Gamma_2\), boundary conditions 2.
We shall define the generalized solution of the boundary-value problems posed. To this end we introduce the Hilbert space \(\mathfrak R\) and the normed space \(\mathfrak R(t_1,t_2)\) as follows. \(\mathfrak R\) is the orthogonal sum of two Hilbert spaces \(L_2(D)\) and \(\mathfrak H(D)\), where \(L_2(D)\) is the Hilbert space of vectors with 6 components and with modulus whose square is integrable over the domain \(D\), while \(\mathfrak H(D)\) is one of the three Hilbert spaces \(H_{\mathrm I}(D), H_{\mathrm{II}}(D), H_{\mathrm{III}}(D)\), introduced in the works \((^{1,2})\). Recall that the scalar product in these spaces is the expression
\[ \begin{aligned} W(\mathbf u,\mathbf v) &= \iiint_D \Bigg[ \lambda \left( \frac{\partial u_1}{\partial x_1} + \frac{\partial u_2}{\partial x_2} + \frac{\partial u_3}{\partial x_3} \right) \left( \frac{\partial v_1}{\partial x_1} + \frac{\partial v_2}{\partial x_2} + \frac{\partial v_3}{\partial x_3} \right) \\ &\quad +2\mu \frac{\partial u_1}{\partial x_1} \frac{\partial v_1}{\partial x_1} + 2\mu \frac{\partial u_2}{\partial x_2} \frac{\partial v_2}{\partial x_2} + 2\mu \frac{\partial u_3}{\partial x_3} \frac{\partial v_3}{\partial x_3} + \mu \left( \frac{\partial u_1}{\partial x_2} + \frac{\partial u_2}{\partial x_1} \right) \left( \frac{\partial v_1}{\partial x_2} + \frac{\partial v_2}{\partial x_1} \right) \\ &\quad + \mu \left( \frac{\partial u_1}{\partial x_3} + \frac{\partial u_3}{\partial x_1} \right) \left( \frac{\partial v_1}{\partial x_3} + \frac{\partial v_3}{\partial x_1} \right) + \mu \left( \frac{\partial u_2}{\partial x_3} + \frac{\partial u_3}{\partial x_2} \right) \left( \frac{\partial v_2}{\partial x_3} + \frac{\partial v_3}{\partial x_2} \right) \Bigg]\, dx_1\,dx_2\,dx_3 . \end{aligned} \tag{4} \]
Here the space \(H_{\mathrm I}(D)\) is obtained by closure, in the scalar product (4), of the linear set of continuously differentiable vectors \(\mathbf u\) vanishing on \(\Gamma\); \(H_{\mathrm{III}}(D)\) is the closure, in the scalar product (4), of the linear set of continuously differentiable vectors vanishing on \(\Gamma_1\), while \(H_{\mathrm{II}}(D)\) is the closure in (4) of the linear set of continuously differentiable vectors satisfying the conditions
\[ \iiint_D \mathbf u\,dx_1\,dx_2\,dx_3=0, \qquad \iiint_D (\mathbf R\times \mathbf u)\,dx_1\,dx_2\,dx_3=0, \]
where \(\mathbf R\) is the radius vector of the point \(x_1,x_2,x_3\). The scalar product in \(\mathfrak R\) will be denoted by \((\, , \,)\), and the norm by \(\|\,\|\). The projection operators from \(\mathfrak R\) onto \(\mathfrak H\) and \(L_2(D)\) will be denoted by \(P_{\mathfrak H}\) and \(P_{L_2}\).
By \(\mathfrak R(t_1,t_2)\) we shall denote the linear normed space of vector-functions with values in \(\mathfrak R\), defined on the interval \(t_1\le t\le t_2\), with norm
\[ \left\|\!\left|\vec\psi\right|\!\right\|_{(t_1,t_2)} = \sup_{t_1\le t\le t_2} \left\|\vec\psi(t)\right\|. \]
\(\mathfrak R(t_1,t_2)\) is a complete space, convergence in which means uniform convergence on the interval \(t_1\le t\le t_2\) of vector-functions in the norm of \(\mathfrak R\).
Multiplying the first equation (1) scalarly by an arbitrary vector \(\vec\varphi\in L_2(D)\), the second by \(\mathbf v\in\mathfrak H\), integrating on the left over the domain \(D\), integrating the latter equation by parts, and denoting by \(\mathbf u_2\) the solution of the equation
\[ \mu\Delta\mathbf u_2+(\lambda+\mu)\operatorname{grad}\operatorname{div}\mathbf u_2+\varrho\mathbf F=0 \tag{5} \]
with one of the boundary conditions
\[ \begin{aligned} &1'.\quad \mathbf u_2|_{\Gamma}=\vec\varphi(s,t),\\ &2'.\quad \sigma_{2x_i x_1}\cos nx_1+\sigma_{2x_i x_2}\cos nx_2+\sigma_{2x_i x_3}\cos nx_3|_{\Gamma} =X_i(s,t),\quad i=1,2,3, \end{aligned} \]
where
\[ \sigma_{2x_i x_j} =\mu\left(\frac{\partial u_{2i}}{\partial x_j} +\frac{\partial u_{2j}}{\partial x_i}\right) +\lambda\delta_{ij}\operatorname{div}\mathbf u_2 . \]
\[ 3'.\quad \Gamma=\Gamma_1+\Gamma_2,\quad \text{on } \Gamma_1 \text{ condition } 1' \text{ holds, and on } \Gamma_2 \text{ condition } 2' \text{ holds.} \]
Denoting by \(\vec\psi\) the pair \(\{\mathbf u_1,\vec\sigma\}\), where \(\mathbf u_1=\mathbf u-\mathbf u_2\), we can give the following
Definition. A solution of equations (1) with one of the boundary conditions 1–3 is a pair \(\{\mathbf u,\vec\sigma\}\) such that the pair \(\vec\psi_1=\{\mathbf u_1,\vec\sigma\}\) satisfies, for every \(\vec\psi\in\mathfrak R\), the functional equation
\[ \begin{aligned} (\vec\psi_1,\vec\psi) &=\iiint_D [N_1P_{\mathfrak H}\vec\psi_1,\;P_{L_2}\vec\psi]\,dx_1dx_2dx_3\\ &\quad-\iiint_D\left[ N_2\int_0^t \exp\left[-\int_\tau^t\frac{ds}{T}\right] \frac{2\mu}{3T}P_{\mathfrak H}\vec\psi_1\,d\tau,\; P_{L_2}\vec\psi \right]\,dx_1dx_2dx_3\\ &\quad+\iiint_D\left[ N_2\int_0^t \exp\left[-\int_\tau^t\frac{ds}{T}\right] \frac{2\mu}{3T}P_{\mathfrak H}\vec\psi_1\,d\tau,\; P_{\mathfrak H}\vec\psi \right]\,dx_1dx_2dx_3\\ &\quad-\iiint_D\left[ N_2\int_0^t \exp\left[-\int_\tau^t\frac{ds}{T}\right] \frac{2\mu}{3T}\vec\varepsilon_2\,d\tau,\; P_{L_2}\vec\psi \right]\,dx_1dx_2dx_3\\ &\quad+\iiint_D\left[ N_2\int_0^t \exp\left[-\int_\tau^t\frac{ds}{T}\right] \frac{2\mu}{3T}\vec\varepsilon_2\,d\tau,\; P_{\mathfrak H}\vec\psi \right]\,dx_1dx_2dx_3, \end{aligned} \tag{6} \]
\[ \vec\varepsilon=\vec\varepsilon_1+\vec\varepsilon_2; \quad \vec\varepsilon_1 \text{ and } \vec\varepsilon_2 \text{ are the columns corresponding to the vectors } \mathbf u_1 \text{ and } \mathbf u_2 . \]
Consider an arbitrary \(t_0\) and divide the interval \(0\le t\le t_0\) into \(n\) equal parts
\(\tau_i\le t\le\tau_{i+1}\), \(i=0,1,2,\ldots,n\), \(\tau_0=0\), \(\tau_{n+1}=t_0\), and set
\[ T(x_1,x_2,x_3,\vec\sigma(t)) = T(x_1,x_2,x_3,\vec\sigma(\tau_i)), \quad \tau_i\le \tau\le \tau_{i+1}. \]
Then the equations with boundary conditions 1–3 become linear on each of the intervals
\(\tau_i\le\tau\le\tau_{i+1}\), and from estimates of the right-hand sides of (6) it follows that the boundary-value problems (1) have a unique solution belonging to \(\mathfrak R(0,t_0)\). The solution \(\vec\psi_{1n}\in\mathfrak R(0,t_0)\) found in this way will be called the Euler polygonal line of the boundary-value problem (1), 1–3.
Lemma 1. The family of Euler polygonal lines is uniformly bounded and equicontinuous in the norm in \(\mathfrak R\) on the interval \(0\le t\le t_0\), if the conditions
\[ \sup_{0\le t\le t_0}\mathrm W(\mathbf u_2,\mathbf u_2)<\infty, \qquad \inf_{\substack{x\in D\\0\le t\le t_0\\-\infty<|\vec\sigma|<\infty}} T(x_1,x_2,x_3,t,\vec\sigma)>0 \tag{7} \]
are satisfied.
The proof of the lemma is based on applying the Cauchy–Bunyakovsky inequality and conditions (7) to the right-hand side of (6).
Theorem 1. Suppose that conditions (7) are satisfied. Suppose, in addition, that
\[ \sup_{0\leq t\leq t_0}\left|\frac{\partial}{\partial t}W(\mathbf u_2,\mathbf u_2)\right|<\infty,\qquad \sup_{\substack{x\in D\\0\leq t\leq t_0\\-\infty<|\vec\sigma|<\infty}} \left|\frac{\partial T}{\partial t}\right|<\infty,\qquad \sup_{\substack{x\in D\\0\leq t\leq t_0\\-\infty<|\vec\sigma|<\infty}} \left|\frac{\partial T}{\partial \vec\sigma}\right|<\infty . \tag{8} \]
Then equation (6) has a unique solution belonging to \(\mathfrak R(0,t_0)\).
Idea of the proof. Choose on the interval \(0\leq t\leq t_0\) an everywhere dense countable sequence \(t_i\); for each \(t_i\) choose a weakly convergent sequence of Euler polygonal lines \(\vec\psi_{1n_1n_2\ldots n_{i-1}n_i}\) from the already chosen subsequence \(\vec\psi_{1n_1n_2\ldots n_{i-1}}\), converging at the points \(t_1,t_2,\ldots,t_{i-1}\). This can always be done by virtue of the uniform boundedness of the family \(\vec\psi_{1n}\). From the sequences \(\vec\psi_{1n_1n_2\ldots n_s}\) choose a diagonal subsequence \(\vec\psi_{1n}\), converging weakly, by virtue of the uniform continuity of the family \(\vec\psi_{1n}\), to some limit \(\vec\psi_1\). Estimates of the right-hand side of (6) imply that this limit is a solution of (6). If, however, there exist two solutions \(\mathbf u_1\) and \(\mathbf v_1\), then from the estimates of the right-hand side of (6) the relation follows
\[ \left|(\vec\psi_1-\vec\varphi_1,\vec\psi)\right| < ct_0\,\left|(\vec\psi_1-\vec\varphi_1,\vec\psi)\right|, \qquad \psi\in\mathfrak R, \]
where the constant \(c\) does not depend on \(t\). Hence the uniqueness of the solution follows.
To find an approximate solution of (6), one may use a Galerkin-type method. Let \(\mathbf g_i\) be a basis in \(\mathfrak R\). The approximate solution
\[ \vec\psi^{(n)}(t)=\sum_{i=1}^{n}c_i^{(n)}(t)\mathbf g_i \]
is found from the system of equations
\[ c_k^{(n)}(t)= \Phi_t\left(\sum_{i=1}^{n}c_i^{(n)}(t)\mathbf g_i,\mathbf g_k\right), \qquad k=1,2,\ldots,n, \tag{9} \]
where \(\Phi_t(\vec\psi,\vec\psi)\) denotes the right-hand side of (6).
Theorem 2. Suppose that conditions (7), (8) are satisfied. Then system (9) has a solution, and moreover a unique one. In addition, for any \(\vec\psi\in\mathfrak R\) and uniformly on the whole interval \(0\leq t\leq t_0\),
\[ (\vec\psi_1-\vec\psi^{(n)}_1,\vec\psi)\to 0,\qquad n\to\infty, \]
where \(\vec\psi_1\) is the solution of equation (6).
Schmidt Institute of Physics of the Earth
Academy of Sciences of the USSR
Received
13 II 1961
CITED LITERATURE
- S. G. Mikhlin, The Problem of the Minimum of a Quadratic Functional, 1952.
- B. M. Naimark, Transactions of the Institute of Physics of the Earth, No. 11 (178) (1959).