THEORY OF ELASTICITY

M. O. BASHELEISHVILI, T. G. GEGELIA

ON THE BASIC SPATIAL BOUNDARY-VALUE PROBLEMS FOR COMPOSITE ISOTROPIC ELASTIC MEDIA*

(Presented by Academician N. I. Muskhelishvili on 25 VI 1964)

1. Let \(D\) be a three-dimensional domain bounded by a surface \(S_0\); let \(D_i\) \((i=1,\ldots,n)\) be a domain lying inside \(D\), together with its closure \(\overline{D}_i\), and bounded by a surface \(S_i\). Let \(\overline{D}_i \cap \overline{D}_j=\varnothing\) \((i,j=1,\ldots,n;\ i\ne j)\),

\[ D_0=D\setminus \bigcup_{i=1}^{n}\overline{D}_i; \]

\(D_{n+1}\) is the complement of \(D\) to the whole space.

Suppose that the domain \(D_i\) \((i=0,\ldots,n)\) is filled with an isotropic homogeneous elastic medium characterized by the constant Lamé parameters \(\lambda_i\) and \(\mu_i\). Introduce the notation:

\[ \Delta_i u=\mu_i\sum_{j=1}^{3}\frac{\partial^2 u}{\partial x_j^2}+(\lambda_i+\mu_i)\operatorname{grad}\operatorname{div}u, \]

\[ T_i u=2\mu_i\frac{\partial u}{\partial \nu}+\lambda_i\nu\operatorname{div}u+\mu_i[\nu\operatorname{rot}u], \]

where \(u=(u_1(x),u_2(x),u_3(x))\) is a vector; \(\nu=(\nu_j)\) is the unit normal exterior with respect to \(D_i\) at the point \(x\), for \(x\in S_i\) \((i=0,\ldots,n)\); \(x=(x_i)\) is a point of three-dimensional Euclidean space; \(x_1,x_2,x_3\) are its Cartesian coordinates.

The basic boundary-value problems of the theory of elasticity for the indicated composite medium are posed as follows:

Find a vector \(u(x)\), continuous together with its first derivatives with respect to the Cartesian coordinates of the point \(x\) in the closed domains \(\overline{D}_i\) \((i=0,\ldots,n)\), possessing continuous second derivatives in the domains \(D_i\) \((i=0,\ldots,n)\), and satisfying the conditions:

\[ \text{1)}\quad \Delta_i u(x)=\psi_i(x)\quad \text{for } x\in D_i\ (i=0,\ldots,n); \]

\[ \text{2′)}\quad u^{+}(x)=f_0(x) \]

or

\[ \text{2″)}\quad (T_0u)^{+}(x)=f_0(x)\quad \text{for } x\in S_0; \]

\[ \text{3)}\quad u^{+}(x)-u^{-}(x)=f_i(x)\quad \text{and}\quad (T_i u)^{+}(x)-(T_0 u)^{-}(x)=\varphi_i(x)\quad \text{for } \]

\[ x\in S_i\ (i=1,\ldots,n). \]

The symbols \(u^{+}(x)\) for \(x\in S_i\) \((i=0,\ldots,n)\) and \(u^{-}(x)\) for \(x\in S_i\) \((i=1,\ldots,n)\) denote the limiting values of \(u\) at the point \(x\) from \(D_i\) and \(D_0\), respectively. \((T_i u)^{+}(x)\) and \((T_i u)^{-}(x)\) for \(x\in S_i\) \((i=0,\ldots,n)\) denote the limiting values of the expressions \(T_i u\) and \(T_0 u\) at the point \(x\) from \(D_i\) and \(D_0\), respectively. \(\psi_i\) \((i=0,\ldots,n)\) are given vectors on \(D_i\); \(f_i\) \((i=0,\ldots,n)\) and \(\varphi_i\) \((i=1,\ldots,n)\) are given vectors on \(S_i\).

In the case when conditions 1), 2′), 3) are fulfilled, we have the first basic problem, and in the case of 1), 2″), 3), the second.

If now the domain \(D_{n+1}\) is occupied by one elastic medium, \(D_0\) by another medium, and \(D_i\) \((i=1,\ldots,n)\) are voids, then the vector \(u\) must satisfy condition 1) for \(i=0\) and \(i=n+1\), and condition 3) for \(i=0\). Conditions 2′) and 2″) are replaced in this case, respectively, by the conditions

\[ \text{2*}\quad u^{-}(x)=f_i(x); \]

\[ \text{2**}\quad (T_0u)^{-}(x)=f_i(x)\quad \text{for } x\in S_i\ (i=1,\ldots,n). \]

* Reported to the Second All-Union Congress on Theoretical and Applied Mechanics in February 1964.

Moreover, the vector \(u\) must be regular at infinity. If the domain \(D\) coincides with the whole space, then conditions \(2')\) and \(2'')\) are replaced by the condition of regularity of the vector \(u\) at infinity.

1. All the problems presented above were studied by V. D. Kupradze \((^{1,2})\) under the assumption that the Poisson coefficients of the media considered are close. In this note another method is proposed for studying these problems. In this method no restrictions are imposed on the Poisson coefficients.

2. Of the problems under consideration, the second boundary-value problem presents the greatest difficulty for investigation. We shall study it in detail. Suppose, for simplicity, that \(\psi_i = 0\), \(n=1\), \(S_0\) is a surface of class \((^3)\) \(\mathcal L(1,\alpha)\), and \(S_1\) is of class \(\mathcal L(2,\alpha)\), \(f_0 \in H(\beta)\) on \(S_0\), \(\varphi_1 \in H(\beta)\) on \(S_1\), \(f_1 \in H(1,\beta)\) on \(S_1\) (see \((^3)\)). \(\alpha\) and \(\beta\) are positive constants, with \(\alpha > \beta\). We choose the origin of coordinates in the domain \(D_0\). All these conditions will henceforth be assumed satisfied.

We seek the solution of the second problem in the form

\[ u(x)=\int_{S_0}\Gamma_0(x,y)g_1(y)\,d_yS+ \]

\[ +\int_{S_1}\bigl[\Gamma_0(x,y)B_0g_3(y)-P_0(x,y)A_0g_2(y)\bigr]\,d_yS \quad \text{for } x\in D_0; \tag{1} \]

\[ u(x)=\int_{S_1}\bigl[\Gamma_1(x,y)B_1g_3(y)-P_1(x,y)A_1g_2(y)\bigr]\,d_yS \quad \text{for } x\in D_1, \tag{2} \]

where
\[ A_k=\mu_1\mu_0[\mu_k(\mu_1+\mu_0)]^{-1};\quad B_k=a_1a_0[a_k(a_1+a_0)]^{-1};\quad a_k=\mu_k(\lambda_k+2\mu_k)^{-1}; \]
\[ \Gamma_k=(4\pi\mu_k r)^{-1}\left\|2\delta_{ij}-b_k r_i r_j\right\|;\quad b_k=(\lambda_k+\mu_k)(\lambda_k+2\mu_k)^{-1}; \]
\[ P_k=(2\pi)^{-1}\left\|\left[(1-c_k)\delta_{ij}+3c_k r_i r_j\right]r_n +(a_1+a_0)^{-1}\left[(-1)^k(a_1-a_0)+2a_1a_0\right](n_i r_j-n_j r_i)r^{-2}\right\|; \]
\[ c_k=2b_k a_1a_0[a_k(a_1+a_0)]^{-1};\quad k=0,1;\quad r^2=\sum (x_i-y_i)^2;\quad y=(y_i); \]
\[ \delta_{ij}\text{ is the Kronecker symbol};\quad n=(n_i)\text{ is the normal at the point }y\text{ exterior with respect to }D_k \text{ for } y\in S_k\ (k=0,1); \]

\[ r_i=\frac{\partial r}{\partial x_i};\qquad r_n=\frac{\partial}{\partial n}\frac{1}{r};\qquad r_{ij}=\frac{\partial^2 r}{\partial x_i\partial x_j}; \]

\(g_i=(g_{ij})\) are the unknown vectors; \(g_1\) and \(g_3\) are sought in the class \(H(\gamma)\) on \(S_0\) and \(S_1\), respectively, and \(g_3\) is sought in the class \(H(1,\gamma)\) on \(S_1\); \(\gamma\) is some positive number.

To determine \(g_1, g_2\), and \(g_3\), the following system of integral equations is obtained:

\[ g_i(x)=\sum_{j=1}^{3}\int_{S_{j-1}} k_{ij}(x,y)g_j(y)\,d_yS=\Phi_i(x); \tag{3} \]

\(x\in S_0\) for \(i=1\) and \(x\in S_1\) for \(i=2,3\). Here \(S_2=S_1\);
\[ k_{11}=(2\pi)^{-1}\left\|(a_0\delta_{ij}+3b_0r_ir_j)r_\nu +a_0(\nu_i r_j-\nu_j r_i)r^{-2}\right\|, \]
\[ k_{12}=\mu_0(2\pi)^{-1}\left\|\nu_i r_n-\nu_j r_{ni} -\delta_{ij}r_{n\nu}-\left\{[r_{ij}\Sigma n_e\nu_e+\nu_i r_{nj}-n_j r_{\nu i} +(n_j\nu_i-n_i\nu_j)r^{-3}-3n_i r^{-1}r_j r_\nu]a_2\right.\right. \]
\[ \left.\left. +12b_0a_1[\nu_j r_i r^{-1}-r(\delta_{ij}-5r_i r_j)r_\nu]r_n -3(3a_1-a_0)n_j r_i r_\nu r^{-1}\right\}(a_0+a_1)^{-1}\right\|, \]
\[ k_{22}=[2\pi(\mu_0+\mu_1)(a_0+a_1)]^{-1} \left\|r_n\left[6(\mu_1a_1b_0-\mu_0a_0b_1)r_ir_j+a_3\delta_{ij}\right]\right. \]
\[ \left. +\left[(a_1-a_0)(\mu_0+\mu_1)+2a_1a_0(\mu_1-\mu_0)\right](\nu_j r_i-\nu_i r_j)\right\|, \]
\[ k_{23}=a_4(2\pi)^{-1}\left\|2(a_0\mu_0-a_1\mu_1)\delta_{ij}r^{-1}-a_5r_{ij}\right\|, \]
\[ k_{32}=a_6\left\|(\nu_jr_n-n_jr_\nu)r_ir^{-1}-(\delta_{ij}-5r_ir_j)rr_\nu r_n\right\|, \]
\[ k_{33}=3(a_0-a_1)[2\pi(a_0+a_1)]^{-1}\left\|r_ir_jr_\nu\right\|; \]
\[ a_3=(\mu_0+\mu_1)(a_0+a_1)+2a_1a_0(\mu_1-\mu_0),\quad a_4=[4\pi\mu_1\mu_0(a_0+a_1)]^{-1}, \]
\[ a_5=a_0\mu_0b_1-a_1\mu_1b_0,\quad a_6=6(a_1-a_0)\mu_1\mu_0[\pi(\mu_0+\mu_1)(a_0+a_1)]^{-1}, \]
\[ k_{13}=k_{11}B_0,\qquad k_{31}=k_{11},\qquad k_{21}=-\Gamma_0,\qquad \Phi_1=f_0,\quad \Phi_2=f_1,\quad \Phi_3=\varphi_1, \]
\[ r_{nj}=\frac{\partial^2}{\partial n\,\partial x_j}\frac{1}{r},\qquad r_{\nu j}=\frac{\partial^2}{\partial \nu\,\partial x_j}\frac{1}{r},\qquad r_\nu=\frac{\partial}{\partial \nu}\frac{1}{r},\qquad r_{n\nu}=\frac{\partial^2}{\partial \nu\,\partial x}\frac{1}{r}. \]

Let \(a_1(x)=1\) for \(x\in S_0\) and \(a_1(x)=0\) for \(x\in S_1\); \(a_2(x)=a_3(x)=0\) for \(x\in S_0\) and \(a_2(x)=a_3(x)=1\) for \(x\in S_1\); \(\Phi_1(x)=\Phi_1(x)\) for \(x\in S_0\) and \(\Phi_1(x)=0\) for \(x\in S_1\); \(\Phi_2(x)=\Phi_3(x)=0\) for \(x\in S_0\) and \(\Phi_2(x)=\Phi_2(x)\), \(\Phi_3(x)=\Phi_3(x)\) for \(x\in S_1\).

Consider the system of equations

\[ h_i(x)+a_i(x)\sum_{j=1}^{3}\int_{S_0\cup S_1} a_j(y)k_{ij}(x,y)h_j(y)\,d_yS=\Phi_i(x), \tag{4} \]

where \(h_i=(h_{ij})\) is the vector sought on \(S_0\cup S_1\) \((i,j=1,2,3)\).

(4) is a system of two-dimensional singular integral equations distributed over \(S_0\cup S_1\). It is easy to establish the connection between the solutions of systems (3) and (4).

Just as in note \((^4)\), the symbolic matrix \(\Phi(x,\vartheta)\) and its determinant are computed:

\[ \det \Phi(x,\vartheta)=[1-a_1(x)a_0^2][1-a_2(x)\chi^2], \]

where

\[ \chi=\frac{a_1-a_0}{a_1+a_0}+\frac{2a_1a_0}{a_1+a_0}\frac{\mu_1-\mu_0}{\mu_1+\mu_0}. \]

Let us study system (4) in the space \(L_p(S_0\cup S_1)\), where \(p\) is an arbitrary number greater than 1. From the inequality

\[ 1-\chi^2=4a_1a_0(a_1+a_0)^{-2}(\mu_1+\mu_0)^{-2} \bigl[\mu_0^2\nu_0(1+a_1)+\mu_1^2\nu_1(1+a_0)+ \]

\[ +2\mu_1\mu_0(1+a_1a_0)>0 \]

it follows that \(\det\Phi(x,\vartheta)>0\). Consequently, system (4) is of normal type in the space \(L_p(S_0\cup S_1)\).

From the arguments given in notes \((^3,^4)\), the validity of the following propositions follows:

For system (4) the Fredholm theorems are valid in the space \(L_p(S_0\cup S_1)\).

Every solution \(h_1,h_2,h_3\) of system (4) belonging to the class \(L_p(S_0\cup S_1)\) satisfies the condition \(h_1\in H(\beta)\), \(h_2\in H(1,\beta)\), \(h_3\in H(\beta)\).

Taking these propositions into account, as well as the boundary properties of the potentials of the theory of elasticity \((^5)\), by a method analogous to that applied in notes \((^{6-8})\), one can make the following conclusion.

For solvability of system (3) it is necessary and sufficient that the conditions

\[ \int_{S_0} f_0(y)\,d_yS-\int_{S_1}\varphi_1(y)\,d_yS=0, \tag{5} \]

\[ \int_{S_0}[R(y)f_0(y)]\,d_yS-\int_{S_1}[R(y)\varphi_1(y)]\,d_yS=0, \tag{6} \]

\[ R(y)=(y_1,y_3,y_3). \]

From the above reasoning the main proposition of this note follows easily:

Theorem. The second basic problem for composite elastic media in the formulation given above is solvable only if conditions (5) and (6) are satisfied. When these conditions are satisfied, which express the vanishing of the principal vector and principal moment of the external forces acting on the boundary of the domain \(D_0\), the solution of the problem is given, up to a rigid displacement, in the form (1) and (2), where \(g_1,g_2\), and \(g_3\) are a solution of system (3).

1. In an analogous way, the fundamental spatial boundary-value problems for composite isotropic elastic media are studied in the case of steady-state oscillations, as well as inhomogeneous problems of diffraction of electromagnetic waves.

2. We note that the problems studied above are boundary-value problems for special elliptic systems with discontinuous coefficients. For a single general elliptic equation this problem has been studied by many authors (see, for example, the work \({}^{9}\), which also contains fairly detailed references on these questions). The method used in the present note is also applicable to a single equation with discontinuous coefficients. In this case one can study not only the Dirichlet and Neumann problems, but also the problem with an oblique derivative, provided that the directions along which the derivatives are prescribed do not lie in the tangent planes to the boundary.

Computing Center
of the Academy of Sciences of the Georgian SSR

Received
20 VI 1964

CITED LITERATURE

\({}^{1}\) V. D. Kupradze, Methods of Potential in the Theory of Elasticity, Moscow, 1963.
\({}^{2}\) W. D. Kupradse, Dynamical Problems in Elasticity, Progress in Solid Mech., III, Amsterdam, 1963.
\({}^{3}\) T. G. Gegelia, Transactions of the Georgian Polytechnic Institute, 1 (81), 69 (1962).
\({}^{4}\) T. G. Gegelia, Transactions of the Tbilisi Mathematical Institute, 28, 53 (1962).
\({}^{5}\) T. G. Gegelia, Transactions of the Computing Center of the Academy of Sciences of the Georgian SSR, 2, 285 (1961).
\({}^{6}\) M. O. Basheleishvili, Transactions of the Computing Center of the Academy of Sciences of the Georgian SSR, 3, 93 (1963).
\({}^{7}\) M. O. Basheleishvili, Transactions of the Computing Center of the Academy of Sciences of the Georgian SSR, 4, 131 (1963).
\({}^{8}\) M. O. Basheleishvili, Transactions of the Computing Center of the Academy of Sciences of the Georgian SSR, 4, 141 (1963).
\({}^{9}\) V. A. Il’in, I. A. Shishmarev, Soviet-American Symposium, Novosibirsk, August 1963.