Abstract
Full Text
MECHANICS
V. A. LOMAKIN
STATISTICAL DESCRIPTION OF THE STRESSED STATE OF A DEFORMABLE BODY
(Presented by Academician A. Yu. Ishlinskii, 20 XII 1963)
When random external forces act on a solid deformable body, or when the body has a statistically specified microinhomogeneous structure, the stress tensor \(\tau_{ij}\) defines a tensor random field. Using methods customary in the theory of random fields \((^{1})\), let us consider the question of the statistical description of the random field of the stress tensor. For simplicity, we shall carry out the consideration in a rectangular Cartesian coordinate system \(x_i\) \((i=1,2,3)\).
We shall say that in some region \(V\) a tensor random field \(\tau_{ij}(M)\) is specified if, to every finite system of points \(M_k(x_s^k)\), \(k=1,\ldots,n\), from the region \(V\), there is assigned an \(Nn\)-dimensional probability distribution law for the system of values \(\tau_{ij}^k=\tau_{ij}(x_s^k)\), where \(N\) is the number of components of the stress tensor. We restrict ourselves to consideration of a symmetric stress tensor and introduce the \(6n\)-dimensional probability density
\[ f^n\left(\tau_{11}^1,\tau_{12}^1,\ldots,\tau_{33}^1,\ldots,\tau_{11}^n,\tau_{12}^n,\ldots,\tau_{33}^n\right) = f^n\left(\tau_{ij}^k\right) \tag{1} \]
for the random variables \(\tau_{ij}^k\), where the functions \(f^n(\tau_{ij}^k)\), \(n=1,2,\ldots\), satisfy certain consistency conditions \((^{1})\).
Let us call the quantity
\[ p^{(m)}_{i_1j_1\ldots k_1l_1\ldots i_nj_n\ldots p_nr_n} = \]
\[ = \overline{(\tau_{i_1j_1}^1-\bar{\tau}_{i_1j_1}^1)\cdots (\tau_{k_1l_1}^1-\bar{\tau}_{k_1l_1}^1)\cdots (\tau_{i_nj_n}^n-\bar{\tau}_{i_nj_n}^n)\cdots (\tau_{p_nr_n}^n-\bar{\tau}_{p_nr_n}^n)} = \]
\[ = \int_{-\infty}^{+\infty}\!\!\cdots\!\!\int (\tau_{i_1j_1}^1-\bar{\tau}_{i_1j_1}^1)\cdots (\tau_{p_nr_n}^n-\bar{\tau}_{p_nr_n}^n) f^n(\tau_{ij}^k)\,d\tau_{i_1j_1}^1\cdots d\tau_{p_nr_n}^n, \tag{2} \]
the \(n\)-point moment of order \(m\), where \(\bar{\tau}_{ij}^k\) is the statistical mean of the quantity \(\tau_{ij}^k\), \(m=m_1+\cdots+m_n\); \(m_k\) is the number of factors on the right-hand side of (2) referring to the point \(M_k\). The moment of order \(m\) (2) for the tensor field \(\tau_{ij}\) is an \(n\)-point tensor \((^{2})\) of rank \(2m\). Knowing the moments (2) of all orders, one can reconstruct the distribution functions (1) \((^{1,3})\); therefore a random tensor field is statistically determined by the totality of the moments (2) of all orders. We shall also say that a statistical description of the field \(\tau_{ij}\) is given with accuracy up to moments of order \(m\) if, for any system of points \(M_1,\ldots,M_n\) \((n\le m)\), the moments (2) up to order \(m\) inclusive are known.
In what follows, the tensor field \(\tau_{ij}\) will be regarded as continuous \((^{1})\). Then all moments (2) can be obtained by a limiting passage from moments whose type and order coincide \((n=m)\); we shall call these moments the fundamental moments. Introducing the mean value \(\sigma_{ij}\) of the stress tensor and the deviation \(p_{ij}\) from the mean value,
\[ \sigma_{ij}=\bar{\tau}_{ij},\qquad p_{ij}=\tau_{ij}-\sigma_{ij}. \tag{3} \]
represent the principal moment of order \(n\) in the form
\[ p_{i_1 j_1 \ldots i_n j_n} = \overline{p_{i_1 j_1}(x_s^1)\ldots p_{i_n j_n}(x_s^n)}. \tag{4} \]
The principal moment of order \(n\) is a function of the \(3n\) coordinates \(x_s^k\) \((s=1,2,3;\ k=1,\ldots,n)\), which we shall regard as independent. In particular, for the principal moment of second order, also called the moment of connection of the field values at two points [1], we have
\[ p_{ijkl} = \overline{p_{ij}(x_s^1)\,p_{kl}(x_s^2)},\qquad p_{ijkl}=p_{jikl}=p_{ijlk}=p_{klij}. \tag{5} \]
In equilibrium of a deformable body and in the absence of body forces, the stress tensor \(\tau_{ij}\) satisfies the equations
\[ \frac{\partial \tau_{ij}}{\partial x_j}=0. \tag{6} \]
Here and below the usual summation convention for tensor quantities over repeated indices from 1 to 3 is used. We shall further apply the operation of statistical averaging (by means of the distributions (1)), which, as is known, has the properties
\[ \overline{f+g}=\bar f+\bar g,\qquad \overline{fg}=\bar f\,\bar g,\qquad \frac{\partial \bar f}{\partial x_i} = \overline{\frac{\partial f}{\partial x_i}},\qquad \bar f'=0\ (f'=f-\bar f), \tag{7} \]
where \(f\) and \(g\) are arbitrary random functions. From (6), (7), and (3) we find
\[ \frac{\partial \sigma_{ij}}{\partial x_j}=0,\qquad \frac{\partial p_{ij}}{\partial x_j}=0. \tag{8} \]
Writing the second equation (8) for the point \(M_k(x_s^k)\) and using (4), we obtain \(3\cdot 6^{\,n-1}n\) differential equations for the principal moments
\[ \frac{\partial p_{i_1 j_1 \ldots i_n j_n}}{\partial x_{j_k}^k}=0,\qquad k=1,\ldots,n. \tag{9} \]
For \(n=2\), relations (9) give 36 equations for the moment of connection
\[ \frac{\partial p_{ijkl}}{\partial x_j^1}=0,\qquad \frac{\partial p_{ijkl}}{\partial x_l^2}=0. \tag{10} \]
Let us consider the special cases of statistically homogeneous and statistically isotropic fields. We shall call a random tensor field \(\tau_{ij}\) statistically homogeneous if the probability distributions (1) are invariant with respect to translations \(x_s^{k'}=x_s^k+a_s\). The principal moments (4) in this case depend only on \((n-1)\) vectors that determine the configuration of the points \(M_k\). The tensor \(\sigma_{ij}\) (3) will then be constant, and the moment of connection (5), called in this case also the correlation tensor, will be a function of one vector \(\xi_s\):
\[ p_{ijkl}=p_{ijkl}(\xi_s),\qquad \xi_s=x_s^2-x_s^1, \tag{11} \]
and the relation
\[ p_{ijkl}(\xi_s)=p_{klij}(-\xi_s). \tag{12} \]
holds.
The equations (10) take the form
\[ \frac{\partial p_{ijkl}(\xi_s)}{\partial \xi_j}=0,\qquad \frac{\partial p_{ijkl}(\xi_s)}{\partial \xi_l}=0, \tag{13} \]
and, by virtue of (12), only one of these two systems of equations is independent.
We shall call a random tensor field \(\tau_{ij}\) statistically isotropic if it is statistically homogeneous and the probability distribution of the components of the tensor \(\tau_{ij}\) in a coordinate system rigidly connected with the system of points \(M_k\) \((k=1,\ldots,n)\) is invariant with respect to rigid rotations and mirror reflections of the configuration determined by this system of points. Following Robertson’s method \((^4)\) and taking (5), (12) into account, we represent the correlation tensor (11) of the statistically isotropic field \(\tau_{ij}\) in the form
\[ \begin{aligned} p_{ijkl}(\xi_s)={}&a_1\delta_{ij}\delta_{kl} +a_2(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\\ &+a_3(\xi_j\xi_k\delta_{il}+\xi_i\xi_l\delta_{jk} +\xi_i\xi_k\delta_{jl}+\xi_j\xi_l\delta_{ik})\\ &+a_4(\xi_i\xi_j\delta_{kl}+\xi_k\xi_l\delta_{ij}) +a_5\xi_i\xi_j\xi_k\xi_l, \end{aligned} \tag{14} \]
where \(a_i=a_i(\rho)\) \((i=1,\ldots,5)\), \(\rho^2=\xi_j\xi_j\), and \(\delta_{ik}\) is the unit tensor of second rank.
Choosing a special coordinate system with its origin at the point \(M_1\) and the \(x_1\)-axis passing through the point \(M_2\), denoting the components of the tensor (11) in this system by \(p^*_{ijkl}\), and setting
\[ p^*_{1111}=p_1,\qquad p^*_{2222}=p_2,\qquad p^*_{1122}=p_3, \]
\[ p^*_{2233}=p_4,\qquad p^*_{1212}=p_5,\qquad p^*_{2323}=p_6, \]
we find
\[ \begin{gathered} p_1=(a_1+2a_2)+2\rho^2(2a_3+a_4)+\rho^4a_5,\\ p_2=a_1+2a_2,\qquad p_3=a_1+a_4\rho^2,\qquad p_4=a_1,\\ p_5=a_2+a_3\rho^2,\qquad p_6=a_2, \end{gathered} \tag{15} \]
with the relation
\[ p_4+2p_6-p_2=0. \tag{16} \]
Solving (15) with respect to \(a_i\), substituting them into (14), and introducing the unit vector
\[ l_i=\frac{\xi_i}{\rho}, \]
we obtain an analogue of the Kármán formula \((^1)\), known in the theory of turbulence,
\[ \begin{aligned} p_{ijkl}={}&p_4\delta_{ij}\delta_{kl} +p_6(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\\ &+(p_5-p_6)(l_jl_k\delta_{il}+l_il_l\delta_{jk} +l_il_k\delta_{jl}+l_jl_l\delta_{ik})\\ &+(p_3-p_4)(l_il_j\delta_{kl}+l_kl_l\delta_{ij}) +(p_1+p_2-2p_3-4p_5)l_il_jl_kl_l. \end{aligned} \tag{17} \]
Putting \(\rho=0\), from (17) and (15) we have
\[ p^0_{ijkl}=p_{ijkl}\big|_{\rho=0} =p_4(0)\delta_{ij}\delta_{kl} +p_6(0)(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}), \]
\[ p_1(0)=p_2(0),\qquad p_3(0)=p_4(0),\qquad p_5(0)=p_6(0), \]
\[ p_4(0)+2p_6(0)-p_2(0)=0. \]
Substituting (17) into the first equation (13), we find
\[ A_1(\rho)\xi_i\delta_{kl} +A_2(\rho)(\xi_l\delta_{ik}+\xi_k\delta_{il}) +A_3(\rho)\xi_i\xi_k\xi_l=0, \tag{18} \]
where
\[ A_1(\rho)=\frac{1}{\rho^2}\,[\rho p'_3+2(p_5-p_6)], \]
\[ A_2(\rho)=\frac{1}{\rho^2}\,[\rho p'_5+(p_5-p_6+p_3-p_4)], \]
\[ A_3(\rho)=\frac{1}{\rho^4}\,[\rho(p'_1-p'_3-2p'_5)+2(p_2-p_3-2p_5)], \]
\[ p'_i=\frac{dp_i}{d\rho}. \]
Since (18) holds for arbitrary \(\xi_j\) (connected only by the relation \(\rho^2=\xi_j\xi_j\)), we have
\[ A_i(\rho)=0,\qquad i=1,2,3, \]
which, together with (16), gives
\[ \rho p'_3+2(p_5-p_6)=0, \]
\[ \rho p'_5+(p_5-p_6+p_3-p_4)=0, \]
\[ \rho(p'_1-p'_3-2p'_5)+2(p_2-p_3-2p_5)=0, \tag{19} \]
\[ p_4+2p_6-p_2=0. \]
Thus, the 6 functions \(p_i(\rho)\), \(i=1,\ldots,6\), are connected by the 4 equations (19) and, consequently, among them there are only 2 independent ones.
Let us note in conclusion that all the results indicated above, in whose derivation the equilibrium equation (6) is not used, hold for the random field of any symmetric tensor of second rank; moreover, they can in an obvious way be generalized to the random field of a tensor of arbitrary rank.
Moscow State University
named after M. V. Lomonosov
Received
20 XII 1963
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