PROCEEDINGS

of the Tomsk Technological Institute

of Emperor Nicholas II.

Vol. 14. 1909. No. 2

I.

S. Yu. Doborzhinsky.

GENERAL EXPRESSIONS

FOR DEFORMATIONS IN A SOLID HOMOGENEOUS BODY,

caused by the action of external forces.

1–8.

GENERAL EXPRESSIONS

for deformations in a solid homogeneous body caused by the action of external forces.

If by \(\sigma_x, \sigma_y, \sigma_z\) we denote the compressive and tensile stresses in the directions of arbitrarily chosen rectangular coordinate axes \(X, Y, Z\), and by \(\tau_x, \tau_y, \tau_z\) the shearing forces tending to rotate an element of the body about an axis whose direction is parallel to one of the above-mentioned \(X, Y, Z\), then the dependence between these quantities at a given point of the body is given by the following equations:

\[ \begin{aligned} \frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_y}{\partial z} + \frac{\partial \tau_z}{\partial y} + X &= 0,\\ \frac{\partial \sigma_y}{\partial y} + \frac{\partial \tau_z}{\partial x} + \frac{\partial \tau_x}{\partial z} + Y &= 0,\\ \frac{\partial \sigma_z}{\partial z} + \frac{\partial \tau_x}{\partial y} + \frac{\partial \tau_y}{\partial x} + Z &= 0; \end{aligned} \tag{1} \]

\(X, Y, Z\) denote in these equations the sums of the components, in the directions of the coordinates, of the external forces acting directly on the given element: these are the so-called body forces. The partial stresses are connected with the changes in the coordinates of a point (deformations), resulting from the action of external forces, by means of the following equalities, in which \(\xi\) denotes the change of the abscissa \(x\), \(\eta\) that of the abscissa \(y\), and \(\zeta\) that of the ordinate \(z\):

\[ \begin{aligned} \sigma_x &= \frac{2G}{m-2} \left[ \frac{\partial \xi}{\partial x}(m-1) + \frac{\partial \eta}{\partial y} + \frac{\partial \zeta}{\partial z} \right],\\ \sigma_y &= \frac{2G}{m-2} \left[ \frac{\partial \xi}{\partial x} + \frac{\partial \eta}{\partial y}(m-1) + \frac{\partial \zeta}{\partial z} \right],\\ \sigma_z &= \frac{2G}{m-2} \left[ \frac{\partial \xi}{\partial x} + \frac{\partial \eta}{\partial y} + \frac{\partial \zeta}{\partial z}(m-1) \right]. \end{aligned} \tag{2} \]

and also

\[ \begin{gathered} \tau_x = G\left(\frac{\partial \eta}{\partial z}+\frac{\partial \zeta}{\partial y}\right),\\ \tau_y = G\left(\frac{\partial \zeta}{\partial x}+\frac{\partial \xi}{\partial z}\right),\\ \tau_z = G\left(\frac{\partial \xi}{\partial y}+\frac{\partial \eta}{\partial x}\right). \end{gathered} \tag{3} \]

The constant \(G\), the coefficient of elasticity of mutual displacement of particles, as well as the constant \(m\), depend on the material of the body under consideration, but do not depend on other conditions.

These equations, in the general form in which they are given above, of course cannot be integrated, and therefore all conclusions begin with the simplest special cases. In fact, there exists only a general theory of beams, derived on the basis of equations (1), (2), and (3). In general, if we wish to make use of these equations as applied to some particular case, we have to introduce into them new additional conditions and integrate new, simpler equations. Connected with the indicated shortcomings is also the fact that many technical problems relating to the strength of materials have not yet been solved theoretically.

From the practical point of view there is no reason to regard the forces \(X, Y, Z\) as variable, as functions of \(x, y, z\); in all technical problems one neglects the forces acting directly on the particles of the body, or transfers their points of application to the surface, as, for example, the forces of gravity. However, for generality we shall regard \(X, Y, Z\) as constants and denote them by:

\[ X = GM,\quad Y = GN,\quad Z = GR. \]

If in equations (1) we substitute expressions (2) and (3), and also use the notation just indicated, then after reduction and simplification we obtain the following three equations:

\[ \begin{gathered} \frac{m-1}{m-2}\frac{\partial^2 \xi}{\partial x^2} +\frac{\partial^2 \xi}{\partial y^2} +\frac{\partial^2 \xi}{\partial z^2} +\frac{m-1}{m-2}\frac{\partial^2 \eta}{\partial x\partial y} +\frac{m-1}{m-2}\frac{\partial^2 \zeta}{\partial x\partial z} +M=0,\\ \frac{m-1}{m-2}\frac{\partial^2 \eta}{\partial y^2} +\frac{\partial^2 \eta}{\partial z^2} +\frac{\partial^2 \eta}{\partial x^2} +\frac{m-1}{m-2}\frac{\partial^2 \zeta}{\partial y\partial z} +\frac{m-1}{m-2}\frac{\partial^2 \xi}{\partial x\partial y} +N=0,\\ \frac{m-1}{m-2}\frac{\partial^2 \zeta}{\partial z^2} +\frac{\partial^2 \zeta}{\partial x^2} +\frac{\partial^2 \zeta}{\partial y^2} +\frac{m-1}{m-2}\frac{\partial^2 \xi}{\partial x\partial z} +\frac{m-1}{m-2}\frac{\partial^2 \eta}{\partial y\partial z} +R=0. \end{gathered} \tag{4} \]

Let us write these equations as follows:

\[ \frac{m-1}{m-2}\frac{\partial}{\partial x} \left( \frac{\partial \xi}{\partial x} + \frac{\partial \eta}{\partial y} + \frac{\partial \zeta}{\partial z} \right) + \frac{\partial^2 \xi}{\partial y^2} + \frac{\partial^2 \xi}{\partial z^2} +\mathrm{M}=0, \]

\[ \frac{m-1}{m-2}\frac{\partial}{\partial y} \left( \frac{\partial \xi}{\partial x} + \frac{\partial \eta}{\partial y} + \frac{\partial \zeta}{\partial z} \right) + \frac{\partial^2 \eta}{\partial z^2} + \frac{\partial^2 \eta}{\partial x^2} +\mathrm{N}=0, \]

\[ \frac{m-1}{m-2}\frac{\partial}{\partial z} \left( \frac{\partial \xi}{\partial x} + \frac{\partial \eta}{\partial y} + \frac{\partial \zeta}{\partial z} \right) + \frac{\partial^2 \zeta}{\partial x^2} + \frac{\partial^2 \zeta}{\partial y^2} +\mathrm{R}=0. \]

To facilitate the subsequent operations, let us denote

\[ \frac{\partial \xi}{\partial x} + \frac{\partial \eta}{\partial y} + \frac{\partial \zeta}{\partial z} = \vartheta,\qquad \frac{m-1}{m-2}=\mu; \tag{5} \]

as a result of this our equations will take the following form:

\[ \begin{aligned} \mu\frac{\partial \vartheta}{\partial x} + \frac{\partial^2 \xi}{\partial y^2} + \frac{\partial^2 \xi}{\partial z^2} +\mathrm{M}&=0,\\ \mu\frac{\partial \vartheta}{\partial y} + \frac{\partial^2 \eta}{\partial z^2} + \frac{\partial^2 \eta}{\partial x^2} +\mathrm{N}&=0,\\ \mu\frac{\partial \vartheta}{\partial z} + \frac{\partial^2 \zeta}{\partial x^2} + \frac{\partial^2 \zeta}{\partial y^2} +\mathrm{R}&=0. \end{aligned} \tag{6} \]

We differentiate the first of these equations with respect to \(x\) and, replacing in it \(\dfrac{\partial \xi}{\partial x}\) by its equal expression from (5), obtain

\[ \mu\frac{\partial^2 \vartheta}{\partial x^2} + \frac{\partial^2 \vartheta}{\partial y^2} + \frac{\partial^2 \vartheta}{\partial z^2} - \frac{\partial^3 \eta}{\partial y^3} - \frac{\partial^3 \eta}{\partial y\,\partial z^2} - \frac{\partial^3 \zeta}{\partial y^2\,\partial z} - \frac{\partial^3 \zeta}{\partial z^3} =0. \]

We differentiate this equation twice with respect to \(x\), and also twice with respect to \(z\),

\[ \mu\frac{\partial^4 \vartheta}{\partial x^4} + \frac{\partial^4 \vartheta}{\partial x^2\partial y^2} + \frac{\partial^4 \vartheta}{\partial x^2\partial z^2} - \frac{\partial^5 \eta}{\partial x^2\partial y^3} - \frac{\partial^5 \eta}{\partial x^2\partial y\,\partial z^2} - \frac{\partial^5 \zeta}{\partial x^2\partial y^2\partial z} - \frac{\partial^5 \zeta}{\partial x^2\partial z^3} =0, \]

\[ \mu\frac{\partial^4 \vartheta}{\partial x^2\partial z^2} + \frac{\partial^4 \vartheta}{\partial y^2\partial z^2} + \frac{\partial^4 \vartheta}{\partial z^4} - \frac{\partial^5 \eta}{\partial z^2\partial y^3} - \frac{\partial^5 \eta}{\partial y\,\partial z^4} - \frac{\partial^5 \zeta}{\partial y^2\partial z^3} - \frac{\partial^5 \zeta}{\partial z^5} =0. \]

We add these two equations and in the result separate out two groups:
\[ \frac{\partial^3}{\partial y^3} \left( \frac{\partial^2 \eta}{\partial x^2} + \frac{\partial^2 \eta}{\partial z} \right) \quad\text{and}\quad \frac{\partial^3}{\partial y\,\partial z^2} \left( \frac{\partial^2 \eta}{\partial x^2} + \frac{\partial^2 \eta}{\partial z^2} \right), \]
in which

we replace the quantity in brackets by the quantity equal to it, taken from equation 2 (6), that is, by the expression \(-\left(\mu \dfrac{\partial \vartheta}{\partial y} + N\right)\). Differentiating the equation thus obtained once more with respect to \(z\), we shall then have

\[ \begin{gathered} \mu \frac{\partial^4}{\partial x^4}\left(\frac{\partial \vartheta}{\partial z}\right) + \frac{\partial^4}{\partial x^2 \partial y^2}\left(\frac{\partial \vartheta}{\partial z}\right) + \mu \frac{\partial^4}{\partial x^2 \partial z^2}\left(\frac{\partial \vartheta}{\partial z}\right) + \mu \frac{\partial^4}{\partial y^4}\left(\frac{\partial \vartheta}{\partial z}\right) \\ + \frac{\partial^4}{\partial x^2 \partial z^2}\left(\frac{\partial \vartheta}{\partial z}\right) + \frac{\partial^4}{\partial z^4}\left(\frac{\partial \vartheta}{\partial z}\right) + (\mu + 1)\frac{\partial^4}{\partial y^2 \partial z^2}\left(\frac{\partial \vartheta}{\partial z}\right) \\ - \frac{\partial^6 \zeta}{\partial x^2 \partial y^2 \partial z^2} - \frac{\partial^6 \zeta}{\partial y^2 \partial z^4} - \frac{\partial^6 \zeta}{\partial x^2 \partial z^4} - \frac{\partial^6 \zeta}{\partial z^6} = 0; \end{gathered} \]

and finally, if by means of the third equation of the group (6) we replace \(\dfrac{\partial \vartheta}{\partial z}\) by the corresponding expression, then ultimately we obtain the following equation

\[ \begin{gathered} \frac{\partial^6 \zeta}{\partial x^6} + \left(1+\frac{1}{\mu}\right)\frac{\partial^6 \zeta}{\partial x^4 \partial y^2} + \left(3+\frac{2}{\mu}\right)\frac{\partial^6 \zeta}{\partial x^2 \partial y^2 \partial z^2} + \left(1+\frac{1}{\mu}\right)\frac{\partial^6 \zeta}{\partial x^2 \partial y^4} \\ \tag{7} + \left(1+\frac{1}{\mu}\right)\frac{\partial^6 \zeta}{\partial x^4 \partial z^2} + \left(1+\frac{1}{\mu}\right)\frac{\partial^6 \zeta}{\partial y^4 \partial z^2} + \left(1+\frac{1}{\mu}\right)\frac{\partial^6 \zeta}{\partial x^2 \partial z^4} \\ + \left(1+\frac{1}{\mu}\right)\frac{\partial^6 \zeta}{\partial y^2 \partial z^4} + \frac{\partial^6 \zeta}{\partial y^6} + \frac{\partial^6 \zeta}{\partial z^6} = 0. \end{gathered} \]

This differential equation of the sixth order with partial derivatives is a symmetric equation of a function of three arguments \(x, y, z\). This equation is satisfied by every algebraic polynomial up to and including the fifth degree, without conditional equalities. In general one may write

\[ \tag{8} W=\sum_0^n (A_3 x + B_3 y + C_3 z)^n, \]

where \(W\) is a particular integral of equation (7), if we assume that the exponent \(n\) indicates to us only the degree of the polynomial, while the summation sign indicates that it includes all values of the exponents from zero to \(n\). It is self-evident that, if \(n>5\), then it is necessary to observe the conditions of equality obtained from substituting \(W\) into (7). Besides this, it should be noted that, if \(n\) is any unrestricted positive integer,

...number, then \(W\) is a symbolic expression for the expansion of any function whatever into a series by the method of undetermined coefficients. If \(W\) is a continuous function—and such must be the function expressing \(\zeta(x,y,z)\)—then the above-mentioned series must either have a finite number of terms, i.e. be an algebraic polynomial, or, for infinite \(n\), be a convergent series.

Let, moreover,

\[ \zeta=\psi(a_3x+b_3y+c_3z)=\psi(w_3), \tag{8} \]

where \(\psi\) denotes an arbitrary function of a trinomial of the first degree. If we substitute for \(\zeta\) this assumed value of it, then equation (7) will become the following:

\[ a_3^6\frac{\partial^6\psi}{\partial w_3^6} +\left(1+\frac{1}{\mu}\right)a_3^4b_3^2\frac{\partial^6\psi}{\partial w_3^6} +\left(3+\frac{2}{\mu}\right)a_3^2b_3^2c_3^2\frac{\partial^6\psi}{\partial w_3^6} +\left(1+\frac{1}{\mu}\right)a_3^2b_3^2\frac{\partial^6\psi}{\partial w_3^6} \]

\[ +\left(1+\frac{1}{\mu}\right)a_3^4c_3^2\frac{\partial^6\psi}{\partial w_3^6} +\left(1+\frac{1}{\mu}\right)b_3^4c_3^2\frac{\partial^6\psi}{\partial w_3^6} +\left(1+\frac{1}{\mu}\right)a_3^2c_3^4\frac{\partial^6\psi}{\partial w_3^6} \]

\[ +\left(1+\frac{1}{\mu}\right)b_3^2c_3^4\frac{\partial^6\psi}{\partial w_3^6} +b_3^6\frac{\partial^6\psi}{\partial w_3^6} +c_3^6\frac{\partial^6\psi}{\partial w_3^6}=0; \]

hence, in order that \(\psi(w_3)\) may be a particular integral of equation (7), it is necessary to assume either

\[ \frac{\partial^6\psi}{\partial w_3^6}=0, \]

or to satisfy the conditional equation

\[ \begin{aligned} a_3^6 &+\left(1+\frac{1}{\mu}\right)a_3^4b_3^2 +\left(3+\frac{2}{\mu}\right)a_3^2b_3^2c_3^2 +\left(1+\frac{1}{\mu}\right)a_3^2b_3^2 +\left(1+\frac{1}{\mu}\right)a_3^4c_3^2 \\ &+\left(1+\frac{1}{\mu}\right)b_3^4c_3^2 +\left(1+\frac{1}{\mu}\right)a_3^2c_3^4 +\left(1+\frac{1}{\mu}\right)b_3^2c_3^4 +b_3^6+c_3^6=0. \end{aligned} \tag{9} \]

But the first condition is nothing more than a particular case of the integral represented by the symbol (8), and therefore (9) remains. Solving it, say with respect to \(c\), we shall find six roots: \(c_3', c_3'', \ldots, c_3^{\mathrm{VI}}\); we shall then have six arbitrary functions of the trinomials

\[ a_3x+b_3y+c_3'z,\quad a_3x+b_3y+c_3''z,\quad \ldots,\quad a_3x+b_3y+c_3^{\mathrm{VI}}z, \]

which satisfy equation (7), and will therefore be particular integrals of this equation.

Reasoning in a wholly analogous manner, we can find the same expressions for \(\eta\) and \(\zeta\), in view of which we shall write:

\[ \begin{aligned} \xi(x,y,z) &= \mathrm{U}+\sum_{1}^{6}\varphi(a_1x+b_1y+c_1z),\\[-2mm] &\hspace{42mm}\underbrace{\hspace{33mm}}_{w_1} \\[2mm] \eta(x,y,z) &= \mathrm{V}+\sum_{1}^{6}\chi(a_2x+b_2y+c_2z),\\[-2mm] &\hspace{42mm}\underbrace{\hspace{33mm}}_{w_2} \\[2mm] \zeta(x,y,z) &= \mathrm{W}+\sum_{1}^{6}\psi(a_3x+b_3y+c_3z); \end{aligned} \tag{10} \]

the summation sign is placed here in order to indicate that there are six different trinomials, whose arbitrary functions are particular integrals of certain equations with \(\xi\) and \(\eta\), analogous to (7).

Since our equations of the 6th order, which served for determining \(\xi,\eta,\zeta\), are obtained by differentiating equations of the 2nd order, it may be supposed that not all the terms satisfying the final equation also satisfy the initial one.

To determine the variables that must be introduced into expressions (10) for \(\xi,\eta,\zeta\), we substitute them into equations (4) and examine the results.

Since each of the particular integrals must satisfy the equations, we must have

\[ (\mu a_1^2+b_1^2+c_1^2)\frac{\partial^2\varphi}{\partial w_1^2} +\mu a_2b_2\frac{\partial^2\chi}{\partial w_2^2} +\mu a_3c_3\frac{\partial^2\psi}{\partial w_3^2}=0, \]

\[ (\mu b_2^2+c_2^2+a_2^2)\frac{\partial^2\chi}{\partial w_2^2} +\mu b_3c_3\frac{\partial^2\psi}{\partial w_3^2} +\mu a_1b_1\frac{\partial^2\varphi}{\partial w_1^2}=0, \tag{11} \]

\[ (\mu c_3^2+a_3^2+b_3^2)\frac{\partial^2\psi}{\partial w_3^2} +\mu a_1c_1\frac{\partial^2\varphi}{\partial w_1^2} +\mu b_2c_2\frac{\partial^2\chi}{\partial w_2^2}=0. \]

The constants \(\mathrm{M}, \mathrm{N}, \mathrm{R}\) are assigned to the equations with \(\mathrm{U}, \mathrm{V}, \mathrm{W}\), in view of the fact that, if the functions \(\varphi,\chi,\psi\) are arbitrary, then they, just like any of their derivatives, may be equal to zero; then it would be necessary to consider

\[ \frac{\partial^2\varphi}{\partial w_1^2}=\mathrm{Const.},\qquad \frac{\partial^2\chi}{\partial w_2^2}=\mathrm{Const.},\qquad \frac{\partial^2\psi}{\partial w_3^2}=\mathrm{Const.} \]

and to divide the arbitrary functions again into two groups, which would only complicate the derivation without making it any more rigorous.

Thus, if \(\varphi, \chi, \psi\) are arbitrary functions, then their second derivatives must also be regarded as arbitrary functions, and then equations (11) can exist independently for any

\[ \frac{\partial^2 \varphi}{\partial w_1^2},\quad \frac{\partial^2 \chi}{\partial w_2^2},\quad \frac{\partial^2 \psi}{\partial w_3^2}, \]

only if

\[ \mu a_1^2 + b_1^2 + c_1^2 = 0,\quad \mu a_2 b_2 = 0,\quad \mu a_3 c_3 = 0, \]

\[ \mu b_2^2 + c_2^2 + a_2^2 = 0,\quad \mu b_3 c_3 = 0,\quad \mu a_1 b_1 = 0, \]

\[ \mu c_3^2 + a_3^2 + b_3^2 = 0,\quad \mu a_1 c_1 = 0,\quad \mu b_2 c_2 = 0. \]

We group these nine conditional equalities as follows:

\[ \mu a_1^2 + b_1^2 + c_1^2 = \mu a_1 b_1 = \mu a_1 c_1 = 0, \]

\[ a_2^2 + \mu b_2^2 + c_2^2 = \mu a_2 b_2 = \mu b_2 c_2 = 0, \]

\[ a_3^2 + b_3^2 + \mu c_3^2 = \mu a_3 c_3 = \mu b_3 c_3 = 0. \]

These equalities can be satisfied either by setting all constants equal to zero, but then the functions themselves will cease to be variable as functions of \(x, y, z\), or by taking

\[ b_1^2 + c_1^2 = 0,\quad a_1 = 0;\quad c_2^2 + a_2^2 = 0,\quad b_2 = 0;\quad a_3^2 + b_3^2 = 0,\quad c_3 = 0, \]

whence

\[ b_1 = \pm i c_1,\quad c_2 = \pm i a_2,\quad a_3 = \pm i b_3. \]

Then, instead of six arbitrary functions for \(\xi, \eta, \zeta\), only two each remain:

\[ \varphi_1(c_1 y + i c_1 z),\quad \varphi_2(c_1 y - i c_1 z), \]

\[ \chi_1(a_2 x + i a_2 z),\quad \chi_2(a_2 x - i a_2 z), \tag{12} \]

\[ \psi_1(b_3 x + i b_3 y),\quad \psi_2(b_3 x - i b_3 y). \]

As for the constants in the polynomials \(U, V, W\), to determine them we have the following determinations, obtained from (4) by replacing \(\xi, \eta, \zeta\) respectively by \(U, V, W\):

\[ \frac{m-1}{m-2}\frac{\partial^2 U}{\partial x^2} +\frac{\partial^2 U}{\partial y^2} +\frac{\partial^2 U}{\partial z^2} +\frac{m-1}{m-2}\frac{\partial^2 V}{\partial x\partial y} +\frac{m-1}{m-2}\frac{\partial^2 W}{\partial x\partial z} +M=0, \]

\[ \tag{13} \frac{m-1}{m-2}\frac{\partial^2 V}{\partial y^2} +\frac{\partial^2 V}{\partial z^2} +\frac{\partial^2 V}{\partial x^2} +\frac{m-1}{m-2}\frac{\partial^2 U}{\partial x\partial y} +\frac{m-1}{m-2}\frac{\partial^2 W}{\partial y\partial z} +N=0, \]

\[ \frac{m-1}{m-2}\frac{\partial^2 W}{\partial z^2} +\frac{\partial^2 W}{\partial x^2} +\frac{\partial^2 W}{\partial y^2} +\frac{m-1}{m-2}\frac{\partial^2 U}{\partial x\partial z} +\frac{m-1}{m-2}\frac{\partial^2 V}{\partial y\partial z} +R=0. \]

Each of these equations reduces to a complete algebraic polynomial of degree \((n-2)\), and therefore—in determining the constants by the method of undetermined coefficients—each of them decomposes into \((n-2)\) homogeneous equations with respect to \(x, y, z\), which again decompose, according to the degrees of these equations, multiplied by three. In view of the fact that, without additional conditions, it is possible to determine neither the degree \(n\) of these equations nor all the constants, we shall not carry out these operations, because of their complexity and uselessness.

Since the constant coefficient in the binomials (12) may be regarded as included in the sign of the function, expressions (10) may finally be represented as follows:

\[ \xi(x,y,z)=\sum_{0}^{n}(A_1x+B_1y+C_1z)^n+\varphi_1(y+iz)+\varphi_2(y-iz), \]

\[ \tag{14} \eta(x,y,z)=\sum_{0}^{n}(A_2x+B_2y+C_2z)^n+\chi_1(x+iz)+\chi_2(x-iz), \]

\[ \zeta(x,y,z)=\sum_{0}^{n}(A_3x+B_3y+C_3z)^n+\psi_1(x+iy)+\psi_2(x-iz). \]

These equations, together with the conditional equations (13), give us the possibility of freely passing to particular cases, for example—to the theory of the stability of beams, and so forth, each of which, without this, requires derivations no less complicated than those given above.* I consider it necessary to note that the notations of the sums are symbolic and express a complete algebraic polynomial of the \(n\)-th degree.

* The initial formulas and notations were taken by me from F. Grashof’s work, Theorie der Elasticität und Festigkeit.