Elasticity theory, planar problem of

The name of a type of problem in which the picture of the phenomenon in an elastic medium is the same in all planes parallel to a certain plane (for example, the $ Ox _ {1} x _ {2} $- plane in a Cartesian coordinate system $ Ox _ {1} x _ {2} x _ {3} $). The mathematical theory of this planar problem also often describes problems of a spatial character (for example, the bending in thin plates).

The planar problem in the theory of elasticity has advanced mainly from the use of formulas expressing the solutions in terms of analytic functions of one complex variable; these formulas were first derived in 1909 by G.K. Kolosov [1], while from the 1920-s onwards, they received a foundation in papers by N.I. Muskhelishvili, and they were used in developing methods for solving numerous boundary value and contact planar problems in the theory of elasticity. The theoretical results obtained in the planar problem have found practical applications.

Complex representations of displacement and stress fields.

One says that an elastic medium is in a state of planar deformation if there exists a Cartesian coordinate system $ Ox _ {1} x _ {2} x _ {3} $ such that relative to it the components of the displacement vector take the form

$$ u _ \alpha = u _ \alpha ( x _ {1} , x _ {2} , t),\ \ \alpha = 1, 2,\ \ u _ {3} = 0, $$

where $ t $ is the time. The components of the stress tensor are

$$ X _ {\alpha \beta } = \lambda \theta \delta _ {\alpha \beta } + 2 \mu e _ { \alpha \beta } ,\ \ X _ {\alpha 3 } = 0,\ \ X _ {33} = \lambda \theta , $$

where $ \lambda $ and $ \mu $ are the Lamé constants, $ \delta _ {\alpha \beta } $ are the Kronecker symbols and $ e _ {\alpha \beta } $ are the components of the deformation tensor: $ e _ {\alpha \beta } = \partial _ \alpha u _ \beta + \partial _ \beta u _ \alpha $; $ \theta = e _ {\alpha \alpha } = \partial _ \alpha u _ \alpha $ being the dilatation ( $ \alpha , \beta = 1, 2 $; the presence of two identical subscripts denotes summation).

Planar deformation is possible in an elastic medium filling a cylinder with generators perpendicular to the $ Ox _ {1} x _ {2} $- plane if the components of the bulk forces take the form $ X _ \alpha = X _ \alpha ( x _ {1} , x _ {2} , t) $, $ X _ {3} = 0 $ and if the lateral forces are independent of the $ x _ {3} $- coordinate and lie in planes perpendicular to the axis of the cylinder. It is necessary to apply normal forces of $ \pm \lambda \theta $ to the ends in order to produce a planar deformation in an elastic cylinder.

With these assumptions, the following are the equations for the dynamics of an elastic body in terms of the components of the displacement vector:

$$ \mu \Delta u _ \alpha + ( \lambda + \mu ) \partial _ \alpha \theta + X _ \alpha = \rho \dot{u} dot _ \alpha ,\ \ \alpha = 1, 2 , $$

where $ \rho $ is the mass distribution, $ p \dot{u} dot _ \alpha $ are the inertial forces and $ \Delta $ is the Laplace operator. If one uses the complex differentiation operations $ 2 \partial _ {\overline{z}\; } = \partial _ {1} + i \partial _ {2} $, $ 2 \partial _ {z} = \partial _ {1} - i \partial _ {2} $ $ ( \partial _ \alpha = \partial / \partial x _ \alpha ) $, then in the absence of inertial forces (the static problem), the system can be written as a single (complex) equation

$$ ( \lambda + 3 \mu ) \partial _ {z \overline{z}\; } ^ {2} u + ( \lambda + \mu ) \partial _ {\overline{z}\; \overline{z}\; } ^ {2} \overline{u}\; + X = 0, $$

where

$$ u = u _ {1} + iu _ {2} ,\ \ X = 2 ^ {-} 1 ( X _ {1} + i X _ {2} ). $$

Let the region $ S $ occupied by the elastic medium be a connected part of the $ Ox _ {1} x _ {2} $- plane bounded by one or more contours $ L _ {0} \dots L _ {m} $ without common points, let $ L = L _ {0} + \dots + L _ {m} $ be the boundary of $ S $ and let the point $ z = 0 $ belong to $ S $.

The solution to the equilibrium equation is expressed by $ u = u _ {0} + TX $, where $ TX $ is some particular solution, which can be put in the form

$$ TX = - \kappa \frac{1}{\mu \pi ( 1+ \kappa ) } \int\limits \int\limits X( \zeta ) \mathop{\rm ln} | \zeta - z | d \zeta _ {1} d \zeta _ {2} + $$

$$ + \frac{1}{2 \mu \pi ( 1+ \kappa ) } \int\limits \int\limits \overline{X}\; ( \zeta - z) \frac{1}{\overline \zeta \; - \overline{z}\; } d \zeta _ {1} d \zeta _ {2} , $$

and $ u _ {0} $ is the general solution to the homogeneous equation $ ( X = 0) $, which is expressed by

$$ u _ {0} = K( \phi , \psi ; \kappa ) = \ \kappa \phi ( z) - z \overline{ {\phi ^ \prime ( z) }}\; - \overline{ {\psi ( z) }}\; , $$

where $ \phi $ and $ \psi $ are arbitrary analytic functions of $ z= x _ {1} + ix _ {2} $ in $ S $( $ \kappa = 3 - 4 \sigma $ where $ \sigma $ is the Poisson constant, $ 0 < \sigma < 0.5 $). If $ X $ is a polynomial in $ x $ and $ y $, it is possible to express $ TX $ in explicit form.

The operator $ K( \phi , \psi ; \kappa ) $ does not alter if the functions $ \phi $ and $ \psi $ are subject to the condition $ \phi ( 0) = 0 $ or $ \psi ( 0) = 0 $. If one of these conditions is fulfilled, any displacement field $ u = u _ {1} + iu _ {2} $ given in $ S $ corresponds to a definite pair of analytic functions $ \phi $ and $ \psi $.

If the constant $ \kappa $ in the previous formulas is replaced by $ \kappa ^ {*} = ( 3- \sigma )/( 1+ \sigma ) $, one obtains a formula for the displacement field of the generalized planar-stressed state.

The complex form

$$ X _ {\alpha \alpha } = 2( \lambda + \mu )( \partial _ {z} u + \partial _ {z bar } u),\ \ X _ {11} - X _ {22} + 2iX _ {12} = 4 \mu \partial _ {\overline{z}\; } u $$

for the components of the stress tensor takes, by virtue of the equality

$$ u = K( \phi , \psi ; \kappa ) + Tx, $$

the form

$$ X _ {\alpha \alpha } = 4 \mathop{\rm Re} \Phi ( z) + T _ {0} X, $$

$$ T _ {11} - T _ {22} + 2iX _ {12} = - 2( z {\Phi ^ { \prime } ( z) } bar + \overline{ {\Psi ( z) }}\; ) + T _ {1} X, $$

where

$$ T _ {0} = 4( \lambda + \mu ) \mathop{\rm Re} \partial _ {z} TX,\ \ T _ {1} X = 4 \mu \partial _ {\overline{z}\; } TX, $$

$$ \Phi ( z) = \phi ^ \prime ( z) ,\ \Psi ( z) = \psi ^ \prime ( z) . $$

Let the elastic medium be subject to a continuous deformation. Then one may assume that the components of the stress and displacement tensor are continuous single-valued functions in $ S $; $ \Phi $ and $ \Psi $ are holomorphic in $ S $, where $ \Phi $ is subject to the condition $ \Phi ^ { \prime } ( 0) = {\Phi ^ { \prime } ( 0) } bar $.

If $ S $ is a bounded simply-connected domain, while the deformation is continuous, then the functions $ \phi $ and $ \psi $ are holomorphic in $ S $. In the case of a bounded multiply-connected domain, $ \phi $ and $ \psi $ will, in general, be multi-valued functions of a particular form.

Basic problems in the planar theory of elasticity.

1) The first basic problem: To determine the elastic equilibrium of a body when external forces are given at its boundary.

A stress force $ ( X _ {n} , Y _ {n} ) $ acting on an arc element $ ds $ of a contour $ L $ with normal $ n $ may be written in complex form as:

$$ ( X _ {n} + iY _ {n} ) ds = \ - 2i \mu d( \phi ( z) + \overline{ {z \phi ^ \prime ( z) }}\; + \overline{ {\psi ( z) }}\; ), $$

and the boundary conditions for the first problem take the form

$$ \tag{1 } \phi ( t) + t \overline{ {\phi ^ \prime ( t) }}\; + \overline{ {\psi ( t) }}\; = f( t) + c( t),\ \ t \in L, $$

where

$$ f( t) = \frac{i}{2 \mu } \int\limits _ { L } ( X _ {n} + iY _ {n} ) ds. $$

Moreover, the arc $ s $ is reckoned on each $ L _ {k} $ from some fixed point $ z _ {k} \in L _ {k} $ in the positive direction; $ c( t) = c _ {k} = \textrm{ const } $ on $ L _ {k} $. One may always assume that $ c _ {0} = 0 $, while the other constants $ c _ {k} $ are determined in the course of solving the problem. If $ m = 0 $, $ \phi $ and $ \psi $ are holomorphic functions on $ S $. Then the equations $ \phi ( 0) = 0 $ and $ \mathop{\rm Im} \phi ^ \prime ( 0) = 0 $ ensure the uniqueness of the solution to (1), while necessary and sufficient conditions for the existence of a solution,

$$ \int\limits _ { L } ( X _ {n} + iY _ {n} ) ds = 0,\ \ 2 \mu \int\limits _ { L } ( x _ {1} Y _ {n} - x _ {2} X _ {n} ) ds - \mathop{\rm Re} \int\limits _ { L } f \overline{dt}\; = 0 , $$

are conditions for static equilibrium in an absolutely rigid body.

If $ m> 0 $, as already noticed, $ \phi $ and $ \psi $ are multi-valued functions of a special form, and they can be expressed in terms of new unknown functions $ \phi ^ {*} $ and $ \psi ^ {*} $ that are holomorphic in $ S $.

2) The second basic problem: To determine the elastic equilibrium of a body from the given displacements of points on the boundary.

This problem leads to a boundary condition of the form

$$ \kappa \phi ( t) - t \overline{ {\phi ^ \prime ( t) }}\; - \overline{ {\psi ( t) }}\; = f( t),\ \ t \in L, $$

where $ f = u _ {1} + iu _ {2} $ is a function given on $ L $.

3) The mixed basic problem: Let $ S $ be the finite simply-connected domain bounded by a closed contour $ L $; let $ L = L ^ \prime + L ^ {\prime\prime} $, where $ L ^ \prime $ consists of a finite number of arcs $ L _ {1} ^ \prime \dots L _ {m} ^ \prime $ of $ L $ that pairwise do not have common points; the external stresses are given on $ L ^ \prime $ and the displacements on $ L ^ {\prime\prime} $. The corresponding boundary conditions may be written as

$$ \gamma ( t) \phi ( t) + t \overline{ {\phi ^ \prime ( t) }}\; + \overline{ {\psi ( t) }}\; = f( t) + c( t),\ \ t \in L, $$

where $ f $ is a given function of the point $ t \in L $; $ \gamma ( t) = 1 $ if $ t \in L ^ \prime $ and $ \gamma ( t) = - \kappa $ if $ t \in L ^ {\prime\prime} $; $ c( t) = c _ {k} = \textrm{ const } $ if $ t \in L ^ \prime $ and $ c( t) = 0 $ if $ t \in L ^ \prime $. The constants $ c _ {k} $( apart from one, which may be chosen arbitrarily) are not given in advance and are determined as the problem is being solved.

4) The third basic problem: The normal component of the displacement vector as well as the tangential component of the external stress vector is given at the boundary of the region.

This problem arises, for example, in the contact of an elastic body with a rigid profile of given shape when the contact between the elastic and rigid bodies occurs over the entire boundary. Other kinds of contact problems have also been encountered. All these problems also lead to boundary value problems for analytic functions.

5) Boundary value problems in the bending of thin plates. Analogous boundary value conditions arise in the bending of thin plates. The deflection $ w $ of the median surface in a thin homogeneous elastic plate subject to a normal load of intensity $ q $ distributed over its surface satisfies the inhomogeneous biharmonic equation

$$ \Delta \Delta w = \frac{q}{D} , $$

where $ D = Eh ^ {3} /12( 1- \sigma ^ {2} ) $ is the cylindrical rigidity, $ h $ is the thickness of the plate and $ E $ is Young's modulus. The general solution to this equation is

$$ w = w _ {0} + \widetilde{T} q , $$

where $ \widetilde{T} q $ is a particular solution, which can be expressed by

$$ \widetilde{T} q = \frac{1}{8 \pi D } \int\limits \int\limits q( \zeta ) | \zeta - t | ^ {2} \mathop{\rm ln} | \zeta - z | d \zeta _ {1} d \zeta _ {2} , $$

and $ w _ {0} $ is the general solution

$$ w _ {0} = \mathop{\rm Re} ( \overline{z}\; \Phi ( z) + \Psi ( z)) $$

to the homogeneous biharmonic equation

$$ \Delta \Delta w _ {0} = 0, $$

where $ \Phi $ and $ \Psi $ are arbitrary analytic functions in $ S $.

If $ q $ is a polynomial in $ x _ {1} $ and $ x _ {2} $, then $ \widetilde{T} q $ can be expressed in explicit form. If $ \Phi ( 0) = 0 $, $ \Psi ( 0) = 0 $, $ \Phi ^ { \prime } ( 0) = {\Phi ^ { \prime } ( 0) } bar $, then $ \Phi $ and $ \Psi $ are expressed uniquely by means of the given biharmonic function $ w _ {0} $.

The solution $ w $ to the equation $ \Delta \Delta w = q/D $ is subject to boundary conditions corresponding to the particular mode of clamping at the plate boundaries. In the case of a plate clamped at the boundaries, the condition $ w = dw/dn = 0 $ should be satisfied at the boundary $ L $ of the region $ S $ occupied by the median surface, where $ n $ is the exterior normal to $ L $. These two conditions can be put in the form of the single equation $ \partial _ {z} w = 0 $( on $ L $). The latter leads to (1).

The solution to this problem always exists, is unique and is given by

$$ w( x _ {1} , x _ {2} ) = \frac{1}{D} {\int\limits \int\limits } _ { S } G( x _ {1} , x _ {2} , \zeta _ {1} ,\ \zeta _ {2} ) q( \zeta _ {1} , \zeta _ {2} ) d \zeta _ {1} d \zeta _ {2} , $$

where $ G $ is a Green function. For the disc $ | z | < 1 $,

$$ G( x _ {1} , x _ {2} , \zeta _ {1} , \zeta _ {2} ) = 2 \ | \zeta - z | ^ {2} \mathop{\rm ln} \frac{| 1- z \overline \zeta \; | }{| \zeta - z | } + $$

$$ -( 1- | z | ^ {2} )( 1- | \zeta | ^ {2} ), $$

$$ z = x _ {1} + ix _ {2} ,\ \zeta = \zeta _ {1} + i \zeta _ {2} . $$

In the case of a free plate, the boundary conditions take the form

$$ \tag{2 } \sigma \Delta w + ( 1- \sigma )( w _ {x _ {1} x _ {2} } \cos ^ {2} v + w _ {x _ {2} x _ {2} } \sin ^ {2} v + $$

$$ + {} w _ {x _ {1} x _ {2} } \sin 2v) = 0, $$

$$ \frac{d \Delta w }{dn} + ( 1- \sigma ) \frac{d}{ds} \left ( \frac{1}{2} ( w _ {x _ {1} x _ {2} } - w _ {x _ {1} x _ {1} } ) \sin 2v + w _ {x _ {1} x _ {2} } \cos 2v \right ) = 0, $$

where $ v $ is the angle constituted by the exterior normal $ n $ with the $ x _ {1} $- axis. The left-hand sides in (2) are, respectively, the bending moment and the generalized shearing force referred to unit length and acting on a lateral element of the plate with normal $ n $. The boundary conditions (2) may be written as

$$ d \left ( \frac{3+ \sigma }{1- \sigma } \Phi - z {\Phi ^ { \prime } } bar - {\Psi ^ { \prime } } bar \right ) = g, $$

where $ g $ is a function given on $ L $.

6) Planar stationary elastic oscillations. When the solutions to the equations for the dynamics of an elastic medium are sought in the form

$$ u = v( x _ {1} , x _ {2} ) e ^ {i \nu t } ,\ \ u _ {3} = 0, $$

where $ \nu $ is the frequency of oscillation, one gets for $ v $ the formula

$$ v = \partial _ {\overline{z}\; } ( w _ {1} + iw _ {2} ) $$

(it is assumed that the external forces are zero, $ X _ \alpha = 0 $). Here $ w _ {1} $ and $ w _ {2} $ are any solutions to the equations:

$$ \Delta w _ {1} + a ^ {2} \nu ^ {2} w _ {1} = 0,\ \ \Delta w _ {2} + b ^ {2} \nu ^ {2} w _ {2} = 0, $$

$$ a ^ {2} = \rho \frac{1}{\lambda + 2 \mu } ,\ b ^ {2} = \frac \rho \mu . $$

The stress field is expressed by

$$ X _ {11} + X _ {22} = -( \lambda + \mu ) \alpha ^ {2} \nu ^ {2a} w _ {1} , $$

$$ X _ {11} - X _ {22} + 2iX _ {12} = 4 \mu \partial _ {\overline{z}\; } ^ {2} ( w _ {1} + iw _ {2} ), $$

$$ X _ {33} = - 2 ^ {-} 1 a ^ {2} \nu ^ {2} w _ {1} . $$

The general solution to

$$ \Delta w + k ^ {2} w = 0,\ \ k ^ {2} = \textrm{ const } , $$

is expressed by

$$ \tag{3 } w = A _ {0} J _ {0} ( k | z | ) + \int\limits _ { 0 } ^ { 1 } \mathop{\rm Re} [ z \Phi ( zt)] J _ {0} ( k | z | \sqrt {1- t } ) dt, $$

where $ A _ {0} $ is an arbitrary real constant, $ \Phi $ is an arbitrary function and $ J _ {0} $ is the Bessel function of the first kind of order zero. From (3) one may derive complex representations for the displacement and stress fields in a planar stationary oscillation of an elastic medium; these can be used to examine boundary value problems and also to construct various complete systems of particular solutions, enabling one to approximate any displacement and stress fields. In particular, these complete systems can be used to construct approximate solutions to boundary value problems.

7) The problem of determining stress concentrations around a hole in an anisotropic or isotropic plate. The basis for approximate methods for solving such problems is also constituted by introducing functions of a complex variable having a special structure as power series, together with various modifications of perturbation theory, as well as the use of theorems on the addition of cylinder and spherical functions with subsequent reduction of the boundary value problems to infinite systems of algebraic equations.

Methods for solving boundary value problems.

The formulas for representing the displacement and stress fields in terms of analytic functions are used to prove existence and uniqueness for the solutions to general boundary value problems, as well as in constructing explicit solutions to certain classes of problems of particular forms.

The method of power series can be used in conjunction with conformal mapping to solve the basic planar problems for domains that can be conformally mapped onto a disc by means of rational functions. The problem is then reduced to a linear algebraic system of equations and to quadratures. This method can be used to solve the basic boundary value problems for any simply-connected domain by the use of an approximate conformal mapping of the domain onto the disc by means of rational functions. When a computer is used, this technique is effective for constructing solutions to basic boundary value problems for planar problems in the theory of elasticity and for plate bending.

The theory of integrals of Cauchy type has been used to reduce these planar problems to well-studied integral equations.

Methods that combine conformal mapping with techniques of integrals of Cauchy type are also useful.

There are also other ways of reducing the boundary value problems in the planar theory of elasticity to integral equations, which enable one to examine existence and uniqueness.

The method of potentials is also used to reduce the boundary value problems in the planar theory of elasticity, without introducing the complex-analytic functions $ \phi $ and $ \psi $.

References

[1]	G.V. Kolosov, "An application of the theory of functions of a complex variable to a planar problem in the mathematical theory of elasticity" , Yur'ev (1909) (In Russian)
[2]	G.V. Kolosov, "The use of complex diagrams and the theory of functions of a complex variable in the theory of elasticity" , Leningrad-Moscow (1935) (In Russian)
[3]	N.I. Muskhelishvili, "Some basic problems of the mathematical theory of elasticity" , Noordhoff (1975) (Translated from Russian)
[4]	N.I. Muskhelishvili, "Singular integral equations" , Wolters-Noordhoff (1972) (Translated from Russian)
[5]	A.N. Vekua, N.I. Muskhelishvili, "Methods of the theory of analytic functions in elasticity theory" , Proc. All-Union congress on theoretical and applied mechanics (1960) , Moscow-Leningrad (1962) (In Russian)
[6]	I.N. Vekua, "New methods for solving elliptic equations" , North-Holland (1967) (Translated from Russian)
[7]	G.N. Savin, "Spannungserhöhung am Rände von Lochern" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[8]	L.A. Galin, "Contact problems in the theory of elasticity" , Moscow (1953) (In Russian)
[9]	I.Ya. Shtaerman, "Contact problems of the theory of elasticity" , Moscow-Leningrad (1949) (In Russian)
[10]	A.I. Kalandiya, "Mathematical methods of two-dimensional elasticity" , MIR (1975) (Translated from Russian)
[11]	I.S. [I.S. Sokolnikov] Sokolnikoff, "Mathematical theory of elasticity" , McGraw-Hill (1946) (Translated from Russian)
[12]	, Three-dimensional problems in the mathematical theory of elasticity , Tbilisi (1968) (In Russian)

References

[a1]	A.H. England, "Complex variable methods in elasticity" , Wiley (Interscience) (1971)
[a2]	J.M. Milne-Thomson, "Plane elastic systems" , Springer (1960)
[a3]	J.M. Milne-Thomson, "Antiplane elastic systems" , Springer (1962)