View source for Contact problems of the theory of elasticity

Problems on the distribution of deformation and stress in a system of solid bodies having parts of their boundaries in common (surfaces of contact). In a general formulation, results on contact problems are restricted to existence theorems and certain approximate methods of solution. More complete results relate to the case when one of the contacting bodies is an elastic half-plane (or half-space), while the other is an absolutely-rigid body impressed into the half-plane (or half-space) by given forces (the problem of stamps). Outside the base of the stamp coming into contact with the elastic body, the boundary conditions on the latter can be given arbitrarily from a number of admissible ones, while on the part underneath the stamp, the boundary conditions are stated in accordance with the nature of contact. Thus, if the elastic body is firmly coupled to the rigid body that is to be pressed, then the displacement under the stamp can be considered prescribed; if on the other hand, the elastic body is allowed to slide along the contact surface of the rigid stamp, then under the stamp the normal component of the displacement is known as well as a certain linear relation between the normal and tangential stresses, depending on the coefficient of friction (Coulomb's law). Other boundary conditions can also be realized. All cases of an elastic half-plane (half-space) reduce to a mixed problem with various boundary conditions on various parts of the boundary. The subject matter of papers devoted to stamp problems consists of the development of methods for solving these problems, including the case when both bodies in contact are elastic. These methods are close to one another and in the planar case, in the final event, reduce to a method of conjugation of piecewise-holomorphic functions (the method of the Riemann–Hilbert problem), by means of which the contact problems are solved in quadratures. The problem of contact of two elastic bodies in the three-dimensional case was first posed and solved by H. Hertz, who considered the area of contact to be very small and the equations of the undeformed surfaces near the place of contact as equations of second-order surfaces. Here it proves possible to use one of the electrostatic analogues, and the function expressing the impression in the region of contact is found in the form of the electrostatic potential of a certain ellipsoid. In the planar case, Hertz' problem reduces to the first-order Fredholm equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c0254401.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c0254402.png" /> is the required stress of one body on another at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c0254403.png" /> of the region of contact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c0254404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c0254405.png" /> is a given function; this problem reduces to a singular integral equation that is solvable in closed form.

In the general formulation the contact problem is stated in the following way.

Problem I) Suppose that in an infinite isotropic elastic body with Lamé constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c0254406.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c0254407.png" /> elastic isotropic isolated cut-ins with constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c0254408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c0254409.png" />, bounded by smooth surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544010.png" /> of arbitrary configuration. Considering the cut-ins to be rigidly fastened to the base medium along the contact surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544011.png" />, it is required to determine the stress condition of the body under the influence of given volumetric pressures.

Problem II) In a finite isotropic elastic body with arbitrary smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544012.png" /> and Lamé constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544013.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544014.png" /> elastic isotropic isolated cut-ins bounded by surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544016.png" />, rigidly fastened to the supporting medium along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544017.png" />. It is required to find the elastic state of the body as a result of the action of given volumetric pressures and given boundary conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544018.png" />.

These same problems can be posed for anisotropic bodies as well as under other assumptions regarding the nature of the contacts along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544020.png" />. Existence theorems have been proved for these problems, in the isotropic case by the method of singular potentials and singular integral equations, and for anisotropic bodies by methods of functional analysis.

In the isotropic case methods of approximate solution in quadratures have also been found. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544021.png" /> be points from the three-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544022.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544023.png" /> be the region bounded by the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544026.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544027.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544028.png" /> matrix of fundamental solutions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544030.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544031.png" /> be the same matrix for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544032.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544033.png" /> be the displacement vector at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544034.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544035.png" /> be the stress operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544036.png" /> the stress vector corresponding to the displacement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544037.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544038.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544040.png" />, be the stress vector corresponding to the displacement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544041.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544042.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544043.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544044.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544045.png" /> matrix with columns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544047.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544048.png" /> be the adjoint matrix. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544049.png" /> matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544051.png" /> are defined as follows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544052.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544053.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544054.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544055.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544056.png" /></td> </tr></table>

Problem I, without loss of generality, is one of determining the displacement from the conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544057.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544058.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544059.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544060.png" /></td> </tr></table>

Let the values of the limits on both sides of the contact boundaries for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544062.png" /> be denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544065.png" />; then for the regular solution:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544066.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1a)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544067.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544068.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1b)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544069.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544070.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544071.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544072.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544073.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544074.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544075.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544077.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544078.png" />.

Formulas

can be written in the form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544079.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544080.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544081.png" /> is a six-dimensional vector. The first of these equations holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544082.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544083.png" />, while the second holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544084.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544085.png" />. The corresponding arbitrarily given values of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544086.png" /> lead to the infinite set of equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544087.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2a)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544088.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544089.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2b)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544090.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544091.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544092.png" /> be the set of six-dimensional vectors

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544093.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544094.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544095.png" /></td> </tr></table>

correspondingly indexed, for example, by the diagonal rule. This set is linearly independent and complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544096.png" />. On the left-hand sides of

are the scalar products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544097.png" /> for any value of the index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544098.png" />, which serve as the components of the given vectors and therefore these products are also given. Because of the completeness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c02544099.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440100.png" />, the unknown vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440101.png" /> can be approximated by a linear combination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440102.png" /> if the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440103.png" /> are found from the condition that the norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440104.png" /></td> </tr></table>

is minimal.

This leads to the system of linear algebraic equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440105.png" /></td> </tr></table>

which is solvable. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440106.png" />-th approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440107.png" /> for the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440108.png" /> is expressed by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440109.png" />.

Substituting the first three components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440110.png" />, as a vector, in place of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440111.png" />, and the second three components in place of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440112.png" /> in the integrands in , one obtains an approximate solution of Problem I in quadratures. The exact solution is the uniform limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440113.png" /> at any interior point of the region.

The formulas for the solution of Problem II are the same apart from one modification: Instead of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440114.png" /> the Green tensor corresponding to the boundary conditions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440115.png" /> is used for the total region bounded by the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025440/c025440116.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.I. Muskhelishvili,   "Some basic problems of the mathematical theory of elasticity" , Noordhoff  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Galin,   "Contact problems in the theory of elasticity" , Moscow  (1953)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.Ya. Shtaerman,   "Contact problems of the theory of elasticity" , Moscow-Leningrad  (1949)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.D. Kupradze,   T.V. Burchuladze,   T.G. Gegelia,   M.O. Basheleishvili,   "Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity" , North-Holland  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G. Fichera,   "Existence theorems in elasticity" , ''Handbuch der Physik'' , '''VIa/2''' , Springer  (1973)  pp. 347–389</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.M.L. Gladwell,   "Contact problems in the classical theory of elasticity" , Sijthoff &amp; Noordhoff  (1980)</TD></TR></table>