Circular Cylinders and Rectangular Prisms under Axial
or All Round Compression (unabridged text)
Jürgen Raasch
Abstract
Exact solutions for some boundary problems in the theory of elasticity are presented which are of
practical interest in connection with hydraulic comminution of ores or other brittle solids of
heterogeneous structure.
1. Introduction
Comminution of solid materials is in any case a process connected with high energy consumption.
This is of great weight, especially in the treatment of mass products as cement, coal or ores. Therefore at all times the grinding machinery has continuously been improved and further developed to
reduce the energy costs. However, it turned out that this can be achieved only on a limited scale. For
this reason alternative methods have already been discussed in the past, to avoid the high energy
losses, basically connected with traditional methods of crushing or grinding by pressure or impact [1].
Special mechanical, electrical and thermal processes were proposed that are applicable, however,
only in the case of brittle solids of heterogeneous structure.
The most simple and most easily realizable method of that kind is hydraulic comminution, whereby
the solid material, submerged in a liquid (mostly water), is subjected to high hydraulic pressure [2,3].
Cracks can thus be produced in the material, if the components differ markedly in their elastic properties. The size and location of the crucial stresses, however, are so far unknown. In order to clarify
these questions, model calculations were made. Linear elasticity was assumed throughout, circular
cylinders and rectangular prisms were chosen as model configurations.
2. Pressure on a Circular Cylinder
2.1 General Equations
It is convenient to use cylindrical coordinates r,ϕ,z for the stress analysis in a circular cylinder. They
are connected to the rectangular coordinates x,y,z by the following relationships
(1)
,
(2)
In case of deformations of rotational symmetry with respect to the z-axis, the stress components are
independent of the angle ϕ and all derivatives with respect to ϕ vanish. Then the equations of equilibrium take the form
(3)
1
(4)
In addition the following compatibility conditions must be fulfilled [4] :
(5)
(6)
= 0
(7)
(8)
thereby using the notations
+
(9)
(10)
A stress function φ(r,z) is introduced, from which the stress components are obtained as follows:
(11)
(12)
(13)
(14)
When inserting Eqs.(11) to (14) in Eq.(3), this equation is completely satisfied, whereas from Eq.(4)
(15)
is obtained. This means that the stress function φ(r,z) must fulfill the biharmonic equation.
In the case of small deformations the connection between the components of stress and strain is
given by the following relations:
(16)
(17)
(18)
(19)
Herein E is the modulus of elasticity and ν Poisson´s ratio.
2
2.2 Uniform Axial Compression
It is assumed that a circular cylinder of radius a is simply loaded by pressure
Fig.1) and that no other forces take effect.
Fig.1
Fig.2
in the z-direction (See
Fig.3
A solution function, describing the state of stresses, has firstly to fulfill the biharmonic equation and
secondly the following boundary conditions:
r = a
= 0
,
= 0
(20)
z = 0
(21)
If a solution function of the form
φ ( r, z ) = A z³ + B r²z
(22)
is chosen, it follows from Eq.(9)
² ( r, z ) = 6 A z + 4 B z
and further from Eq.(11) to (14)
= 0
The constants A and B are determined by the boundary conditions as
(23)
The state of stresses in the cylinder is uniform and described by
(24)
This solution is trivial and well-known. Unfortunately it is only of limited technical importance. For
instance, it does not describe the state of stresses in an elastic cylinder compressed between two
3
rigid plates, as it is realized in an ordinary testing machine, because the frictional forces between the
cylinder and the plates of the testing machine are not included in the stress analysis.
2.3 Axial Compression against a Rigid Plate
It is assumed that a circular cylinder of radius a is pressed in the z-direction against a rigid plate (See
Fig.2) in the course of which no slip occurs between cylinder and plate. The solution function, as in
the previous case, must fulfill the biharmonic equation and special boundary conditions, here:.
(25)
z = 0
(26)
The additional condition z = 0 ,
makes the solution much more difficult compared to the
previous case. Muskhelishvili [5] was the first to investigate problems of this kind, where the displacements are prescribed on one part of the boundaries and the stresses on the other part. Here
his method (Fourier series expansions) was not used. Instead, a new and uncommon solution was
developed that solves this special problem. The stress function was chosen in the following form:
(27)
Herein k is the notation for the integration variable and
and
are Bessel
functions of the first kind and of zero and first order. Their arguments are imaginary, their derivations
are given by the relations
,
(28)
Inserting Eq.(27) in Eq.(9) yields
(29)
From this result it can be concluded that Eq.(27) satisfies the biharmonic equation.
The equations for the stress components are obtained from Eq.(27) making use of Eq.(28):
(30)
+ 6 ν A - 2 (1 - 2 ν) B
(31)
(32)
(33)
4
On the axis of the cylinder the shear stresses must vanish. This is granted by
(34)
The free parameters A , B , ρ(k) and f(k) are determined by the boundary conditions Eq.(25) and (26).
The stress components in the contact plane of the cylinder are obtained from Eqs.(30) to (32):
(35)
Inserting Eqs(35) in Eq.(26) gives
=
(36)
(37)
The boundary condition
zero for r = a :
is satisfied by equating the integrand in Eq.(33) with
(38)
The function f(k) is determined by the last boundary condition
known integral
. To that end the
(39)
is made use of. If the function f(k) is defined as
(40)
it follows from Eq.(25)
(41)
Thus a complete and exact solution of the problem described in Fig.(2) was achieved. From Eqs.(30)
to (33) the stress components for arbitrary radii r and arbitrary distances z from the rigid plate can be
calculated, if Poisson´s ratio ν is known. Here a value ν = 0.2 was chosen, which is more or less valid
for many crystalline materials such as basalt, diamond, quartz, glass and concrete. Using computer
programs the stress components were evaluated point by point. In Figs.(4) to (7) these values are
represented as distributions of r/a for different cross sections z/a. As can be seen from these
diagrams the stress components
attain their maximum values in cross section z = 0
and decrease with increasing distance z. The stress component
is uniform and identical to
at
z = 0 . This is not a precondition but part of the solution. This finding might be surprising, but is not an
indication for a defect or for incompleteness of the solution function
in Eq.(27). This function fulfills the biharmonic equation and all conceivable boundary conditions.
5
Fig. 4 Stress component
Fig. 5 Stress component
Fig. 6 Stress component
Fig. 7 Stress component
The stress component is not uniform for smaller distances z/a , but becomes uniform and identical
to
for z → ∞ . The dependence of on
for smaller distances z/a has the consequence that
the cross sections do not remain even. Therefore the above solution is not an exact solution for an
elastic cylinder compressed between two rigid plates, but probably a very good approximation. The
state of stresses in an elastic cylinder, compressed between two rigid plates, was for the first time
explored by Filon [6]. He used a Fourier series expansion for the stress function. Herewith he could
satisfy the boundary conditions at the cylinder jacket as exactly as desired, but not that of no slip in
the contact planes. Later Picket [7] tried to achieve this with a second Fourier series expansion for
the displacements. Because of the inter-dependency of these series the solution becomes very
complicated. The convergence is poor for r/a > 0.8 . The stress distributions depicted in his paper
nevertheless show a certain conformity with the above Figs.(4) to (7).
The applicability of the solution presented in this paper as well as those of Filon [6] and Pickett [7]
depends on the fulfillment of the presumption, that there is no slip between elastic cylinder and
rigid plate. The answer to the question concerning whether or not this can be fulfilled, can be taken
from Fig.(4). The coefficient of adhesive friction µ between cylinder and plate must satisfy the simple
condition
/
, in this case µ > 0.16 .
6
A solution function in the form of an integral as in Eq.(27) was already applied by Barton [8]. He used
it for the stress analysis in a solid shaft on which a collar is shrunk. Here the same solution function,
in combination with a fictitious pressure loading on the cylinder jacket, is used to fulfill the boundary
condition
in the contact plane of the cylinder.
The solution function (27) describes the contact between an elastic body and a rigid plane wall. The
inverse problem, the contact between a rigid body (punch) and an elastic half space was investigated
by Gladwell [9], Fabrikant [10] and many other authors. However, their work does not include the
case treated in this paper.
2.4 Composition of two Circular Cylinders of Different Elastic Materials
The solution function Eq.(27) also solves the main problem of this paper described in the
introduction. That is the state of stress in a model cylinder composed of two parts of different
materials and subjected to all-round pressure (See Fig.3). It is assumed that the two parts are firmly
connected (grown together) in the z = 0 -plane.
The modulus of elasticity and Poisson´s ratio for the upper part of the model cylinder are referred to
by
, those of the lower part by
.The direction of the coordinate z in the
upper part is upwards, that in the lower part downwards.
The boundary conditions at the cylinder jacket are the same for the two parts, i.e.:
(42)
In the cross section z = 0 stress and strain of the two parts must fit into one another:
(43)
(44)
The solution function of the upper part is defined by
and that of the lower part by
The two solution functions are identical to Eq.(27). The same is true for the stress components
derived there from, i.e. for the equations (30) to (33).
For the upper part of the cylinder the boundary conditions (42) are satisfied by defining
7
and for the lower part correspondingly
Herein
are additional constants, determined by the boundary conditions.
From Eqs.(42) and (43) it follows
(45)
(46)
The boundary condition
leads to the relation
(47)
For the shear stresses in the plane z = 0 it follows from Eq.(33):
It is obvious that the boundary condition
cannot be fulfilled for arbitrary radii
r , because the values of the integrals depend on the particular value of Poisson´s ratio. In this case a
complete and exact solution can only be obtained, if
is assumed. With this restriction it
follows for the constants
(48)
Now with Eqs.(45) to (48) all the other constants can be calculated:
(49)
(50)
8
(51)
For the constants
the corresponding relations can be obtained.
3. Pressure on a Rectangular Prism
3.1 General Equations in Rectangular Coordinates
A state of plane stress in the xz-plane is assumed. In this case the stress components
are zero and the equations of equilibrium take the form
(52)
(53)
By introducing a stress function
the equations of equilibrium are satisfied with
(54)
The compatibility conditions require that
(55)
i.e. the stress function must fulfill the biharmonic equation.
If in Eqs.(52) to (55) the coordinate x is replaced by the coordinate y, a state of plane stress in the
yz-plane can be described.
In rectangular coordinates the connection between the components of stress and strain is given by
the following relations:
(56)
(57)
(58)
3.2 Uniform Axial Compression and Axial Compression against a Rigid Plate
It is assumed that a prism of width 2b and depth 2c is simply loaded by pressure in the z-direction
(See Fig.8). The state of stress in the prism can be regarded as a special case of plane stress, either in
the xz- or in the yz-plane. In the first case from the solution function
(59)
9
the following stresses are derived
(60)
In the second case from
(61)
the stress components
(62)
In both cases the constant A has the value
(63)
Fig.8
Fig.9
Fig.10
When it is assumed that the same prism is pressed against a rigid plate (See Fig.9) without slip
between prism and plate, the biharmonic equation (55) and the following boundary conditions must
be fulfilled by the solution function
:
(64)
The solution function is composed, in analogy to Eq.(27), in the form:
(65)
Here from the stress components are obtained as
(66)
(67)
(68)
The constant A is determined by the boundary condition (64). With Eq.(67) it´s value is found to be
10
(69)
From the boundary condition
it follows with Eqs.(66) and (67) :
(70)
The boundary condition
for x = b :
is satisfied by equating the integrand in Eq.(68) with zero
(71)
If the function f(k) is defined as
(72)
for x = b , the integral in Eq.(66) is reduced to the known integral
(39)
From the boundary condition
it follows herewith
(73)
The extension of the solution function (65) compared with Eq.(59) was necessary to be able to fulfill
the boundary condition
. Hereby sliding of the prism on the rigid plate is excluded in
the x-direction, but not so in the y-direction. This gave rise to the idea to superimpose upon the state
of plane stress in the xz-plane an equal one in the yz-plane in order to exclude slip in any direction.
For this second state of plane stress a formally equal solution function was chosen:
(74)
The stress components are:
(75)
(76)
(77)
The boundary conditions
(78)
(79)
ensure that the outer surface of the prism is unstressed. in the first case this is attained with the
definitions (71) and (72),in the second case likewise with
(80)
11
(81)
The boundary condition
leads to
(82)
corresponding to Eq.(73). In the contact plane of the prism, i.e. for z= 0 , the strain components in
the x- and y-direction must vanish :
(83)
(84)
Furthermore the balance of forces in the z-direction requires that
(85)
From Eq. (85) it follows with Eqs. (66) and (75)
(86)
and from Eqs.(83) and (84)
(87)
and further from Eqs.(73) and (82)
(88)
Herewith all of the free constants are determined. Again the result is a complete and exact solution
from which the stress components for arbitrary distances x, y and z can be calculated.
0,3
0,18
0,16
0,25
0,14
0,2
0,1
-s x/pA
t xz /pA
0,12
0,08
0,06
0,04
0,02
0,15
0,1
0,05
0
-0,02
0
0
0,2
0,4
0,6
0,8
1
0
0,2
0,4
x/b
z=0
z=0,1b
0,6
0,8
x/b
z=0,2b
z=0
12
z=0,1b
z=0,2b
1
Fig. 11 Stress component
Fig. 12 Stress component
1,1
1,05
-s z /pA
1
0,95
0,9
0,85
0,8
0,75
0
0,2
0,4
0,6
0,8
1
x/b
z=0
z=0,1b
z=0,2b
Fig. 13 Stress component
Fig.14 Dependence of
on
For all plane sections through a circular cylinder, containing the axis of the cylinder, the same stress
distributions are obtained. This does not hold for a prism. In a plane section through the edges of the
prism the stress distributions are markedly different from those in the xz-plane. The distributions of
the stress components σ_x ,σ_z ,τ_xz depicted in Figs.11 to 13 are valid only for y = 0. They were
calculated for the special case b = c , i.e. for a prism with quadratic cross section. For Poisson´s ratio
again the value ν = 0.2 was chosen. The distributions of the stress components were calculated for
the same relative distances from the rigid plane, to facilitate a direct comparison with the results for
a circular cylinder.
3.3 Composition of two Prisms of Different Elastic Materials
It is assumed that a prism of width 2b and depth 2c consists of two parts of different materials, which
are firmly connected (grown together) in the z = 0 -plane, and that the composed prism is subjected
to all- round pressure (See Fig.10).
As in the case of the circular cylinder the modulus of elasticity and Poisson´s ratio for the upper part
are referred to as
, those of the lower part by
. The direction of the
coordinate z in the upper part is upwards, that in the lower part downwards.
The boundary conditions at the side-faces of the prism are the same for the two parts, i.e. :
(89)
(90)
In the cross section z = 0 stress and strain of the two parts must fit into one another:
(91)
13
(92)
(93)
The state of plane stress in the xz-plane of the upper part is defined by
and that of the lower part by
The state of plane stress in the yz-plane is correspondingly described by
Apart from the indices in the brackets these equations agree with Eq.(65) and Eq.(74). For the stress
components derived herewith the same is true. From the boundary conditions (89) to (93) the
following relations for the constants were obtained:
(94)
(95)
(96)
(97)
(98)
4. Conclusions
All solution functions, presented in sections 2 and 3, are exact analytical solutions of the theory of
elasticity under the respective boundary conditions.
The stress distributions in a circular cylinder pressed against a rigid plate, as depicted in Figs.4 to 7,
show a remarkable consistency with those in Figs.11 to 13 , calculated for a rectangular prism of
quadratic cross section. The analytical solutions for cylinder and prism in a way confirm each other.
Therefore, not the stress functions should be object of criticism, but probably the boundary condi14
tions which cause an unsteady change of the radial stresses for z = 0 , as Pickett [7] already pointed
out. However, this fault is confined to the fringe of the contact area between model and rigid plate,
as can be seen from Figs.4 to 7 , and likewise to that of the contact area between the different
materials in the composed model configurations.
The main subject of this paper is not the axial compression against a rigid wall, but the state of stress
in a model composed of different materials and subjected to all-round pressure. It was assumed that
the different materials are joined together in plane areas. This can be expected especially, when at
least one component consists of crystals. For cylinders and prisms, containing such plane areas
perpendicular to their longitudinal axis, the state of stress in the adjoining parts could be calculated.
The stress distributions are similar to those in Figs.4 to 11 and differ from these only by a constant
factor, dependent on the values of the modulus of elasticity E and Poisson's ratio ν of the two different materials. Obviously nowhere in the model configurations tensile stresses can be brought about
by all-round pressure. Thus for fracture initiation only shear stresses come into question. As can be
concluded from Figs.4 to 11 the highest shear stresses are found at the fringe of the boundary areas
between the different materials.
When an elastic circular cylinder is pressed against a rigid plate, the maximum value of the shear
stresses is approximately 0.16 for any point of the perimeter, as can be seen from Fig.4 . In the
case of a rectangular prism various values are found for different section planes. For the plane y = 0
the maximum value is nearly the same as that for the circular cylinder. In a plane section through the
edges of the prism shear stresses are obtained in both the x- and y-direction. For a prism of quadratic
cross section this leads to a resultant shear stress of approximately 0.22
. The shear stresses are
proportional to the quantity C. For circular cylinder and rectangular prism the same expression for C
was found:
(41, 88)
In the case of models composed of different materials and subjected to all-round pressure, the
quantity C is to be replaced by
:
For a circular cylinder this expression was derived on the rigorous assumption
, for a
rectangular prism without any restrictions. The value of
strongly depends on the relation of the
two moduli of elasticity as can be seen from Fig.14 . In this diagram
was plotted against
. For Poisson´s ratio the value ν
was chosen. For the contact plane between the two
parts of a prism of quadratic cross section, composed of two different materials, the highest shear
stresses can be calculated from
(49, 94)
Example: On the assumption that the relation of the moduli of elasticity is
, from
Fig.14 the value
is taken. Herewith the maximum shear stress is calculated as
(99)
15
This value seems to be small and the question arises whether these stresses are sufficient to initiate
cracks. However, it has to be taken into account that especially in the contact areas innumerablle
defects of the texture, foreign substance inclusions and incipient cracks exist, that can reduce the
strength of the material by orders of magnitude.
From the model calculations some decisive conclusions for hydraulic comminution can be drawn:
1) Cracks are always caused by shear stresses.
2) The highest shear stresses are always found at the fringe of the contact areas between the
different materials.
3) In sharp corners of the perimeter, as in the case of a prism, the shear stresses are higher than at
straight boundaries.
4) There is no dependence on the absolute size of the model configurations and of the contact areas.
5) The technical applicability of hydraulic comminution of brittle materials can be decided using the
characteristic number
that only depends on the elastic properties of the components.
5. References
[1] H. Rumpf, Chem.-Ing.-Techn. 37 (1965), S.187/202
[2] A. Müller, E. Linß, G. Wollenberg, H.P. Scheibe, Steinbruch und Sandgrube 96 (2003), S.24/25
[3] G. Linke, Chem.-Ing.-Techn. 40 (1968), S.117/120
[4] S. Timoshenko and J.N. Goodier, Theory of Elasticity, Second Edition, McGraw-Hill (1951), S.346
[5] N.I. Muskhelishvili, Some Basic Problems of the Mathem. Theory of Elasticity, Noordhoff (1963)
[6] L.N.G. Filon, Trans. Roy. Soc. (London) A 198 (1902)
[7] G. Pickett, J. Appl. Mechn. 11 (1944), S. 176/182
[8] M.V. Barton, J. Appl. Mechn. 8 (1941)
[9] L.A.Galin, G.M.L.Gladwell, Contact Problems, Springer (2008)
[10] V.I.Fabrikant, Mixed Boundary Value Problems, Dordrecht (1991)
16
17
18
19
20