Plane elasticity solution for a half-space weakened by a circular hole and loaded by a concentrated force
-
Alexei V. Proskura
Abstract
We present a complete analytical solution for the plane elasticity problem of a concentrated force acting on a half-space weakened by a circular hole. The solution is based on the algorithm proposed by G.B. Jeffery in 1920 for elasticity problems, using bipolar coordinates to characterize the geometry. We found the biharmonic stress function describing stresses and strains in the half-space and the associated biharmonic function that allows determining the displacement field. Both functions are given in the form of Fourier series with quite compact coefficients. Particular attention is paid to the displacements at the plane boundary of the half-space. It is discussed how these displacements can be used in an inverse procedure for the identification of a circular hole diameter and position.
1 Introduction
We address the task of finding the elastic field that arises under the action of a concentrated load applied to the boundary of the half-space weakened by a circular hole. The algorithm for solving the plane elasticity problems with the geometry corresponding to the bipolar coordinate system, in particular, the modified algorithm for the half-space with a hole, was given by G.B. Jeffery as early as in 1920 [1]. In 1962, R.M. Evan-Iwanowski published the stress functions for this problem [2]. The author applied the algorithm by G.B. Jeffery and considered the normal and tangential forces acting on the boundary of the half-space. However, in practice, the use of the solution by Evan-Iwanowski [2] is not possible because of the evident mistakes in published formulas. Later, this problem has been solved in other ways by applying numerical methods and extended, for example, to the case with an elliptical hole (see e.g., ref. [3]). In any case, published and known solutions do not allow us to determine the displacement field in the loaded half-space with a hole, in particular, the displacements at the surface of the half-space.
To obtain valid analytical stress functions and elastic fields and, in particular, to determine the surface displacements, we readdressed this classic elasticity problem.
The practical motivation for revisiting such a classic problem of linear elasticity is related, for example, to the use of tactile sensors [4]. Such sensors, for example biologically inspired sensors, are able to detect force in a specific direction, in particular when the force is acting on the surface of an elastic body. Additional equipment may allow simultaneous measurements of the displacements at various points at the surface of this body. The displacements of surface points induced by the force depend on the state of near-surface layers of the body and may be sensitive to defects of various natures that are situated in the surface vicinity. Thus, the information on how the defect affects the surface displacement induced by the given force is required to elaborate the approach to the defect identification. This is the subject of the inverse problem for the determination of defect parameters from known surface displacement fields generated by the given force.
However, one cannot state the inverse problem without having in hand the solution of the direct problem. In the following, we focus on the direct problem in the simplest case of a plane elasticity problem for an isotropic half-space with a cylindrical hole. To our best knowledge, there exists no published correct analytical solution for such an elasticity problem.
2 Statement of the problem, geometry, and basic equations
Let us consider an elastic half-space y≥0 with an inner circular cylindrical hole. The linear force f=fTex+fNey is applied at point (x0, 0) (Figure 1). It is required to find the analytical solution of this elasticity boundary value problem, i.e., to determine the biharmonic stress function and the elastic fields in the half-space with a hole, and to determine the displacements at the half-space boundary.
An elastic half-space y≥0 containing a circular hole and loaded by linear force f=fTex+fNey. The model uses bipolar (α, β) and Cartesian (x, y) coordinate systems. The hole and the boundary of a half-space have coordinates α=α1 and α=α2=0, correspondingly. Force is applied at point (0, β0), which corresponds to the Cartesian coordinates(x0, 0).
A convenient coordinate system to obtain solutions to an elasticity problem in such a geometry is the bipolar coordinate system (α, β). The relationship among bipolar (α, β) coordinates, Cartesian coordinates (x, y), and the polar radius r are listed below. We also provide useful formulas relating the radius of the hole p, the coordinate of the center of the hole d, their aspect ratio 
and the shortest distance between boundary of the hole and half-space boundary h with the bipolar coordinate of the hole α1 (α1>0):
Here -∞<α<∞, -π≤β≤π. The discontinuity of coordinate β takes place on segment (-a, +a) of axis 0y [Figure 1 shows part of segment (-a, +a) along the positive direction of the y-axis]. Above the point +a and below the point -a along the y-axis bipolar coordinate, β=0. The axis 0x is the line α=0; the Cartesian points (0, ±a) correspond to α=±∞. The point at infinity has coordinate (α=0, β=0).
In the bipolar coordinate system, stresses (σαα, σββ, ταβ) and strains (εαα, εββ, εαβ) can be found from the biharmonic function Φ, and the displacements (u, v) are obtained by using two interconnected biharmonic functions – main function Φ and associated function Ψ [1]. Then, the stresses are given as follows [1]:
where 
The strains can be determined through Hooke’s law.
The relations between functions Φ and Ψ are defined as follows [1]:
where μ, λ are Lame constants.
The displacements have the following representations in terms of the biharmonic functions Φ and Ψ [1]:
where u, v are displacements in the directions normal to the lines of constant coordinates α and β, respectively [1]; in other words, u≡uα, v≡uβ.
According to Eq. (3b), the associated biharmonic function Ψ is determined up to some functions of α and β. As correctly pointed out by J.B. Jeffery [1], the only possible arbitrary terms in gΨ that do not affect the stresses are given by
or in Cartesian coordinates
The terms in Eqs. (5a) and (5b) (except the second one) correspond to the rotation about the origin and pure translation of a rigid body [1].
Strains can be determined from the displacements in Eq. (4) [1]:
Hooke’s law allows one to define stresses:
A comparison of stresses obtained with Eqs. (2) and (7) helps verify the displacements calculated from Eq. (4).
3 Biharmonic function Φ and stresses
The required biharmonic function Φ is sought as a sum of the known function 
giving the solution for normal (fNey) or tangential (fTex) force applied at the uniform half-space boundary, and an additional function 
due to the presence of the hole:
The biharmonic functions 
have the following standard forms [5]:
where 
and x0 is a coordinate of the applied force.
The stresses corresponding to the function 
satisfy conditions at the straight boundary of the half-space:
It is therefore required to find 
which provides the stresses satisfying the boundary conditions
where a1 is a coordinate of the free surface of the hole (Figure 1).
The general expressions for the biharmonic functions 
which give the stresses that satisfy the conditions at the straight boundary [Eq. (11a,b)], are known [1, 6]. In order not to overload the subscripts N and T, in the formulas below we write the expression for the biharmonic functions 
as follows:
It can be easily demonstrated that the stress function Φ* of Eq. (12) with arbitrary coefficients 
gives the stresses of Eq. (2) satisfying the boundary conditions of Eq. (11a,b) automatically [1].
The algorithm to find the coefficients 
are defined as follows: in the boundary condition equations [Eqs. (11c,d)], the known stresses 
and 
are represented in the form of Fourier series with respect to the variable β, and the unknown
and 
are expressed with the help of Eq. (2) through the biharmonic function Φ* in the form of Eq. (12). Equations (11c,d) are solved for the unknown coefficients of the series of Eq. (12).
In Appendix 1, Fourier series for stress 
and 
are given for the normal and tangential forces applied at the boundary of the half-space. In Appendix 2, the detailed algorithm for determining the coefficients 
is demonstrated for the normal and tangential forces applied at the planar boundary.
Then, stresses σij under consideration are the sum of stresses 
caused by the force in the uniform half-space and the additional stresses 
due to the hole:
The terms in the sums in Eqs. (13a,b,c) are calculated from the known stress functions 
of Eqs. (9a) and (9b) and the additional stress functions 
of Eq. (12) on the basis of the relations in Eqs. (2a)–(2c).
3.1 A concentrated force with a magnitude of fN is normal to the planar boundary
For the normal force acting on the straight boundary of the half-space, the stress function 
caused by the hole is represented by the Fourier series Eq. (12) with the following coefficients:
The stress function 
given by means of Eqs. (12) and (14), has quite a compact form and allows to find the stresses 
and strains 
through the series representation with analytical coefficients. Stresses 
caused by the normal force that acts on the uniform half-space can be represented in the bipolar coordinates α and β of Eq. (1a,b) with functions 
of Eq. (9a) and relations of Eq. (2):
For completeness, we give a formula for the stress components σββ, arising at the boundary of the hole, with the coordinate α=α1:
where
The behavior of the stress component σββ at the boundary of the hole is demonstrated by plots in Figure 2.
Stress component σββ at the boundary of the hole.
(A) Stress component σββ as a function of the polar angle θ associated with the hole. Stresses are given for the boundary of the hole with coordinate α1=1 (1), which corresponds to the aspect ratio 
For comparison, the same component is shown for a uniform half-space on the line α=α1=1 (2). (B) Stress diagram at the boundary of the hole. Concentrated force is applied normally to the straight boundary of the half-space at point β0=±π. The stresses are shown in units of 
We did not study the convergence of the obtained series, representing the stresses 
in full details. However, we can note that for the parameters of the hole α1<1, the accuracy of the boundary conditions begins to fall in orders. Acceptable accuracy remains for the parameter α1≈0.3. For smaller parameters α1, it is necessary to use special techniques to sum the series.
3.2 A concentrated force with a magnitude of fT is tangential to the planar boundary
The stress function 
caused by the hole is represented by the Fourier series of Eq. (12) with the following coefficients:
As a result, the stress function 
given by Eqs. (12) and (17), allows to find the stresses 
and strains 
through the series with analytic coefficients.
The stresses 
caused by the tangential force applied to the surface of the uniform half-space can be represented in the bipolar coordinates α and β of Eqs. (1a,b) with the help of functions 
in Eq. (9b) and the relations in Eqs. (2a)–(2c):
Finally, the total stresses can be found from Eq. (13).
As an example, the stress component σββ behavior at the boundary of the hole is shown in Figure 3. From Figures 2 and 3, one can conclude that in the case of normal load, stresses at the boundary of the hole increase significantly compared with how it would be in a uniform medium. There are areas of tension and compression. In the case of the tangential force, the effect of the hole is not so noticeable.
Stress component σββ at the boundary of the hole.
(A) Stress component σββ as a function of the polar angle θ associated with the hole. Stresses are given for the boundary of the hole with coordinate α1=1 (1), which corresponds to the aspect ratio 
For comparison, the same component is shown for uniform half-space on the line α=α1=1 (2). (B) Stress diagram at the boundary of the hole. The concentrated force is tangential to the straight boundary of the half-space at point β0=±π.The stresses are shown in units of 
4 Biharmonic function Ψ and displacements
The displacements u, v for the considered geometry are determined as a sum of the displacements in a uniform half-space u0, v0 and the additional displacements caused by the hole u*, v*:
where, as before, u, v are the displacements in the directions normal to the lines of constant α and β, respectively [1],
To derive the displacements u*, v* from Eq. (4) in addition to the function Φ*, it is necessary to know the associated biharmonic function Ψ*. This function is constructed on the basis of the biharmonic function Φ* [1]:
where
where the coefficients 
are the same as in the expression for the function Φ* of Eq. (12).
A special term, 
is added in Eq. (20) to eliminate the rotation and pure translation of the body as a whole [see Eq. (5)]:
Here, the coefficients 
are found from the condition of vanishing displacements at infinity α=0, β=0.
4.1 Displacements due to a concentrated normal force
For the case of the loading of a normal force, the biharmonic function 
caused by the presence of a hole is given by Eqs. (20) and (21), taking into account the coefficients of Eq. (14) and the following supplementary coefficients:
Under the normal concentrated load, the displacements of the straight boundary caused by a hole have the following simple form:
where
where
Equation (23a) shows that the displacements have the property illustrating the principle of reciprocity of the work:
Figure 4 illustrates the displacements of a half-space caused by the presence of the hole.
Contour maps of the normal 
(A) and the tangential 
(B) surface displacements due to hole under the action of the normal force. Coordinate x is a coordinate of the measured displacement; x0 is a coordinate of applied force. The linear quantities are expressed in units of the radius of the hole ρ; the displacements are presented in units of 
where fN is a value of the applied normal force; and λ, μ are the elastic modules. The distance between the center of the hole and the planar boundary is d=1.54ρ.
Figure 5 shows the effect of the characteristic hole aspect ratio for the boundary displacement. In this case, the normal force is applied at the origin (see Figure 1) and the normal displacement is measured at the point of the force application.
Normal surface displacement 
due to the hole as a function of the aspect ratio δ=d/ρ. The concentrated normal force is applied at the origin (see Figure 1), and the displacement is measured at the point of the force application. Here, d is a distance between the center of the hole and the straight surface and ρ is a radius of the hole. The displacement is represented in units of 
where fN is a value of the applied normal force and λ, μ are the elastic modules.
4.2 Displacements due to a concentrated tangential force
For the case of a tangential load, the biharmonic function 
due to the presence of a hole is given by Eqs. (20) and (21), taking into account the coefficients of Eq. (17) and the following supplementary coefficients:
Under a tangential concentrated load, the displacements at the boundary of the half-space caused by a hole have the following form:
where
and
where
Equation (26b) shows that the displacements have the following property:
Comparison of 
and 
demonstrates the expected correlation
Figure 6 shows the straight boundary displacements due to the hole.
Contour maps of the normal 
(A) and tangential 
(B) surface displacements due to the hole under the action of the tangential force. Coordinate x is a coordinate of the measured displacement; x0 is a coordinate of applied force. The linear quantities are expressed in units of the radius of the hole ρ; the displacements are presented in units of 
where fT is a value of the applied tangential force; and λ, μ are the elastic modules. The distance between the center of the hole and the planar surface is d=1.54ρ.
From Figures 4 and 6, it becomes obvious that the hole has the greatest impact on the normal displacement of planar boundary under the action of the normal force.
5 Discussion
In this section, we briefly outline the approach to the solution of the inverse problem that can be formulated as follows: given a displacement produced at the external plane boundary by the given force – find the hole position and size under the assumption of circular geometry of the hole. We will not provide a complete solution to this difficult problem, but only give a simple example of the solution with more details to be published elsewhere.
Let us assume that there is a possibility to measure the surface displacements. One can measure (i) the surface displacements for the body with a defect (presumably hole) loaded by a concentrated linear force and (ii) the surface displacements for the solid body without a hole loaded by a concentrated linear force. In other words, the displacements 
for the body with a hole and 
for the standard body without a hole are assumed to be known. Coordinates of points of force applications and the response points (i.e., the places where the displacements were measured) are the same for the body with a hole and for the standard body. Hence, we possess the information on the additional displacements 
due to the hole in the tested body. Note that for the plane elasticity problem, instead of the points we mean the lines.
We are interested in a possibility to determine the parameters of the hole, namely the hole size ρ and the hole depth h (Figure 1), on the basis of the measured displacement maps 
similar to the maps shown in Figures 4 and 6. The algorithm will include a number of steps. From the maps, one can easily determine the place on the surface, under which a hole is localized (see Figure 4). This will serve as the origin of the Cartesian coordinate system associated with a hole (see Figure 1). Then, the distance measured along the surface from the origin |x| should be related to the bipolar coordinate β by the following equation:
If the force is applied at the origin of the Cartesian coordinate x=0, the bipolar coordinate for the force β0 is equal to =±π in formulas for surface displacements [Eqs. (23) and (26)]. Then, the bipolar coordinate of the hole α1 and bipolar coordinate of the response point β can be extracted from two equations, for instance Eqs. (23a) and (23b) or from Eqs. (23a) and (26b) on the basis of the measured displacements, the known magnitude of the force, and the known elastic modules μ, λ.
The above can be illustrated by a particular example. Suppose we have measured
From Eqs. (23a) and (26b), we find α1≅1.1 and β≅1.57≅π/2. Note that exactly those parameters were put initially into Eqs. (23a) and (26b) to get Eqs. (30a) and (30b). Information on parameter α1 allows one to obtain the aspect ratio of the hole δ=d/ρ=coshα1. The bipolar parameter α (Figure 1) is derived from Eq. (29). After that one extracts the radius of the hole ρ=α/sinhα1, the coordinate of the center of the hole d=acothα1, and the shortest distance between boundary of the hole and half-space boundary h=atanh(α1/2).
6 Conclusions
We have presented an analytical solution to the problem of finding the plane elastic fields arising in the half-space weakened by a hole, in the case of a concentrated load applied to a planar boundary. To our best knowledge, the correct analytical solution to this problem was not given before in the correct closed form. The solution operates with biharmonic functions that allow calculating all elastic fields, in particular, the planar surface displacements, which are modified by the presence of a hole. Biharmonic functions are given in terms of Fourier series. For the first time, the surface displacements are presented in the analytical form of series with compact coefficients.
The information on the surface displacements makes it possible to determine the hole size and the depth of its bedding under the surface, i.e., to analyze the so-called inverse problem. This, in turn, may be useful for developing procedures of defect identification based on force-surface displacement sensing. We also believe that the analytical solution obtained can be useful for the verifications of numerical procedures.
This research was in part supported by European Regional Development Fund (Centre of Excellence “Mesosystems: Theory and Applications”, TK114) and “Nano-Com” 3.2.1101.12-0010 project. The support of the European Commission (Marie Curie Program Project FP7-220419) for A.E.R. is gratefully acknowledged.
Appendix
Appendix 1. Expansion of the stresses 
and 
in the Fourier series
Our task is to express the stresses 
and 
included in the boundary conditions of Eqs. (11c,d) in the form of Fourier series:
1. In the case of the concentrated force fNey (Figure 1), the stresses 
and 
have the following form:
The representations in Eqs. (A1a) and (A1b) for the stresses of Eqs. (A2a) and (A2b) have the following coefficients:
2. In the case of the concentrated force fTex (Figure 1), the stresses 
and 
have the following form:
The representations in Eqs. (A1a) and (A1b) for the stresses of Eqs. (A4a) and (A4b) have the following coefficients:
Appendix 2. Algorithm by G.B. Jeffery: calculation of the sums included in the algorithm
The biharmonic function Φ* providing the stresses that satisfy the conditions at the straight boundary, Eqs. (11a,b), is given by the expression [1, 6]
The algorithm for computing the coefficients 
has the form of systems of algebraic equations [1, 6]. These equations are obtained from the boundary conditions in Eqs. (11c,d). The coefficients 
are the solutions of the following system of equations:
With the help of the coefficients 
the coefficients 
can be found; they are the solutions of the following system of equations:
where
ap, cp, and dp are the coefficients of Fourier series for stresses 
and 
in Eq. (A1a) and (A1b).
1. In the case of the concentrated force fNey (Figure 1), the sums in Eq. (A7b) are given by
2. In the case of the concentrated force fTex (Figure 1), the sums in Eq. (A7b) are given by
Appendix 3. How to eliminate the rotation and pure translation of the half-space as a rigid body
According to G.B. Jeffery, the associated biharmonic function Ψ is obtained from the stress function Φ by integration of Eq. (3b). Without the details described by Jeffery [1], for our problem, the biharmonic function 
is given by
where B0 and the Fourier coefficients are determined from Eqs. (12) and (20), and λ, μ are Lame constants.
Displacements 
can be found with the help of functions Φ* and 
as follows:
where 
Preliminary calculations based on Φ* in Eq. (12) and 
in Eq. (A10) showed that the found displacements contain those corresponding to rotation and pure translation of the half-space as a rigid body, and the function 
is responsible for the rotation. The stresses calculated with the help of the displacements in Eq. (A11), strains in Eq. (6), and Hooke’s law in Eq. (7) coincide with the stresses calculated by using the stress function Φ*.
The function 
must be corrected so that it gives the displacements u*, v*. satisfying the conditions at infinity:
A. The elimination of rotation from the displacements.
Let us represent the displacements in the form of a series:
The rotation is enclosed in the displacement 
and caused by a part of the biharmonic function ΨA [see Eq. (5)]:
The function ΨA corresponds to the following displacement at the planar surface 
It is necessary to eliminate the rotation from each term in Eq. (A13a), i.e., for each term of the series to find the corresponding coefficient of rotation 
The coefficients 
give the coefficient A in the sum
Taking into account Eq. (A16a), the biharmonic function 
is transformed to
B. The elimination of pure translation along axis 0x from the displacements.
The pure translation is included in the displacement 
and caused by a part of the biharmonic function ΨC [see Eq. (5)]:
The function ΨC is responsible for the following displacement at the planar surface 
The coefficient C is a sum of the coefficients 
which are calculated from Eq. (A12b) for every term of the series Eq. (A13b):
As a result, the biharmonic function 
is transformed as follows:
C. The elimination of pure translation along axis 0y from the displacements.
The pure translation is enclosed in the displacement 
and caused by a part of the biharmonic function ΨD [see Eq. (5)]:
The function ΨD corresponds to the following displacement at the planar surface 
The coefficient D is a sum of the coefficients 
which are calculated from Eq. (A12a) for every term of the series Eq. (A13a):
Finally, the associated biharmonic function Ψ* takes the following form:
In our problem, the biharmonic function gΨ* gives the required correct displacements caused by the presence of the hole.
References
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- Editorial Viewpoint
- Small-scale plasticity critically needs a new mechanics description
- Plane elasticity solution for a half-space weakened by a circular hole and loaded by a concentrated force
- More hyperelastic models for rubber-like materials: consistent tangent operators and comparative study
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Artikel in diesem Heft
- Masthead
- Masthead
- Aims & Scope
- Editorial Viewpoint
- Small-scale plasticity critically needs a new mechanics description
- Plane elasticity solution for a half-space weakened by a circular hole and loaded by a concentrated force
- More hyperelastic models for rubber-like materials: consistent tangent operators and comparative study
- Modeling and simulation of size effects in metallic glasses with a nonlocal continuum mechanics theory
- Multiscale unified prediction of size/scale and Hall-Petch effects in the mechanics of polycrystalline materials
- Shift in failure modes in foam core sandwich composites subject to repeated slamming on water
