INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VIII, August 2024
www.ijltemas.in Page 24
Equilibrium Solution of Two – Dimensional Non-Homogeneous
Equations in the Theory of Elastic Mixtures
Udoh, Paul J. And Udoh, Inemesit J.
Department of Mathematics, University of Uyo, Nigeria
DOI: https://doi.org/10.51583/IJLTEMAS.2024.130803
Received: 30 July 2024; Accepted: 10 August 2024; Published: 27 August 2024
Abstract: The problem of plane elasticity for a doubly connected body with inner and outer boundaries in a regular polygonal
form with common centre and parallel sides has been studied. The sides of the polygon were exposed to external forces. The
nature of the force term was determined by application of complex variable theory. Kolosov’s method of solution was applied to
obtain the biharmonic equation of the forcing term. The forces on the particle were studied under 2-dimensions from which the
compatibility and equilibrium equations were derived. The compatibility and equilibrium equations were used to derive the force
– stress relations. The results shows that there is a significant relationship between the angle of the force term on the plane of the
particle and the stress state of the particle, which is in conformity with existing experimental results.
Keywords: Elasticity, equilibrium, forcing term, biharmonic, compatibility, stress state.
I. Introduction
The theory of elasticity describes deformable materials such as rubber, cloth, paper and flexible metals. It is often used to model
the behavior of non-rigid curves, surfaces and solids as a function of time. Elasticity deformable models are active and respond
naturally to applied forces, constraints, ambient, media and impenetrable obstacles, Terzopoulos et al, (1987).
One of the most efficient and elegant techniques of solving problems in the linear theory of elasticity is the method of complex
stress functions which is mainly associated with Kolosov, (1909), Muskhelishvili (1966), Bock, and Gurlebeck, (2009). In
particular, application of complex variable theory in solving elasticity problems resulted from the complex potential, which is
peculiar to analytic complex functions. It is exceedingly fruitful for effective solution of boundary value problems and general
functions that relates theoretically with Cauchy’s integral formula and conformal mappings, Kapanadze and Gulna (2016). Chou
and Pagano (2001) opined that one of the major problems in the theory of elasticity is that of determining the full strength of
surfaces which aid in controlling the stress concentration both on the surface and at the boundaries of surfaces.
Recently, construction and engineering practices suffer major setbacks resulting from negligence, poor analysis and examination
of materials and the stress strength of surfaces and contours on which the load/stress are imposed. Odishelidze and Kriado (2006)
further established that investigation of stresses concentration near the contour of surfaces is one of the major problems in plane
elasticity theory, especially in plate with a hole where the tangential-normal stresses and the tangential-normal moments can
reach such values that cause destruction of plates or formation of plastic zones near the hole at some points. In cases of infinite
domains, the minimum of maximum values of tangential-normal stresses will be obtained on such holes, where these values
maintain constant (full strength holes).
A mixed problem of plane elasticity theory for doubly-connected domain with partially unknown boundary conditions was solved
in Odishelidze et al (2015). The problem of plane elasticity theory for a doubly connected domain with partially unknown
boundary was solved in Odishelidze (2015) using the methods of the theory of analytic complex functions. Boundary valued
equation for force term in non-homogeneous equation of statics in the theory of elastic mixtures was solved in Udoh and Ndiwari,
(2018) using Kolosov-Muskhelishvili formula for a displacement vector in an elastic mixture of homogeneous body. Biharmonic
solution for a force term in a non-homogeneous equation of statics in the theory of elastic mixtures was provided by Ndiwari and
Ongodiebi. (2020) using complex variable theory, where constant introduction of the force term at a fixed point on the plane
directly affected the stability of the particle.
In this work, we considered a problem of plane elasticity for a doubly connected domain with inner and outer boundaries in a
regular polygonal form with common center and parallel sides. The sides of the body were exposed to external unknown force
and the boundary conditions were determined at equilibrium in order to ascertain the impact of the forcing term and its
relationship with the stability of the isotropic elastic material. We derived the forcing term from the non-homogeneous equation
of statics in the theory of elastic mixture. The unknown forces were analyzed in two-dimensions from stress function to derive the
equilibrium, compatibility and biharmonic equations. The basic equation of elasticity was obtained using the compatibility
equation and the stress-strain relation. The boundary equation of the unknown forcing term was derived and graphs generated to
illustrate and explain the relationship between the angle of the forcing term and the stress state of the isotropic elastic material.
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VIII, August 2024
www.ijltemas.in Page 25
Mathematical Formulation
We considered a homogeneous isotropic elastic body in a doubly connected domain D on the complex plane z = x + iy. Its outer
and inner boundaries are L
o
and L
1
respectively and form a rectangle with a common center z = 0 and parallel sides. The
neighbourhood of the vertices of the inner rectangle are equal smooth arcs which are symmetric angles of equidistance from the
centre as in Figure 1. We assumed that the edges of the isotropic elastic body are exposed to the external force in the form of
load. We further assumed that both boundaries, L
0
of the elastic body and that of the hole are smooth and free from frictional
forces. Under these assumptions, the normal displacements of the outer and inner boundaries are constant respectively, while the
unknown arcs are exposed to external force. Our aim was to determine the equilibrium solution for the force term, F and the
relationship between the angle of the force term on the plane of the particle and the stress state of the particle.
Figure 1. Isotropic elastic body
II. Method of Solution
To determine the force term F, we applied the non-homogeneous equation in the theory of elastic mixtures as our governing
equation and adopt [Kolosov, (1909) and Muskhelishvili. (1966)]. The displacement components of the vector are represented in
this theory by means of four arbitrary analytic functions as in [Ndiwari and Ongodiebi (2020)]. The basic non-homogeneous
equations governing the theory of elastic mixture in 2-dimensions [Kapanadze and Gulna (2016)] is given by
Where Δ is the 2-dimensional Laplacian, grad and div are the principal operators of the field theory,
and
are the partial
densities (positive constants of the mixture), and are the mass forces, respectively; u' = (u
1
', u
2
') = w' and u'' = (u
1
'', u
2
'') =
w'' are the displacement vectors, Ψ' and Ψ'' denote the product of the partial density ρ and the mass force F
,
respectively, a
1
,a
2
, b
1
,
b
2
, c and dare combination of constitutive constants characterizing the physical properties of the mixtures specified as
a
1
= μ
1
–λ
5
, a
2
= μ
2
‒ λ
5,
b= μ
2
+ λ
2
+ λ
5
+ p
-1
α
2
ρ
1
, b
1
= μ
1
+ λ
1
+ λ
5
+ p
-1
α
2
ρ
2,
c = μ
3
+ λ5, d = μ
3
+ λ
4
– λ
5
– p
-1
α
2
ρ
1,
p = ρ
1
+ ρ
2
, (2)
a
2
=μ
2
– λ
2
+λ
5
+ p
-1
a
2
ρ
1
α
2
=λ
3
–λ
4
where μ
i
, (i = 1,2,3) is the mixture’s permeability constant and λ
i
, (i =1,2,3,4,5) is the mixture’s thermal conductivity constant.
Applying Complex Variables Theory
Applying complex variable theory, we solve Equation (1) as follows:
=
1
+ x
2
(3)
And its conjugate as
=
1
– x
2
(4)
Adding (3) and (4) gives
2
1
= + (5)
′
′
′′
′′
′ ′
′
′
′′
′′
′′
′′
1
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VIII, August 2024
www.ijltemas.in Page 26
Subtracting (4) from (3) gives
2ix
2
= z - (6)
Expressing (5) in partial differential equation, gives
(7)
Expressing (6) in partial differential equation, gives
(8)
Adding (7) and (8) gives
(9)
Subtracting (8) from (9) we obtain
(10)
Multiplying (9) and (10) we have
(11)
Equating the two right terms of (11) to the right part of (7) and (8) respectively, we obtain
(12)
(13)
Replacing the two right hand term of (11) by the two right terms of (12) and (13) respectively, gives
(14)
Let the displacement vectors
be represented in their complex form [12] by
(15)
(16)
Operating (14) on (15) and (16) respectively, we have obtained
(17)
(18)
where the displacement vectors
and
depend on the elastic and plastic regions.
We adopt [12] in order to make (1) solvable.
Let
(19)
and
= 2ϴ' (20)
Substituting (19) and (20) for Δu' and div u' in (1)
we obtain
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VIII, August 2024
www.ijltemas.in Page 27
and
(21)
where our Laplacian here is defined as
(22)
Substituting (22) in (21) we obtain
and
(23)
(24)
(25)
From [13], Integrating (24) and (25) wrt we obtain
(26)
(27)
Where
are the analytic (non-homogeneous) terms and
is the displacement function in the transformed
state as a result of contact with external force and
, is the complex conjugate function.
For the Non-Homogeneous Term
Comparing the non-homogeneous part of (26) and (1), we have that
(28)
So that
(29)
Expressing the above in partial differential equation gives:
(30)
Equating the real and imaginary parts of (30), gives
(31)
(32)
We adopt [12] by Introducing new variables Let the new variables be given by
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VIII, August 2024
www.ijltemas.in Page 28
(33)
(34)
Substituting (33) and (34) in (32), we obtain
(35)
(36)
From (35) and (36) we have that
(37)
(38)
From (37) and (38),
(Neglecting the imaginary part) (39)
Newtonian Gravitation and Gravitational Force
In the classical field theory, [5] describes the Newtonian gravitation which describes the gravitational force F, as a mutual
interaction between two masses, M
1
and M
2
expressed as:
(40)
In this context, M
1
is the isotropic elastic body (Figure1), M
2
is the object of our forcing term, G is the Earth gravitational
constant and r is the distance between the centre of the two masses M
1
and M
2
respectively. The massive body M
1
has a
gravitational field g. Since the gravitational force F, is conservative, the field g can be written as a gradient of a gravitational
scalar potential as
(41)
Gauss’ Law and Poisson Equation for Gravity
Gauss’ law of gravity is equivalent to Newton’s law of universal gravitation. The differential form of Gauss’ law is given as
(42)
Where is the divergence, G is the universal constant and ρ is the mass density at each point. Gauss’ law is also given in
integral form as
(43)
where V is a closed region bounded by a simple closed oriented surface which is the infinitesimal piece of the volume and g is
the gravitational field. Also, in the case of a gravitational field due to attracting massive objects of density ρ, Gauss’ law for
gravity in differential form can be used to obtain the corresponding Poisson equation for gravity:
(44)
Substituting (41) in (44), gives
(45)
(45) is the Poisson equation for gravity [11]. Hence, (39) is equivalent to (45) because it involves the mutual interaction between
the isotropic elastic body (M
1
)
in Figure1 and the object of our force term (M
2
).
That is
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VIII, August 2024
www.ijltemas.in Page 29
(46)
Hence, our forcing term is a gravitational force, and it is Poisson in nature, as such it is restricted to a plane.
Equation of Equilibrium
We consider the surface area of Figure 1 as follows:
Figure 3: - Surface area of Figure1 and its force distribution
where
F = mg is the gravitational force (mass x gravity)
S = Surface area (length (x breadth (β)).
Figure 4: - Stress distribution on a rectangular Plane
where:
σ
x
= Normal stress in the x-direction
σ
у
= Normal stress in the y-direction
τ
xy
= Shear stress in the x-direction
τ
yx
= Shear stress in the y-direction
[6] deduced that
(47)
(Symmetric).
(48) is the Equation of balance or Equilibrium equation in 2-dimensions.
d
a
c
b
τ
x
y
τ
y
x
k
σ
x
σ
x
h
Y
0
X
F = mg
S
β
α
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VIII, August 2024
www.ijltemas.in Page 30
Compatibility Equation
Equation (47) shows two equations in three unknowns, with stress components
. For compatibility, we adopt [6]
for the strain - displacement relation of the deformation process by introducing
(48)
(49)
(50)
Where
are displacement vectors in the transformed state
Differentiating (48), (49) and (50) twice with respect to y, x, and xy respectively, we obtain
(51)
(52)
53)
From (51), (52) and (53), we obtain
(54)
(54) is the compatibility equation in 2-dimensions.
Biharmonic Equation
To solve (54), we apply the stress – strain relationship [11] for plane stress to obtain
(55)
(56)
(57)
where: v = Poisson ratio, E = Young modulus, G = Modulus of rigidity.
Substituting (55), (56) and (57) into (54), we obtain
(58)
(59)
(60)
We differentiate (47) with respect to x and y respectively to eliminate the shearing stress,
and obtain
(61)
(62)
Adding (61) and (62), we obtain
(63)
(64)
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VIII, August 2024
www.ijltemas.in Page 31
Substituting (64) in (60), gives
Such that
and
(65)
To solve (65), we introduce a new function φ called Airy’s stress function [6].
For the case under consideration, φ can be defined, such that:
(66)
Substituting (66) in (65), we have
(67)
That is
So that
(68)
(68) is called the Biharmonic Equation.
Comparing (46) and (68), our force term, F becomes
= 0
Hence, (69)
(69) shows that the force component is Biharmonic in nature.
Stress State of the Force Term on the Plane
We now consider our force (F), to act on a rectangular plane of area.
Figure 5: Surface area and force distribution
Generally, stress is the force per unit area of a body/particle and can be expressed as
(70)
Where = stress, Force = force and = area.
Normal stress (fig. 4) in the x-direction is:
F
α
β
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VIII, August 2024
www.ijltemas.in Page 32
, (71)
Shear stress in the x-direction is given as:
, (72)
While the normal stress in the y-direction is
(73)
From Airy’s stress function [6], we have
(74)
(75)
+ k (76)
Adding (74), (75) and (76)
(77)
(77) is the solution that satisfies a typical Biharmonic Equation in 2-dimensions [6].
Stress Distribution on the X and Y Coordinates
Figure 6: Stress distributions in a 2-dimensional rectangular plane
Figure 7: Stress distribution on the x- direction
ϴ
0
ϴ
0
F
X
A.σsinϴ
A.τ
xy
cosϴ
ϴ
ϴ
0
ϴ
ϴ
0
ϴ
0
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VIII, August 2024
www.ijltemas.in Page 33
Figure 8: Stress distribution on the y – direction
Deducing from figure 7, the force impact (stress) on the particle in the x-coordinate is obtained as
F
x
= A (τ
xy
cosϴ + σ
x
sinϴ) (78)
(79)
Similarly, from Figure 8, the force impact (stress) on the particle in the y – coordinate is as
(80)
(81)
Hence, we have the system:
(82)
(82) is the stress impact on the coordinates.
Boundary Condition of the Force/stress on the Plane
The boundary condition is obtained from the requirement that the total stress on the planes of the particle is zero when no force
was introduced. That is, the magnitude of the total stress component,
(83)
Then,
(84)
(84) is our third result showing the magnitude of the ratio of the normal stressese on the particle.
III. Results
From (68) and (69), the magnitude of the force components on the particle is zero at equilibrium.
From [10], the magnitude of the force term is given by
F = s (cos
2
a - sin
2
a).
From (84), our graph is given by:
A.σ
y
cosϴ
F
Y
A
A.τ
xy
sinϴ
ϴ
0
ϴ
0
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue VIII, August 2024
www.ijltemas.in Page 34
Figure 9: Graph of the ratio of normal stresses against the angle ϴ.
IV. Discussion of Results
The graph in Figure 9 above results from the relationship between the ratio of the normal forces and the angle of the force. The
curve rises from the origin, 0 when no force term was introduced and rises uniformly to the point when the force acted at angle
60
o
and increases rapidly when the force acted above 60
o
. The magnitude of the ratio of the normal stresses on the particle
attained its maximum value when the force acted at angle 75
o
. Beyond 75
o
, the particle demonstrates its elastic nature and the
curve drops rapidly downward to the origin, 0.When the force acted at 90
0
on the plane of the particle, the body regains its elastic
potential and returns to its equilibrium state as the stress resolves itself to zero due to the perfect angular formation which allows
even distribution of the stresses on both coordinates. When the force acted beyond 90
o
, the magnitude of the ratio of the normal
stresses on the particle oscillates back to its maximum point, which shows that the elastic potential of the particle has been
weakened.
V. Conclusion
In this paper, the problem of non-homogeneous equation of statics in the theory of elastic mixture was considered using complex
variable theory. Our theoretical solution for the stress state of the isotropic elastic body examined was found to be consistent with
the experimental existing result.
References
1. Terzopoulos, D., Platt, J., Barr, A., and Fliesher, K. (1987). Elastically Deformable Models. Computer Graphics, 21(4):
205-213.
2. Kolosov, G. V. (1909). An Application of the Theory of Functions of Complex Variable to the Problems of
Mathematical Elasticity Theory. Yur’ev. (in Russian).
3. Muskhelishvili, N. I. (1966). Some Basic Problems of Mathematical Elasticity Theory, Nauka (in Russian).
4. Bock, S. and Gurlebeck, K. (2009). On a Spatial Generalization of the Kolosov-Muskhelishvili Formulae. Mathematical
Method in the Applied Science, (32): 223-240.
5. Kapanadze, G. A. and Gulna, B. (2016). About One Problem of Plane Elasticity for a Polygonal Domain with a
Curvilinear Hole. American Institute of Mathematics, 21: 21-29.
6. Chou, T. and Pagano, G. (2001). Two and Three Dimensional Stress Function (2nd Edition), Institute of General
Mechanics, Aache, Germany.
7. Odishelidze, N. T. and Kriado, F. F. (2006). A Mixed Problem of Plane Elasticity for a domain with Partially Unknown
Boundary. International Applied Mechanics, 42(3): 342-349.
8. Odishelidze, N., Criado, F., and Sanchez, J. M. (2015). Stress Concentration in an Elastic Square plate with full Strength
hole. Mathematics and Mechanics of Solids, 21(5): 552-561.
9. Odishelidze, N. T. (2015). A Mixed Problem of Plane Elasticity Theory for a Multiply Connected Domain with Partially
Unknown Boundary: The Case of a Rhombus. Zeits Chrift fur angewandte Mathematik and Physik, 66(5): 2899-2907.
10. Udoh, P. J. and Ndiwari, E. (2018). Boundary-valued Equations for the Force Term in Non-Homogeneous Equation of
Statics in the Theory of Elastic Mixtures. Asian Research Journal of Mathematics, 8(1): 1-11.
11. Ndiwari, N. and Ongodiebi, Z. (2020). Biharmonic Solution for the Forcing Term in a Non-Homogeneous Equation of
Statics in the Theory of Elastic Mixtures. International Journal of Engineering Research and Technology. 9(7): 1428 –
1438.
12. Kapanadze, G. A. and Gulna, B. (2016). About One Problem of Plane Elasticity for a Polygonal Domain with a
Curvilinear Hole. American Institute of Mathematics, 21: 21-29.
0
2
4
6
8
10
12
14
16
0 20 40 60 80 100 120
Ratio of Normal Stresses
Angle of
force (ϴ)
Angle of the force ϴ.
