Original article Biomath 1 (2012), 1211225, 1–4

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Mathematical Problems in the Theory of Bone
Poroelasticity

Merab Svanadze∗, Antonio Scalia†
∗Institute for Fundamental and Interdisciplinary Mathematics Research

Ilia State University, Tbilisi, Georgia
svanadze@gmail.com

†Dipartimento di Matematica e Informatica
Università di Catania, Catania, Italy

scalia@dmi.unict.it

Received: 15 July 2012, accepted: 22 November 2012, published: 29 December 2012

Abstract—This paper concerns with the quasi-static
theory of bone poroelasticity for materials with double
porosity. The system of equations of this theory based on
the equilibrium equations, conservation of fluid mass, the
effective stress concept and Darcy’s law for material with
double porosity. The internal and external basic boundary
value problems (BVPs) are formulated and uniqueness of
regular (classical) solutions are proved. The single-layer
and double-layer potentials are constructed and their basic
properties are established. Finally, the existence theorems
for classical solutions of the BVPs are proved by means of
the potential method (boundary integral method) and the
theory of singular integral equations.

Keywords-bone poroelasticity; double porosity; bound-
ary value problems.

I. INTRODUCTION

The concept of porous media is used in many areas of
applied science (e.g., biology, biophysics, biomechanics)
and engineering. Poroelasticity is a well-developed the-
ory for the interaction of fluid and solid phases of a fluid
saturated porous medium. It is an effective and useful
model for deformation-driven bone fluid movement in
bone tissue [1], [2], [3].

The theory of consolidation for elastic materials with
double porosity was presented by Aifantis and his co-
workers [4], [5]. The Aifantis’ theory unifies the earlier
proposed models of Barenblatt’s for porous media with

double porosity [6] and Biot’s for porous media with
single porosity [7].

However, Aifantis’ quasi-static theory ignored the
cross-coupling effects between the volume change of the
pores and fissures in the system. The cross-coupled terms
were included in the equations of conservation of mass
for the pore and fissure fluid by several authors [8], [9],
[10].

The double porosity concept was extended for mul-
tiple porosity media by Bai et al. [11]. The theory
of multiporous media, as originally developed for the
mechanics of naturally fractured reservoirs, has found
applications in blood perfusion. The double porosity
model would consider the bone fluid pressure in the
vascular porosity and the bone fluid pressure in the
lacunar-canalicular porosity [1], [2], [3]. An extensive
review of the results in the theory of bone poroelasticity
can be found in the survey papers [1], [2]. For a history
of developments and a review of main results in the
theory of porous media, see de Boer [12].

The investigation of BVPs of mathematical physics by
the classical potential method has a hundred year history.
The application of this method to the 3D BVPs of the
theory of elasticity reduces these problems to 2D singular
integral equations [13]. Owing to the works of Mikhlin
[14], Kupradze and his pupils (see [15], [16]), the the-
ory of multidimensional singular integral equations has
presently been worked out with sufficient completeness.

Citation: M Svanadze, A Scalia, Mathematical Problems in the Theory of Bone Poroelasticity, Biomath 1
(2012), 1211225, http://dx.doi.org/10.11145/j.biomath.2012.11.225

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M Svanadze at al., Mathematical Problems in the Theory of Bone Poroelasticity

This theory makes it possible to investigate 3D problems
not only of the classical theory of elasticity, but also
problems of the theory of elasticity with conjugated
fields. For an extensive review of works on the potential
method, see Gegelia and Jentsch [17].

This paper concerns with the quasi-static theory of
bone poroelasticity for materials with double porosity
[8], [9], [10]. The system of equations of this theory
based on the equilibrium equations, conservation of fluid
mass, the effective stress concept and Darcy’s law for a
material with double porosity. The internal and external
basic BVPs are formulated and uniqueness of classical
solutions are proved. The single-layer and double-layer
potentials are constructed and their basic properties are
established. Finally, the existence theorems for classical
solutions of the BVPs are proved by means of the
boundary integral method and the theory of singular
integral equations.

II. BASIC EQUATIONS

Let x = (x1,x2,x3) be a point of the Euclidean three-
dimensional space R3, let t denote the time variable,
t ≥ 0, u(x, t) denote the displacement vector in solid,
u = (u1,u2,u3); p1(x, t) and p2(x, t) are the pore
and fissure fluid pressures, respectively. We assume that
subscripts preceded by a comma denote partial differ-
entiation with respect to the corresponding Cartesian
coordinate, repeated indices are summed over the range
(1,2,3), and the dot denotes differentiation with respect
to t.

In the absence of body force the governing system
of field equations of the linear quasi-static theory of
elasticity for solid with double porosity consists of the
following equations [8], [9], [10].

1) The equilibrium equations

tlj,j = 0, l = 1, 2, 3, (1)

where tjl is the component of the total stress tensor.
2) The equations of fluid mass conservation

div v(1) + ζ̇1 + β1ėrr + γ(p1 −p2) = 0,

div v(2) + ζ̇2 + β2ėrr −γ(p1 −p2) = 0,
(2)

where v(1) and v(2) are fluid flux vectors for the pores
and fissures, respectively; elj is the component of the
strain tensor,

elj =
1

2
(ul,j + uj,l) , l,j = 1, 2, 3, (3)

β1 and β2 are the effective stress parameters, γ is the
internal transport coefficient and corresponds to a fluid

transfer rate respecting the intensity of flow between the
pores and fissures, γ > 0; ζ1 and ζ2 are the increment of
fluid in the pores and fissures, respectively, and defined
by

ζ1 = α1 p1 + α3 p2, ζ2 = α3 p1 + α2 p2, (4)

α1 and α2 measure the compressibilities of the pore and
fissure systems, respectively, α3 is the cross-coupling
compressibility for fluid flow at the interface between the
two pore systems at a microscopic level [8], [9], [10].
However, the coupling effect (α3) is often neglected in
the literature [4], [5], [6].

3) The equations of the effective stress concept

tlj = t
′
lj − (β1p1 + β2p2) δlj, l,j = 1, 2, 3, (5)

where t′lj = 2µelj +λerrδlj is the component of effective
stress tensor, λ and µ are the Lamé constants, δlj is the
Kronecker’s delta.

4) The Darcy’s law for material with double porosity

v(1) = −
κ1
µ′

grad p1, v
(2) = −

κ2
µ′

grad p2, (6)

where µ′ is the fluid viscosity, κ1 and κ2 are the
macroscopic intrinsic permeabilities associated with the
matrix and fissure porosity, respectively. We note that
in the real porous media the fissure permeability κ2
is much greater than the matrix permeability κ1, while
the fracture porosity is much smaller than the matrix
porosity.

Substituting equations (3)-(6) into (1) and (2), we ob-
tain the following system of homogeneous equations in
the linear quasi-static theory of elasticity for solids with
double porosity expressed in terms of the displacement
vector u, pressures p1 and p2.

µ∆u + (λ + µ)∇div u −β1∇p1 −β2∇p2 = 0,

k1∆p1 −α1ṗ1 −α3ṗ2 −γ(p1 −p2) −β1divu̇ = 0,

k2∆p2 −α3ṗ1 −α2ṗ2 + γ(p1 −p2) −β2divu̇ = 0,
(7)

where ∆ and ∇ are the Laplacian and gradient operators,
respectively, and kj =

κj
µ′

(j = 1, 2).

In the follows we assume that the inertial energy
density of solid with double porosity is a positive definite
quadratic form. Thus, the constitutive coefficients satisfy
the conditions:

µ > 0, 3λ + 2µ > 0, k1 > 0, k2 > 0,

α1 > 0, α1α2 −α23 > 0.

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M Svanadze at al., Mathematical Problems in the Theory of Bone Poroelasticity

If the displacement vector u, the pressures p1 and p2
are postulated to have a harmonic time variation, that is,

{u,p1,p2}(x, t) = Re
[{

u′,p′1,p
′
2

}
(x)e−iωt

]
,

then from system (7) we obtain the following system of
homogeneous equations of steady vibrations in the linear
quasi-static theory of elasticity for solids with double
porosity

µ∆u′ + (λ + µ)∇divu′ −β1∇p′1 −β2∇p
′
2 = 0,

(k1∆ + a1)p
′
1 + a3p

′
2 + iω β1 div u

′ = 0,

a3p
′
1 + (k2 ∆ + a2)p

′
2 + iω β2 div u

′ = 0,
(8)

where aj = iω αj −γ, a3 = iω α3 + γ (l,j = 1, 2); ω
is the oscillation frequency, ω > 0.

III. BOUNDARY VALUE PROBLEMS
Let S be the closed surface surrounding the finite

domain Ω+ in R3, S ∈ C2,λ0, 0 < λ0 ≤ 1, Ω̄ =
Ω ∪S, Ω− = R3\Ω̄+.
Definition 1. A vector function U = (u′,p′1,p

′
2) =

(U1,U2, · · · ,U5) is called regular in Ω− (or Ω+) if

1) Ul ∈ C2(Ω−)∩C1(Ω̄−) (or Ul ∈ C2(Ω+)∩C1(Ω̄+)),

2)
Ul(x) = O(|x|−1), Ul,j(x) = o(|x|−1),

where |x|� 1, l = 1, 2, · · · , 5, j = 1, 2, 3.
The basic BVPs of steady vibrations in the linear

quasi-static theory of elasticity for solid with double
porosity are formulated as follows.

Find a regular (classical) solution U = (u′,p′1,p
′
2) to

system (8) satisfying the boundary condition

lim
Ω+3x→ z∈S

U(x) ≡{U(z)}+ = f (z)

in the Problem (I)+f , and

lim
Ω−3x→ z∈S

U(x) ≡{U(z)}− = f (z)

in the Problem (I)−f , where f is the known five-
component vector function.

IV. UNIQUENESS THEOREMS
We have the following results.

Theorem 1. The internal homogeneous BVP (I)+f admits
at most one regular solution.

Theorem 2. The external BVP (I)−f admits at most one
regular solution.

Theorems 1 and 2 can be proved similarly to the
corresponding theorems in the classical theory of ther-
moelasticity (for details see [13, Chapter III]).

V. BASIC PROPERTIES OF ELASTOPOTENTIALS
The system (8) may be written as B(Dx) U(x) = 0,

where B(Dx) is the matrix differential operator corre-
sponding left-hand side of (8) and Dx = ( ∂∂x1 ,

∂
∂x2

, ∂
∂x3

).
We introduce the following notations:
1) Z(1)(x, g) =

∫
S

Γ(x − y)g(y)dyS is the single-

layer potential,
2) Z(2)(x, g) =

∫
S

[P̃(Dy, n(y))Γ
>(x − y)]>g(y)dyS

is the double-layer potential, where Γ = (Γlj)5×5 is
the fundamental matrix of the operator B(Dx),
P̃ =

(
P̃lj
)

5×5
is the matrix differential operator of the

first order, g is five-component vector, the superscript
> denotes transposition.
Remark 1. In the Aifantis’ quasi-static theory (α3 = 0),
the fundamental matrix Γ(x) is constructed by Svanadze
[18].

We have the following basic properties of elastopo-
tentials.

Theorem 3. If S ∈ C2,λ0, g ∈ C1,λ
′
(S), 0 < λ′ <

λ0 ≤ 1, then:
(a)

Z(1)(·, g) ∈ C0,λ
′
(R3) ∩C2,λ

′
(Ω̄±) ∩C∞(Ω±),

(b)
B(Dx) Z

(1)
(x, g) = 0, x ∈ Ω±,

(c) P(Dz, n(z)) Z
(1)

(z, g) is a singular integral,
(d)

{P(Dz, n(z)) Z(1)(z, g)}± = ∓
1

2
g(z)

+P(Dz, n(z)) Z
(1)

(z, g), z ∈ S,
where P(Dz, n(z)) is the stress operator in the linear
theory of elasticity for solids with double porosity.

Theorem 4. If S ∈ C2,λ0, g ∈ C1,λ
′
(S), 0 < λ′ <

λ0 ≤ 1, then:
(a)

Z(2)(·, g) ∈ C1,λ
′
(Ω̄±) ∩C∞(Ω±),

(b)
B(Dx) Z

(2)
(x, g) = 0, x ∈ Ω±,

(c) Z(2)(z, g) is a singular integral,
(d)

{Z(2)(z, g)}± = ±
1

2
g(z) + Z(2)(z, g), z ∈ S.

Theorems 3 and 4 can be proved similarly to the
corresponding theorems in the classical theory of ther-
moelasticity (for details see [13, Ch. X]).

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M Svanadze at al., Mathematical Problems in the Theory of Bone Poroelasticity

VI. EXISTENCE THEOREM

We introduce the notation

K1 g(z) ≡
1

2
g(z) + Z(2)(z, g),

K2 g(z) ≡−
1

2
g(z) + Z(2)(z, g)

for z ∈ S. Obviously, on the basis of theorem 4, K1 and
K2 are the singular integral operators.
Lemma 1. The singular integral operators K1 and K2
are of the normal type with an index equal to zero.

Lemma 1 leads to the following existence theorems.

Theorem 5. If S ∈ C2,λ0, f ∈ C1,λ
′
(S), 0 < λ′ < λ0 ≤

1, then a regular (classical) solution of the internal BVP
(I)+f exists, is unique and is represented by double-layer
potential

U(x) = Z(2)(x, g) for x ∈ Ω+,

where g is a solution of the singular integral equation

K1 g(z) = f (z) for z ∈ S

which is always solvable for an arbitrary vector f .

Theorem 6. If S ∈ C2,λ0, f ∈ C1,λ
′
(S), 0 < λ′ < λ0 ≤

1, then a regular (classical) solution of the external BVP
(I)−f exists, is unique and is represented by sum

U(x) = Z(2)(x, g) − i Z(1)(x, g) for x ∈ Ω−,

where g is a solution of the singular integral equation

Kg(z) − i Z(1)(z, g) = f (z) for z ∈ S

which is always solvable for an arbitrary vector f .

Theorem 5 and 6 are proved by means of the potential
method and the theory of singular integral equations (for
details see [13]).

VII. CONCLUSION

1. By the above mentioned method it is possible
to prove the existence and uniqueness theorems in the
modern linear theories of elasticity and thermoelasticity
for materials with microstructure.

2. On the basis of theorems 1 to 6 it is possible to
obtain numerical solutions of the BVPs of the quasi-
static theory of elasticity for solids with double porosity
by using of the boundary element method.

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http://dx.doi.org/10.1016/S0021-9290(98)00161-4
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http://dx.doi.org/10.1016/j.jmps.2012.01.013
http://dx.doi.org/10.1016/0020-7225(82)90036-2
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http://dx.doi.org/10.11145/j.biomath.2012.11.225

	Introduction
	Basic Equations
	Boundary value problems
	Uniqueness theorems
	Basic properties of elastopotentials 
	Existence theorem 
	Conclusion
	References