The constitutive equations in linear elasticity are represented by the generalized Hooke′s law. If the state of vanishing strain corresponds to zero stress, then in Cartesian co-ordinates, the generalized Hooke′s law can be written as:
kl ijkl
ij c ε
σ = ( i, j, k, l=x, y, z) (3.01)
where c are components of a fourth-rank tensor, repressenting the properties of a ijkl material, which generally varies from one point to another in the material. The c ijkl are called elastic stiffness constants. Since Eq. (3.01) contains nine equations (corresponding to all possible combinations of the subscripts i and j and each equation contains nine strain variables), there are 81 elastic stiffness constants. These are not all independent hower. It will be seen that cijkl =cjikl =cijlk =cjilk, which reduces the number of independent constants to 36. In addition, the cijkl =cklij, and this means the constants are further reduced to 21. This is the maxium number of constants for any medium.
In a Cartesian co-ordinate system, (x,'y,'z'), the Eq. (3.01) can then be expressed as:
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣
⎡
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣
⎡
=
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣
⎡
' '
' '
' '
' '
' '
' '
66 65 64 63 62 61
56 55 54 53 52 51
46 45 44 43 42 41
36 35 34 33 32 31
26 25 24 23 22 21
16 15 14 13 12 11
' '
' '
' '
' '
' '
' '
y x
x z
z y
z z
y y
x x
y x
x z
z y
z z
y y
x x
C C C C C C
C C C C C C
C C C C C C
C C C C C C
C C C C C C
C C C C C C
γ γ γ ε ε ε
τ τ τ σ σ σ
(3.02)
The number of elastic constants c for describing their deformability is 21, 9, 5, ijkl and 2 for generally anisotropic, orthotropic, transversely isotropic, and isotropic material, respectively. Thus, for a general anisotropic elastic material, there are 21 independent elastic stiffness constants. If there exist three orthogonal planes of elastic symmetry at any point in a solid, then there are 9 independent elastic stiffness constants, and the material is said to be orthotropic. If at any point there is an axis of symmetry
such that the elastic properties in any direction within a plane perpendicular to the axis are all the same, the number of independent elastic stiffness constants will reduce to 5.
The plane is called an isotropic plane and the material is called a transversely isotropic material. If any plane in the material is a plane of elastic symmetry, then the material is isotropic, and has only 2 independent elastic stiffness constants.
Fig. 3.1 displays a transversely isotropic medium, in which the 'z axis is the rotation axis of elastic symmetry, the x' and 'y axes are in the plane of transversely isotropy.
In the co-ordinate system (x,'y,'z') , the corresponding matrix form is
{ }
σ x'y'z' =[ ]
c x'y'z'{ }
ε x'y'z'.z<0 (region 2)
z>0 (region 1)
H
P x,x'
φ
φ
y' y' z'
z
y
Fig. 3.1 ( Px, Py, Pz ) acting in an inclined transversely isotropic full-space
Regarding the different co-ordinate system (x,y,z), the constitutive equations will have the same form as
{ }
σ xyz =[ ]
a xyz{ }
ε xyz. Hence, the generalized Hooke′s law for the transversely isotropic material can be expressed as:⎥⎥
1. E and 'E are Young′s moduli in the plane of transverse isotropy and in a direction normal to it, respectively.
2. υ and υ are Poisson′s ratios characterizing the lateral strain response in the plane ' of transverse isotropy to a stress acting parallel or normal to it, respectively.
3. G' is the shear modulus in planes normal to the plane of transverse isotropy.
If a new co-ordinate system (x, y, z) is obtained from the original system (x', y', 'z) by rotation through an angle φ about the common axis x=x' (the axis of x and
'
x parallel to the strike of transverse plane). The matrix of direction cosines lij for the transformation formulae of the elastic constants are:
[ ]
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
−
=
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
=
φ φ
φ φ
cos sin 0
sin cos
0
0 0
1
l l l
l l l
l l l l
33 32 31
23 22 21
13 12 11
ij (3.05)
where i, j=1-3.
The elastic stiffnesses matrix in the old co-ordinate system (x,'y,'z') is
[ ]
c x' zy'' . Therefore, the elastic stiffnesses matrix [ ]a xyz in the new co-ordinate system (x,y,z) can be expressed as:[ ] [ ] [ ] [ ]
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣
⎡
=
=
66 65 64 63 62 61
56 55 54 53 52 51
46 45 44 43 42 41
36 35 34 33 32 31
26 25 24 23 22 21
16 15 14 13 12 11
' ' '
a a a a a a
a a a a a a
a a a a a a
a a a a a a
a a a a a a
a a a a a a
q c q
a xyz ij T ij
z y
x (3.06)
where T is the transpose matrix;
[ ]
qij are written as follows.[ ]
The new elastic constants of
[ ]
a xyz obtained directly from the old elastic conctant a1-a5 and φ. It is important to note that[ ]
a xyz exist 13 elastic constants under the plane of elastic symmetry system, and the expressions of the elastic constants aij(i, j=1-6) with respect to a1-a5 and φ are presented in Appendix A. Appendix A show thatThen, the generalized Hooke′s law for a transversely isotropic material is:
{ }
σ xyz =[ ]
a xyz{ }
ε xyz (3.08)where
{ }
σ xyz and{ }
ε xyz are vectors of stress and engineering strain, respectively. In Cartesian co-ordinates, they are{ }
σ xyz =[
σxx σyy σzz τyz τzx τxy]
T (3.09){ }
ε xyz =[
εxx εyy εzz γyz γzx γxy]
T. (3.10)Strain-displacement relations
When a sign convention for soil and rock problem is required, it is customary to define compressive stresses as positive and tensile stresses as negative. The
strain-displacement relationship under small strain condition in a Cartesian coordinate co-ordinate system.
Generalized Hooke′s law in terms of the derivative of displacements
Hence, from Eqs. (3.08), (3.09), (3.10) and (3.11), the generalized Hooke′s law equations for a transversely isotropic medium in a Cartesian coordinate system can be expressed as:
)
)
Equilibrium equations
In Cartesian coordinates, the equations of motion can be expressed by a tensor form as: volume in i-direction, and finally the double dot indicates the second order partial differentiation with respect to time t. If the motion of the solid does not involve acceleration, then Eq. (3.13) reduces to the equilibrium equation as:
⎥⎥
where Fx, Fy, and Fz stand for the components of the body forces per unit volume in the co-ordinate directions, x, y and z, respectively. Substituting Eqs. (3.12a)-(3.12f) (σxx,
σyy, σzz, τyz, τzx, τxy) into Eq. (3.14) enables the equations to be regrouped as Navier-Cauchy equations for an inclined transversely isotropic material as:
z x
Point loads
For a dynamic elastic problem, an arbitrary time-harmonic body force in z-direction with angular frequency ω can be expressed as (Eringen and Suhubi, 1975; Rahman, 1995):
where Fz*(x,y ,z) is the complex amplitude of the body force. Following the suggestions of Eringen and Suhubi (1975) and Rahman (1995), a concentrated force in z-direction (Fz) can be represented as the form of a body force:
t
e i
(z) (y)
(x)δ δ ω
zδ
z p
F = (3.17)
where δ ( ) is the Dirac delta function.
Nevertheless, for static problems, the terms associated with time t in Eq. (3.17) should be removed. As this research concerning about the static problems, ω in Eq.
(3.17) will be zero. Hence, three-dimensional static point loads with components (Fx, Fy, Fz) acting at the origin of the co-ordinate can be expressed as the form of body forces:
) z ( ) y ( ) x ( P
Fx = xδ δ δ (3.18a)
) z ( ) y ( ) x ( P
Fy = yδ δ δ (3.18b)
) z ( ) y ( ) x ( P
Fz = zδ δ δ (3.18c)
Then, the point loads (Fx, Fy, Fz) applied at the point (0, 0, h) of the co-ordinate system can be described as the form of body forces as:
) ( ) ( )
(x y z h P
Fx = xδ δ δ − (3.19a)
) ( ) ( )
(x y z h P
Fy = yδ δ δ − (3.19b)
) ( ) ( )
(x y z h
P
Fz = zδ δ δ − (3.19c)
The Dirac detla function is a mathematical artifice for representing an extremely localized function with a finite total area. For example, δ(x−ξ) is the limit of a spike-like function of x, which is zero almost everywhere expect very near x=ξ. That is, the Dirac detla function has an extremely sharp peak in such a way that its area
above the x axis is unity, i.e.,
⎪⎩
⎪⎨
⎧
∉
= ∈
∫
ab (x− )dx 10 ifif ξ ((aa,,bb),), ξ ξδ
(3.20) where (a, b) stands for the open interval between a and b, excluding the end point. As
long as the point of concentration ξ lies between a and b, the upper and lower limits can be replaced by (−∞,∞). Therefore, an alternative definition of the Dirac detla function is:
1 )
( − =
∫
−∞∞ δ x ξ dx(3.21) In mechanics, δ(x−ξ) may symbolize a concentrated force, i.e., the limit of a
pressure distribution with a sharply peaked intensity around x=ξ and a unit body forces per unit volume (Chiang,1997).