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Fundamentals of the Mechanics of Composites
1 CHAPTER 1
F U N D A M E N T A L S OF THE MECHANICS OF COMPOSITES 1.1
INTRODUCTION
The Hooke's law for an elastic material is given by σ c € ϋ = ijk£ k£
U-la)
where ay and ey are the stress and stain components, respectively, and Cy k£ are the elastic stiffnesses. The repetition of a Latin index (whether superscript or subscript) in a term will denote a summation. The inverse relation (1.1a) is given by e S ij = ijk£ °U 0-lb) where Sy k£ are the elastic compliances. The relationship between C i j£k and Sjjk£ is of the form S C S S 52 ijpq pqk£ = (*ik }1 + ii j k ) / The elastic stiffnesses of isotropic materials are given by C i j£k = λ iy Ski + μ (% 5 j£ + Sd
fijk )
(1.2)
where λ and μ are the Lamé constants and Sy is the Kronecker delta. The Young's modulus E, Poisson's ratio ν and bulk modulus Κ are related in the form Ε = μ (3λ+2μ)/(λ+μ) ν = λ/[2(λ+μ)] Κ = Λ + 2μ/3 Defining the deviatoric stress and strain by σ s ij = α - ^ k V e e e 5 ij = ij - kk i j / 3
(1.3)
3
the Hooke's law for isotropic materials can be written in the form
0-4)
2
Let us introduce the following notation for Cy k^: (11) — 1 , (22) — 2 , (33) — 3 (12) = (21) — 4 , (13) = (31) — 5 , (23) = (32) —
6
(1.5)
The Hooke's law for a transversely isotropic material, with the axis of sym metry oriented in the x 1-direction, is C 0 0 0 €l l '71 1 "11 0 c 0 0 C2 2 ^22 22 ^23 0 0 0 33 '73 3 0 0 C44 23 *2 1 1 e 0 c symm. i3 '13
44
2
'12
^22~^2s)
i' L 2
(1.6) involving five independent constants. The elements Cy can be expressed in terms of the engineering constants in the form c E 4 i i
=
A
+
* "A >
C
2 V
12 =
C 22
*
* K
= + 5 0
.
E
T / ( ! + ^T) »
C 23 = / c - 0 . 5 E x/ ( l + i / x) ,
C 66 - ( C 22 - C 2 )3/ 2 .
(1.7a)
Here E A, y A are the axial Young's modulus and Poisson's ratio, G A is the axial shear modulus, and Ε τ, νΎ are respectively, the transverse Young's modulus and Poisson's ratio. Here κ is given by κ = 0.25 E A/ [ 0 . 5 ( l - i / T) ( E A/ E T) Equivalently, E A
C =
l l
"
2 CC 2 12/( 22
+ C 2l)
" A = c 1 / 2( c 22 + C 2 )3 ,
»
v\\
3 Ε τ = [ C 1 (1C 22 + C 2 )3 - 2 C 1 ]2( C 22 - C 2 )3/ ( C 1 C1 22 2 ν C ) Ύ ~ (^11^23 " C i ) / ( C C 2 n 22 GA= C
44
C 1 )2 ,
(1.7b)
The transverse shear modulus is given by G T = E x/ [ 2 ( l + i ^ x) ] For an orthotropic material in which a symmetry exists with respect to three mutually orthogonal planes we have e 0 0 0 €n c 12 c 13 7' 1 1 c 0 0 0 £22 ^ 2 2 22 ^23 c 0 0 0 (1.8) 233ê 7' 3 3 ^33 0 0 2 C 1 2 12 symm. 0 2 13 e 7'13 2 3 ^66 23 which involves nine independent constants. 1.2 REPRESENTATIVE VOLUME ELEMENT This is a sample of the composite material that (a) is structurally entirely typical of the whole composite on average, and (b) contains a sufficient number of material phases, i.e., it should be large compared to the scales of microstructure, but is still small compared to the entire body. A representative volume element as defined would retain and represent the properties of the composite medium, and furthermore, these properties would be insensitive to values of boundary conditions (surface values of traction and displacement) as long as these values are macroscopically uniform. That is, they fluctuate about a mean with a wavelength small compared with the dimensions of the volume, and the effects of such fluctuations become insignificant within a few wavelengths of the surface. The reason for emphasizing the concept of representative volume element is that it appears to provide a valuable dividing boundary between continuum theories and microscopic theories. For scales larger than the representative volume element we can use continuum mechanics and reproduce properties of the material as a whole. Below the scale of the representative volume element we must consider the microstructure of the material.
4 1.3 VOLUMETRIC AVERAGING Under conditions of an imposed macroscopically homogeneous stress or deformation field on the representative volume element, the average stress and strain are respectively defined by
~°i>
=7
J
a y dV
(1.9)
where V is the volume of the representative volume element. 1.4 HOMOGENEOUS BOUNDARY CONDITIONS Homogeneous boundary conditions applied on the surface of a homogene ous body will produce a homogeneous field there. Such boundary condi tions are obtained by imposing displacements at the boundary S in the form
Ui(S)-4
jX
(1.11)
where £y are constant strains. Alternatively, tractions can be imposed on S such that ti = σ?, HJ where
(1.12)
are constant stresses and η is the unit outward normal vector to
1.5 AVERAGE STRAIN THEOREM To calculate the average strains in a composite material it appears that one must solve the elasticity problem of the representative volume element sub jected to the displacements homogeneous boundary conditions (1.11). It turns out that the average strains ëy are identical to the constant strains ey applied on S. This is the average strain theorem which can be established as follows. The strain-displacement relations are
5 (1.13) Substituting (1.13) in (1.10) yields x)
2 V i „J «
(uf
J vT
J + u(V) dV J+
'
2)
(uf
Jv
1
J
2) + u ( ) dV h
(1.14)
2
where " Γ and "2" denote phase 1 and 2 of the two phased composites with V ls V 2 being the volumes occupied by the two phases. The use of the Gauss theorem
f u i Pl dV = f uj n p dS
Jv
(1.15)
Js
implies that
2 Vé„
«
(1 1} 2) J 1 (u?> n= + u ( n:) dS (u. ) n: + u ( n.) dS + ι J JS
(1.16)
where S x and S 2 are the bounding surfaces of phases 1 and 2, respectively. The surfaces S x and S 2 contain the interfaces S 12 and the external surface S. Assuming perfect contact between the phases, i.e., UW = u . « on S 12 , 1
1
12
(1.17)
'
it follows that the contributions from S 12 in the two integrals in eqn.(1.16) cancel each other. This leads to £ y
2V
(u ; n 3 +
Uj
n ;) dS
(1.18)
Substituting in eqn. (1.18) the homogeneous boundary conditions (1.11), yields the final relations (1.19)
6 In the case when there is no perfect bonding between the two phases of the composite such that the jump of the displacement across S 12 denoted by [ u j is not zero: [Uil-uW-
upUo
o n S 1 ,2
(1.20)
the average strain in the composite would be still given by (1.18), but with (1.10) replaced by
'
i j
=
Vj i €i
d V
" 2V I
s
y
( [ U i l Dj
"
t U j ] D i )
^
( L 2 1 )
For perfect contact [ u j = 0 at S 12 and eqn. (1.21) reduces to (1.10). 1.6 AVERAGE STRESS THEOREM The homogeneous boundary conditions with tractions applied on S, produce a stress field in the composite whose average, σ^·, is identical to the con stant stress <7y. This follows from the average stress theorem. To this end, consider the equilibrium equations in the absence of body forces: *Uj = 0 which implies that
CT x ( ik j),k
(1.22)
x σ x - °ik,k fij + ϋ j,k
- °ij jk = "α
23
0· )
Substituting this relation in (1.9) provides
Vây =
J yv
(σΛ
"
)Xk dV j
(1.24)
By Gauss theorem (1.15) we have
J 'Sx
s2
Since the tractions are continuous at the interfaces S 1 ,2 i.e.,
7 1 οί. ) nW = - S y
j
2 n( )
υ
on S,. ,
j
12
(1.26)
the contributions from S 12 to the two integral cancel each other and eqn. (1.25) reduces to
v
°ii -
ik
Jss
CT xn j k dS
.f
=
s
n j ! kd S
Js
x
dV
(1.27)
Jv
Thus
1.7 EFFECTIVE ELASTIC MODULI Consider a representative volume element of the composite and subject it to homogeneous displacement boundary conditions (1.11), which produce a uniform strain e~ in a homogeneous material. By the average strain theo rem, eg are also in the composite. A computation of σΥ} would yield the effective elastic constants. To this end, suppose that k the induced dis placement field due to e\i = 1 at S is denoted by u | ^ ( x ) . By a linear superposition k£ ( x ) = e° uj ) (x) U i ki
(1.29)
where a summation should be carried over k and I. The resulting strain is given by
= "2 Al (u[f
+ u j f ))
The stress at point χ is determined from the Hooke's law in the form
(1.30)
8 σ
ϋ « = \ At («g? * »2?)
0.31)
where C y ^ x ) are the space dependent elastic moduli of the composite material. The elastic moduli can assume the values of the moduli 09
ypq
ijpq
and
of the phases,
The volume average of the stresses over the representative volume ele ment is determined from eqn. (1.9). It follows that ^U-C^iki
(1-32)
where
and i k£ = e\i by the average strain theorem. Thus the average stress is related to the average strain by the effective elastic moduli C . ^ . By imposing on the composite material homogeneous traction boundary conditions (1.12), the following relation can be similarly obtained
where S * t/ are the effective compliances.
1.8 RELATIONS BETWEEN AVERAGES - A DIRECT APPROACH Let us consider the homogeneous boundary conditions (1.11) according to which displacements are applied on the surface S of a representative volume element V: Ui(S) - 6°y Xj
(1.35)
For a two-phase composite with perfect bonding between the constituents 1 U
ij
2 U
(1.36)
9 where ca with α = 1,2 denote the volume fractions of the phases, and
; («) » _L f W J(< dV ,t ij
v
« JvV
ij
(1.37)
which is the average strain in the phase ot. Similarly, α (τί ) dV
(1.38)
ά.. = lC â.W + r *(?)
(1.39)
â.W = - i τ
α and
By the average strain theorem we have
êy=4
(1.40)
Using (1.32), (1.40), (1.39) and (1.36) yields
where
H - S £? e)
c
i
(1
·
42)
has been used for the constitutive law in the phase. Eqn. (1.41) implies that the effective moduli can be readily determined from the elastic moduli of the phases provided the average strain e ^ in the phase a=2 is known. A similar derivation using traction boundary conditions (1.12) would provide that
^ - ^ • « . « Φ - Φ ^
(1.43)
10 1.9 RELATIONS BETWEEN AVERAGES - E N E R G Y APPROACH The effective properties have been defined in terms of the explicit relations (1.32) involving average stress and strain in the representative volume element. Presently, the effective properties are derived from energy equivalence. Consider an elastic medium denoted by index 1 containing an elastic inclusion denoted by 2 (the proof is established for one inclusion, but can readily be generalized to the case of multiple inclusions). The external surface is denoted by S, and the matrix - inclusion interface by S 1 .2 Under the homogeneous boundary conditions Ui(S) = 6° u Xj ,
(1.44)
the external work is given by W = i j ti uj dS = \ 4
j^ ik
n k j XdS
(1.45)
Using Gauss's theorem in conjunction with
it follows that
W = - €*>·
[
ay dV +
f
tit Xj dS
(1.46)
Consider now the same geometry but this time replace the inclusion by the matrix material. Under boundary conditions (1.44), the displacement 0 field in this all matrix medium is homogeneous and given throughout by xx{ where the superscript in the displacement indicates that it = ey Xj 5 belongs to the all matrix geometry. Using (1.47) (U
+
u
4k>
n
Vi
i '
·
4 )8
11
W
= \ 4 $ £ } K,£ + u ) dV C
v
£ik
+
I |
s
q u° dS
Applying again the Gauss's theorem to the first integral yields,
K,£
+u dV € *,k> = °kr U
n £ dS + t%
x r n k dS
Js
Jvx - J
Js
( u k n £ + u £ n k) dS S 12
(1.49)
where perfect contact at the interface is assumed. Noting that
1
x r nt dS = Stt V S
(1.50)
and defining o\t = C g , 4
(1.51)
which is the stress field produced by (1.44) in a homogeneous all matrix medium, yields W
= "2
4
V +i f S (t, u° - t? U ) dS i
(1.52)
12
This work should be equal to the strain energy denoted by U. Let U o -
\
C $£
4
«°K£
V
(1-53)
be the strain energy in the all matrix homogeneous material. It follows that
12
U = U 0+ i f S ( t j u î - t ° U ) di S
(1.54)
12
This is a version of Eshelby's formula for the special case of homogeneous boundary conditions (1.44). Under the homogeneous boundary conditions (1.44), the work integral calculated from surface tractions and displacements can be written as w
=
\ 1 Οe = j U
dS
=u ï \% 4
*
J
d Sx
i
σ n x \ O £- IT σ * k j = 2 ϋ ϋ
V
It follows from eqn. (1.53) and (1.54) that
C 4 At - $£ 4 4*
+ u u f S (*i i - i t?) ^
(1.56)
12
which provides a definition of the effective moduli from energy considera tions (i.e. through the equality of the strain energy stored in the hetero geneous media to that stored in the equivalent homogeneous media). The definition of the effective moduli on the bases of the direct approach (eqn.(1.41)) and energy considerations (eqn.(1.56)) are equivalent. This can be seen by contracting (1.41) with e~ yielding
c,;< «I «·* -
4
At * «. (c& - <$>
; g .·
Using eqns. (1.37), (1.42) and (1.51), this can be transformed to
<.. 57)
13 Using
= 0 , U j = £ ; J Xj and the Gauss theorem in eqn. (1.58) yields
(1.56). REFERENCES Christensen, R.M., 1979, Mechanics of Composite Materials (Wiley, New York). Hashin, Z., 1972, Theory of fiber reinforced materials, NASA CR-1974. Hill, R., 1963, J. Mech. Phys. Solids 1 1 , 357. Hill, R., 1964, J. Mech. Phys. Solids 12, 199. Mura, T., 1987, Micromechanics of Defects in Solids (Martinus Nijhoff, The Hague).
F U N D A M E N T A L S OF THE MECHANICS OF COMPOSITES 1.1
INTRODUCTION
The Hooke's law for an elastic material is given by σ c € ϋ = ijk£ k£
U-la)
where ay and ey are the stress and stain components, respectively, and Cy k£ are the elastic stiffnesses. The repetition of a Latin index (whether superscript or subscript) in a term will denote a summation. The inverse relation (1.1a) is given by e S ij = ijk£ °U 0-lb) where Sy k£ are the elastic compliances. The relationship between C i j£k and Sjjk£ is of the form S C S S 52 ijpq pqk£ = (*ik }1 + ii j k ) / The elastic stiffnesses of isotropic materials are given by C i j£k = λ iy Ski + μ (% 5 j£ + Sd
fijk )
(1.2)
where λ and μ are the Lamé constants and Sy is the Kronecker delta. The Young's modulus E, Poisson's ratio ν and bulk modulus Κ are related in the form Ε = μ (3λ+2μ)/(λ+μ) ν = λ/[2(λ+μ)] Κ = Λ + 2μ/3 Defining the deviatoric stress and strain by σ s ij = α - ^ k V e e e 5 ij = ij - kk i j / 3
(1.3)
3
the Hooke's law for isotropic materials can be written in the form
0-4)
2
Let us introduce the following notation for Cy k^: (11) — 1 , (22) — 2 , (33) — 3 (12) = (21) — 4 , (13) = (31) — 5 , (23) = (32) —
6
(1.5)
The Hooke's law for a transversely isotropic material, with the axis of sym metry oriented in the x 1-direction, is C 0 0 0 €l l '71 1 "11 0 c 0 0 C2 2 ^22 22 ^23 0 0 0 33 '73 3 0 0 C44 23 *2 1 1 e 0 c symm. i3 '13
44
2
'12
^22~^2s)
i' L 2
(1.6) involving five independent constants. The elements Cy can be expressed in terms of the engineering constants in the form c E 4 i i
=
A
+
* "A >
C
2 V
12 =
C 22
*
* K
= + 5 0
.
E
T / ( ! + ^T) »
C 23 = / c - 0 . 5 E x/ ( l + i / x) ,
C 66 - ( C 22 - C 2 )3/ 2 .
(1.7a)
Here E A, y A are the axial Young's modulus and Poisson's ratio, G A is the axial shear modulus, and Ε τ, νΎ are respectively, the transverse Young's modulus and Poisson's ratio. Here κ is given by κ = 0.25 E A/ [ 0 . 5 ( l - i / T) ( E A/ E T) Equivalently, E A
C =
l l
"
2 CC 2 12/( 22
+ C 2l)
" A = c 1 / 2( c 22 + C 2 )3 ,
»
v\\
3 Ε τ = [ C 1 (1C 22 + C 2 )3 - 2 C 1 ]2( C 22 - C 2 )3/ ( C 1 C1 22 2 ν C ) Ύ ~ (^11^23 " C i ) / ( C C 2 n 22 GA= C
44
C 1 )2 ,
(1.7b)
The transverse shear modulus is given by G T = E x/ [ 2 ( l + i ^ x) ] For an orthotropic material in which a symmetry exists with respect to three mutually orthogonal planes we have e 0 0 0 €n c 12 c 13 7' 1 1 c 0 0 0 £22 ^ 2 2 22 ^23 c 0 0 0 (1.8) 233ê 7' 3 3 ^33 0 0 2 C 1 2 12 symm. 0 2 13 e 7'13 2 3 ^66 23 which involves nine independent constants. 1.2 REPRESENTATIVE VOLUME ELEMENT This is a sample of the composite material that (a) is structurally entirely typical of the whole composite on average, and (b) contains a sufficient number of material phases, i.e., it should be large compared to the scales of microstructure, but is still small compared to the entire body. A representative volume element as defined would retain and represent the properties of the composite medium, and furthermore, these properties would be insensitive to values of boundary conditions (surface values of traction and displacement) as long as these values are macroscopically uniform. That is, they fluctuate about a mean with a wavelength small compared with the dimensions of the volume, and the effects of such fluctuations become insignificant within a few wavelengths of the surface. The reason for emphasizing the concept of representative volume element is that it appears to provide a valuable dividing boundary between continuum theories and microscopic theories. For scales larger than the representative volume element we can use continuum mechanics and reproduce properties of the material as a whole. Below the scale of the representative volume element we must consider the microstructure of the material.
4 1.3 VOLUMETRIC AVERAGING Under conditions of an imposed macroscopically homogeneous stress or deformation field on the representative volume element, the average stress and strain are respectively defined by
~°i>
=7
J
a y dV
(1.9)
where V is the volume of the representative volume element. 1.4 HOMOGENEOUS BOUNDARY CONDITIONS Homogeneous boundary conditions applied on the surface of a homogene ous body will produce a homogeneous field there. Such boundary condi tions are obtained by imposing displacements at the boundary S in the form
Ui(S)-4
jX
(1.11)
where £y are constant strains. Alternatively, tractions can be imposed on S such that ti = σ?, HJ where
(1.12)
are constant stresses and η is the unit outward normal vector to
1.5 AVERAGE STRAIN THEOREM To calculate the average strains in a composite material it appears that one must solve the elasticity problem of the representative volume element sub jected to the displacements homogeneous boundary conditions (1.11). It turns out that the average strains ëy are identical to the constant strains ey applied on S. This is the average strain theorem which can be established as follows. The strain-displacement relations are
5 (1.13) Substituting (1.13) in (1.10) yields x)
2 V i „J «
(uf
J vT
J + u(V) dV J+
'
2)
(uf
Jv
1
J
2) + u ( ) dV h
(1.14)
2
where " Γ and "2" denote phase 1 and 2 of the two phased composites with V ls V 2 being the volumes occupied by the two phases. The use of the Gauss theorem
f u i Pl dV = f uj n p dS
Jv
(1.15)
Js
implies that
2 Vé„
«
(1 1} 2) J 1 (u?> n= + u ( n:) dS (u. ) n: + u ( n.) dS + ι J JS
(1.16)
where S x and S 2 are the bounding surfaces of phases 1 and 2, respectively. The surfaces S x and S 2 contain the interfaces S 12 and the external surface S. Assuming perfect contact between the phases, i.e., UW = u . « on S 12 , 1
1
12
(1.17)
'
it follows that the contributions from S 12 in the two integrals in eqn.(1.16) cancel each other. This leads to £ y
2V
(u ; n 3 +
Uj
n ;) dS
(1.18)
Substituting in eqn. (1.18) the homogeneous boundary conditions (1.11), yields the final relations (1.19)
6 In the case when there is no perfect bonding between the two phases of the composite such that the jump of the displacement across S 12 denoted by [ u j is not zero: [Uil-uW-
upUo
o n S 1 ,2
(1.20)
the average strain in the composite would be still given by (1.18), but with (1.10) replaced by
'
i j
=
Vj i €i
d V
" 2V I
s
y
( [ U i l Dj
"
t U j ] D i )
^
( L 2 1 )
For perfect contact [ u j = 0 at S 12 and eqn. (1.21) reduces to (1.10). 1.6 AVERAGE STRESS THEOREM The homogeneous boundary conditions with tractions applied on S, produce a stress field in the composite whose average, σ^·, is identical to the con stant stress <7y. This follows from the average stress theorem. To this end, consider the equilibrium equations in the absence of body forces: *Uj = 0 which implies that
CT x ( ik j),k
(1.22)
x σ x - °ik,k fij + ϋ j,k
- °ij jk = "α
23
0· )
Substituting this relation in (1.9) provides
Vây =
J yv
(σΛ
"
)Xk dV j
(1.24)
By Gauss theorem (1.15) we have
J 'Sx
s2
Since the tractions are continuous at the interfaces S 1 ,2 i.e.,
7 1 οί. ) nW = - S y
j
2 n( )
υ
on S,. ,
j
12
(1.26)
the contributions from S 12 to the two integral cancel each other and eqn. (1.25) reduces to
v
°ii -
ik
Jss
CT xn j k dS
.f
=
s
n j ! kd S
Js
x
dV
(1.27)
Jv
Thus
1.7 EFFECTIVE ELASTIC MODULI Consider a representative volume element of the composite and subject it to homogeneous displacement boundary conditions (1.11), which produce a uniform strain e~ in a homogeneous material. By the average strain theo rem, eg are also in the composite. A computation of σΥ} would yield the effective elastic constants. To this end, suppose that k the induced dis placement field due to e\i = 1 at S is denoted by u | ^ ( x ) . By a linear superposition k£ ( x ) = e° uj ) (x) U i ki
(1.29)
where a summation should be carried over k and I. The resulting strain is given by
= "2 Al (u[f
+ u j f ))
The stress at point χ is determined from the Hooke's law in the form
(1.30)
8 σ
ϋ « = \
0.31)
where C y ^ x ) are the space dependent elastic moduli of the composite material. The elastic moduli can assume the values of the moduli 09
ypq
ijpq
and
of the phases,
The volume average of the stresses over the representative volume ele ment is determined from eqn. (1.9). It follows that ^U-C^iki
(1-32)
where
and i k£ = e\i by the average strain theorem. Thus the average stress is related to the average strain by the effective elastic moduli C . ^ . By imposing on the composite material homogeneous traction boundary conditions (1.12), the following relation can be similarly obtained
where S * t/ are the effective compliances.
1.8 RELATIONS BETWEEN AVERAGES - A DIRECT APPROACH Let us consider the homogeneous boundary conditions (1.11) according to which displacements are applied on the surface S of a representative volume element V: Ui(S) - 6°y Xj
(1.35)
For a two-phase composite with perfect bonding between the constituents 1 U
ij
2 U
(1.36)
9 where ca with α = 1,2 denote the volume fractions of the phases, and
; («) » _L f W J(< dV ,t ij
v
« JvV
ij
(1.37)
which is the average strain in the phase ot. Similarly, α (τί ) dV
(1.38)
ά.. = lC â.W + r *(?)
(1.39)
â.W = - i τ
α and
By the average strain theorem we have
êy=4
(1.40)
Using (1.32), (1.40), (1.39) and (1.36) yields
where
H - S £? e)
c
i
(1
·
42)
has been used for the constitutive law in the phase. Eqn. (1.41) implies that the effective moduli can be readily determined from the elastic moduli of the phases provided the average strain e ^ in the phase a=2 is known. A similar derivation using traction boundary conditions (1.12) would provide that
^ - ^ • « . « Φ - Φ ^
(1.43)
10 1.9 RELATIONS BETWEEN AVERAGES - E N E R G Y APPROACH The effective properties have been defined in terms of the explicit relations (1.32) involving average stress and strain in the representative volume element. Presently, the effective properties are derived from energy equivalence. Consider an elastic medium denoted by index 1 containing an elastic inclusion denoted by 2 (the proof is established for one inclusion, but can readily be generalized to the case of multiple inclusions). The external surface is denoted by S, and the matrix - inclusion interface by S 1 .2 Under the homogeneous boundary conditions Ui(S) = 6° u Xj ,
(1.44)
the external work is given by W = i j ti uj dS = \ 4
j^ ik
n k j XdS
(1.45)
Using Gauss's theorem in conjunction with
it follows that
W = - €*>·
[
ay dV +
f
tit Xj dS
(1.46)
Consider now the same geometry but this time replace the inclusion by the matrix material. Under boundary conditions (1.44), the displacement 0 field in this all matrix medium is homogeneous and given throughout by xx{ where the superscript in the displacement indicates that it = ey Xj 5 belongs to the all matrix geometry. Using (1.47) (U
+
u
4k>
n
Vi
i '
·
4 )8
11
W
= \ 4 $ £ } K,£ + u ) dV C
v
£ik
+
I |
s
q u° dS
Applying again the Gauss's theorem to the first integral yields,
K,£
+u dV € *,k> = °kr U
n £ dS + t%
x r n k dS
Js
Jvx - J
Js
( u k n £ + u £ n k) dS S 12
(1.49)
where perfect contact at the interface is assumed. Noting that
1
x r nt dS = Stt V S
(1.50)
and defining o\t = C g , 4
(1.51)
which is the stress field produced by (1.44) in a homogeneous all matrix medium, yields W
= "2
4
V +i f S (t, u° - t? U ) dS i
(1.52)
12
This work should be equal to the strain energy denoted by U. Let U o -
\
C $£
4
«°K£
V
(1-53)
be the strain energy in the all matrix homogeneous material. It follows that
12
U = U 0+ i f S ( t j u î - t ° U ) di S
(1.54)
12
This is a version of Eshelby's formula for the special case of homogeneous boundary conditions (1.44). Under the homogeneous boundary conditions (1.44), the work integral calculated from surface tractions and displacements can be written as w
=
\ 1 Οe = j U
dS
=u ï \% 4
*
J
d Sx
i
σ n x \ O £- IT σ * k j = 2 ϋ ϋ
V
It follows from eqn. (1.53) and (1.54) that
C 4 At - $£ 4 4*
+ u u f S (*i i - i t?) ^
(1.56)
12
which provides a definition of the effective moduli from energy considera tions (i.e. through the equality of the strain energy stored in the hetero geneous media to that stored in the equivalent homogeneous media). The definition of the effective moduli on the bases of the direct approach (eqn.(1.41)) and energy considerations (eqn.(1.56)) are equivalent. This can be seen by contracting (1.41) with e~ yielding
c,;< «I «·* -
4
At * «. (c& - <$>
; g .·
Using eqns. (1.37), (1.42) and (1.51), this can be transformed to
<.. 57)
13 Using
= 0 , U j = £ ; J Xj and the Gauss theorem in eqn. (1.58) yields
(1.56). REFERENCES Christensen, R.M., 1979, Mechanics of Composite Materials (Wiley, New York). Hashin, Z., 1972, Theory of fiber reinforced materials, NASA CR-1974. Hill, R., 1963, J. Mech. Phys. Solids 1 1 , 357. Hill, R., 1964, J. Mech. Phys. Solids 12, 199. Mura, T., 1987, Micromechanics of Defects in Solids (Martinus Nijhoff, The Hague).