The classical way to treat incompressible linear elastic materials is to use the inverse constitutive relationship (strain as a function of the stress), based on the compliance tensor, in place of the direct constitutive equation (stress as a function of strain), based on the elasticity (stiffness) tensor. This is because the elasticity tensor is affected by a diverging bulk modulus, requiring in order to allow the material to sustain any hydrostatic load, and is therefore not defined. In this work we show that there is a part of the elasticity tensor that can be saved also for incompressible materials, by filtering the components that deal with hydrostatic loads. The procedure is based on the treatment of incompressibility by means of the constant of isochoric motion, i.e. of conservation of volume, and fourth-order tensor algebra

Federico, S.

Grillo, Alfio

Wittum, G.

PORTO Publications Open Repository TOrino

Politecnico di Torino
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[Article] Considerations on incompressibility in Linear Elasticity
Original Citation:
Federico S.; Grillo A.; Wittum G. (2009). Considerations on incompressibility in Linear Elasticity. In:
IL NUOVO CIMENTO C, vol. 32 n. 1, pp. 81-87. - ISSN 2037-4909
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DOI 10.1393/ncc/i2009-10336-5
Colloquia: GCM8
IL NUOVO CIMENTO Vol. 32 C, N. 1 Gennaio-Febbraio 2009
Considerations on incompressibility in linear elasticity
S. Federico(1)(∗), A. Grillo(2) and G. Wittum(2)
(1) Department of Mechanical and Manufacturing Engineering, The University of Calgary
2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada
(2) G-CSC, Goethe Universita¨t Frankfurt
Kettenhofweg 139, D-60325 Frankfurt am Main, Germany
(ricevuto il 3 Aprile 2009; pubblicato online il 29 Maggio 2009)
Summary. — The classical way to treat incompressible linear elastic materials is
to use the inverse constitutive relationship (strain as a function of the stress), based
on the compliance tensor, in place of the direct constitutive equation (stress as a
function of strain), based on the elasticity (stiﬀness) tensor. This is because the
elasticity tensor is aﬀected by a diverging bulk modulus, required in order to allow
the material to sustain any hydrostatic load, and is therefore not deﬁned. In this
work we show that there is a part of the elasticity tensor that can be “saved” also for
incompressible materials, by “ﬁltering” the components that deal with hydrostatic
loads. The procedure is based on the treatment of incompressibility by means of
the constraint of isochoric motion, i.e. of conservation of volume, and fourth-order
tensor algebra.
PACS 46.05.+b – General theory of continuum mechanics of solids.
PACS 46.25.-y – Static elasticity.
1. – Introduction
Materials for which the stiﬀness under hydrostatic loads is very high compared to that
under distortional loads are often assumed to be incompressible. In the realm of linear
elasticity, the bulk elastic modulus, κ, of incompressible materials is thought to diverge,
so that the elasticity (stiﬀness) tensor L is not deﬁned. Therefore, “incompressible”
materials are typically represented by means of the compliance tensor Z (see, e.g. [1]),
so that the diverging modulus is reduced to a compliance approaching zero.
In this work, we aim at giving a representation of the (stiﬀness) elasticity tensor L
that is suitable for the description of incompressible materials, with no diverging terms.
In order to achieve this, we treat incompressibility by imposing the kinematical constraint
of isochoric motion (e.g., [2]). In linear elasticity, the linearised kinematical constraint
(∗) E-mail: salvatore.federico@ucalgary.ca
c© Societa` Italiana di Fisica 81
82 S. FEDERICO, A. GRILLO and G. WITTUM
is expressed by prescribing the inﬁnitesimal strain ε to be purely deviatoric, i.e. such
that tr(ε) = 0, and by adding the term −pI to the stress, with the hydrostatic pressure
p having the meaning of Lagrange multiplier associated with the kinematical constraint
tr(ε) = 0.
With no loss of generality, we show that, for the case of isotropic materials, this
procedure automatically ﬁlters the diverging terms in the elasticity tensor L, and provides
the correct components of the deviatoric part of the elasticity tensor.
2. – General framework
Let us denote by E2 and E4 the spaces of second- and fourth-order tensors, respec-
tively, and let Q : E2 → R be a quadratic form deﬁned by
(2.1) Q(A) =
1
2
A : Q : A,
where A ∈ E2 is a symmetric second-order tensor, and Q ∈ E4 is a fourth-order tensor
equipped with both diagonal and pair symmetry [3, 4]. The full-symmetric identity in
the space of fourth-order tensors, I ∈ E4, can be decomposed as [3]
(2.2) I = K+M,
where K and M are fourth-order tensors deﬁned by
K =
1
3
I ⊗ I,(2.3)
M = I− 1
3
I ⊗ I,(2.4)
respectively, and I is the identity in the space of second-order tensors E2. It can be
shown [3] that tensors K and M are orthogonal, in the sense that K : M = M : K = O
(O ∈ E4 being the null fourth-order tensor), and idempotent, i.e. K : K = K, and
M : M = M.
For any symmetric second-order tensor A ∈ E2, the application of tensors K and M
onto A gives the spherical and deviatoric part of A, respectively, i.e.
(2.5) K : A = AsI =
1
3
tr(A)I, and M : A = Ad = A− 1
3
tr(A)I.
It can be proven (cf., for example, [3]) that tensors K and M constitute a basis
spanning the subspace of E4 of symmetric, isotropic fourth-order tensors. Therefore, if
Q is assumed to be isotropic, it is possible to write
(2.6) Q = qKK+ qMM,
where qK and qM are the scalar coeﬃcients of the representation of Q in the basis {K,M}.
Because of the orthogonality of this representation, it is easy to show that the inverse of
Q is given by
(2.7) Q−1 =
1
qK
K+
1
qM
M.
CONSIDERATIONS ON INCOMPRESSIBILITY IN LINEAR ELASTICITY 83
Use of the decomposition of I given in eq. (2.2) leads to the identities
A = I : A = K : A +M : A,(2.8)
Q = I : Q : I.(2.9)
By use of eqs. (2.2) and (2.9), and the symmetry of A, the quadratic form Q(A) in
eq. (2.1) can be rewritten as
(2.10) Q(A) =
1
2
A : [K : Q : K+K : Q : M +M : Q : K+M : Q : M] : A.
This result can be recast in a more compact notation by introducing the following de-
composition of Q(A):
QK(A) =
1
2
A : [K : Q : K] : A,(2.11)
Qmixed(A) =
1
2
A : [K : Q : M +M : Q : K] : A,(2.12)
QM (A) =
1
2
A : [M : Q : M] : A.(2.13)
Indeed, use of deﬁnitions (2.11)–(2.13) allows for rewriting eq. (2.10) as
(2.14) Q(A) = QK(A) + Qmixed(A) + QM (A).
Under the assumption of isotropy of Q (cf. eq. (2.6)), the quantity Qmixed(A) vanishes
identically because tensors K and M are orthogonal to each other. The orthogonality of
tensors K and M also implies that the products K : Q : K and M : Q : M, which feature in
the deﬁnitions of QK(A) and QM (A) (cf., eqs. (2.11) and (2.13), respectively), become
(2.15) K : Q : K = qKK, and M : Q : M = qMM.
Finally, the quantity Q(A) in eq. (2.14) becomes
(2.16) Q(A) =
1
2
3qK(As)2 +
1
2
qMA
d : Ad,
where As = 13 tr(A) is the spherical component of tensor A, and A
d : Ad = tr[(Ad)T Ad].
If we give Q, Q and A the meaning of strain energy potential W , linear elasticity
tensor L and inﬁnitesimal strain ε, respectively, the quadratic form (2.1) reads
(2.17) W (ε) =
1
2
ε : L : ε.
By assuming that L is isotropic, it is possible to decompose it into
(2.18) L = LKK+ LMM = 3κK+ 2μM,
84 S. FEDERICO, A. GRILLO and G. WITTUM
where κ and μ in LK = 3κ and LM = 2μ are the bulk modulus and the shear modulus,
respectively. Following the procedure outlined above, we can write eq. (2.17) in the same
form as (2.16), i.e.
(2.19) W (ε) =
1
2
[9κ(εv)2 + 2μ εd : εd],
where εv = εs is the volumetric (spherical) component of the strain. If we instead give
Q, Q and A the meaning of complementary potential Ψ (the Legendre transform of W ),
linear compliance tensor Z and Cauchy stress σ, respectively, the quadratic form (2.1)
reads
(2.20) Ψ(σ) =
1
2
σ : Z : σ,
and an isotropic compliance tensor Z can be decomposed into
(2.21) Z = ZKK+ ZMM =
1
3κ
K+
1
2μ
M,
where the components ZK = 13κ and ZM =
1
2μ are the algebraic inverses of LK = 3κ
and LM = 2μ, according to (2.7). Following again the same procedure, eq. (2.20) can be
written as
(2.22) Ψ(σ) =
1
2
[
1
κ
(σh)2 +
1
2μ
σd : σd
]
,
where σh = σs = 13 tr(σ) represents the hydrostatic (spherical) contribution of the stress.
In eqs. (2.19) and (2.22), the volumetric contribution to W (ε) and the hydrostatic
contribution Ψ(σ) are completely decoupled from their deviatoric counterparts. Equa-
tion (2.22) becomes useful, for example, for a treatment of incompressibility based on
the algebraic properties of fourth-order tensors. For the sake of brevity, this paper has
been limited to the case of isotropic elastic materials.
3. – Constrained strain energy function
According to eq. (2.5), the strain tensor ε ∈ E2 admits the additive decomposition
given by tensors K and M:
(3.1) ε = K : ε +M : ε = εvI + εd.
It is therefore possible to introduce two functions, denoted by K : E2 → R and M :
E2 → E2, respectively, such that K(A) = As, andM(A) = M : A = Ad.
If we let W : E2 → R denote the strain energy, then there exists a function U :
R× E2 → R such that the following identity holds:
(3.2) W (ε) = U(K(ε),M(ε)).
CONSIDERATIONS ON INCOMPRESSIBILITY IN LINEAR ELASTICITY 85
3.1. Incompressible case. – In the theoretical framework of linear elasticity, incom-
pressibility is modelled by imposing the kinematic constraint tr(ε) = 0. This constraint
can be accounted for by introducing a new function V : E2 → R such that eq. (3.2) is
re-deﬁned as
(3.3) W (ε, π) = U(K(ε),M(ε), π) = V (M(ε))− πK(ε),
where π is the unknown scalar Lagrange multiplier associated with the constraint
K(ε) = 0.
Within the quadratic approximation of the strain energy density, and under the as-
sumption of isotropic material, the procedure shown in sect. 2 leads to express V (M(ε))
in eq. (3.3) as
(3.4) V (M(ε)) = 1
2
LMM(ε) :M(ε) = 12LMε
d : εd,
where, according to the isotropic decomposition of the elasticity tensor given in eq. (2.6),
LM = 2μ denotes the coeﬃcient of M, i.e. twice the shear modulus.
By substituting eq. (3.4) back into eq. (3.3), the strain energy density reads
(3.5) W (ε, π) =
1
2
(2μ)M(ε) :M(ε)− πK(ε),
and the Cauchy stress is given by
(3.6) σ =
∂W
∂ε
(ε, π) = −1
3
πI + 2μεd.
If we now set π/3 = p, eq. (3.6) becomes
(3.7) σ = −pI + 2μεd,
where the deviatoric part of stress is given by σd = 2μεd. This result can be used in
order to perform the Legendre transformation of V (M(ε)) = V (εd) into a function of
the stress, Ψ(σd). Indeed, by introducing the compliance ZM , and writing εd = ZMσd,
we obtain
(3.8) Ψ(σd) =
1
2
ZMσ
d : σd.
For eq. (3.8) to be consistent with the expression of the elastic energy as a function of the
Cauchy stress, given in eq. (2.22), it is necessary that ZM = 12μ , and that the material
constant ZK vanishes identically, i.e. ZK = 0 (cf., for example, [1]). Thus, for the case
of an incompressible linear elastic material, there can be no hydrostatic contribution
to the compliance. This result can be seen as the outcome of a “ﬁltering procedure”
that, applied to Ψ(σ) in eq. (2.22), gives the correct expression of the elastic energy
to be used in the case of an incompressible linear elastic material. Finally, the inverse
Legendre transformation of the “ﬁltered” energy, i.e. Ψ(σd), gives the strain energy for
incompressibility, i.e. V (εd).
86 S. FEDERICO, A. GRILLO and G. WITTUM
3.2. Comparison with the case of an incompressible ﬂuid . – We assumed that func-
tion V depends only on the deviatoric part of the strain tensor, M(ε), on the basis
of considerations similar to those made by Eringen in [5], when treating the case of an
incompressible ﬂuid. Indeed, by writing the Helmholtz free-energy density of an incom-
pressible ﬂuid in a form analogous to eq. (3.3), the energy term denoted by V cannot
depend on the ﬂuid mass density. This result, however, can be generalised to the case
of a ﬂuid whose mass density, rather than being constant (this would be the case of a
truly incompressible ﬂuid), is prescribed through a constitutive (or state) function of a
set of thermodynamic state variables other than the pressure. If the mass density is inde-
pendent of ﬂuid pressure, the Helmholtz free-energy density cannot be transformed into
the Gibbs free-energy density by performing a Legendre transformation between mass
density and pressure. On the other hand, by writing
(3.9) W (, ϕ) = V (ϕ)− π
[
1

− 1
0(ϕ)
]
,
where ϕ denotes a generic thermodynamic variable (other than pressure) assigned to the
ﬂuid, and 0 a given function of ϕ, it is possible to deﬁne the Gibbs-like potential
(3.10) G(π, ϕ) := V (ϕ) +
π
0(ϕ)
,
where the Lagrange multiplier π is identiﬁed with the ﬂuid pressure. The case of incom-
pressible ﬂuid is recovered if the constitutive law (ϕ) reduces to requiring that the ﬂuid
mass density equals a reference constant, i.e.  = ¯0.
4. – Filtering procedure
In subsect. 3.1, we assumed that function V depends only on the deviatoric part εd
of the strain tensor. In general, however, we may let V depend on the whole strain
tensor, ε, under the assumption that a “ﬁctitious” elasticity tensor, L∗, exists. Under
this assumption, function V becomes
(4.1) V (ε) =
1
2
ε : L∗ : ε.
However, this expression can be also written as
(4.2) V (ε) =
1
2
3L∗K(ε
v)2 +
1
2
L∗Mε
d : εd.
Now, for consistency with the theory shown in subsect. 3.1, the material constant L∗K has
to be chosen equal to zero, i.e. L∗K = 0, while the material constant L
∗
M has to coincide
with the “true” one, and is therefore deﬁned as L∗M = LM = 2μ.
The result L∗K = 0 does not contradict the fact that, for an incompressible material,
the bulk modulus diverges. Indeed, consistency with previous theories is retrieved by
admitting that the “true” bulk modulus is given by
(4.3) κ =
1
3
L∗K + κ0.
CONSIDERATIONS ON INCOMPRESSIBILITY IN LINEAR ELASTICITY 87
If this is the case, requiring L∗K = 0 and that κ0 diverges, implies that the true bulk
modulus, κ, diverges too, as the theory predicts.
5. – Conclusion
The procedure shown in sect. 4 applies in a straightforward way when it is used
directly on Ψ, for it leads to setting ZK = 0. When this procedure is applied to the
strain energy density function, it should be used in order to identify function Ψ. Consis-
tency with previous theories is retrieved by admitting that the “true” volumetric elastic
coeﬃcient, LK , diverges (whereas the compliance ZK is zero). This result is not in con-
tradiction with the fact that LK contains additively the ﬁctitious coeﬃcient L∗K which,
in turn, vanishes identically.
∗ ∗ ∗
This study was partially funded by: Schulich School of Engineering, University of
Calgary, Canada, and Alberta Ingenuity Fund, Canada (SF); Johann-Wolfgang-Goethe
Universita¨t Frankfurt am Main, Germany, and “Bundesministerium fu¨r Wirtschaft und
Technnologie–BMWi” (German Ministry for Economy and Technology) under contract
02E10326 (AG and GW).
REFERENCES
[1] Destrade M., Martin P. A. and Ting T. C. T., J. Mech. Phys. Solids, 50 (2002) 1453.
[2] Ogden R. W., Non-Linear Elastic Deformations (Dover Publications, Inc., Mineola, NY,
USA) 1984.
[3] Walpole L. J., Adv. Appl. Mech., 21 (1981) 169.
[4] Federico S., Grillo A. and Herzog W., J. Mech. Phys. Solids, 52 (2004) 2309.
[5] Eringen A. C., Mechanics of Continua (John Wiley and Sons, Inc., New York, NY, USA)
1980.


Considerations on incompressibility in Linear Elasticity

Federico S.

GRILLO, ALFIO

Wittum G.

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

Federico, Salvatore

Grillo,, Alfio

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2009-10336-5Colloquia: GCM8IL NUOVO CIMENTO Vol. 32 C, N. 1 Gennaio-Febbraio 2009Considerations on incompressibility in linear elasticityS. Federico(1)(∗), A. Grillo(2) and G. Wittum(2)(1) Department of Mechanical and Manufacturing Engineering, The University of Calgary2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada(2) G-CSC, Goethe Universita¨t FrankfurtKettenhofweg 139, D-60325 Frankfurt am Main, Germany(ricevuto il 3 Aprile 2009; pubblicato online il 29 Maggio 2009)Summary. — The classical way to treat incompressible linear elastic materials isto use the inverse constitutive relationship (strain as a function of the stress), basedon the compliance tensor, in place of the direct constitutive equation (stress as afunction of strain), based on the elasticity (stiffness) tensor. This is because theelasticity tensor is affected by a diverging bulk modulus, required in order to allowthe material to sustain any hydrostatic load, and is therefore not defined. In thiswork we show that there is a part of the elasticity tensor that can be “saved” also forincompressible materials, by “filtering” the components that deal with hydrostaticloads. The procedure is based on the treatment of incompressibility by means ofthe constraint of isochoric motion, i.e. of conservation of volume, and fourth-ordertensor algebra.PACS 46.05.+b – General theory of continuum mechanics of solids.PACS 46.25.-y – Static elasticity.1. – IntroductionMaterials for which the stiffness under hydrostatic loads is very high compared to thatunder distortional loads are often assumed to be incompressible. In the realm of linearelasticity, the bulk elastic modulus, κ, of incompressible materials is thought to diverge,so that the elasticity (stiffness) tensor L is not defined. Therefore, “incompressible”materials are typically represented by means of the compliance tensor Z (see, e.g. [1]),so that the diverging modulus is reduced to a compliance approaching zero.In this work, we aim at giving a representation of the (stiffness) elasticity tensor Lthat is suitable for the description of incompressible materials, with no diverging terms.In order to achieve this, we treat incompressibility by imposing the kinematical constraintof isochoric motion (e.g., [2]). In linear elasticity, the linearised kinematical constraint(∗) E-mail: salvatore.federico@ucalgary.cac© Societa` Italiana di Fisica 8182 S. FEDERICO, A. GRILLO and G. WITTUMis expressed by prescribing the infinitesimal strain ε to be purely deviatoric, i.e. suchthat tr(ε) = 0, and by adding the term −pI to the stress, with the hydrostatic pressurep having the meaning of Lagrange multiplier associated with the kinematical constrainttr(ε) = 0.With no loss of generality, we show that, for the case of isotropic materials, thisprocedure automatically filters the diverging terms in the elasticity tensor L, and providesthe correct components of the deviatoric part of the elasticity tensor.2. – General frameworkLet us denote by E2 and E4 the spaces of second- and fourth-order tensors, respec-tively, and let Q : E2 → R be a quadratic form defined by(2.1) Q(A) =12A : Q : A,where A ∈ E2 is a symmetric second-order tensor, and Q ∈ E4 is a fourth-order tensorequipped with both diagonal and pair symmetry [3, 4]. The full-symmetric identity inthe space of fourth-order tensors, I ∈ E4, can be decomposed as [3](2.2) I = K+M,where K and M are fourth-order tensors defined byK =13I ⊗ I,(2.3)M = I− 13I ⊗ I,(2.4)respectively, and I is the identity in the space of second-order tensors E2. It can beshown [3] that tensors K and M are orthogonal, in the sense that K : M = M : K = O(O ∈ E4 being the null fourth-order tensor), and idempotent, i.e. K : K = K, andM : M = M.For any symmetric second-order tensor A ∈ E2, the application of tensors K and Monto A gives the spherical and deviatoric part of A, respectively, i.e.(2.5) K : A = AsI =13tr(A)I, and M : A = Ad = A− 13tr(A)I.It can be proven (cf., for example, [3]) that tensors K and M constitute a basisspanning the subspace of E4 of symmetric, isotropic fourth-order tensors. Therefore, ifQ is assumed to be isotropic, it is possible to write(2.6) Q = qKK+ qMM,where qK and qM are the scalar coefficients of the representation of Q in the basis {K,M}.Because of the orthogonality of this representation, it is easy to show that the inverse ofQ is given by(2.7) Q−1 =1qKK+1qMM.CONSIDERATIONS ON INCOMPRESSIBILITY IN LINEAR ELASTICITY 83Use of the decomposition of I given in eq. (2.2) leads to the identitiesA = I : A = K : A +M : A,(2.8)Q = I : Q : I.(2.9)By use of eqs. (2.2) and (2.9), and the symmetry of A, the quadratic form Q(A) ineq. (2.1) can be rewritten as(2.10) Q(A) =12A : [K : Q : K+K : Q : M +M : Q : K+M : Q : M] : A.This result can be recast in a more compact notation by introducing the following de-composition of Q(A):QK(A) =12A : [K : Q : K] : A,(2.11)Qmixed(A) =12A : [K : Q : M +M : Q : K] : A,(2.12)QM (A) =12A : [M : Q : M] : A.(2.13)Indeed, use of definitions (2.11)–(2.13) allows for rewriting eq. (2.10) as(2.14) Q(A) = QK(A) + Qmixed(A) + QM (A).Under the assumption of isotropy of Q (cf. eq. (2.6)), the quantity Qmixed(A) vanishesidentically because tensors K and M are orthogonal to each other. The orthogonality oftensors K and M also implies that the products K : Q : K and M : Q : M, which feature inthe definitions of QK(A) and QM (A) (cf., eqs. (2.11) and (2.13), respectively), become(2.15) K : Q : K = qKK, and M : Q : M = qMM.Finally, the quantity Q(A) in eq. (2.14) becomes(2.16) Q(A) =123qK(As)2 +12qMAd : Ad,where As = 13 tr(A) is the spherical component of tensor A, and Ad : Ad = tr[(Ad)T Ad].If we give Q, Q and A the meaning of strain energy potential W , linear elasticitytensor L and infinitesimal strain ε, respectively, the quadratic form (2.1) reads(2.17) W (ε) =12ε : L : ε.By assuming that L is isotropic, it is possible to decompose it into(2.18) L = LKK+ LMM = 3κK+ 2μM,84 S. FEDERICO, A. GRILLO and G. WITTUMwhere κ and μ in LK = 3κ and LM = 2μ are the bulk modulus and the shear modulus,respectively. Following the procedure outlined above, we can write eq. (2.17) in the sameform as (2.16), i.e.(2.19) W (ε) =12[9κ(εv)2 + 2μ εd : εd],where εv = εs is the volumetric (spherical) component of the strain. If we instead giveQ, Q and A the meaning of complementary potential Ψ (the Legendre transform of W ),linear compliance tensor Z and Cauchy stress σ, respectively, the quadratic form (2.1)reads(2.20) Ψ(σ) =12σ : Z : σ,and an isotropic compliance tensor Z can be decomposed into(2.21) Z = ZKK+ ZMM =13κK+12μM,where the components ZK = 13κ and ZM =12μ are the algebraic inverses of LK = 3κand LM = 2μ, according to (2.7). Following again the same procedure, eq. (2.20) can bewritten as(2.22) Ψ(σ) =12[1κ(σh)2 +12μσd : σd],where σh = σs = 13 tr(σ) represents the hydrostatic (spherical) contribution of the stress.In eqs. (2.19) and (2.22), the volumetric contribution to W (ε) and the hydrostaticcontribution Ψ(σ) are completely decoupled from their deviatoric counterparts. Equa-tion (2.22) becomes useful, for example, for a treatment of incompressibility based onthe algebraic properties of fourth-order tensors. For the sake of brevity, this paper hasbeen limited to the case of isotropic elastic materials.3. – Constrained strain energy functionAccording to eq. (2.5), the strain tensor ε ∈ E2 admits the additive decompositiongiven by tensors K and M:(3.1) ε = K : ε +M : ε = εvI + εd.It is therefore possible to introduce two functions, denoted by K : E2 → R and M :E2 → E2, respectively, such that K(A) = As, andM(A) = M : A = Ad.If we let W : E2 → R denote the strain energy, then there exists a function U :R× E2 → R such that the following identity holds:(3.2) W (ε) = U(K(ε),M(ε)).CONSIDERATIONS ON INCOMPRESSIBILITY IN LINEAR ELASTICITY 853.1. Incompressible case. – In the theoretical framework of linear elasticity, incom-pressibility is modelled by imposing the kinematic constraint tr(ε) = 0. This constraintcan be accounted for by introducing a new function V : E2 → R such that eq. (3.2) isre-defined as(3.3) W (ε, π) = U(K(ε),M(ε), π) = V (M(ε))− πK(ε),where π is the unknown scalar Lagrange multiplier associated with the constraintK(ε) = 0.Within the quadratic approximation of the strain energy density, and under the as-sumption of isotropic material, the procedure shown in sect. 2 leads to express V (M(ε))in eq. (3.3) as(3.4) V (M(ε)) = 12LMM(ε) :M(ε) = 12LMεd : εd,where, according to the isotropic decomposition of the elasticity tensor given in eq. (2.6),LM = 2μ denotes the coefficient of M, i.e. twice the shear modulus.By substituting eq. (3.4) back into eq. (3.3), the strain energy density reads(3.5) W (ε, π) =12(2μ)M(ε) :M(ε)− πK(ε),and the Cauchy stress is given by(3.6) σ =∂W∂ε(ε, π) = −13πI + 2μεd.If we now set π/3 = p, eq. (3.6) becomes(3.7) σ = −pI + 2μεd,where the deviatoric part of stress is given by σd = 2μεd. This result can be used inorder to perform the Legendre transformation of V (M(ε)) = V (εd) into a function ofthe stress, Ψ(σd). Indeed, by introducing the compliance ZM , and writing εd = ZMσd,we obtain(3.8) Ψ(σd) =12ZMσd : σd.For eq. (3.8) to be consistent with the expression of the elastic energy as a function of theCauchy stress, given in eq. (2.22), it is necessary that ZM = 12μ , and that the materialconstant ZK vanishes identically, i.e. ZK = 0 (cf., for example, [1]). Thus, for the caseof an incompressible linear elastic material, there can be no hydrostatic contributionto the compliance. This result can be seen as the outcome of a “filtering procedure”that, applied to Ψ(σ) in eq. (2.22), gives the correct expression of the elastic energyto be used in the case of an incompressible linear elastic material. Finally, the inverseLegendre transformation of the “filtered” energy, i.e. Ψ(σd), gives the strain energy forincompressibility, i.e. V (εd).86 S. FEDERICO, A. GRILLO and G. WITTUM3.2. Comparison with the case of an incompressible fluid . – We assumed that func-tion V depends only on the deviatoric part of the strain tensor, M(ε), on the basisof considerations similar to those made by Eringen in [5], when treating the case of anincompressible fluid. Indeed, by writing the Helmholtz free-energy density of an incom-pressible fluid in a form analogous to eq. (3.3), the energy term denoted by V cannotdepend on the fluid mass density. This result, however, can be generalised to the caseof a fluid whose mass density, rather than being constant (this would be the case of atruly incompressible fluid), is prescribed through a constitutive (or state) function of aset of thermodynamic state variables other than the pressure. If the mass density is inde-pendent of fluid pressure, the Helmholtz free-energy density cannot be transformed intothe Gibbs free-energy density by performing a Legendre transformation between massdensity and pressure. On the other hand, by writing(3.9) W (, ϕ) = V (ϕ)− π[1− 10(ϕ)],where ϕ denotes a generic thermodynamic variable (other than pressure) assigned to thefluid, and 0 a given function of ϕ, it is possible to define the Gibbs-like potential(3.10) G(π, ϕ) := V (ϕ) +π0(ϕ),where the Lagrange multiplier π is identified with the fluid pressure. The case of incom-pressible fluid is recovered if the constitutive law (ϕ) reduces to requiring that the fluidmass density equals a reference constant, i.e.  = ¯0.4. – Filtering procedureIn subsect. 3.1, we assumed that function V depends only on the deviatoric part εdof the strain tensor. In general, however, we may let V depend on the whole straintensor, ε, under the assumption that a “fictitious” elasticity tensor, L∗, exists. Underthis assumption, function V becomes(4.1) V (ε) =12ε : L∗ : ε.However, this expression can be also written as(4.2) V (ε) =123L∗K(εv)2 +12L∗Mεd : εd.Now, for consistency with the theory shown in subsect. 3.1, the material constant L∗K hasto be chosen equal to zero, i.e. L∗K = 0, while the material constant L∗M has to coincidewith the “true” one, and is therefore defined as L∗M = LM = 2μ.The result L∗K = 0 does not contradict the fact that, for an incompressible material,the bulk modulus diverges. Indeed, consistency with previous theories is retrieved byadmitting that the “true” bulk modulus is given by(4.3) κ =13L∗K + κ0.CONSIDERATIONS ON INCOMPRESSIBILITY IN LINEAR ELASTICITY 87If this is the case, requiring L∗K = 0 and that κ0 diverges, implies that the true bulkmodulus, κ, diverges too, as the theory predicts.5. – ConclusionThe procedure shown in sect. 4 applies in a straightforward way when it is useddirectly on Ψ, for it leads to setting ZK = 0. When this procedure is applied to thestrain energy density function, it should be used in order to identify function Ψ. Consis-tency with previous theories is retrieved by admitting that the “true” volumetric elasticcoefficient, LK , diverges (whereas the compliance ZK is zero). This result is not in con-tradiction with the fact that LK contains additively the fictitious coefficient L∗K which,in turn, vanishes identically.∗ ∗ ∗This study was partially funded by: Schulich School of Engineering, University ofCalgary, Canada, and Alberta Ingenuity Fund, Canada (SF); Johann-Wolfgang-GoetheUniversita¨t Frankfurt am Main, Germany, and “Bundesministerium fu¨r Wirtschaft undTechnnologie–BMWi” (German Ministry for Economy and Technology) under contract02E10326 (AG and GW).REFERENCES[1] Destrade M., Martin P. A. and Ting T. C. T., J. Mech. Phys. Solids, 50 (2002) 1453.[2] Ogden R. W., Non-Linear Elastic Deformations (Dover Publications, Inc., Mineola, NY,USA) 1984.[3] Walpole L. J., Adv. Appl. Mech., 21 (1981) 169.[4] Federico S., Grillo A. and Herzog W., J. Mech. Phys. Solids, 52 (2004) 2309.[5] Eringen A. C., Mechanics of Continua (John Wiley and Sons, Inc., New York, NY, USA)1980.

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Considerations on incompressibility in linear elasticity

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Considerations on incompressibility in Linear Elasticity

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