STEP-BY-STEP SIMPLIFICATION OF THE MICROPOLAR ELASTICITY THEORY TOTHE COUPLE-STRESS AND CLASSICAL ELASTICITY THEORIES
Soroosh Hassanpour∗Mechanical and Mechatronics Engineering
University of WaterlooWaterloo, ON, Canada N2L 3G1Email: [email protected]
G. R. HepplerSystems Design Engineering
University of WaterlooWaterloo, ON, Canada N2L 3G1Email: [email protected]
ABSTRACTThe micropolar elasticity theory provides a useful mate-
rial model for dealing with fibrous, coarse granular, and largemolecule materials. Though being a well-known and well-developed elasticity model, the linear theory of micropolar elas-ticity is not without controversy. Specially simplification of themicroppolar elasticity theory to the couple-stress and classicalelasticity theories and the required conditions on the materialelastic constants for this simplification have not been discussedconsistently. In this paper the linear theory of micropolar elas-ticity is reviewed first. Then the correct approach for a consistentand step-by-step simplification of the micropolar elasticity modelwith six elastic constants to the couple-stress elasticity modelwith four elastic constants and the classical elasticity model withtwo elastic constants is presented. It is shown that the classicalelasticity is a special case of the couple-stress theory which itselfis a special case of the micropolar elasticity theory.
INTRODUCTIONThe classical theory of linear elasticity has a long history
of development and verification and produces acceptable resultsin numerous engineering problems with various structural ma-terials. However, for the cases with large stress gradients (e.g.,in the vicinity of holes and cracks) or materials with significantmicrostructure contribution (e.g. composites, polymers, soil, andbone) the classical theory of elasticity fails to produce acceptableresults. To improve the results of the classical theory of elastic-
∗Address all correspondence to this author.
ity Voit [1] incorporated the effects of couple stresses and gen-eralized the symmetric classical theory of elasticity to the asym-metric couple-stress theory. This was then extended by E. andF. Cosserat [2] who considered a body microrotation field, in-dependent of the body displacement field. Eringen [3] furtherdeveloped Cosserat’s model by including the body microinertiaand renamed it as the micropolar theory of (asymmetric) elastic-ity. Nowacki [4] provides an extensive description of the lineartheory of micropolar elasticity.
Broadly speaking these newer, more elaborate, materialmodels are useful when dealing with materials that have a de-fined internal structure; e.g. fibrous materials such as bone,coarse granular materials such as soil, and large molecule poly-mers such as foams. It is noteworthy that experimental verifi-cation of the micropolar theory for these materials is not fullyaccomplished yet and one is faced with a situation when theoryprecedes experiment.
Although being a well-known and well-developed elasticitymodel, the linear theory of micropolar elasticity is not withoutcontroversy. Especially simplification of the micropolar elas-ticity to the classical elasticity theory and determination of themicropolar material parameters are labeled as inconsistent. Inparticular, whereas the micropolar elasticity model with zero mi-cropolar elastic constants (including a zero micropolar couplemodulus) is considered by some authors to coincide with theclassical elasticity model (e.g. [5], [6], and [7]), there are otherauthors who observed some inconsistencies in the micropolarelasticity model with a zero micropolar couple modulus (e.g. [8]and [7, 9]).
Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014
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This paper suggests that the apparent inconsistencies in themicropolar theory of linear elasticity are mainly a result of theapproach taken or assumed for simplification of the micropo-lar theory of elasticity to the classical theory of elasticity andmay be resolved provided a different simplification approach istaken. We will show, assuming a general linear theory of mi-cropolar elasticity applied to a homogeneous, isotropic, and cen-trally symmetric material, that a new approach, in which a zeromicropolar couple modulus is not required, can be taken for sim-plification of the micropolar theory of elasticity to the classicaltheory of elasticity.
MICROPOLAR ELASTICITY THEORYTo provide a brief overview of the three-dimensional linear
theory of micropolar elasticity, consider a general homogeneous,isotropic, and centrally symmetric elastic body occupying a vol-ume domain V in R3 , bounded by surface S. Assume that thebody undergoes a motion and deformation due to the action ofexternal volume force and moment
→f V and →m
V . A body frameFb and a position vector
→p (with respect to the inertial frame
Fo ) correspond to each representative infinitesimal element ofthe body (see Fig. 1).
o
bp
V
S
dV
1
o x
2
o x
3
o x
1
b x
2
b x
3
b x
dV
FIGURE 1. A GENERAL ELASTIC BODY AND ITS REPRESEN-TATIVE ELEMENT.
In the micropolar elasticity model, the (classical) displace-ment field vector→u is complemented by a microrotation field vec-tor →ϑ (independent of the displacement field). Consequently, thetranslational velocity and acceleration vectors are →u and →u, andthe angular velocity and acceleration vectors are →ϑ and →ϑ [10].
The micropolar deformation is fully described by (asymmet-ric) strain and twist tensors,↔ε and↔τ , which are defined as:
εi j = u j , i− ei jk ϑk,
τi j = ϑ j , i,(1)
where ei jk is the third-order Levi-Civita or permutation tensor.Based on these definitions the following relations can be derived:
ϑi =12
ei jk
(uk , j− ε jk
),
τi j =12
e jkl
(ul ,ki− εkl , i
),
τii = −12
ei jk ε jk , i.
(2)
It is also useful to define the (classical) macrorotation vector →θand the (classical) macrorotation tensor↔θ
× such that:
θi =12
ei jkuk , j,
θ×i j = − ei jk θk = −
12
(u j , i−ui, j
).
(3)
Then the strain and twist tensors can be decomposed into theirsymmetric and antisymmetric (skew-symmetric) parts as:
εi j = εsi j + ε
ai j,
εsi j =
12
(u j , i +ui, j
),
εai j =
12
(u j , i−ui, j
)− ei jkϑk = ei jk
(θk−ϑk
),
(4)
and:
τi j = τsi j + τ
ai j,
τsi j =
12
(ϑ j , i +ϑi, j
),
τai j =
12
(ϑ j , i−ϑi, j
),
(5)
where note that↔εa is a representation of the difference between
the (classical) macrorotation and the (micropolar) microrotation.As shown in Fig. 2, in a micropolar continuum, the (classical
force) stress field tensor ↔σ is completed by a (micropolar) cou-ple stress field tensor
↔χ . Considering the free body diagram in
Fig. 2, for a micropolar elastic body under the action of a generalvolume force
→f V and a general volume moment →m
V the balanceof linear and angular momenta can be written in the followingdifferential form:
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where ρV is the material volume mass density and ıV is the ma-
terial volume microinertia density. Note that a more generalcase is when the material has a tensor of microinertia density
↔ıV , however this paper is confined to the isotropic case where
↔ıV = ıV
↔1 [4].
o
1
o x
2
o x
3
o x
Vm
Vf
2dx
1dx
3dx ij
ij
FIGURE 2. FREE BODY DIAGRAM OF A REPRESENTATIVEELEMENT IN A MICROPOLAR ELASTIC BODY.
Analogous to the decomposition of strain and twist tensors,the force and couple stress tensors can be decomposed into theirsymmetric and antisymmetric parts as:
σi j = σsi j +σ
ai j,
σsi j =
12
(σi j +σ ji
), σ
ai j =
12
(σi j−σ ji
),
(7)
and:
χi j = χsi j +χ
ai j,
χsi j =
12
(χi j +χ ji
), χ
ai j =
12
(χi j−χ ji
).
(8)
Utilizing Eqn. (7) and noting the fact that for any symmetricsecond-order tensor ↔d , ei jkd jk = 0, the equilibrium relations inEqn. (6) can be rewritten as:
σsji, j +σ
aji, j + f V
i = ρV ui,
χ ji, j + ei jk σajk +mV
i = ıVϑ i.
(9)
Solving the second relation of Eqn. (9) for the antisymmetricforce stress tensor ↔σ
a and substituting into the first relation ofEqn. (9) one can rewrite the balance relations as:
σsji, j +
12
ei jk
(χlk , l j +mV
k , j− ıVϑ k , j
)+ f V
i = ρV ui,
ei jk
(χlk , l +mV
k − ıVϑ k
)= 2σ
aji.
(10)
As can be concluded from Eqn. (9) the antisymmetric part of theforce stress tensor, ↔σ
a , couples the linear and angular momentabalance relations.
The linear theory of micropolar elasticity proposed by Erin-gen results in a set of two constitutive relations with six elas-tic constants for a general homogeneous, isotropic, and centrallysymmetric elastic body. These relations have the following form:
σi j =(
µ +κ
)εi j +
(µ−κ
)ε ji +λ εkk1i j,
χi j =(
γ +β
)τi j +
(γ−β
)τ ji +α τkk1i j,
(11)
where 1i j is the Kronecker delta tensor. Among the six elasticconstants denoted in Eqn. (11), µ and λ are the classical Lameparameters (µ is also called Lame shear modulus or shear mod-ulus). The other four constants κ , γ , β , and α are the new elasticconstants usually referred to as the micropolar or Cosserat elas-tic constants. Note that κ is usually called the micropolar couplemodulus. The micropolar constants represent the contribution ofthe material microstructure to the elastic properties of the body.
By decomposing the strain and twist tensors, as given byEqns. (4) and (5), the constitutive relations in Eqn. (11) can berewritten as:
σi j = 2 µ εsi j +λ εkk1i j +2κ ε
ai j,
χi j = 2γ τsi j +α τkk1i j +2β τ
ai j.
(12)
Now a similar decomposition for the force and couple stress ten-sors, as given by Eqns. (7) and (8), gives rise to the followingrelations:
σsi j = 2 µ ε
si j +λ εkk1i j, σ
ai j = 2κ ε
ai j,
χsi j = 2γ τ
si j +α τkk1i j, χ
ai j = 2β τ
ai j,
(13)
where the first relation is identical with the constitutive relationof classical elasticity.
Considering the original constitutive relations in Eqn. (11)and applying the Einstein summation convention on stress ten-sors, ↔σ and
↔χ , it can be shown that:
σkk = 3Bεkk, B = λ +23
µ,
χkk = 3B τkk, B = α +23
γ,
(14)
where B is known as the bulk modulus, and B as dual of the bulkmodulus can be called the tortile or torsional bulk modulus.
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Simplifying the first constitutive relation given by Eqn. (11)for the simple force stress state of uniform tension along axis ox1,where the only nonzero element of the force stress tensor is σ11 ,results in definitions of the (classical) strain Poisson’s ratio ν andthe (classical) Young’s modulus E as:
ν = − ε22
ε11= − ε33
ε11=
λ
2(
µ +λ
) ,E =
σ11
ε11=
µ
(2 µ +3λ
)µ +λ
= 2 µ
(1+ν
).
(15)
The second constitutive relation in Eqn. (11) can be simplifiedto account for the simple couple stress state of uniform torsionalong axis ox1 , where the only nonzero element of the couplestress tensor is χ11 . Then the micropolar twist Poisson’s ratio ξ
and the micropolar tortile or torsional modulus E can be definedas:
ξ = − τ22
τ11= − τ33
τ11=
α
2(
γ +α
) ,E =
χ11
τ11=
γ
(2γ +3α
)γ +α
= 2γ
(1+ξ
).
(16)
Utilizing the constitutive relations in Eqn. (11) and the def-initions of the strain and twist tensors in Eqn. (1) to replace theforce and couple stresses in the balance of momenta relationsgiven by Eqn. (6), the system of partial differential equations(PDEs) representing the equations of motion for a micropolarcontinuum are derived as:(
µ +κ
)ui, j j +
(µ−κ +λ
)u j , ji
+2κ ei jk ϑk , j + f Vi = ρ
V ui,(γ +β
)ϑi, j j +
(γ−β +α
)ϑ j , ji
+2κ
(ei jk uk , j−2ϑi
)+mV
i = ıVϑ i.
(17)
Considering Eqns. (9) and (10) and substituting from the as-sociated constitutive equations and the definitions of the strainand twist tensors into them, the following alternative forms forthe dynamic equations can be attained respectively:
µ ui, j j +(
µ +λ
)u j , ji−2κ ei jk
(θk , j−ϑk , j
)+ f V
i = ρV ui,(
γ +β
)ϑi, j j +
(γ−β +α
)ϑ j , ji +4κ
(θi−ϑi
)+mV
i = ıVϑ i,
(18)
and:
µ ui, j j +(
µ +λ
)u j , ji
+12
ei jk
((γ +β
)ϑk , ll j +mV
k , j− ıVϑ k , j
)+ f V
i = ρV ui,(
γ +β
)ϑi, j j +
(γ−β +α
)ϑ j , ji
+mVi − ıV
ϑ i = −4κ
(θi−ϑi
),
(19)
where it is recalled that for any symmetric second-order tensor
↔d , ei jkd jk = 0.In the linear micropolar elasticity theory the strain energy
density U Ve is expressed as:
2U Ve = σi j εi j +χi j τi j. (20)
By decomposing the force stress, couple stress, strain, and twisttensors into their symmetric and antisymmetric parts this expres-sion can be rewritten as:
2U Ve = σ
si j ε
si j +σ
ai j ε
ai j +χ
si j τ
si j +χ
ai j τ
ai j. (21)
Correspondingly substitutions from the constitutive relations inEqns. (11) and (13) into Eqns. (20) and (21) result in the follow-ing strain energy density expressions:
2U Ve =
(µ +κ
)εi j εi j +
(µ−κ
)ε ji εi j +λ εii ε j j
+(
γ +β
)τi j τi j +
(γ−β
)τ ji τi j +α τii τ j j,
(22)
and:
2U Ve = 2 µε
si j ε
si j +λ εii ε j j +2κε
ai j ε
ai j
+2γ τsi j τ
si j +α τii τ j j +2βτ
ai j τ
ai j.
(23)
The fact that the strain energy density expression shouldhave a positive definite quadratic form imposes the following re-strictions on the material elastic constants [4]:
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volume force→f V and moment →m
V . For such a body, utilizingthe linear micropolar theory of elasticity results in the relationsand definitions given by Eqns. (1)–(24). Now by taking into ac-count the constitutive relations, especially the first relation givenin Eqn. (12), and letting the micropolar couple modulus κ tend toinfinity while noting that the force stress tensor ↔σ should remainfinite, one can conclude that the antisymmetric part of the straintensor, i.e.↔ε
a , should vanish:
εai j = ei jk
(θk−ϑk
)= 0,
ϑi = θi =12
ei jk uk , j.(25)
Consequently, the kinematic relations given by Eqns. (1)–(5) canbe simplified to:
εi j = εsi j + ε
ai j =
12
(u j , i +ui, j
),
εsi j =
12
(u j , i +ui, j
),
εai j = 0,
τi j = τsi j + τ
ai j =
12
e jkl ul ,ki,
τsi j =
14
e jkl ul ,ki +14
eikl ul ,k j,
τai j =
14
e jkl ul ,ki−14
eikl ul ,k j,
τii = 0.
(26)
For such a case (i.e. when κ → ∞ and thus εai j = 0 and τii =
0), the constitutive relations given by Eqn. (13) take the form:
σsi j = 2 µ εi j +λ εkk1i j, σ
ai j = ∞×0 = ¿,
χsi j = 2γ τ
si j, χ
ai j = 2β τ
ai j,
where ¿ is a symbol that represents a numerical quantity whosemagnitude cannot be determined (an indeterminate quantity).However, as τii = 0 this form impose an unnecessary constrainton the couple stress tensor, that is χii = 0. To remove this con-straint it can be assumed that in the constitutive relations givenby Eqn. (13) (in addition to the micropolar couple modulus κ )the micropolar twist coefficient α goes to infinity as well. Usingthe second relation of Eqn. (14), this assumption gives rise to thefollowing form for the constitutive relations in Eqn. (13):
σsi j = 2 µ εi j +λ εkk1i j, σ
ai j = ∞×0 = ¿,
χsi j−
13
χkk1i j = 2γ τsi j, χ
ai j = 2β τ
ai j,
χii = ∞×0 = ¿,
(27)
or equivalently:
σi j−σai j = 2 µ εi j +λ εkk1i j,
χi j−13
χkk1i j =(
γ +β
)τi j +
(γ−β
)τi j,
σai j = ∞×0 = ¿, χii = ∞×0 = ¿.
(28)
It is worthwhile to note here that the indeterminacy of the asym-metric force stress tensor↔σ
a and the summation of normal couplestresses χii means they cannot be obtained from the constitutiverelations and (if possible) one should use the kinetic balance re-lations to determine them.
Whereas the bulk modulus B, strain Poisson’s ratio ν , andYoung’s modulus E defined in Eqns. (14) and (15) remain un-changed as κ and α tend to infinity, by letting α in Eqns. (14)and (16) go to infinity one can derive the tortile bulk modulus B,twist Poisson’s ratio ξ , and tortile modulus E which correspondto the current case, that is:
B = λ +23
µ, ν =λ
2(
µ +λ
) ,E =
µ
(2 µ +3λ
)µ +λ
= 2 µ
(1+ν
),
B = ∞, ξ =12, E = 3γ.
(29)
By letting κ and α go to infinity, however, the kinetic rela-tions will remain unchanged as no restriction is imposed on theforce and couple stress tensors and, therefore, one can repeat, forexample, the equilibrium relations given by Eqn. (6):
σ ji, j + f Vi = ρ
V ui,
χ ji, j + ei jk σ jk +mVi = ıV ei jk uk , j,
(30)
or more properly the equilibrium relations in Eqn. (10):
σsji, j +
12
ei jk
(χlk , l j +mV
k , j− ıV eklm um, l j
)+ f V
i = ρV ui,
ei jk
(χlk , l +mV
k − ıV eklm um, l
)= 2σ
aji.
(31)
Substitution from the constitutive relations of Eqn. (28) intothe balance relations given by Eqn. (31) (or revision of the mo-tion equations in Eqn. (19) for the case when κ→∞ and α→∞)
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results in the corresponding equations of motion:
µ ui, j j +(
µ +λ
)u j , ji +
14
ei jk
(γ +β
)eklm um, lnn j
+12
ei jk
(mV
k , j−12
ıV eklm um, l j
)+ f V
i = ρV ui,
2σaji−
13
ei jk χll ,k =12
ei jk
(γ +β
)eklm um, lnn
+ ei jk
(mV
k −12
ıV eklm um, l
).
(32)
Here the first relation of Eqn. (32) corresponds to a set of threePDEs, enough for determination of the displacement vector.However, the second relation of Eqn. (32) also corresponding toa set of three PDEs does not provide enough information to com-pute the undetermined parts of the force and couple stress ten-sors, i.e. ↔σ
a and χii , from a known displacement vector. Indeed,there are only three equations that should be used to determinefour unknowns (three elements of ↔σ
a and the scalar χii).Finally when κ → ∞ and α → ∞, the strain energy density
U Ve can be written as (compare this to Eqns. (20) and (21) given
previously for a general micropolar case):
2U Ve =
(σi j−σ
ai j
)εi j +
(χi j−
13
χkk1i j
)τi j
= σsi j εi j +
(χ
si j−
13
χkk1i j
)τ
si j +χ
ai j τ
ai j.
(33)
Now substituting from Eqn. (27) or (28) into Eqn. (33) results inthe following expression for strain energy density U V
e (in com-parison with Eqn. (22) or (23)):
2U Ve = 2 µεi j εi j +λ εii ε j j +
(γ +β
)τi j τi j +
(γ−β
)τ ji τi j
= 2 µεi j εi j +λ εii ε j j +2γ τsi j τ
si j +2βτ
ai j τ
ai j,
(34)which have a positive definite form provided (compared to theconditions in Eqn. (24)):
µ > 0, 2 µ +3λ > 0,γ > 0, β > 0.
(35)
The relations given by Eqns. (25)–(35) (especially after ig-noring the terms containing the material microinertia density ıV )are known as the relations of the indeterminate couple-stress the-ory [11] (since, as mentioned previously, the number of equa-tions in Eqn. (32) are not enough for complete determination ofengaged unknowns, the couple-stress theory is usually called theindeterminate couple-stress theory).
To sum up, one can conclude that the couple-stress theorywith four material elastic constants µ, λ , γ, and β is a special caseof the more general micropolar theory of elasticity consisting ofsix material elastic constants µ , κ , λ , γ , β , and α which can beobtained mathematically as κ → ∞ and α → ∞ (it is also usualto neglect the microinertia effects by assuming ıV → 0).
SIMPLIFICATION TO CLASSICAL ELASTICITYConsider the relations of the couple-stress theory, given in
the previous section by i.e. Eqns. (25)–(35), which were obtainedfrom the relations of the micropolar elasticity theory by assumingκ,α→∞. By taking another step and letting the micropolar twistcoefficients γ and β and the microinertia density ıV go to zero,the couple-stress theory relations will further simplify to a set ofrelations in which the effects of the couple stresses are almostnegligible. For such a case the constitutive relations will be:
σi j−σai j = 2 µ εi j +λ εkk1i j,
χi j−13
χkk1i j = 0,
σai j = ¿, χii = ¿.
Although one can continue while keeping the indeterminate por-tion of the couple stress, i.e. χii , it is more useful to neglect thecouple stresses completely and consequently write the constitu-tive relations as:
σi j−σai j = 2 µ εi j +λ εkk1i j,
σai j = ¿,
χi j = 0.
(36)
This implies that the couple stresses do not exist and as a resultthere is no need to define the twist tensor and derive the equa-tions related to it. One can accordingly simplify Eqns. (25)–(35)to obtain; the definitions of the strains and the micro or macrorotations:
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the definitions of the bulk modulus, strain Poisson’s ratio, andYoung’s modulus:
B = λ +23
µ, ν =λ
2(
µ +λ
) ,E =
µ
(2 µ +3λ
)µ +λ
= 2 µ
(1+ν
),
(39)
the relations of the balance of linear and angular momenta:
σ ji, j + f Vi = ρ
V ui,
ei jk σ jk +mVi = 0,
(40)
σsji, j + f V
i +12
ei jk mVk , j = ρ
V ui,
ei jk mVk = 2σ
aji,
(41)
µ ui, j j +(
µ +λ
)u j , ji +
12
ei jk mVk , j + f V
i = ρV ui,
2σaji = ei jk mV
k ,(42)
the strain energy density definitions:
2U Ve =
(σi j−σ
ai j
)εi j = σ
si j εi j, (43)
2U Ve = 2 µεi j εi j +λ εii ε j j, (44)
and the conditions under which the strain energy has a positivedefinite quadratic form:
µ > 0, 2 µ +3λ > 0. (45)
The set of relations given by Eqns. (37)–(45) correspond tothe asymmetric theory of classical elasticity [12]. Compared tothe (well-known) symmetric classical elasticity, in the asymmet-ric theory of classical elasticity, although the strain tensor is sym-metric, the force stress tensor can be asymmetric in the presenceof a volume moment distribution. There is no constitutive rela-tion for the antisymmetric part of the stress tensor (as given by
Eqn. (38)) and the antisymmetric stress tensor is instead deter-mined by the angular momentum balance equation (as given byEqn. (41) or (42)). Also, the volume moment distribution ap-pears as an equivalent force distribution in the linear momentumbalance equation (as in Eqn. (41) or (42)).
Accordingly, one can conclude that the asymmetric classicaltheory of elasticity with two material elastic constants µ and λ
is a special case of the more general couple-stress theory includ-ing four material elastic constants µ , λ , γ , and β which can beobtained mathematically as γ → 0, β → 0, and ıV → 0 [13].
Recalling that the couple-stress theory is itself a special caseof the micropolar theory of elasticity, one can obtain the classicaltheory of elasticity directly from the micropolar theory of elas-ticity by letting κ,α → ∞ and γ,β , ıV → 0. In other words, thecouple-stress theory is an intermediate theory derived throughthe process of recovering the classical elasticity theory from themicropolar elasticity theory. This is summarized in the flowchartshown in Fig. 3.
Micropolar
Theory of
Elasticity
Classical
Theory of
Elasticity
0
0Couple-Stress
Theory
,
,
,
,,
FIGURE 3. SEQUENTIAL STRUCTURE FOR SIMPLIFICATIONOF THE MICROPOLAR ELASTICITY THEORY TO THE COUPLE-STRESS AND CLASSICAL ELASTICITY THEORIES.
SUMMARY AND CONCLUSIONSThe controversial nature of the well-developed theory of mi-
cropolar elasticity is a drawback for this more elaborate and com-prehensive material model. This controversy is mainly about therelationships between the micropolar elasticity theory and theclassical elasticity theory and their corresponding material elas-tic constants. To be specific , the micropolar elastic model withzero micropolar elastic constants including a zero couple modu-lus κ , which is traditionally known to coincide with the classicalelastic model, bears (physical) difficulties [8].
One should note that the micropolar couple modulus κ de-termines the strength of coupling between the displacement andlocal rotation fields [14]. Though, simplifying the micropolar
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elasticity for the case κ = 0 is more straightforward, this cor-responds to a decoupling of the rotational and translational de-grees of freedom (DOFs) [15]. Therefore, a micropolar elasticitymodel with κ = 0 corresponds to an elastic continuum in whichthe constitutive particles or cells are free to rotate and indeed inthe presence of a volume moment rotate infinitely (a singularityoccurs in the presence of a volume moment).
This paper presented an alternative approach for a step-by-step simplification of the micropolar elasticity model to the clas-sical elasticity model in which letting κ → 0 was not required.Indeed, it was shown that the micropolar elasticity model withsix elastic constants µ , κ , λ , γ , β , and γ will be simplified tothe couple-stress elasticity model with four elastic constants µ ,λ , γ , and β provided κ,α → ∞ (and usually ıV → 0). Then let-ting γ,β , ıV → 0, will further simply the model to the classicalelasticity model with two elastic constants µ and λ .
The presented approach is beneficial as, first, it does not en-force a zero micropolar couple modulus, and second, it suggestsa sequential relationship between the micropolar, couple-stress,and classical elasticity models (as shown in Fig. 3). Note that inthe conventional approach the classical and couple-stress elas-ticity models corresponds to two different simplified cases ofthe micropolar elasticity model, respectively where κ → 0 andκ → ∞. This conventional approach is inconsistent with the factthat the couple-stress elasticity theory is a generalized form ofthe classical elasticity theory [16].
It should be noted that the presented approach and suggestedconditions for recovery of the classical elasticity model from themicropolar elasticity model are consistent with those suggestedin [17–19] which were obtained from a different point of view,that is considering the relationships between the structural char-acteristics of classical and micropolar gyroelastic materials.
ACKNOWLEDGMENTThis work was partially supported by the Natural Science
and Engineering Research Council of Canada (NSERC).
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