A rational approach to Cosserat continua, with application to plate and beam theories

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A rational approach to Cosserat continua, with application to plate and beam theories Gianpietro Del Piero a,b,∗ a b

Dipartimento di Ingegneria, Università di Ferrara, Via Saragat 1, 44100 Ferrara, Italy International Research Center M&MoCS, Palazzo Caetani, 04012 Cisterna di Latina, Italy

a r t i c l e

i n f o

Article history: Received 13 June 2013 Received in revised form 22 October 2013 Accepted 3 November 2013 Available online xxx Keywords: Generalized continua Cosserat continua Plate theories Beam theories

a b s t r a c t In a recently proposed approach, a generalized continuum is defined by the specification of the form of the external power, plus some regularity assumptions on the system of the contact actions. Under these assumptions the power can be expressed as a volume integral, the internal power. The conditions of indifference to rigid virtual velocities lead to a reduced form of the internal power, which determines the internal forces and generalized deformations to be related by constitutive equations and to be specified in the boundary conditions. Further restrictions, imposed by kinematic constraints, determine special subclasses of continua. In this context, the equations of some classical plate and beam theories are deduced from those of the three-dimensional Cosserat continuum in a quite simple and natural way. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction In the papers Del Piero (2009, 2014), an alternative to the traditional approaches to the mechanics of generalized continua has been proposed. Both in the new and in the traditional approaches, the starting point is the definition of a list of primary variables. They usually are, but need not be, of kinematic nature. For example, in a continuum with microstructure, the list consists of a macroscopic variable, the displacement vector, plus a number of order parameters, scalar, vectorial or tensorial, describing microstructural properties. With each variable are associated two dual variables, a distance action and a contact action. Their products by virtual variations of the corresponding primary variable define the external power. The traditional approaches are based either on energy minimization (Toupin, 1964; Mindlin and Eshel, 1968) or on the principle of virtual power (Toupin, 1962; Germain, 1973, 1973). In both cases, with each primary variable is associated a generalized deformation. In the approach based on the principle of virtual power, the balance equations of linear and angular momentum are assumed. With each generalized deformation, they associate an internal force. In the energetic approach, the energy is assumed to be a function of the generalized deformations, the internal forces are the partial derivatives of the energy, and the balance equations are deduced from the stationarity condition imposed to the energy.

∗ Tel.: +39 3485121548. E-mail address: [email protected]

A weak point of the energetic approach is that it presumes the existence of an energy. In practice, this restricts its application to hyperelastic materials. For the approach based on the principle of virtual power, a major inconvenience is that ad hoc supplementary balance equations are required for each order parameter, see Del Piero (2009, 2014) for a detailed discussion. The innovative aspect of the proposed theory is that, as a consequence of some regularity assumptions made on the system of contact actions, the internal forces and generalized deformations are deduced from the primary and dual variables directly, that is, without specifying the balance equations or assuming the existence of an energy. The regularity assumptions consist in assuming that the contact actions are ˇ bounded Cauchy fluxes, according to the definition given by Silhav y´ (2008) and based on earlier work of Gurtin and Martins (1976). An illustration of this assumption would require large use of measuretheoretic concepts, and this falls out of the scope of the present communication. For a detailed exposition, the interested reader is addressed to the paper Del Piero (2014). As shown in the following sections, a consequence of the assumed regularity of the contact actions is the possibility of transforming the external power into a volume integral, of the same form of the internal power of the traditional approaches. This integral is subject to indifference requirements, which determine the balance equations for each specific generalized continuum. Thus, in the proposed approach the balance equations are determined by the assumed form of the external power, by the regularity of the contact actions, and by the indifference requirements. In particular, the indifference requirements determine a reduced form of the internal power, in which only the products of the active

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Please cite this article in press as: Del Piero, G., A rational approach to Cosserat continua, with application to plate and beam theories. Mech. Re. Commun. (2013), http://dx.doi.org/10.1016/j.mechrescom.2013.11.003

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internal forces by the unconstrained, or free, generalized deformations appear. That is, they determine the variables to be involved in the constitutive equations. A class of continua, determined by the form of the external power and by the indifference requirements, may include a large variety of materials, each one defined by a specific constitutive equation. In the present paper, after a short introduction to micropolar continua in Section 2, in Section 3 the proposed approach is applied to the class of continua defined by the external power and indifference requirements usually assumed for the Cosserat continua (Mindlin and Tiersten, 1962; Toupin, 1962). Section 4 deals with the constrained Cosserat continuum obtained by imposing the coincidence between the local rotations associated with the macroscopic and microscopic deformations. The rest of the paper is devoted to the deduction of the field equations of the classical theories of plates, Section 5, and beams, Section 6, from the equations of the three-dimensional Cosserat continuum. This is done simply by imposing supplementary kinematic constraints. This deduction is more simple and direct than the traditional ones, see e.g. Erbay (2000), Eringen (1999), Ies¸an (2009), Neff (2003), Rubin (2000) and the large bibliography reported in Altenbach et al. (2010). 2. Micropolar continua A micropolar continuum is a continuum whose primary variables are the macroscopic displacement u plus a finite number of vector fields d˛ , the directors. The latter represent material directions which characterize the body’s response at the microscopic level.1 The distance actions associated with u and d˛ are the body forces b and the body microforces ˇ˛ , and the corresponding contact actions are the surface tractions s and the surface microtractions  ˛ . The integral2



Pext (, v, ˛ ) =



(b(x) · v(x) + ˇ˛ (x) · ˛ (x)) dV +

(s(x) · v(x) ∂



+  ˛ (x) · ˛ (x)) dA

(2.1)

is the virtual power exerted on the portion  of the body under virtual variations v, ˛ of u and d˛ , respectively. In Del Piero (2014) it has been proved that, if the contact actions are bounded Cauchy fluxes, not only they have surface densities s and  ˛ , but also volume densities, here denoted by −f and −˛ . This double characterization of the contact actions is expressed by the pseudobalance equations







f (x) dV + 

˛ (x) dV +

s(x) dA = 0, ∂





 ˛ (x) dA = 0, ∂

(2.2)

which hold for all parts  of the body. In spite of their similarity with the balance equation of the linear momentum, these are not balance equations, since they only express the identity of the representation of the same contact action as a surface and a volume integral. However, by Noll’s theorem on the dependence of the contact forces on the normal3 and by Cauchy’s tetrahedron theorem, these equations imply the existence of second-order tensor fields T and ˛ such that s(x) = T (x) n,

 ˛ (x) = ˛ (x) n,

(2.3)

where n is the exterior unit normal to ∂ at x. Thus, in the present approach, the existence of the stress tensors is not a consequence of the balance of linear momentum, but of the regularity assumed for the contact forces. Substituting Eqs. (2.3) into the expression of the external power and using the divergence theorem, we get the local forms of the pseudobalance equations4 divT + f = 0,

div˛ + ˛ = 0,

(2.4)

with which the right-hand side of (2.1) takes the form



((b − f ) · v + T · ∇ v + (ˇ˛ − ˛ ) · ˛ + ˛ · ∇ ˛ ) dV .

(2.5)



This integral is called the internal power, and is denoted by Pint (, v, ˛ ). In contrast with the similar integral appearing in the method of virtual power, it does not come from an independent assumption on the structure of the internal forces, but is an alternative form of the virtual power (2.1), deduced using the pseudobalance equations. The internal power tells us that, for a micropolar continuum, the generalized deformations are v, ∇ v, ˛ and ∇ ˛ , and the corresponding internal forces are (b − f), T, (ˇ˛ − ˛ ), and ˛ . The indifference requirements impose that the internal power be insensitive to rigid virtual velocities v, ˛ . They restrict the internal power to products of active internal forces by free generalized deformations. The restrictions may differ from one class of continua to another, since the definition of rigid depends on the physical nature of the order parameters. In the following Sections, the restrictions appropriate to Cosserat continua are explicitly determined. 3. The Cosserat continuum A Cosserat continuum is a micropolar continuum equipped with an orthonormal triplet of directors d˛ , which stays orthonormal in all deformed configurations. In a virtual deformation, at any point x the triplet performs an infinitesimal rigid rotation, described by a vector ω(x). The corresponding virtual velocities are ˛ (x) = ω(x) × d˛ (x),

(3.1)

and, again omitting the argument x, the corresponding virtual powers are ˇ˛ · ˛ = ˇ˛ · ω × d˛ = d˛ × ˇ˛ · ω,

(3.2)

 ˛ · ˛ =  ˛ · ω × d˛ = d˛ ×  ˛ · ω. After defining the body couple and the surface couple c = d˛ × ˇ ˛ ,

m = d˛ ×  ˛ ,

the external power takes the form



Pext (, v, ω) =

(3.3)



(b · v + c · ω) dV + 

(s · v + m · ω) dA.

(3.4)

∂

Comparison with (2.1) shows that a Cosserat continuum is a micropolar continuum with a single order parameter, whose virtual variation is ω. By consequence, there is only one microscopic pseudobalance equation (2.2)2 . Eqs. (2.3) reduce to s = T n,

m = M n,

(3.5)

with T the Cauchy stress tensor and M the couple-stress tensor. Moreover, there are only two local pseudobalance equations 1 For example, the orientations of the crystalline lattice or the directions of crystal defects. 2 Here and in the following, summation over repeated indices is assumed. 3 Noll (1959). For a proof in the context of measure theory, see Del Piero (2014, Theorem 4.3).

divT + f = 0,

4

divM +  = 0,

(3.6)

For simplicity of notation, from here onwards the argument x is omitted.

Please cite this article in press as: Del Piero, G., A rational approach to Cosserat continua, with application to plate and beam theories. Mech. Re. Commun. (2013), http://dx.doi.org/10.1016/j.mechrescom.2013.11.003

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with −f and − the volume densities of the contact actions associated with the macroscopic deformation and with the order parameter, respectively. The internal power (2.5) then reduces to



((b − f ) · v + T · ∇ v + (c − ) · ω + M · ∇ ω) dV .

Pint (, v, ω) = 

(3.7)

For a Cosserat continuum, the rigid virtual velocities are the rigid translations of the body, v(x) = w, and the simultaneous rigid rotations of the body and of the directors

v(x) = w × x,

ω(x) = w.

(3.8)

Pint (, w, 0) = 0,

Pint (, w × x, w) = 0.

(div M + c). The balance equation (3.16) can be expressed in terms of the active internal forces. Indeed, using the identity curl t = −divT W , which follows from (3.11), Eq. (3.16) takes the form divT S − curl t + b = 0,

(3.9)

S Tij,j − eijk tk,j + bi = 0.

ˆ M(x) n = m(x).

(3.19)

Moreover, due to the general relation between a vector and the corresponding skew-symmetric tensor

They must be satisfied by all vectors w. By the arbitrariness of , from the translational indifference condition it follows that

T W n = t × n,

b = f.

the first condition can be given the form

(3.10)

That is, the volume density −f of the contact action must be the opposite of the applied body force b. Moreover, denoting by t the vector associated with the skew-symmetric part of T ti =

1 e T , 2 ijk kj

(3.11)

TijW nj = eijk tj nk ,

T S (x) n + t × n = sˆ(x).

(3.20)

In view of the specialization to plates and beams, it is convenient to write the boundary conditions in components. In a local orthonormal reference system {ei } with e3 = n, the tangent and normal components are

the identity

S +e t =s ˆ˛ , T˛n ˛ˇ ˇ

ˆ ˛, M˛n = m

T · ∇ (w × x) = 2t · w,

S Tnn

ˆ n, Mnn = m

Tkj ekih wi xh,j = Tkj eijk wi = 2ti wi ,

holds. Then the rotational indifference condition requires that 2t + c −  = 0.

(3.12)

To get the reduced form of the internal power (3.7), decompose T and ∇ v into the sums of their symmetric and skew-symmetric parts T = TS + TW ,

∇ v = ∇ S v + ∇ W v.

(3.13)

Using the identity T W · ∇ v = t · curl v,

TijW vi,j = ti eijk vk,j ,

which is a consequence of (3.11), the internal power (3.7) takes the form



(T S · ∇ S v + t · (curl v − 2ω) + M · ∇ ω) dV .

Pint (, v, ω) =

(3.14)



It shows that the active generalized forces are T S , 2t, and M, and that the free generalized deformations are ∇ S v, ϕ and ∇ ω, where ϕ=

1 curl v − ω 2

(3.15)

is the relative rotation between the body and the triad of the directors. From (3.10) and (3.6)1 , the local form of the balance equation of linear momentum divT + b = 0,

(3.16)

follows. Moreover, (3.12) and (3.6)2 determine the local form of the balance equation of angular momentum5 divM + c + 2t = 0,

Mij,j + ci + eijk Tkj = 0.

(3.17)

The latter says that in a Cosserat continuum the Cauchy stress is not symmetric, and that its skew-symmetric part is determined by

5

In both balance equations, inertia forces can be included as particular body forces and microforces, see Noll (1963, Section 7).

(3.18)

At the free part of the boundary, the boundary conditions of traction ˆ of the contact force s and of the contact prescribe the values sˆ, m couple m. By Eqs. (3.5), this is the same as to prescribe the normal components of the tensors T and M T (x) n = sˆ(x),

Therefore, the indifference conditions are

3

= sˆn ,

(3.21)

respectively, with e˛ˇ = e˛ˇn . The incremental equilibrium problem consists in determining the displacements from a given equilibrium configuration, due to increments ıb, ıc of the body forces at the interior points of , to displacements vˆ , ω ˆ at the constrained part of the boundary, and ˆ of the contact actions at the free part of the to increments ıˆs, ım boundary. To avoid technical complications, the given equilibrium configuration is taken as the reference configuration. The active internal forces T S , t, M acting on this configuration are supposed to be known. The unknowns are the displacements v, ω and the increments ıT S , ıt, ıM of the internal forces. The field equations are the balance equations (3.17) and (3.18), and the boundary conditions of traction are Eqs. (3.21), all written in the incremental form, that is, with the internal and external forces replaced by the corresponding increments. At the constrained part of the boundary, the boundary conditions of place prescribe the values vˆ , ω ˆ of the displacements and rotations

v(x) = vˆ (x),

ω(x) = ω(x). ˆ

(3.22)

For the incremental problem, the constitutive equations are relations between the increments ıTS , ıt, ıM and the free generalized deformations ∇ S v, ϕ, ∇ ω. They describe the response of a specific material in the class of the Cosserat continua. The material can be elastic, dissipative, elastic–plastic, rate dependent, and so on. In particular, the constitutive equations for an elastic Cosserat continuum are functional relations of the type ıT S = ˚(∇ S v, ϕ, ∇ ω),

ıt = X(∇ S v, ϕ, ∇ ω),

ıM =  (∇ S v, ϕ, ∇ ω),

(3.23)

in which the constitutive functions ˚, X and  depend on the current configuration. In most cases, the simpler relations ıT S = ˚(∇ S v),

ıt = X(2ϕ),

ıM =  (∇ ω),

(3.24)

are used. More special forms are required if the constitutive functions have a potential, the elastic energy, which is subjected to conditions of physical plausibility (Jeong and Neff, 2010). The

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equations of the incremental equilibrium problem are collected in Table 1 at the end of the paper. The proposed theory is not restricted to infinitesimal deformations. Indeed, with the incremental equilibrium problem it is possible to follow the quasi-static evolution of the body under varying data, starting from a given equilibrium configuration. This can be done by solving a sequence of incremental equilibrium problems, in each of which the current equilibrium configuration is updated by the solution of the preceding problem. The incremental solution technique has two main advantages over energy minimization: (i) following the evolution of the deformation under varying data, most of the inconveniences due to nonuniqueness of the solution in finite deformations are eliminated; (ii) the method works also in the presence of dissipation.6

This is a combination of the balance equations (3.17) and (3.18) of the unconstrained continuum. Next, equating the surface integrals in (4.4) we get



∂

+

((si − TijS nj +

1 (m − Mkh nh ) eikj vi,j ) dA = 0. 2 k

(mk − Mkh nh ) eikj vi,j = (mk − Mkn )( eikn vi,n + eik˛ vi,˛ ) = (mˇ − Mˇn ) e˛ˇ v˛,n + (mk − Mkn ) eik˛ vi,˛ .

The constrained Cosserat continuum is obtained by imposing the kinematic constraint7

(4.7) Then, a further application of the divergence theorem





(4.1)

which requires the vanishing of the relative rotation (3.15) at all points of the body and in all virtual deformations. By consequence, the external power (3.4) takes the form8



(mk − Mkn ) eik˛ vi,˛ dA = − ∂

By the pseudobalance equations (2.2), there are second-order tensor fields T, M satisfying Eqs. (3.5). In the expression (3.7) of the internal power, ω is replaced by (1/2) curl v. The indifference conditions (3.9) provide the balance equations (3.16) and (3.17), and the internal power (3.14) reduces to9



(T S · ∇ S v + M · 

1 ∇ curl v) dV . 2

(4.3)

The free generalized deformations are now ∇ S v and (1/2)∇ curl v, and the active internal forces are T S and M. To get the equations of the incremental equilibrium problem, write equations (4.2) and (4.3) for the whole body , and transform their right-hand sides with the aid of the divergence theorem





∂

(mk − Mkn ),˛ eik˛ vi dA,

(4.8)

followed by substitution into (4.6), yields





1 1 Pext (, v) = (b · v + c · curl v) dV + (s · v + m · curl v) dA. 2 2  ∂ (4.2)

Pint (, v) =

(4.6)

The tangential components of the gradient vi,j are determined by the boundary values of vi . Taking on ∂ a local orthonormal reference system {ei } with e3 = n, rewrite the last boundary term with separated normal and tangential components

4. The constrained Cosserat continuum

1 ω = curl v, 2

1 (c + Mkh,h ) eikj nj ) vi 2 k

∂

+

S ((si − Tin +

1 1 (c + Mkh,h ) eikn − (mk − Mkn ),˛ eik˛ ) vi 2 k 2

1 (m − Mˇn ) e˛ˇ v˛,n ) dA = 0. 2 ˇ

(4.9)

At the free part of the boundary, this equation is satisfied by imposing the boundary conditions of traction S + T˛n

1 1 ˆ n,ˇ ), − Mˇh,h ) = sˆ˛ + e˛ˇ (cˇ + m e (M 2 ˛ˇ nn,ˇ 2

S + Tnn

1 1 ˆ ˇ,˛ , = sˆn + e˛ˇ m e M 2 ˛ˇ ˇn,˛ 2

(4.10)

ˆ ˛. M˛n = m There are no conditions on the normal derivative vn,n and on ˆ˛ the surface traction mn .10 Moreover, the last condition M˛n = m ˆ ˇ,˛ . Therefore, the second of the boundary conimplies Mˇn,˛ = m ditions (4.10) reduces to11





1 1 1 1 1 ck eikj vi,j ) dV + (si vi + mk eikj vi,j ) dA = (bi − ck,j eikj ) vi dV + ((si + ck eikj nj ) vi + mk eikj vi,j ) dA, 2 2 2 2 2 ∂ 

∂



 (4.4) 1 1 1 1 S S S (Tij vi,j + Mkh eikj vi,jh ) dV = (−Tij,j + Mkh,hj eikj ) vi dV + ((Tij − Mkh,h eikj ) vi nj + Mkh eikj vi,j nh ) dA. Pint ( , v) = 2 2 2 2



∂

Pext ( , v) =

(bi vi +

Equating the two expressions of the power and using the arbitrariness of v, from the volume integrals we get 1 div T S + curl (div M + c) + b = 0, 2 S Tij,j +

1 e (M + ck,j ) + bi = 0. 2 ijk kh,hj

S Tnn = sˆn .

(4.11)

The incremental versions of (4.5), (4.10) and (4.11), together with the boundary conditions of place (4.5)

v˛ (x) = vˆ ˛ (x),

vn (x) = vˆ n (x),

v˛,n (x) = vˆ ˛,n (x),

(4.12)

1 curl v). 2

(4.13)

and the constitutive equations 6

For physical examples of nonuniqueness in finite elasticity, see e.g. Ciarlet (1988, Section 5.8). For the quasi-static evolution of bodies made of rate-independent materials, see Mielke (2011) and the papers cited therein. 7 Toupin (1964, Section 11). 8 Due to the constraint (4.1), the only kinematic variable left in the expression of the external power is v. However, the presence of a hidden variable ω is revealed by the fact that the external power has an extra term with respect to the power of a classical continuum. In the terminology introduced in Capriz (1985), this circumstance characterizes a continuum with latent microstructure. 9 The term ∇ curl v characterizes the constrained continuum as a particular second gradient continuum, see e.g. Germain (1973, Section 6.3).

ıT S = ˚(∇ S v,

1 curl v), 2

ıM =  (∇ S v,

form the incremental equilibrium problem for the constrained elastic Cosserat continuum.

10 That is, the boundary conditions are five, instead of the six of the unconstrained theory. See Schaefer (1967). 11 See Bleustein (1967).

Please cite this article in press as: Del Piero, G., A rational approach to Cosserat continua, with application to plate and beam theories. Mech. Re. Commun. (2013), http://dx.doi.org/10.1016/j.mechrescom.2013.11.003

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5. Plate theories

and the constitutive equations

A plate can be viewed as a body with a cylindrical shape, made of a Cosserat continuum subjected to the kinematic constraints 3

v(x) = v3 (x1 , x2 ) e ,

˛

ω(x) = ω˛ (x1 , x2 ) e ,

˛ ∈ {1, 2},

(5.1)

e3

with the direction of the cylinder’s axis. The constraints require that, at all points x, the displacement v(x) be parallel to e3 and the rotation ω(x) of the directors be about an axis orthogonal to e3 . By consequence, the three-dimensional vectors v and ω degenerate into a scalar and a two-dimensional vector, respectively. The same do the associated vectors b, s and c, m. The external power then takes the form





Pext (, v, ω) =

(b3 v3 + c˛ ω˛ ) dV +

(s3 v3 + m˛ ω˛ ) dA. (5.2) ∂



m˛ = M˛ˇ nˇ .

(5.3)

The component b3 of the body force is now viewed as a transverse load q. Thus, the internal power (3.7) takes the form



Pint (S, v, ω) =

S

ıQ˛ = ˚˛ (ϕ, ∇ ω),

ω˛ = e˛ v3, ,

(5.11)

under which the relative rotation (5.7) becomes zero. With this restriction, the external power (5.2) reduces to





Pext (S, v) =

(b3 v3 + c˛ e˛ v3, ) dA +

(5.5)

S

((Q˛,˛ + b3 ) v3 + Q˛ v3,˛ + (M˛ˇ,ˇ + c˛ ) e˛ v3,

+ M˛ˇ e˛ v3, ˇ ) dA, and setting b3 = q c ∗ = e ˛ c˛ ,



S

∗ ((Q˛,˛ + q) v3 + Q˛ v3,˛ − (Mˇ ,ˇ + c ∗ ) v3,

∗ − Mˇ v3, ˇ ) dA.

(5.7)

which is the two-dimensional counterpart of the relative rotation (3.15). Eqs. (5.5) are the equilibrium equations at the internal points of

. The boundary conditions of traction are Eqs. (5.3), rewritten in the form ˆ ˛, M˛n = m

(5.8)

with the subscript n denoting the component in the direction of the normal to ∂ . The incremental versions of (5.5) and (5.8), the boundary conditions of place



 

S

Pint (S , v) = −

v3 (x) = vˆ 3 (x),

ω˛ (x) = ω ˆ ˛ (x),

(q v3 − c˛∗ v3,˛ ) dA +

S



∗S M˛ˇ,˛ˇ v3 dA −

(5.15)

The indifference requirements (3.9) now provide the balance equations ∗ Mˇ˛,ˇ + c˛∗ − Q˛ = 0,

(5.16)

and the internal power reduces to

(5.6)

ϕ˛ = v3,˛ + e˛ˇ ωˇ ,

Pext (S , v) =

(5.14)

we finally get

Pint (S, v) = −

(Q˛ (v3,˛ + e˛ˇ ωˇ ) + M˛ˇ ω˛,ˇ ) dA.

The active internal forces are Q˛ and M˛ˇ , and the free generalized deformations are the rotation gradient ω˛,ˇ and the vector

Qn = sˆ3 ,

∗ Mˇ = e ˛ M˛ˇ ,





S

(5.13)

and12

m∗ = e ˛ m˛ ,

and the reduced form of the internal power is Pint (S, v, ω) =

(5.12)



Pint (S, v) =

Q˛,˛ + q = 0, M˛ˇ,ˇ + c˛ + e˛ˇ Qˇ = 0,

(s3 v3 + m˛ e˛ v3, ) d . ∂S

From Eqs. (5.3), after an integration by parts, we get the internal power

(5.4)

In the indifference conditions (3.9), w is now a vector orthogonal to e3 . Accordingly, the balance equations (3.16) and (3.17) become

(5.10)

form the incremental equilibrium problem for the Reissner theory of plates (Reissner, 1944). Just as the constrained Cosserat continuum was obtained by introducing the kinematic constraint (4.1), the Kirchhoff–Love theory of plates (Kirchhoff, 1876; Love, 1927) can be deduced from Reissner’s theory by imposing the kinematic constraint

Pint (S, v) =

((Q˛,˛ + q) v3 + Q˛ v3,˛ + (M˛ˇ,ˇ + c˛ ) ω˛

+ M˛ˇ ω˛,ˇ ) dA.

Q˛,˛ + q = 0,

ıM˛ˇ = ˛ˇ (ϕ, ∇ ω),

S

The constraints (5.1) also require that both v and ω be independent of x3 . Then, the body can be identified with the cylinder’s cross section. The parts  of the body reduce to plane surfaces S, the volume elements dV reduce to area elements, and the area elements dA reduce to line elements d . Moreover, the stress tensor Tij degenerates into the vector Q˛ of the internal shearing forces, and the couple-stress tensor Mij degenerates into the 2 × 2 tensor of the internal moments M˛ˇ . Then Eqs. (3.5) take the form s3 = Q˛ n˛ ,

5

∗ Mˇ˛ v3,˛ˇ dA.

(5.17)

Thus, in the constrained plate theory there is a single free generalized deformation, the curvature tensor −v3,˛ˇ . Moreover, by the ∗S of symmetry of the second derivative, only the symmetric part M˛ˇ ∗ contributes to the power. Therefore, the active internal force M˛ˇ

can be identified with the symmetrized tensor M*S . For an elastic material, the constitutive equation is a relation between the increment ıM*S and the curvature tensor ıM ∗S =  (−∇∇ v3 ).

(5.18)

For the formulation of the incremental equilibrium problem, we follow the same procedure adopted for the three-dimensional constrained continuum. We write the powers (5.12) and (5.17) for the cross section S of the cylinder . Then, using the divergence theorem, in the volume integrals we transform the terms involving the derivatives of v3



∂S

∂S

S

(s3 v3 − m∗˛ v3,˛ ) d =

S



∗ (q + c˛,˛ ) v3 dA +

∂S

((s3 − c˛∗ n˛ ) v3 − m∗˛ v3,˛ ) d , (5.19)

∗S ∗S (M˛ˇ v3,˛ nˇ − M˛ˇ,ˇ v3 n˛ ) d .

(5.9)

∗ ∗ ∗ ∗ 12 The moments M11 and M22 are bending moments, and M12 and M21 are twisting moments. They are the moments currently used in textbooks on plate theories.

Please cite this article in press as: Del Piero, G., A rational approach to Cosserat continua, with application to plate and beam theories. Mech. Re. Commun. (2013), http://dx.doi.org/10.1016/j.mechrescom.2013.11.003

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Comparing the volume terms we get the field equation ∗S M˛ˇ,˛ˇ

∗ + c˛,˛

+ q = 0,

(5.20)

which is a combination of the balance equations (5.16). Moreover, comparing the boundary terms we get



and the bending moments, and Q3 and M3 are the axial force and the torsional moment. Substituting into (6.2) and integrating by parts one gets the internal power



b

((qi + Qi ) vi + Qi vi + (ci + Mi ) ωi + Mi ωi ) dx3 ,

Pint ((a, b), v, ω) = a

∂S

∗S ∗S ((M˛ˇ,ˇ n˛ − s3 + c˛∗ n˛ )v3 − (M˛ˇ nˇ − m∗˛ )v3,˛ ) d = 0. (5.21)

Like in the three-dimensional constrained theory, only the component of v3,˛ normal to the boundary is independent, the tangential component being determined by the values of v3 at the boundary. Therefore, keeping e3 in the direction of the cylinder’s axis, we take a local reference system with e1 , e2 coincident with the outward normal n and the tangent vector  to the lateral surface, respectively. With a further use of the divergence theorem, Eq. (5.21) yields



0= ∂S

∗S ∗S ∗S ((Mnˇ,ˇ −s3 +cn∗ ) v3 −(Mn −m∗ ) v3, − (Mnn − m∗n ) v3,n ) d

∂S

∗S ((Mnˇ,ˇ

∗S + Mn,

− s3 + cn∗

− m∗, ) v3

∗S − (Mnn

− m∗n ) v3,n ) d . (5.22)

The last integral shows that the boundary conditions of place are

v3 (x) = vˆ 3 (x),

v3,n (x) = vˆ 3,n (x).

(5.23)

Moreover, observing that ∗S Mnˇ,ˇ

where a prime denotes differentiation with respect to x3 . The indifference conditions have again the form (3.9). They provide the balance equations qi + Qi = 0,

c˛ + M˛ − e˛ˇ Qˇ = 0,

∗S + Mn,

=

∗S Mnn,n

∗S ∗S ˆ ∗, , Mnn,n + 2Mn, + cn∗ = sˆ3 + m

∗S ˆ ∗n . Mnn =m

(5.24)

Like in the three-dimensional case, the number of the boundary conditions is smaller than in the unconstrained theory. Indeed, the three scalar conditions (5.8) are replaced by the two conditions (5.24).



b

 (Q˛ (v˛ − e˛ˇ ωˇ ) + Q3 v3 + M˛ ω˛ + M3 ω3 ) dx3

Pint ((a, b), v, ω) = a

(6.6)

is obtained. It shows that the free generalized deformations are

v3 , ω˛ , ω3 , and the difference

A beam can be viewed as a body with a cylindrical shape, made of a Cosserat continuum subjected to the kinematic constraints

v(x) = v(x3 ),

ω(x) = ω(x3 ),

(6.1)

where x3 is the component of x parallel to the cylinder’s axis. Under such constraints, each cross section of the cylinder undergoes a rigid displacement, with the translation determined by v and the rotation determined by ω. The body can be reduced to the cylinder’s axis, with the parts  of the body replaced by intervals (a, b) and the boundary ∂ replaced by the endpoints a, b. The external power has the form



Pext ((a, b), v, ω) =

(6.7)

which is the counterpart of the relative rotation (3.15). For an elastic material the constitutive equations have the form ıQi = ˚i (ϕ˛ , v3 , ω˛ , ω3 ),

vi (0) = vˆ i0 ,

Qi (l) = Pˆ i ,

Ci = Mi n.

(6.3) e3

The exterior unit normal n to the cross section is at x3 = b and −e3 at x3 = a. The components Q˛ and M˛ are the shearing forces

(6.9)

(6.10)

Mi (0) = −Cˆ i0 ,

(6.11)

for concentrated couples Cˆ i , Cˆ i0 applied at the same points. The Euler–Bernoulli beam theory is obtained by imposing the kinematic constraint ω˛ = −e˛ˇ vˇ ,

(6.12)

that is, by assuming that the relative rotation (6.7) is identically zero. Under this constraint, the external power (6.2) becomes



b

Pext ((a, b), vi , ω3 ) = a

(qi vi − c˛ e˛ˇ vˇ + c3 ω3 ) dx3

+ [Pi vi − C˛ e˛ˇ vˇ + C3 ω3 ]

b

+ [Pi vi − C˛ e˛ˇ vˇ

a

b

=

+ C3 ω3 ]

(qi vi + c˛∗ v˛ + c3 ω3 ) dx3

a

+ [Pi vi + C˛∗ v˛ + C3 ω3 ]b

where qi and ci are distributed forces and couples per unit length, and Pi and Ci are concentrated couples and forces, representing the contact actions between (a, b) and the rest of the beam. The pseudobalance equations now imply the existence of two fields of internal forces Qi , Mi , such that

ωi (0) = ω ˆ i0 ,

for concentrated forces Pˆ i , Pˆ i0 applied at the endpoints of the beam, and

 (6.2)

ωi (l) = ω ˆ i ,

Qi (0) = −Pˆ i0

(qi vi + ci ωi ) dx3 + [Pi vi + Ci ωi ]b + [Pi vi + Ci ωi ]a ,

(6.8)

and the incremental versions of the boundary conditions of traction, they form the incremental equilibrium problem for the Timoshenko beam theory. By (6.3) with n3 = ±1, the boundary conditions of traction are

b

a

ıMi = i (ϕ˛ , v3 , ω˛ , ω3 ).

Together with the incremental versions of the equilibrium equations (6.5), the boundary conditions of place

Mi (l) = Cˆ i , 6. Beam theories

(6.5)

from which the reduced form of the internal power

vi (l) = vˆ i ,

∗S + 2Mn, ,

the boundary conditions of traction take the form

Pi = Qi n,

c3 + M3 = 0,

ϕ˛ = v˛ − e˛ˇ ωˇ ,

 =

(6.4)

+ [Pi vi + C˛∗ v˛ + C3 ω3 ]a ,

(6.13)

with c˛∗ = e˛ˇ cˇ and C˛∗ = e˛ˇ Cˇ like in (5.14). After setting M˛∗ = e˛ˇ Mˇ , from Eqs. (6.3) we get the internal power



Pint ((a, b), vi , ω3 ) = a

b

((Qi + qi ) vi + Q˛ v˛ + Q3 v3 + (M˛∗ + c˛∗ ) v˛

+ M˛∗ v ˛ + (M3 + c3 ) ω3 + M3 ω3 ) dx3 .

(6.14)

Please cite this article in press as: Del Piero, G., A rational approach to Cosserat continua, with application to plate and beam theories. Mech. Re. Commun. (2013), http://dx.doi.org/10.1016/j.mechrescom.2013.11.003

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Table 1 The equilibrium equations, incremental constitutive equations, and boundary conditions of place and traction for 3D, 2D, and 1D unconstrained and constrained elastic Cosserat continua. The numbers on the left column refer to the place occupied by each equation in the present paper.

The indifference conditions are Pint ((a, b), w, 0) = 0,

Pint ((a, b), 0, w) = 0,

Pint ((a, b), w × x, 0) = 0,

3D: Unconstrained Cosserat continuum S (3.18) and (3.17) Tij,j − eijk tk,j + bi = 0, Mij,j + ci + eijk Tkj = 0

with w an arbitrary vector. They provide the restrictions Qi + qi = 0,

Q˛ + M˛∗ + c˛∗ = 0,

M3 + c3 = 0,

(3.23)

(6.15)

from which the reduced form of the internal power



b

Pint ((a, b), vi , ω3 ) =

(Q3 v3 + M˛∗ v ˛ + M3 ω3 ) dx3

(6.16)

a

follows. The free generalized deformations are the axial stretching v3 , the curvature vector v ˛ , and the angle of twist ω3 , and the corresponding active internal forces are the axial force Q3 , the bending moments M˛∗ , and the torsional moment M3 . The constitutive equations have the form ıQ3 = ˚3 (v ˛ , v3 , ω3 ),

ıM˛∗ = ˛ (v ˛ , v3 , ω3 ),

ıM3 = 3 (v ˛ , v3 , ω3 ).

(6.17)

l

Pext ((0, l), vi , ω3 )=

vi = vˆ i ,

(3.21)

S T˛n + e˛ˇ tˇ = sˆ˛ ,

2D: Reissner plate (5.5) (5.10)

Q˛,˛ + q = 0, M˛ˇ,ˇ + c˛ + e˛ˇ Qˇ = 0 ıQ˛ = ˚˛ (ϕ, ∇ ω), ıM˛ˇ =  ˛ˇ (ϕ, ∇ ω)

(5.9)

v3 = vˆ 3 ,

ω˛ = ω ˆ˛

(5.8)

Qn = sˆ3 ,

ˆ˛ M˛n = m

+ [(P˛ − c˛∗ ) v˛ + P3 v3 + C˛∗ v˛ + C3 ω3 ]0 , b

(6.18)



(M˛∗ v˛ − Q3 v3 − M3 ω3 ) dx3

(6.5)

− [(Q3 v3 − M˛∗ v˛ + M˛∗ v˛ + M3 ω3 ]0 . Equating the two integrals, from the arbitrariness of vi and ω3 we get the equilibrium equations 

M3 + c3 = 0,

Q3 + q3 = 0.

v˛ (l) = vˆ ˛ ,

v˛ (0) = vˆ ˛0 ,

v˛ (l) = vˆ ˛ ,

v˛ (0) = vˆ ˛0 ,

v3 (l) = vˆ 3 ,

v3 (0) = vˆ 30 ,

ω3 (l) = ω ˆ 3 ,

ω3 (0) = ω ˆ 30 ,

(6.20)

and the boundary conditions of traction are obtained by prescribing the increments ıPˆ ˛ of the concentrated forces and ıCˆ ˛ of the concentrated couples applied at the endpoints. Comparing the boundary terms of Eqs. (6.18), we get

M3 (l) = Cˆ 3 ,

∗ , M˛∗ (l) = Cˆ ˛

Q3 (l) = Pˆ 3 ,

(6.21)

at x = l, and M˛∗ (0) + c˛∗ (0) = Pˆ ˛0 , M3 (0) = −Cˆ 30 , at x = 0.

ˆ ˛, M˛n = m

ˆn Mnn = m

Qi + qi = 0,

M3 + c3 = 0,

M˛ + c˛ − e˛ˇ Qˇ = 0

ıQi = ˚i (ϕ˛ , v3 , ω˛ , ω3 ), ıMi = i (ϕ˛ , v3 , ω˛ , ω3 ) vi (l) = vˆ i , vi (0) = vˆ i0 , ωi (l) = ω ˆ i , ωi (0) = ω ˆ i0

(6.10) and (6.11)

Qi (l) = Pˆ i ,

Mi (l) = Cˆ i ,

Qi (0) = −Pˆ i0 ,

Mi (0) = −Cˆ i0

3D: Constrained Cosserat continuum S Tij,j + bi +

1 e (Mkh,hj 2 ijk

(4.12)

v˛ = vˆ ˛ ,

vn = vˆ n ,

(4.10)

S T˛n +

(4.10) and (4.11)

S Tnn = sˆn ,

(4.13)

+ ck,j ) = 0

ıTijS = ˚ij (∇ S v, 12 curl v), 1 e (Mnn,ˇ 2 ˛ˇ

ıMij = ij (∇ S v, 12 curl v)

v˛,n = vˆ ˛,n

− Mˇh,h − cˇ ) = sˆ˛ +

1 ˆ e m 2 ˛ˇ n,ˇ

ˆ˛ M˛n = m

∗ , M˛∗ (0) = −Cˆ ˛0

Q3 (0) = −Pˆ 30 ,

∗S ∗ M˛ˇ,˛ˇ + c˛,˛ +q=0

(5.18)

∗S ıM˛ˇ = ˛ˇ (−∇∇ v3 )

(5.23)

v3 = vˆ 3 ,

(5.24)

∗S ∗S ˆ ∗, , Mnn,n + 2Mn, + cn∗ = sˆ3 + m

v3,n = vˆ 3,n ∗S ˆ ∗n Mnn =m

1D: Euler-Bernoulli beam 

M3 + c3 = 0,

(6.19)

M˛∗ + c˛∗ − q˛ = 0,

(6.17)

ıM˛∗ = ˛ (v ˛ , v3 , ω3 ), ıM3 = 3 (v ˛ , v3 , ω3 )

Q3 + q3 = 0

ıQ3 = ˚3 (v ˛ , v3 , ω3 ),

(6.20)

v˛ (l) = vˆ ˛ , v˛ (0) = vˆ ˛0 , v˛ (l) = vˆ ˛ , v˛ (0) = vˆ ˛0 v3 (l) = vˆ 3 , v3 (0) = vˆ 30 , ω3 (l) = ω ˆ 3 , ω3 (0) = ω ˆ 30

(6.21)

M˛∗ (l) + c˛∗ (l) = −Pˆ ˛ ,

(6.22)

∗ M˛∗ (l) = Cˆ ˛ ,

∗ M˛∗ (0) = −Cˆ ˛0

(6.21) and (6.22)

M3 (l) = Cˆ 3 , −Pˆ 30

Q3 (l) = Pˆ 3 ,

M˛∗ (0) − c˛∗ (0) = Pˆ ˛0 M3 (0) = −Cˆ 30 ,

Q3 (0) =

(6.19)

The boundary conditions of place are obtained by prescribing the displacements v˛ , v3 and the rotations v˛ , ω3 at the endpoints

M˛∗ (l) + c˛∗ (l) = −Pˆ ˛ ,

S Tnn = sˆn ,

(6.8) (6.9)

a

+ [(Q3 v3 − M˛∗ v˛ + M˛∗ v˛ + M3 ω3 ]

M˛∗ + c˛∗ − q˛ = 0,

ωi = ω ˆi

1D: Timoshenko beam

(5.20)

+ c3 ω3 ) dx3

+ [(P˛ + c˛∗ ) v˛ + P3 v3 + C˛∗ v˛ + C3 ω3 ]

Pint ((a, b), vi , ω3 )=

ıti = Xi (∇ S v, 2ϕ, ∇ ω)

2D: Kirchhoff–Love plate

(q˛ v˛ + q3 v3 − c˛∗ v˛

0



ıTijS = ˚ij (∇ S v, 2ϕ, ∇ ω), ıMij = ij (∇ S v, 2ϕ, ∇ ω)

(3.22)

(4.5)

The equilibrium equations and boundary conditions are obtained writing the powers (6.13) and (6.16) for the whole beam, and then reducing the derivatives in the integral terms by integration by parts. The result is



7

(6.22)

7. Conclusion Using the properties of bounded Cauchy fluxes, in the two preceding Sections the classical equations of the incremental equilibrium problems for plates and beams have been deduced from those of the unconstrained and constrained three-dimensional Cosserat continuum. This has been done by imposing the kinematic constraints (5.1) and (6.1), respectively, to the body’s deformation. The equilibrium equations, the constitutive equations, and the boundary conditions of each problem, all written in components, are collected in Table 1. It is worth noting that the plate and beam theories do not enjoy the same level of generality of the three-dimensional theory. Indeed, their range of application is limited by the assumed cylindrical shape of the reference configuration, which here coincides with the current configuration. In general, the cylindrical shape assumed for the load-free configuration is lost in the subsequent deformation. Therefore, the formulations of the incremental equilibrium problems for plates and beams given above apply only to the rare cases in which the deformed configuration maintains a cylindrical geometry.

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Please cite this article in press as: Del Piero, G., A rational approach to Cosserat continua, with application to plate and beam theories. Mech. Re. Commun. (2013), http://dx.doi.org/10.1016/j.mechrescom.2013.11.003