Minimum Principles for Interacting Cosserat Elastic Continua

Abstract

This paper is concerned with the linear theory of composites modelled as mixtures of two Cosserat elastic continua. First, we present a minimum principle in the case of equilibrium. Then, we consider the dynamic theory and establish a minimum principle of Reiss type for a mixed problem.

1. Background

In order to describe adequately the behaviour of some kinds of mixtures it is necessary to introduce into the continuum theory terms reflecting the microstructure of the materials. In a realistic continuum model, each particle of a granular material or a solid containing microscopic components (e.g., nanocomposites) possesses six degrees of freedom. Important among such materials are animal bones, solid with microcracks, and other synthetic materials with pores or microreinforcements. The origin of the modern theories of mixtures of materials with microstructure goes back to the papers of Allen and Kline [1], Twiss and Eringen [2, 3], and Dunwoody [4]. For a review of the literature on mixtures with microstructure the reader is referred to [5, 6]. In many papers the mechanical behaviour of composites is modelled as mixtures of interacting continua (see, e.g., [7–10] and references therein). A general theory of mixtures with microstructure has been established in [2, 3]. In [3], the nonlinear constitutive equations for mixtures of micromorphic and micropolar elastic bodies are derived. The results are used to derive the linear theory of mixtures composed of two Cosserat elastic constituents. The equations presented in [2, 3, 6] are sufficiently general to apply to any constituents, whether solid or fluid, in eulerian description. A theory of a mixture of two Cosserat elastic solids in lagrangian description has been established in [10]. This theory allows us to formulate the boundary conditions in the reference configuration.

In the present paper we consider the linear theory of mixture of two Cosserat elastic solids. In Section 2 we present the notations and the basic equations. Section 3 is devoted to a minimum principle in the equilibrium theory. In Section 4 we consider the dynamic theory and establish a minimum principle of Reiss type. Variational characterizations of solutions in nonpolar theories of mixtures have been presented in various papers (see, e.g., [5, 11–15] and references therein).

2. Methods Mathematical Formulation

We consider a body which is made up of two interpenetrating elastic solids of Cosserat type and . We assume that the body, at some instant, occupies the region with the piecewise smooth surface . The motion of the body is referred to a fixed system of rectangular cartesian axes () and to the reference configuration . We denote by the outward unit normal of . Letters in boldface stand for tensors of an order , and if has the order , we write ( subscripts) for the components of in the cartesian coordinate frame. We will employ the usual summation and differentiation conventions. Latin subscripts (unless otherwise specified) are understood to range over the integers , summation over repeated subscripts is implied, and subscripts preceded by a comma denote partial differentiation with respect to the corresponding cartesian coordinate. We use a superposed dot to denote the partial differentiation with respect to the time . First, we consider the linear theory of elastostatics. We assume that the constituents and are each elastic solids of Cosserat type. We consider a theory for binary mixtures where the typical particles of and occupy the same position in the reference configuration. We denote by and the displacement vector fields associated with the constituents and , respectively. Let be the mass density of the constituent in the reference configuration. We denote by and the microrotation vectors associated with the constituents and , respectively. Let and denote the stress tensors associated with the constituents and , respectively. Further, let and denote the partial couple-stress tensors associated with and , respectively. The equations of equilibrium can be expressed in the form on . Here we have used the following notations: is the diffusive force, is the diffusive couple, is the body force per unit mass acting on the constituent , is the alternating symbol, and is the body couple per unit mass acting on the constituent . We introduce the functions , , , , , and by The constitutive equations are The constitutive coefficients have the following symmetries:

We assume that (a) and are strictly positive and continuous on ; (b) and , , are continuous on ; (c) the constitutive coefficients are continuously differentiable on and satisfy relations (4).

We say that the array is an admissible deformation field on provided , , , . An admissible stress field on is an ordered array of functions with the properties , , , , , and , , , . By an admissible state on we mean an ordered array , , , with the following properties:() is an admissible deformation on ;();() is an admissible stress field on .

If we define addition and multiplication of an admissible state by a scalar through then the set of all admissible states is a linear vector space.

We say that , is an elastic state on corresponding to the body loads if is an admissible state that satisfies (1)–(3) on . Let , , be the subsets of so that , . By an external data system on we mean an ordered array , with the following properties: , , , ; ; are piecewise regular on ; and are continuous on ; ; are piecewise regular on ; and are continuous on .

The mixed problem of the equilibrium theory consists in finding an elastic state that corresponds to the body loads and satisfies the boundary conditions

We note that a rigid deformation field is characterized by where and are arbitrary constants.

3. Minimum Principle

In this section we establish a minimum principle of elastostatics which characterize the solution of the mixed problem.

Let , be an admissible state on . The internal energy density corresponding to is defined by The strain energy corresponding to is By a kinematically admissible state we mean an admissible state that satisfies (2) and the boundary conditions

Theorem 1. Assume that the internal energy is a positive definite form. Let denote the set of all kinematically admissible states, and let be the functional on defined by for every , . Further, let be a solution of the mixed problem. Then for every , and equality holds only if modulo a rigid displacement.

Proof. Let , and define Then , is an admissible state with the properties It follows from (4), (8), (9), (11), and (13) that If we take into account (14) and use (1), then we obtain By using the divergence theorem, (15), and (17) we get In view of (18), relation (16) can be written as Since is a solution of the mixed problem, (19) implies Since the internal energy is positive definite, we obtain
Moreover, only if , , , , , and . Thus, only if modulo a rigid displacement.

Theorem 1 extends the principle of minimum potential energy from the classical elasticity (see, e.g., [16]).

4. Dynamic Theory

In this section we establish a minimum principle of Reiss type (see [17, 18]) in the dynamic theory. The equations of motions are on , where and are coefficients of inertia and . To the field equations (22), (2), and (3) we must add boundary conditions and initial conditions. The initial conditions are where , , , , , , , and are prescribed functions. We consider the boundary conditions where , , , , , , , and are given.

We assume that(i), , , and are continuous on ;(ii), , , , , and are continuous on ;(iii)the constitutive coefficients are continuously differentiable on ;(iv)the constitutive coefficients satisfy relations (4), and the coefficients of inertia are symmetric;(v) and are strictly positive on , and and are positive definite tensors on ;(vi) are continuous on , and are continuous on ;(vii) and are continuous on and , respectively;(viii) and are continuous in time and piecewise regular on and , respectively.

Let and bo nonnegative integers. We say that is of class on if is continuous on and the functions exist and are continuous on . We denote by .

We say that is a dynamically admissible deformation field on provided that ; , , , , , , , are continuous on . By a dynamically admissible stress field on we mean an ordered array of functions with the following properties: (a) , and are of class on ; (b) on ; (c) , , , and are continuous on .

An admissible process on is an ordered array , , with the following properties:(1) is a dynamically admissible deformation field;(2) on ;(3), and are continuous on ;(4) is a dynamically admissible stress field on .

We say that , , is an elastic process corresponding to the body loads if is an admissible process that satisfies (22), (2), and (3) on . Given an external data system , , on , the dynamic problem consists in finding an elastic process corresponding to the body loads which satisfies the initial conditions (23) and the boundary conditions (24). We call such an elastic process a solution of the mixed problem.

Let be the function defined by

Let and be functions on that are continuous in time. We denote by the convolution of and :

We define the functions , and on by

Following [16] we can prove the following.

Theorem 2. Let , , . Then , , and satisfy (22) and the initial conditions (23) if and only if

The next theorem gives an alternative characterization of the mixed problem in which the initial conditions are incorporated into the field equations. This result is useful in the derivation of minimum principle.

Theorem 3. Let be an admissible process. Then is a solution of the mixed problem if and only if satisfies (2), (3), and (29) on and the boundary conditions (24).

In what follows we denote by or the Laplace transform with respect to the time of the function : for some .

In addition to the assumptions made previously, we assume that(), , , and possess Laplace transforms;()the internal energy is positive for any , and .

We say that is bounded at infinity if exists for each in the domain of definition of . We will assume that the functions used to describe the mixed problem are bounded at infinity.

We write for the th derivative of with respect to holding fixed. Following [17] we introduce the set of admissible weight functions. We say that if is a function on with the following properties:() exists for ;(), ,where is a continuous and positive function on and has a finite limit at infinity. An example of a weight function is with (cf. [17]).

By a kinematically admissible process we mean an admissible process that satisfies (2) and the boundary conditions

Let denote the set of all kinematically admissible processes such that the functions of and their first derivatives with respect to cartesian coordinates possess Laplace transforms.

Theorem 4. Assume that hypotheses and hold. Let be the functional on defined by for every . Further, let be a solution of the mixed problem. Then for every .

Proof. We consider and introduce by . If we denote , , then we have We note that If we use the definition of the weight functions we get By using (35) and (36) the functional can be expressed in the form where, for convenience, we have suppressed the argument .
If we take the Laplace transform of relations (29) and use (35) we get
In view of (3), (4), (34), and (38), we have With the aid of the divergence theorem and taking into account that is a solution of the problem, from (32), (37), (39), and (34) we find that From (40) and hypotheses and (v) we conclude that (33) holds.

In a similar way we can establish a minimum principle for the problem characterized by the following boundary conditions: where , and are prescribed functions.

5. Conclusions

In the present work, a theory of composites modelled as mixtures of two elastic Cosserat continua is investigated. A counterpart of the principle of minimum potential energy of the classical elastostatics is presented. In the dynamic theory, the solution of the boundary-initial-value problem is characterized by equations which incorporate the initial conditions. A minimum principle for the solution of dynamic problems is established.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

The authors equally contributed to this work.

Acknowledgment

The authors express their gratitude to the referees for their helpful suggestions.