In continuum mechanics, the theory of constitutive equations founded on the axioms introduced by Noll in. 1958 (Truesdell and Noll 1965) leads, in the case of ...
Continuum Mech. Thermodyn. (1997) 9:61–72
c Springer-Verlag 1997
On universal relations in continuum mechanics Edvige Pucci1 and Giuseppe Saccomandi2 1 2
Istituto di Energetica,Universit`a degli Studi di Perugia, Via G. Duranti, 06125 Perugia, Italy MEMOMAT, Universit`a La Sapienza, Via A. Scarpa 16, 00161, Roma, Italy
Received: August 16, 1996
This paper is devoted to a systematic study of local universal relations in continuum mechanics. We show that it is possible to determine the complete set of independent universal relations whose characterization is obtained by linear universal rules. A historical review of the literature on the topic and various significant examples are given. 1 Introduction In continuum mechanics, the theory of constitutive equations founded on the axioms introduced by Noll in 1958 (Truesdell and Noll 1965) leads, in the case of simple materials, to the following representation relative to a reference configuration for the stress tensor T(χ(X, t), t) = G(Gradχt ; X, t) .
(1.1)
In this equation T is the Cauchy stress at time t in the material particle E of the body B which has position X in the Euclidian space E at t = 0 and current position x given by x = χ(X, t) at time t; G is a symmetric response functional which satisfies the principle of material objectivity, and χt (X, s) = χ(X, t − s), s ≥ 0, denotes the history of the motion up to time t. The history of the deformation gradient F will be denoted by Ft . The constitutive equation of a simple material embraces almost all the mechanical constitutive laws usually used in physics and engineering, such as the Newtonian fluids or the Hookean solids. In particular in (1.1) we recover the theory of elastic materials when the functional G depends only on F at time t and not on its history. We remark that, in static theory, every simple material has the response of an elastic material. Once G is assigned, the balance equations, boundary conditions and equation (1.1) allow one to determine the admissible motions (kinematical unknowns) and the corresponding stress field of the body (dynamical unknowns). To determine the kinematical unknowns, (1.1) is introduced into the balance equations, which, usually, can be solved under suitable regularity conditions. It is then a trivial matter, once that a motion has been determined, to compute by (1.1) the corresponding stress field in any point and for any time. The constitutive equations (1.1) can be specialized to specific classes of materials, each class different in the invariance properties assigned to G. According to the admitted symmetry group Gˆ , i.e. according to the set of all second order invertible tensor H for which G(Ft H) = G(Ft ) ,
(1.2)
we distinguish the different classes of materials. In the sequel G will denote the corresponding class of materials.
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This classification is important since it leads to mathematical definitions for the various types of real materials (Truesdell and Noll 1965); and, once Gˆ is specified, i.e. the material symmetry is selected, particular restrictive conditions on the form of the response functional result. The choice of the type of simple material, along with the imposed material symmetry, leads to very useful representation theorems for the tensor G. It is then possible to specify the response of the material by means of a certain number of independent scalar functions of the history of deformation, called the response coefficients which characterize the different materials. Although a priori information about them can be derived at the physical level considering the empirical inequalities and at the mathematical level by requiring the wellposedness of boundary value problems, these quantities are mainly to be observed in suitable experiments. The mathematical results sought by experimenters to guide their design of tests and loading devices for a practical evaluation of the response coefficients are mainly those concerning problems investigated without further specifications on the response functional. These analytical results are known as the universal solutions and the universal relations. A vectorial function x = χ(X, t) which satisfies the balance equations with zero body force is called a controllable solution and, in equilibrium, is supported by suitable surface tractions alone. A controllable solution which is the same for all materials in a given class G is a universal solution. For example, if we consider homogeneous elastic materials and we assume isotropy as material symmetry, the homogeneous deformations are the set of all universal static solutions (Ericksen 1955). Any controllable solution which is the same for all materials in a family formed by a proper subset of a class G (obviously we do not consider singletons) will be called a relatively universal solution (deformation or motion). Besides universal solutions, other kinds of universal results, involving also the dynamical unknowns, exist. In correspondence with a given deformation or motion, a local universal relation is an equation between the stress components and the position vector components which holds in any X and t and which is the same for any material in an assigned family (or class). This relation, obviously, becomes an identity when also the stress components are expressed as functions of X and t. Sometimes directly from (1.1), apart from the balance equations, it is possible to derive a rule between stress components and the kinematical quantities which does not depends on the response coefficients. These rules are universal relations when the kinematical quantities are valued in correspondence with a universal or relatively universal solution. For example, from the constitutive equation for isotropic elastic materials we deduce the rule TB = BT, where B is the left Cauchy-Green strain tensor. From this rule, as discussed in Beatty (1987a), a great variety of universal relations are generated when B corresponds to a universal solution. Again, from this rule, universal relations are deduced by using relatively universal deformations for subclasses of isotropic elastic materials (Currie and Hayes 1982) . Only in very special cases is it possible to generate universal rules from TB = BT when B corresponds to true non-universal solutions, as it has been described by means of examples in Pucci and Saccomandi (1996a). As remarked by Beatty (1987b): “... universal results of various kinds are road signs posted to direct and to warn the experimenter in his exploration of the constitutive properties of real materials”. Driven by these considerations, here we investigate general questions on universal relations in the framework of some types of simple materials. Our aim is to determine when universal relations exist and then to give their complete characterization. In the next Section, we will present a historical survey of some results about local universal relations to emphasize the starting point of our systematic investigation. In Sect. 3, we specify the form of G for which our results are valid and give a list of materials of interest in application. In Sect. 4, restricting our attention to universal or relatively universal solutions, we introduce the idea of the universal manifold by reading (1.1) in a suitable Euclidian space from a geometrical point of view. This framework allows us not only to unify the approach to universal relations for the various forms of G, but also to find an interesting characterization of the complete set of these relations as the determining equations of the universal manifold. From the space dimension we deduce the number of linear independent universal
On universal relations in continuum mechanics
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rules for the case under consideration. Moreover, we give a representation for nonlinear universal relations that follows from the condition of annihilation of these rules on the universal manifold. In Sect. 5, the results obtained are developed for various meaningful cases with the aim of obtaining the general characterization of the universal rules. Examples are presented in illustration of our procedure by considering constitutive equations proposed in the literature as theoretical models of real materials. In the last Section, further remarks are presented to enlighten the importance of the hypothesis here assumed to obtain the general results.
2 Historical background In this Section we present a survey of results about local universal relations to outline the developments of the research in both the theoretical and applied approach to the topic. We specify that the term local is used to distinguish the results here studied from the ones related to global universal relations (Truesdell and Noll 1965). Moreover, we note that, in the literature, many other rules are identified as universal, as for example some relations between the velocities of propagation of elastic waves (Boulanger and Hayes 1991). Perhaps, the first example of a local universal relation is Rivlin’s rule for simple shear in isotropic elastic materials (Truesdell and Noll 1965). This rule relates normal and shear stress components with the amount of shear. If F = I + 2k e1 ⊗ e2 is the gradient of deformation in a simple shear, then Rivlin’s universal relation is T11 − T22 = kT12 . The first theoretical approach to universal relations appeared in Hayes and Knops (1966). They consider universal deformations for isotropic elastic materials (i.e. the homogeneous deformations) and determine the universal relations of the form tr(AT) = 0 where A is a symmetric matrix independent of the response coefficients. Their results point out that, in general, only three independent universal relations which express the property that the stress tensor T and strain tensor B are coaxial, can be obtained. The method used underlines the reason why also trivial universal relations, i.e. the ones that say that components of the stress tensor are zero, are important. It is also remarked that, when the eigenvalues of the strain have multiplicity 2 or 3, new universal relations, expressing the same property for the eigenvalues of T, can be found. In Wang (1969), universal relations for a subclass of elastic fluid crystals are found. After the determination of universal solutions, Wang presents what he calls the complete set of universal relations corresponding to a particular solution. For simple shear, this set consists of four linearly independent universal relations from which any other linear universal relation can be obtained by means of a combination. We point out that two of these rules are trivial. In the paper by Currie and Hayes (1982) we find, as a by product of a research on controllable solutions for classes of isotropic elastic materials, the first examples of universal relations corresponding to relatively universal solutions. Later on, results about universal relations for isotropic elastic materials are presented by Wineman and Ghandi (1984). The method of Hayes and Knops is used to find local universal relations associated with the universal deformation made by a shear superposed on a triaxial extension. These results are then used in Rajagopal and Wineman (1987) to examine some interesting facts in applications. In 1987, several papers dealing with the topic considered here have been published. The paper of Beatty (1987a) organizes the results about universal relations in isotropic elasticity associated with universal or relatively universal solutions. The subject is the tensorial rule TB = BT (expression of the coaxiality of the two tensors) and the three scalar equations which are derived from this one in terms of physical components. These equations are recognized as the generators of many universal relations, but no mention is made about the completeness of this set. In the review paper of Beatty (1987b), not only many results are summarized, but other examples for specific classes of isotropic elastic materials of interest in applications are given; among others, there is also mention of universal relations for constrained Bell materials. Universal relations for constrained isotropic materials are presented in Beatty (1989) in terms of the extra stress. Besides their theoretical importance, it is unlikely that they can have applications in laboratory tests. In this paper, as in Beatty and Hayes (1992) for constrained Bell materials, and in all the applications dedicated to incompressible materials, the undetermined scalar constraint function associated with the reaction stress is
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considered as a response coefficient and therefore all universal relations are generated in the same way as for unconstrained materials. The existence of universal relations peculiar to constrained materials is first recognized in Pucci and Saccomandi (1996b). For isotropic elastic materials subject to an isotropic constraint, a fourth linear universal relation independent from TB = BT can be found, for example, in correspondence to any universal deformation with constant principal invariants (and this is true also when all the eigenvalues are different). In the literature, we find also interesting results about universal relations related to true nonuniversal solutions, i.e. to controllable solutions whose kinematics or geometry is affected by response coefficients (Ogden, Chadwick and Haddon 1972, Beatty 1988). The paper of Pucci and Saccomandi (1996a) is dedicated to this specific subject. Outside the theory of elastic materials we notice that universal relations are presented for mixtures in Gandhi, Rajagopal and Wineman (1985), for viscoelastic fluids by Wineman and Rajagopal (1988), for elasticplastic materials in Rubin and Chen (1991), and for simple materials (first-gradient materials) by Negahban and Gandhi (1993). 3 Constitutive equations We will restrict our attention to all the simple materials for which the constitutive equations for the stress tensor (1.1) can be represented as follows T=
m X
µk P(k ) ,
(3.1)
k =1
where the tensors P(k ) ∈ Sym are kinematical quantities and the response coefficients µk are certain scalar functions of x, t and P(k ) . In Eq. (3.1) we also include the cases of internally constrained materials for which some µk represent the undetermined constraint reaction scalars. Then all the results we shall present are still valid for constrained materials, but as it has been shown in Pucci and Saccomandi (1996b), they are not exhaustive. In (3.1) we identify unconstrained hyperelastic materials i.e. all the materials whose elastic potential energy is given by a strain energy function Σ(X, t) = Σ(C, X), where C = FT F, and for which T = 2(detF)−1 F
∂Σ(C, X) T F . ∂C
(3.2)
Indeed, material symmetry for a hyperelastic solid tells us that the corresponding strain energy function takes the form (3.3) Σ(C, X) = Σ(Ik (C, X), X), k = 1, . . . , m where {Ik , k = 1, . . . , m} is an integrity basis made up of a finite number of invariants relative to the symmetry group. For example, when we consider a homogeneous isotropic solid, we have Σ = Σ(I1 , I2 , I3 ) ≡ Σ(trC, 12 {(trC)2 − trC2 }, detC) and then (3.2) becomes T = µ1 I + µ2 B + 2µ3 B−1 , −1/2
−1/2
(3.4)
1/2
[I2 Σ2 + I3 Σ3 ], µ2 = 2I3 Σ1 , µ3 = −2I3 Σ2 with Σi = ∂Σ/∂Ii . where the µ1 = 2I3 For a hyperelastic hemitropic solid, if the axis of hemitropy is in the direction of a unit vector a0 in the reference configuration, we have Σ = Σ(I1 , . . . , I6 ) ≡ Σ(trC, 12 {(trC)2 − trC2 }, detC, a0 · Ca 0 , a0 · C2 a0 , a0 · Ca0 × C2 a0 ) and, with a = Fa0 , T =µ1 I + µ2 B + µ3 B−1 + µ4 a ⊗ a + µ5 [a ⊗ Ba]sym +µ6 [Ba ⊗ F(a0 × FT a) + a ⊗ BF(a0 × FT a) + a ⊗ F(FT Ba × a0 )]sym .
(3.5)
If we consider transverse isotropy, I6 is no longer an invariant; and µ6 in (3.5) is taken zero (Spencer 1982; Zheng 1994).
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Another example is given by an orthotropic solid. We have Σ = Σ(I1 , . . . , I7 ) ≡ Σ(trC, 12 {(trC)2 − trC2 }, detC, a0 · Ca0 , a0 · C2 a0 , b0 · Cb0 , b0 · C2 b0 ), where a0 and b0 are the unit vectors normal to the two planes of reflectional symmetry in the reference configuration, and T = µ1 I + µ2 B + µ3 B−1 + µ4 a ⊗ a + µ5 [a ⊗ Ba]sym + µ6 b ⊗ b + µ7 [b ⊗ Bb]sym ,
(3.6)
where b = Fb0 . In (3.5) and (3.6), as in (3.4), the µk are functions of the corresponding invariants and the Σi derivatives. Obviously, in (3.1) we recover other special constitutive equations reported in the literature; these characterize particular families of materials through specific particular analytical forms of the strain energy. Moreover, hyperelastic solids are not the only simple materials for which the constitutive equation for stress tensor can be represented by (3.1). For example, the constitutive equation for the subclass of linearly isotropic viscous materials described by T = µ1 I + µ2 B + µ3 B−1 + µ4 D ,
(3.7)
where D is the stretching tensor and µi are functions of I1 , I2 and I3 , recently studied by Beatty and Zhou (1991), are again represented by (3.1). Also the grade n fluids are simple materials with constitutive equation of the form (3.1) (Truesdell and Noll 1965). The results to be presented below can be read both in the Lagrangian and in the Eulerian representation; the particular application will indicate the best approach.
4 The universal manifold We will consider a universal (or relatively universal) solution and compute the corresponding tensors P(k ) and response coefficients µk . When a time t and a position X in the reference configuration are fixed the (k ) components of the tensors P = P(k ) (X, t) become fixed real numbers, whereas the µk = µk (X, t) still could vary in subsets of R, the real line, since these coefficients change depending on the material. Let us denote with Tij the components of T with respect to an arbitrarily chosen basis {ei } and introduce the (6 + m) dimensional space S 6+m with coordinates τ ≡ {τ1 , . . . , τ6 } and µ ≡ {µ1 , . . . , µm } where τ1 = T11 ,
τ2 = T12 ,
τ3 = T13 ,
the relations T=
m X
τ4 = T22 , (k )
µk P
τ5 = T23 ,
τ6 = T33 ;
,
(4.1)
k =1
could be read as a representation of a linear m-dimensional homogeneous manifold, Vm , parametrized by the µ. Our interest is on the projection, Π(Vm ), of Vm on the subspace S 6 (τ ). In deriving such a projection we consider the following matrix (1) (1) (1) (1) (1) (1) P 11 P 12 P 13 P 22 P 23 P 33 (2) (2) (2) (2) (2) (2) P 11 P 12 P 13 P 22 P 23 P 33 (4.2) ... ... ... ... ... ... (m) (m) (m) (m) (m) (m) P 11 P 12 P 13 P 22 P 23 P 33 (k )
where the entries are the components of the tensors P with respect to {ei }. If the rank r of (4.2) is less than 6, then the manifold Π(Vm ) is not empty and its dimension is r; in the following it will be indicated as Π(Vm )r . This opportunity surely happens when m < 6. The linear manifold Π(Vm ) is the universal manifold represented by 6−r independent linear homogeneous equations in the variables τ , namely, a set of equations of the form 6 X (1) (1) (m) (m) τp φpq P , . . . , P ≡ = 0, Hq τ , P , . . . , P p=1
q = 1, . . . , 6 − r ,
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Then the equations Hq (T, P(1) , . . . , P(m) ) = 0,
q = 1, . . . , 6 − r ,
are a set of linear universal relations we were searching for. Obviously these are not the only possible linear universal relations since there are many ways to represent Π(Vm )r , but every linear universal relation can be obtained as a linear combination of these. universal simple approach is well illustrated by considering a non linear relation, The power of this (1) (m) = 0 which R T, P(1) , . . . , P(m) = 0. This rule can be read in S 6+m as an equation R τ , P , . . . , P must be an identity in Vm . Since R does not depend on µ, obviously, R ≡ 0 also in Π(Vm )r ⊂ Vm . This means that certain functions λh (T, P(1) , . . . , P(m) ) exist such that R(T, P(1) , . . . , P(m) ) ≡
6−r X
λh Hh
(4.3)
h=1
holds identically in Π(Vm )r , and hence the meaningful core of universal relations is given by the linear ones. Moreover, let us consider a specific class of materials, and let us choose a family of materials in this class characterized, for example, by a particular form of the strain energy function. In so doing we establish some links between the µh . It may happen that these links can be read as a manifold Ws in S 6+m (6 < s < 6 + m). ∗ ∗ ∗ )r 0 , where Vs−6 = Vm ∩ Ws and r 0 = s − 6. The Π(Vs−6 )r 0 Now the projection manifold of interest is Π(Vs−6 0 is the new universal manifold and since r < r, in this case, new universal relations arise. If the manifold ∗ )r 0 is also non linear and there is at least a true non linear universal relation Ws is non linear, then Π(Vs−6 i.e. not representable in terms of a basis of linear universal relations as in (4.3). General references about ideas of algebraic geometry here used, are in Segre (1972) and in Lang (1958).
5 Representations of the universal manifold We have just seen that, in correspondence to all the universal or relatively universal solutions, when m < 6 in (3.1), the dimension of the universal manifold is not zero. In this Section, we will examine these particular cases from a general point of view, determining a characterization of the manifold i.e. an intrinsic representation of the linear universal relations. Several applications are also presented.
5.1 The m = 1 case Let us consider the simplest case when m = 1 i.e. T = µP .
(5.1)
Now the universal manifold has dimension r = 1 and we have five linear independent universal relations. These rules express the property that T and P are coaxial and that their eigenvalues are proportional and can obviously be included in the vector equation Tu × Pu = 0 , which must hold for any vector u. An important example of a material for which the stress tensor is given in the form (5.1) is the Blatz-Ko −1/2 −1 B ]. foamed polyurethane elastomers. In this case µ = µ0 is a negative real constant and P = [1 − I3 If we consider a homogeneous triaxial stretch such that B = λ21 e1 ⊗ e1 + λ22 e2 ⊗ e2 + λ23 e3 ⊗ e3 , the universal manifold is represented by
(5.2)
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T11 [1 − (λ1 λ2 λ33 )−1 ] = T33 [1 − (λ31 λ2 λ3 )−1 ], T22 [1 − (λ1 λ2 λ33 )−1 ] = T33 [1 − (λ1 λ32 λ3 )−1 ] ,
(5.3)
T12 = T13 = T23 = 0 . −1/4
When λ1 = λ2 = λ3 , we have T11 = T22 = 0 which are the universal relations reported in (Beatty 1987b). Another class of materials which is represented in (5.1) are the isotropic subfluids presented in Wang (1970) for which µ = α and P = I. 5.2 The m = 2 case Now we have T = µ1 P(1) + µ2 P(2) ,
(5.4)
the dimension of the universal manifold is r = 2. Hence four universal linear independent relations are found. They can be determined by considering that the equation Tu · (P (1) u × P(2) u) = 0 ,
(5.5)
must be verified for any vector u. This equation is homogeneous and cubic in the components of u, and is to be identically satisfied. We thus generate ten linear universal relations, but, obviously, only four of them are independent. If P(1) and P(2) are coaxial it is convenient to express three of these universal relations as TP (1) = P(1) T, and the fourth rewriting (5.5) in the eigenbasis. In (5.4), we recover several interesting constitutive equations. In nonlinear elasticity, for example we have the general Blatz-Ko model for the solid polyurethane rubber (Beatty 1987b): −1/2
T = Σ3 (I3 )I + µˆ 0 I3
B,
(5.6)
where µˆ 0 is a positive constant, or the generalized neo-Hookean solid (Ogden 1984): n−1 b T = −pI + µˆ 1 + (I1 − 3) B, n
(5.7)
where p is the indeterminate scalar pressure, µˆ is a positive constant and n ≥ 1/2 ensures ellipticity for all deformations (Knowles and Stenberg 1975). If we consider the stretch (5.2) again, the complete set of universal relations for both (5.6) and (5.7) is given by T12 = T13 = T23 = 0 , (5.8) (T11 −
T22 )(λ22
−
λ23 )
= (T22 −
T33 )(λ21
−
λ22 )
.
Even some types of simple elastic fluid crystals possessing one preferred line or one preferred plane may be included in (5.4). Their constitutive equation is (Wang 1970) T = f0 (I3 )I + f1 (I3 )w ⊗ w ,
(5.9)
where w is the direction of the preferred unit vector in the deformed configuration, and f0 , f1 are arbitrary smooth functions of their argument. The complete set of universal relations for simple shear F = I + k e2 ⊗ e2 , is T13 = T23 = 0, T11 − T33 = h1 T12 , T22 − T33 = h2 T12 , where h1 = k , h2 = −1/k when the material has a preferred plane {e1 , e3 }, and h1 = 1/k , h2 = k if when the material has e2 as preferred direction. As a last example, we consider Navier-Stokes incompressible fluids for which
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T = −pI + 2νD ,
(5.10)
where ν is the constant kinematic viscosity. In accordance with controllable shearing flows with stretching tensor (5.11) 2D = fy (y, z )(e1 ⊗ e2 + e2 ⊗ e1 ) + fz (y, z )(e1 ⊗ e3 + e3 ⊗ e1 ) , where f (x , y) is a solution of the Poisson equation ∆f = C arbitrary constant, we have the following complete set of universal relations (5.12) T11 = T22 = T33 , T23 = 0, fz T12 = fy T13 . 5.3 The m = 3 case Now we have T = µ1 P (1) + µ2 P(2) + µ3 P(3) ,
(5.13)
and the dimension of the universal manifold is r = 3 and three linear independent universal relations exist. These rules can be computed by the tensorial relation (P(1) u × P(2) u · P(3) u)T =
(P(2) u × P(3) u · Tu)P(1) + +(P(3) u × P(1) u · Tu)P(2) + (P(1) u × P(2) u · Tu)P(3)
(5.14)
which must be verified for every vector u. This equation is obtained by considering that µi =
(P (j ) u × P(k ) u · Tu) , (P(i ) u × P(j ) u · P(k ) u)
where (i , j , k ) is an even permutation of (1,2,3). When P(1) , P(2) , P(3) are coaxial, (5.14) simplifies to the celebrated relation TP(1) = P(1) T (Beatty 1987a). In this case, we recover the constitutive Eq. (3.4) along with nearly all the universal relations that have been presented in the literature. Another constitutive equation of the form (5.13) is the one for the incompressible second grade fluids model compatible with thermodynamics. Dunn and Fosdick (1974) have shown for this model that the stress is determined by T = −pI + µA ˜ 1 + α1 (A2 − A21 ) where A1 = 2D and A2 are the first and second Rivlin-Ericksen tensors, and µ˜ ≥ 0, α1 ≥ 0 are constant constitutive response coefficients. The complete set of universal relations for the controllable shearing flow with stretching tensor (5.11) are given by fz T12 = fy T13 , fy fz (T33 − T11 ) = (2fy2 + fz2 )T23 , fy fz (T22 − T11 ) = (fy2 + 2fz2 )T23 . 5.4 The m = 4, 5 cases If m = 4 we have the constitutive equation T = µ1 P(1) + µ2 P(2) + µ3 P(3) + µ4 P(4) .
(5.15)
For any controllable solution, the universal manifold has dimension r = 4 and is represented by two linear independent universal relations. In (5.15) we find the subclass of linearly isotropic viscous materials (3.7) for which the two independent universal relations are given by w×ω =0, (5.16) Here w and ω are the axial vectors corresponding to the skew-symmetric tensors TB − BT and DB − BD, respectively.
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We also recover the class of elastic fiber reinforced materials for which T = µ1 I + µ2 B + µ3 B−1 + qa ⊗ a ,
(5.17)
where q is the undetermined constraint reaction scalar and a is the fiber direction in the deformed configuration. Universal relations for this case have been discussed in Beatty (1989), but only in terms of extra stress S = T − qa ⊗ a. For (5.17) two universal relations in terms of the Cauchy stress tensor are found w · a = 0,
Bw · a = 0 ,
where w is the axial vector already introduced. These rules are a different way to express the relation (5.16) where now ω must be read as the axial vector corresponding to (a ⊗ a)B − B(a ⊗ a). When the second grade fluid model is assumed as an approximation of more complex models, apart from the restrictions imposed by thermodynamics, we have T = −pI + µA1 + α1 A2 + α2 A21 , ˆ × ωˆ = 0, where w ˆ and ωˆ are the axial vectors corresponding The two universal relations are again given by w to TA1 − A1 T and A2 A1 − A1 A2 , respectively. For the same universal shearing motion consider earlier we have the two universal relations fz T12 = fy T13 ,
fy fz (T22 − T33 ) = (fy2 − fz2 )T23 .
The case m = 5 is the only remaining case for which it is possible to find universal relations for all universal or relatively universal solutions. The universal manifold has dimension r = 5 and only one linear independent universal rule is found. For example, if we consider hyperelastic transversely isotropic materials, obtained from (3.5) setting µ6 = 0, the universal rule is given by w·a=0. (5.18) The cases we have just examined are those for which always, for any P(i ) , we have a non empty linear universal manifold. Therefore at least one universal linear relation exists in correspondence to a universal or relatively universal solution. Given a material and the computed rank r of the matrix (4.2), the complete set of linear universal relations can be deduced from the characteristic formulae in the corresponding subsection.
6 Complementary results and concluding remarks The previous results apply regardless of restrictions on the underlying universal, or relatively universal, solutions. Obviously if we limit our attention to particular materials or particular solutions, we can obtain universal relations also when m ≥ 6 and new universal rules independent from those obtained above when m ≤ 5. To this end, it is sufficient that the rank of the matrix is not maximum. This situation has been observed by Hayes and Knops (1966) who recognized this result for isotropic elastic materials. Let us choose the deformation associated with the stretch (5.2) and rewrite the matrix (4.2) as 1 0 0 1 0 1 λ21 0 0 λ22 0 λ23 . −2 −2 λ1 0 0 λ2 0 λ−2 3 Then, when two stretches coincide, i.e. two proper numbers are equal, the rank of this matrix is r = 2. Consequently, a fourth universal relation is obtained. This new rule expresses the property that T also has two corresponding equal proper numbers. The hemitropic materials are an interesting example where the naive formula 6 − m (number of stress components minus number of constitutive coefficients) fails to give the right number of independent linear universal relations. In this case, although six are the coefficients, as a consequence of the form of P(6) , the
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rank of the 6 × 6 matrix (4.2) is r = 5, therefore, for all universal solutions there is one universal relation which is again expressed by (5.18). Let us consider an orthotropic material with constitutive equation (3.6) for which the two unit vectors a0 and b0 are assumed independent from X. Then all the homogeneous deformations are universal. Obviously, we cannot have universal relations, by assuming a general universal solution, since the rank of the matrix (4.2) is r = 7. On the other hand, if we choose the homogeneous spherical deformation with strain tensor B = α2 I the rank of the matrix becomes r = 3. Hence the universal manifold is no longer empty, and three independent linear universal relations exist. These can be deduced from the corresponding formula (5.14). Another important situation for which we can obtain further universal independent rules results from our considering particular kinds of universal, or relatively universal, solutions for constrained materials (Pucci and Saccomandi 1996b). In the above results, we have always considered undetermined constraint reaction scalar as a response coefficient, but this is restrictive. In reality, this scalar must be determined from the balance equations, and this fact is very useful in obtaining universal results . When the reaction scalar is determined as a linear and homogeneous form in the response coefficients, the rank of the matrix (4.2) decreases. It is shown that, for example, this is the case for isotropic elastic materials subject to an isotropic constraint when the deformation has constant principal invariants. Additional interesting cases are given for all families of materials for which the response coefficients are constant. As an example, we show the existence of a fourth universal relation for the classical Mooney-Rivlin materials defined by the strain energy 2Σ = α(I1 − 3) + β(I2 − 3). For any relatively universal deformation, the reaction scalar has the form p = απ1 (x) + βπ2 (x), where π1 and π2 are certain functions to be determined from the balance equations, and such that the total stress is always expressed as T = α(B − π1 (x)I) − β(B−1 + π2 (x)I) . Besides the coaxiality conditions, we have also the universal rule Tu · {(B − π1 (x)I)u × (B−1 + π2 (x)I)u} = 0 .
(6.1)
For the Navier Stokes fluids (5.10), we have the same situation. Indeed for the controllable shearing flow in (5.11), the corresponding pressure is p(x ) = −νρCx , where we have assumed p(0) = 0. To obtain the complete set, we must add the following independent universal relation to those in (5.12): T11 fy + CxT12 = 0 . When we consider true non-universal solutions particular remarks must be made. In this case, the theory here presented is no longer suitable, since nothing can be said a priori on the manifold (4.1) in the space S 6+m . However, it is again possible to show the existence of universal relations and this has been done in Pucci and Saccomandi (1996a). To illustrate this situation, we use an example deduced from the results of Ogden, Chadwick and Haddon (1972) for isotropic elastic materials. The geometry of the deformation considered is r = R,
θ = Θ + φ(R),
z = Z + f (R) ,
(6.2)
and represents combined axial and rotational shear of a tube. As shown in Rivlin (1949) this deformation is admitted by any incompressible elastic material, but not universally, because the functions φ(R) and f (R) exist for any material, but can be determined only when the particular material is specified. Indeed, from the balance equations, we deduce that φ and f must be solutions of the equations R 3 φR µ(k 2 ) = c1 R 2 + c2 ,
RfR µ(k 2 ) = c3 R 2 + c4 ,
(6.3)
where k 2 = R 2 φ2R + fR2 , and µ = µ2 − µ3 . When the strain B is computed from (6.2), it is easy to check that TB = BT, with some trivial algebra, yields RφR fR (T33 − T22 ) + (R 2 φ2R − fR2 )T23 = 0 , fR (T11 − T33 ) + fR2 T13 + RφR fR T12 − RφR T23 = 0 , fR T12 − RφR T13 = 0 .
(6.4)
On universal relations in continuum mechanics
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Then, although stress and strain are coaxial, the above scalar relations cannot be considered true universal relations since they depend on the derivatives of f (R) and φ(R) and, by (6.3), on the constitutive coefficient µ. In this case, as recognized in Ogden, Chadwick and Haddon (1972), we have the nonlinear universal relation 2 2 − T12 )=0. T13 T12 (T33 − T22 ) − T23 (T13
(6.5)
It is worth nothing that, for the subclass of Mooney Rivlin materials (µ = α + β constant), the deformation (6.2) where φ(R) = B /R 2 and f (R) = D ln(R/R0 ), is relatively universal. In this case, it is again possible to use our arguments and to build the complete set of linear independent universal relations, given by the rule TB = BT and (6.1). Moreover, as shown in Sect. 4, the nonlinear universal relation (6.5) can be now represented by the formula (4.3), where the linear universal relations are H1 ≡ 2BT13 + DRT12 = 0 ,
H2 ≡ 2BDR(T33 − T22 ) + (D 2 R 2 − 4B 2 )T23 = 0 ,
and λ1 = −T23
T12 T13 − , 2B DR
λ2 =
T13 T12 . 2BDR
The principal aim in this paper was to recognize more general properties that concern existence and determination of universal relations in continuum mechanics. All illustrative examples relate to constitutive equations proposed in the literature as meaningful models of real materials, so we achieve the double purpose of both explaining the theory and of providing results that may be useful in applications. Acknowledgements. This work has been partially supported by GNFM of Italian CNR. We thank Professor Beatty for stimulating our interest in this problem and for his constructive criticisms.
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