Royal Society Publishing

Some topics on a new class of elastic bodies

Roger Bustamante


In this paper, we study the problem of prescribing deformation as a function of stresses. For the particular case of small deformations, we find a weak formulation, from which we define the constitutive equation of a Green-like material, where an energy function that depends on the Cauchy stress tensor is proposed. Constraints on the deformation are studied for this new class of elastic bodies.


1. Introduction

In a recent paper, Rajagopal (2007) showed that the class of elastic bodies (bodies that do not dissipate energy) is larger than Cauchy or Green elastic materials (see also Rajagopal 2003; Rajagopal & Srinivasa 2007). A new class of a constitutive relationship between the Cauchy stress Embedded Image and the left Cauchy stretch tensor Embedded Image is1 (Rajagopal 2003)Embedded Image(1.1)from where we have, as special classes,Embedded Image(1.2)

In this paper, we investigate the class of materials described by (1.2)2; in particular, we consider the special case when the displacement gradient is small, but Embedded Image is, in general, a nonlinear function of Embedded Image.

This problem has already been studied in a short note by Bustamante & Rajagopal (in press), where for a constitutive equation of the form2 Embedded Image, they studied boundary-value problems for plane strain and stresses. They also found a weak formulation for such problems.

Here, we extend the results shown in that paper. In §2, we have a short review of some basic aspects of the theory of elasticity. We also give some comments on the theory of implicit constitutive equations developed by Rajagopal and co-workers (Rajagopal 2003, 2007; Rajagopal & Srinivasa 2007). In §3a, we study briefly a weak formulation for the boundary-value problem associated with the constitutive equation Embedded Image; this weak formulation is obtained for the plane strain problem using a general system of orthogonal coordinates. In §3b, we propose a weak formulation for the general three-dimensional problem. From these results, we define a new kind of Green-like (or hyperelastic-like) material in §4. In §5, we study briefly the problem of constraints on deformations and how they can be incorporated in materials described by Embedded Image=Embedded Image(Embedded Image), and in particular for the subclass Embedded Image. In §6, we comment about the results obtained, we speak in particular on the boundary conditions obtained from the weak formulation, and also on the complementary energy function in finite elasticity.

2. Fundamental relations

(a) Kinematics

Let Embedded Image denote a body and X denote a particle in this body in the reference configuration κr(Embedded Image). In the current configuration κt(Embedded Image), the position of the same particle is denoted by x. We assume that there exists a mapping Χ, such that (Truesdell & Noll 2004)Embedded Image(2.1)The displacement field u and the deformation gradient F are defined asEmbedded Image(2.2)The left and right Cauchy–Green stretch tensors Embedded Image and Embedded Image are defined asEmbedded Image(2.3)We define c asEmbedded Image(2.4)The Green–St Venant strain E and the linearized strain ϵ are defined, respectively, asEmbedded Image(2.5)

(b) Balance and compatibility equations

The Cauchy stress tensor Embedded Image must satisfy the balance equation (in the absence of body forces)Embedded Image(2.6)If Tij are the contravariant components of Embedded Image, the above equation is equivalent to (Truesdell & Toupin 1960)Embedded Image(2.7)

It is well known that to have deformations that satisfy (2.3)2, (2.4) or (2.5)1, some integration conditions must be satisfied (e.g. Truesdell & Toupin 1960; Fosdick 1966). These conditions areEmbedded Image(2.8)where Embedded Image is Riemann's tensor, whose components are defined as (Fosdick 1966)Embedded Image(2.9)whereEmbedded Image(2.10)The tensor Embedded Image can be either Embedded Image, c or E.

(c) Constitutive equations

As mentioned in §1, Rajagopal (2003, 2007) considered a new class of elastic bodies, whose constitutive relation is of the form Embedded Image(Embedded Image,Embedded Image)=0.

Assuming Embedded Image to be isotropic, Rajagopal (2007) obtained the implicit constitutive relationEmbedded Image(2.11)where βi, for i=1, …, 8, are scalar functions that depend on the invariants obtained from the pair Embedded Image, Embedded Image.

From the above relation, it is possible to see that what we know as Cauchy isotropic elastic material, Embedded Image, is in fact a special case of (2.11). Moreover, a constitutive equation of the form (1.2)2Embedded Image(2.12)is also a subclass of (2.11), and can be seen as a new class of elastic bodies.

Let us assume that (see Rajagopal 2007) Embedded Image with δ≪1, this implies that Embedded Image. Note that, in general, the converse is not true. From (2.2) and (2.3)1, we haveEmbedded Image(2.13)which, as a result of the previous assumption, gives the approximationEmbedded Image(2.14)and, from (2.12), we obtain the following constitutive equation valid for an isotropic material in the case Embedded Image with δ≪1:Embedded Image(2.15)In (2.12) and (2.15), we can assume that α0, α1, α2, γ0, γ1 and γ2 are only scalar functions of the invariants of Embedded Image.

Invariance under superposed rigid motions should be studied directly, for example, from (2.12). We need QEmbedded ImageQT=QEmbedded Image(Embedded Image)QT=Embedded Image(QEmbedded ImageQT) for all orthogonal Q with det Q=1. Using (2.13), for example, in (2.12), after some manipulations we haveEmbedded Image(2.16)If Embedded Image, we see from the above equation that (2.15) is approximately invariant to rigid motions. Note that, in general, this is not the case if we would just assume ‖ϵ‖ to be small.

On assuming Embedded Image, Rajagopal (2007) showed that the appropriate approximation, when we consider small deformations but arbitrarily large stresses, is of the formEmbedded Image(2.17)and not of the form Embedded Image, which is the classical form used to model nonlinear behaviour of solids for infinitesimal deformations.

A final remark about Embedded Image and the scalar functions γ1, γ2 and γ3 that appear in (2.15). The constitutive equations (2.15) and (2.17) are valid for the case where we have small deformations and arbitrarily ‘large’ stresses; therefore, Embedded Image, γ1, γ2 and γ3 should be such that ϵ remains small for arbitrary Embedded Image. This could be seen as a restriction on the constitutive equation.

There is another more subtle and important restriction to consider. As we saw previously, ‖ϵ‖ small does not imply necessarily Embedded Image small. The condition Embedded Image is important in order to have invariance to rigid motions; as a result, this condition should be seen as an additional restriction on the constitutive equations.

(d) Boundary-value problem

In general, from (1.2)2 or (2.17), we will not be able to obtain explicit expressions for Embedded Image as a function of Embedded Image or ϵ. We could still try to obtain Embedded Image numerically from (1.2)2 or (2.17), and then to solve (2.6) to find Embedded Image using the usual procedures. However, we can try a different method, as shown in the paper by Bustamante & Rajagopal (in press, §1), where we can express Embedded Image as a function of Airy's stress function, and then from (1.2)2 or (2.17), we could obtain either Embedded Image or ϵ as, in general, nonlinear functions on this stress potential. But if Embedded Image or ϵ (or eventually Embedded Image or c) are known and given from (2.3), (2.4) or (2.5), then to have continuous displacement fields, the compatibility equation (2.8) must be satisfied.

For the plane strain and stress cases, using Cartesian coordinates and working with the constitutive equation (2.15), Bustamante & Rajagopal (in press, §1) obtained a highly nonlinear partial differential equation for the stress potential (eqn (36) of that paper). That equation is a generalization of the well-known biharmonic equation, which is found following a similar procedure for the linearized theory (Muskhelishvili 1953).

Owing to the present difficulties to solve (2.8) for ϵ using, for example, (2.17), Bustamante and Rajagopal developed a weak formulation for the plane problem, only considering Cartesian coordinates. This weak formulation will be used to solve some boundary-value problems working with the finite-element method. In this paper, we study more general cases.

An important question about equation (2.8) is: what boundary conditions do we have to use to solve this problem? We will partially answer this question in the following sections.

3. Weak formulations

(a) Plane case, general coordinates

We work with an arbitrary system of orthogonal coordinates in a general plane strain problem. In such a case, the compatibility equations (2.8) for E, using Eϵ with ‖ϵ‖ small, reduces to (Eringen 1980; Saada 1993)Embedded Image(3.1)

Consider an Airy stress function Φ, the equilibrium equation (2.7) (plane strain) is satisfied if Embedded Image, Embedded Image and Embedded Image (Love 1944). Taking the product of (3.1) with Embedded Image (a virtual Airy stress function), integrating in the volume Embedded Image and using the divergence theorem, after some manipulations, we obtainEmbedded Image(3.2)where Embedded Image are the components of the virtual stress tensor associated with the virtual Airy stress function through Embedded Image, Embedded Image and Embedded Image. On the surfaces Embedded Image and Embedded Image, we have assumed ϵij,k and ϵij as given data, respectively. We have Embedded Image and Embedded Image.

(b) Three-dimensional case

The equilibrium equations (2.7) in the three-dimensional problem (no body forces) are satisfied if (see §227 of Truesdell & Toupin (1960), and the paper by Finzi (1934))Embedded Image(3.3)where eijk is the permutation symbol and ars are the components of a tensor potential a. This tensor is symmetric. From (3.3), we haveEmbedded Image(3.4)Embedded Image(3.5)Embedded Image(3.6)Embedded Image(3.7)Embedded Image(3.8)andEmbedded Image(3.9)

If we consider Embedded Image with δ≪1, the general form of the compatibility equation (2.8) (E)Embedded Image, with Eϵ, is (Eringen 1980; Saada 1993)Embedded Image(3.10)

Only six of the above equations are independent; these areEmbedded Image(3.11)Embedded Image(3.12)Embedded Image(3.13)Embedded Image(3.14)Embedded Image(3.15)andEmbedded Image(3.16)

Consider a virtual tensor potential Embedded Image. Consider the following integral, where we have taken the multiplication of the above equations by different components of Embedded Image, adding them in one expression and integrating:Embedded Image(3.17)We can prove, for example, that after some manipulations, the multiplication of equation (3.11) by Embedded Image becomesEmbedded Image(3.18)Similar expressions can be found for the multiplication of equation (3.12) by Embedded Image, equation (3.13) by Embedded Image, equation (3.14) by Embedded Image, equation (3.15) by Embedded Image and equation (3.16) by Embedded Image. These are the expressions that appear in (3.17). As a result, (3.17) is equivalent toEmbedded Image(3.19)where the contravariant components di of the vector d that appear in the second integral areEmbedded Image(3.20)Embedded Image(3.21)andEmbedded Image(3.22)

From (3.19), using (3.4)–(3.9) and the divergence theorem, we obtainEmbedded Image(3.23)where Embedded Image is the virtual stress tensor, whose components are calculated using Embedded Image from (3.4)–(3.9).

The components di of the vector d can be expressed asEmbedded Image(3.24)where Embedded Image and Embedded Image are the components of the second- and third-order tensors (i)Λ and (i)ϒ, respectively. The matrix forms of the tensors (i)Λ areEmbedded Image(3.25)Embedded Image(3.26)andEmbedded Image(3.27)The non-zero components of the third-order tensors (i)ϒ areEmbedded Image(3.28)Embedded Image(3.29)Embedded Image(3.30)Embedded Image(3.31)Embedded Image(3.32)Embedded Image(3.33)Embedded Image(3.34)Embedded Image(3.35)Embedded Image(3.36)Embedded Image(3.37)Embedded Image(3.38)Embedded Image(3.39)Embedded Image(3.40)Embedded Image(3.41)andEmbedded Image(3.42)

In coordinate-free notation, (3.23) can be written asEmbedded Image(3.43)From (3.24), we see thatEmbedded Image(3.44)In (3.43) and (3.44), the products Embedded Image, Embedded Image and Embedded Image are interpreted as Embedded Image, Embedded Image and Embedded Image, respectively.

In (3.44), the term Embedded Image could be viewed as the dot product Λ.n of a third-order tensor Λ with n. The components of this tensor are defined asEmbedded Image(3.45)The dot product Λ.n is equivalent to Embedded Image.

Regarding the term Embedded Image, this could be viewed as the dot product of a fourth-order tensor ϒ with n, where the components of this tensor are defined asEmbedded Image(3.46)The dot product ϒ.n is interpreted as Embedded Image.

Therefore, (3.43) can be written asEmbedded Image(3.47)Finally, if Embedded Image on Embedded Image and Embedded Image on Embedded Image, with Embedded Image and Embedded Image, where Embedded Image and Embedded Image are prescribed values of a and ∇a on Embedded Image, then Embedded Image on Embedded Image and Embedded Image on Embedded Image, (3.47) becomesEmbedded Image(3.48)

4. A Green-like material

As in the classical theory of nonlinear elasticity (Ogden 1997; Truesdell & Noll 2004), from (3.48) we could assume that there might be some materials for which there exists an ‘energy’ function3 W=W(Embedded Image), such thatEmbedded Image(4.1)From the above definition, we have, in the first integral on the left-hand side of (3.48),Embedded Image(4.2)where now Embedded Image could be interpreted as the variation in a of W; Embedded Image as the variation in a of Embedded Image; and Embedded Image as the variation of a.

For an isotropic material, we have that W should be a function of three invariants. We choose two different sets of invariantsEmbedded Image(4.3)andEmbedded Image(4.4)These two sets are not independent.

For an isotropic material, we have Embedded Image. Using the chain rule in (4.1), considering (4.3) and (4.4) we obtain, respectively,Embedded Image(4.5)andEmbedded Image(4.6)In the above expressions, W1, W2 and W3 are the partial derivatives of W in I1, I2 and I3, respectively.

In the classical theory of elasticity, for Green materials, the independent variable is the deformation gradient F. The class of deformations that, in general, are admitted are such that det F>0; as a result, F−1 exists. But in our case, in which W=W(Embedded Image), it is perfectly possible from the physical point of view to have a stress tensor such that det Embedded Image=0; in fact, Embedded Image=0 for all Embedded Image would correspond to the case that there is no external load and no residual stress. Therefore, for some problems, Embedded Image−1 might not exist, and this could create problems in the evaluation of the constitutive equation (4.6); however, we have that Embedded Image, where cof Embedded Image is the cofactor matrix associated with Embedded Image, and this matrix can always be defined.

For later reference, we also illustrate the case of a transversely isotropic material, which is symmetric with respect to a field e, with ‖e‖=1. In such a case, we have W=W(Embedded Image,e)=W(I1, I2, I3, I4, I5), where the invariants Ii, i=1, …, 5, are defined as (e.g. Spencer 1971; Zheng 1994)Embedded Image(4.7)From (4.1) and the chain rule, we obtainEmbedded Image(4.8)where Wi are the partial derivatives of W in Ii, i=1, …, 5.

5. Constraints

Constraints are defined as restrictions on deformations, which can be described as (Truesdell & Noll 2004)Embedded Image(5.1)The function γ must be frame indifferent; therefore, (5.1) is written asEmbedded Image(5.2)

In the case where we work with the constitutive equation (1.2)2, from (5.1) an isotropic constraint could be written asEmbedded Image(5.3)where we must impose the restriction Embedded Image for all Q orthogonal with det Q=1.

Now, for a function Embedded Image=Embedded Image(Embedded Image), an isotropic constraint of the form (5.3) would just introduce a restriction in the form of Embedded Image in a direct way, which means that we would not need to assume some sort of decomposition of a stress in two parts, with one of them that does not do work with any deformation compatible with the constraint, which is the classical, somehow controversial, argument used in the classical theory of elasticity (Rajagopal & Saccomandi 2005).

Consider, for example, an isotropic material (2.12), where Embedded Image. Let us study, as illustration, the incompressibility constraint det F−1=0, which can be described asEmbedded Image(5.4)Using (2.12) in (5.4), we obtainEmbedded Image(5.5)which could be seen as a cubic equation we could use to find, for example, α0 as a function of α1, α2 and Embedded Image.

Let us study now how constraints can be incorporated in our constitutive equation (2.17), which is a special case of (1.2)2. If Embedded Image, δ≪1, we have Embedded Image and Embedded Image, then (5.2) is equivalent toEmbedded Image(5.6)Expanding the function λ as a Taylor series, we getEmbedded Image(5.7)which, if we neglect the terms O(δ2), becomesEmbedded Image(5.8)Let us study two examples where we apply (5.8).

(a) Incompressibility

Consider an incompressible material, from λ(Embedded Image)=det Embedded Image−1, we have λ(I)=0, Embedded Image, and so Embedded Image; therefore, (5.8) is equivalent to I:ϵ=0, which is equal toEmbedded Image(5.9)This is a classical result, equivalent to div u=0, which could have been obtained directly from det F=1, using F=I+∇u, with Embedded Image and δ≪1.

If we apply (5.9) to the constitutive equation of our new elastic isotropic material (2.15) Embedded Image, we obtainEmbedded Image(5.10)where, for example γ0 has been expressed explicitly as a function of γ1, γ2 and Embedded Image.

Interesting results are found for a Green-like material. Consider the set of invariants (4.3) and its corresponding constitutive equation (4.5). The restriction (5.9) implies thatEmbedded Image(5.11)

The solution of this equation isEmbedded Image(5.12)where Embedded Image is a function in two variables Embedded Image and Embedded Image.

For this case, we use the set of invariants (4.4), from (4.6) and (5.9) we obtain the partial differential equationEmbedded Image(5.13)whose solution is Embedded Image.

(b) Inextensibility

Let us consider an additional case, where we have a material that is inextensible in the direction e with ‖e‖=1. This is an example of a non-isotropic constraint. In this case, (5.2) becomes (e.g. Truesdell & Noll 2004)Embedded Image(5.14)In this case, we have λ(I)=0 and Embedded Image; therefore, (5.8) is equivalent to (e.g. §17.2 of Podio-Guidugli (2000))Embedded Image(5.15)

Let us consider only a Green-like material. If the direction in which this material is inextensible is the same field e we used in the definition of the transversely isotropic material (4.8), then the restriction (5.15) impliesEmbedded Image(5.16)The solution of this equation is of the formEmbedded Image(5.17)whereEmbedded Image(5.18)Embedded Image(5.19)

6. Some further remarks

In this paper, we explored some issues concerning a constitutive equation of the form Embedded Image. We discussed the boundary-value problem that we should solve for such a constitutive law. We obtained a weak formulation for the general three-dimensional case, extending some results presented in a recent paper by Bustamante & Rajagopal (in press, §1). From this weak formulation, for the case in which ‖ϵ‖ is small, we were able to define a Green-like material, through the use of an energy function W=W(Embedded Image). Additional issues concerning constraints were studied, in particular how constraints on deformations impose restrictions on either Embedded Image or W(Embedded Image).

(a) Boundary conditions

An important issue that appeared in this paper corresponds to the boundary conditions found from the weak formulation (3.48). In that expression, we have the terms Λ.n and ϒ.n product Embedded Image and Embedded Image, respectively; these terms are integrated over the surface Embedded Image of the body. From there, we could assume that boundary conditions associated with the partial differential equations (3.11)–(3.16) could beEmbedded Image(6.1)where Embedded Image and Embedded Image would be the prescribed values of Λ and ϒ, which from (3.25)–(3.27) to (3.28)–(3.42) would be calculated from prescribed values of ϵ and its first derivatives in x on Embedded Image.

In the classical theory of elasticity, the two important boundary conditions considered in most problems are a prescribed external traction Embedded Image and a prescribed displacement Embedded Image on the boundary of the body (Truesdell & Noll 2004). These are not the only boundary conditions we can find, other types can be found, for example, in §§5.1 and 5.2 of Ciarlet (1988) and in the paper by Noll (1978).

The condition (6.1) would correspond to a new kind of boundary conditions for the strains. In fact, even for the linearized theory, where, for (2.15), we would have Embedded Image, Embedded Image and γ2=0, working, for example, in the plane case, for the third integral on the left-hand side of (3.2) we would getEmbedded Image(6.2)We see that only for the second term inside the square brackets on the right-hand side of (6.2) could we identify the condition Embedded Image.

From the physical point of view: could we actually apply, for example, a prescribed strain on the boundary of a body?

Bustamante & Rajagopal (in press, §1) developed explicitly the partial differential equation that must be solved to find the Airy stress function Φ (see §3a) for the plane case, when we use Cartesian coordinates (see eqn (36) of that paper). A boundary condition for that equation could be the classical traction on the boundary Embedded Image, where Embedded Image is expressed in terms of the second derivatives of Φ. We then see that if we want to solve the strong form of the boundary-value problem, we do not have difficulties imposing directly Embedded Image on Embedded Image.

(b) On the Green-like material

We developed a weak formulation for the case Embedded Image; from this formulation in §4, we were able to define a new class of Green-like elastic material. From those results, we could be tempted to do the same for the more general nonlinear problem Embedded Image; however, this has been proved to be much more difficult.

In the case Embedded Image with δ≪1, we have Eϵ and from (2.9), Embedded Image, with Γkls calculated from (2.10) with Embedded Image=ϵ. The equation Γkls,r−Γkrs,l=0 was multiplied by appropriate components of the virtual stress potential Embedded Image, from where we obtained (3.48), which was used to propose the existence of an energy function W(Embedded Image), such that Embedded Image.

Now, for the nonlinear case Embedded Image=Embedded Image(Embedded Image), we have tried to do the same; however, in the expression for the Riemann tensor, we have to deal with nonlinear terms of the form Embedded Image. We have not been able, so far, either to incorporate these terms in Embedded Image, or to incorporate them in boundary conditions of the form (6.1). Of course, we could leave these terms aside, considering for Embedded Image only the contribution of the linear terms that come from Γkls,r, and leaving terms that come from Embedded Image as some sort of extra contribution for the virtual work. But evidently if we do so, any function W=W(Embedded Image) will not actually consider the whole energy stored by the body.

In the classical theory of nonlinear elasticity, there has been a long discussion on the existence of complementary energy functions (e.g. §5.4.3 of Ogden (1997); see also Ogden (1975), Dill (1977) and Zhong-Heng (1980)). This complementary energy function, which can be denoted as Ωcc(S), should be such that Embedded Image, where S would be the nominal or first Piola–Kirchhoff stress tensor. This complementary energy function has been proposed directly from the classical energy function Ω=Ω(F), through the use of a Legendre transformation. The important assumption made is the invertibility of the function Ω(F), which in general is not possible (Dill 1977).

Although the strong forms (2.7) and (2.8) are different, and so we cannot expect that weak forms derived from them could be directly comparable; it is remarkable to note that similarly we have not been able to find an equation for a Green-like elastic material, for the case in which, for example, Embedded Image is given as a function of Embedded Image (or for any other pair of strains and stresses); in the general case, we work with large deformations and displacements.

(c) Final comments

When we work with Embedded Image, assuming Embedded Image with δ≪1, the displacement field u can be calculated easily from (2.5)2 and is unique if (2.8) holds. But it is interesting to note that when we work with the more general case Embedded Image, once the compatibility equation (2.8) (with Embedded Image=c=Embedded Image−1) would be satisfied, we could say that Χ(X,t) is unique, but in order to calculate explicitly this field, we would need to solve (2.3)1 Embedded Image. This problem is not trivial (e.g. Blume 1989).

In a future communication, we expect to show solutions of some simple boundary-value problems, solving directly (2.8) with Embedded Image=ϵ, for the plane stress case, and for an isotropic material (2.15) ϵ=γ0I+γ1Embedded Image+γ2Embedded Image2, considering some simple forms for γi, i=0, 1, 2.


  • For an application, see Mollica et al. (2007).

  • This equation was derived from (1.2)2. The symbol ϵ is the linearized strain.

  • In the classical linear theory of elasticity, from the volume integral of the dot product of a virtual displacement Embedded Image field with the balance equation div Embedded Image+b=0, we obtainEmbedded Imagewhere Embedded Image would be the virtual deformation. The above equation can be seen as a balance of internal and external virtual work.

    Equation (3.48) could be seen as an equivalent expression for the balance of virtual work, working now with a virtual tensor potential Embedded Image. In (3.48), the surface integrals Embedded Image and Embedded Image would contribute to the external virtual work; however, their exact physical meanings are not clear yet. This is why we put the word ‘energy’ in quotation marks.

    • Received October 23, 2008.
    • Accepted December 22, 2008.


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