## Abstract

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 and the left Cauchy stretch tensor is1 (Rajagopal 2003)(1.1)from where we have, as special classes,(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 is, in general, a nonlinear function of .

This problem has already been studied in a short note by Bustamante & Rajagopal (in press), where for a constitutive equation of the form2 , 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 §3*a*, we study briefly a weak formulation for the boundary-value problem associated with the constitutive equation ; this weak formulation is obtained for the plane strain problem using a general system of orthogonal coordinates. In §3*b*, 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 =(), and in particular for the subclass . 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 denote a body and **X** denote a particle in this body in the reference configuration *κ*_{r}(). In the current configuration *κ*_{t}(), the position of the same particle is denoted by **x**. We assume that there exists a mapping ** Χ**, such that (Truesdell & Noll 2004)(2.1)The displacement field

**u**and the deformation gradient

**F**are defined as(2.2)The left and right Cauchy–Green stretch tensors and are defined as(2.3)We define

**c**as(2.4)The Green–St Venant strain

**E**and the linearized strain

**are defined, respectively, as(2.5)**

*ϵ*### (b) Balance and compatibility equations

The Cauchy stress tensor must satisfy the balance equation (in the absence of body forces)(2.6)If *T*^{ij} are the contravariant components of , the above equation is equivalent to (Truesdell & Toupin 1960)(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 are(2.8)where is Riemann's tensor, whose components are defined as (Fosdick 1966)(2.9)where(2.10)The tensor can be either , **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 (,)=0.

Assuming to be *isotropic*, Rajagopal (2007) obtained the implicit constitutive relation(2.11)where *β*_{i}, for *i*=1, …, 8, are scalar functions that depend on the invariants obtained from the pair , .

From the above relation, it is possible to see that what we know as Cauchy isotropic elastic material, , is in fact a special case of (2.11). Moreover, a constitutive equation of the form (1.2)_{2}(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) with *δ*≪1, this implies that . Note that, in general, the converse is not true. From (2.2) and (2.3)_{1}, we have(2.13)which, as a result of the previous assumption, gives the approximation(2.14)and, from (2.12), we obtain the following constitutive equation valid for an isotropic material in the case with *δ*≪1:(2.15)In (2.12) and (2.15), we can assume that *α*_{0}, *α*_{1}, *α*_{2}, *γ*_{0}, *γ*_{1} and *γ*_{2} are only scalar functions of the invariants of .

Invariance under superposed rigid motions should be studied directly, for example, from (2.12). We need **Q****Q**^{T}=**Q**()**Q**^{T}=(**Q****Q**^{T}) for all orthogonal **Q** with det **Q**=1. Using (2.13), for example, in (2.12), after some manipulations we have(2.16)If , 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 , Rajagopal (2007) showed that the appropriate approximation, when we consider small deformations but arbitrarily large stresses, is of the form(2.17)and not of the form , which is the classical form used to model nonlinear behaviour of solids for infinitesimal deformations.

A final remark about and the scalar functions *γ*_{1}, *γ*_{2} and *γ*_{3} 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, , *γ*_{1}, *γ*_{2} and *γ*_{3} should be such that * ϵ* remains small for arbitrary . 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 small. The condition 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 as a function of or *ϵ*. We could still try to obtain numerically from (1.2)

_{2}or (2.17), and then to solve (2.6) to find 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 as a function of

*Airy's stress*function, and then from (1.2)

_{2}or (2.17), we could obtain either or

**as, in general, nonlinear functions on this stress potential. But if or**

*ϵ***(or eventually or**

*ϵ***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 **E**≈** ϵ** with ‖

**‖ small, reduces to (Eringen 1980; Saada 1993)(3.1)**

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

### (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))(3.3)where *e*^{ijk} is the permutation symbol and *a*_{rs} are the components of a *tensor* potential **a**. This tensor is symmetric. From (3.3), we have(3.4)(3.5)(3.6)(3.7)(3.8)and(3.9)

If we consider with *δ*≪1, the general form of the compatibility equation (2.8) ^{(E)}, with **E**≈** ϵ**, is (Eringen 1980; Saada 1993)(3.10)

Only six of the above equations are independent; these are(3.11)(3.12)(3.13)(3.14)(3.15)and(3.16)

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

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

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

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

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

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

Therefore, (3.43) can be written as(3.47)Finally, if on and on , with and , where and are prescribed values of **a** and ∇**a** on , then on and on , (3.47) becomes(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*(), such that(4.1)From the above definition, we have, in the first integral on the left-hand side of (3.48),(4.2)where now could be interpreted as the variation in **a** of *W*; as the variation in **a** of ; and 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 invariants(4.3)and(4.4)These two sets are not independent.

For an isotropic material, we have . Using the chain rule in (4.1), considering (4.3) and (4.4) we obtain, respectively,(4.5)and(4.6)In the above expressions, *W*_{1}, *W*_{2} and *W*_{3} are the partial derivatives of *W* in *I*_{1}, *I*_{2} and *I*_{3}, 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*(), it is perfectly possible from the physical point of view to have a stress tensor such that det =**0**; in fact, =**0** for all would correspond to the case that there is no external load and no residual stress. Therefore, for some problems, ^{−1} might not exist, and this could create problems in the evaluation of the constitutive equation (4.6); however, we have that , where cof is the cofactor matrix associated with , 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*(,**e**)=*W*(*I*_{1}, *I*_{2}, *I*_{3}, *I*_{4}, *I*_{5}), where the invariants *I*_{i}, *i*=1, …, 5, are defined as (e.g. Spencer 1971; Zheng 1994)(4.7)From (4.1) and the chain rule, we obtain(4.8)where *W*_{i} are the partial derivatives of *W* in *I*_{i}, *i*=1, …, 5.

## 5. Constraints

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

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

Now, for a function =(), an isotropic constraint of the form (5.3) would just introduce a restriction in the form of 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 . Let us study, as illustration, the incompressibility constraint det **F**−1=0, which can be described as(5.4)Using (2.12) in (5.4), we obtain(5.5)which could be seen as a cubic equation we could use to find, for example, *α*_{0} as a function of *α*_{1}, *α*_{2} and .

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 , *δ*≪1, we have and , then (5.2) is equivalent to(5.6)Expanding the function *λ* as a Taylor series, we get(5.7)which, if we neglect the terms *O*(*δ*^{2}), becomes(5.8)Let us study two examples where we apply (5.8).

### (a) Incompressibilit*y*

Consider an incompressible material, from *λ*()=det −1, we have *λ*(**I**)=0, , and so ; therefore, (5.8) is equivalent to **I**:* ϵ*=0, which is equal to(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 and

*δ*≪1.

If we apply (5.9) to the constitutive equation of our *new elastic isotropic* material (2.15) , we obtain(5.10)where, for example *γ*_{0} has been expressed explicitly as a function of *γ*_{1}, *γ*_{2} and .

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 that(5.11)

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

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

### (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)(5.14)In this case, we have *λ*(**I**)=0 and ; therefore, (5.8) is equivalent to (e.g. §17.2 of Podio-Guidugli (2000))(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) implies(5.16)The solution of this equation is of the form(5.17)where(5.18)(5.19)

## 6. Some further remarks

In this paper, we explored some issues concerning a constitutive equation of the form . 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*(). Additional issues concerning constraints were studied, in particular how constraints on deformations impose restrictions on either or

*W*().

### (a) Boundary condition*s*

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 and , respectively; these terms are integrated over the surface of the body. From there, we could assume that boundary conditions associated with the partial differential equations (3.11)–(3.16) could be(6.1)where and 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 .

In the classical theory of elasticity, the two important boundary conditions considered in most problems are a prescribed external traction and a prescribed displacement 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 , and *γ*_{2}=0, working, for example, in the plane case, for the third integral on the left-hand side of (3.2) we would get(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 .

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 §3*a*) 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 , where 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 on .

### (b) On the Green-like material

We developed a weak formulation for the case ; 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 ; however, this has been proved to be much more difficult.

In the case with *δ*≪1, we have **E**≈** ϵ** and from (2.9), , with Γ

_{kls}calculated from (2.10) with =

**. The equation Γ**

*ϵ*_{kls,r}−Γ

_{krs,l}=0 was multiplied by appropriate components of the virtual stress potential , from where we obtained (3.48), which was used to propose the existence of an

*energy*function

*W*(), such that .

Now, for the nonlinear case =(), 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 . We have not been able, so far, either to incorporate these terms in , or to incorporate them in boundary conditions of the form (6.1). Of course, we could leave these terms aside, considering for only the contribution of the linear terms that come from Γ_{kls,r}, and leaving terms that come from as some sort of *extra* contribution for the *virtual work*. But evidently if we do so, any function *W*=*W*() 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 Ω_{c}=Ω_{c}(**S**), should be such that , 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, is given as a function of (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 , assuming 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 , once the compatibility equation (2.8) (with =**c**=^{−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}. 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 =** ϵ**, for the plane stress case, and for an isotropic material (2.15)

**=**

*ϵ**γ*

_{0}

**I**+

*γ*

_{1}

*+**γ*

_{2}

^{2}, considering some simple forms for

*γ*

_{i},

*i*=0, 1, 2.

## Footnotes

↵For an application, see Mollica

*et al*. (2007).↵This equation was derived from (1.2)

_{2}. The symbolis the linearized strain.*ϵ*↵In the classical linear theory of elasticity, from the volume integral of the dot product of a virtual displacement field with the balance equation div +

**b**=**0**, we obtainwhere 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 . In (3.48), the surface integrals and 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.

- © 2009 The Royal Society