Rotational invariance conditions in elasticity, gradient elasticity and its connection to isotropy

# Rotational invariance conditions in elasticity, gradient elasticity and its connection to isotropy

## Abstract

For homogeneous higher gradient elasticity models we discuss frame-indifference and isotropy requirements. To this end, we introduce the notions of local versus global SO(3)-invariance and identify frame-indifference (traditionally) with global left SO(3)-invariance and isotropy with global right SO(3)-invariance. For specific restricted representations, the energy may also be local left SO(3)-invariant as well as local right SO(3)-invariant. Then we turn to linear models and consider a consequence of frame-indifference together with isotropy in nonlinear elasticity and apply this joint invariance condition to some specific linear models. The interesting point is the appearance of finite rotations in transformations of a geometrically linear model. It is shown that when starting with a linear model defined already in the infinitesimal symmetric strain , the new invariance condition is equivalent to isotropy of the linear formulation. Therefore, it may be used also in higher gradient elasticity models for a simple check of isotropy and for extensions to anisotropy. In this respect we consider in more detail variational formulations of the linear indeterminate couple stress model, a new variant of it with symmetric force stresses and general linear gradient elasticity.

Key words: invariance conditions, frame-indifference, covariance, isotropy, orthogonal group, strain gradient elasticity, couple stress, polar continua, symmetric stress, hyperstresses, modified couple stress model, rotational invariance, form-invariance, Rayleigh product

AMS Math 74A10 (Stress), 74A35 (polar materials), 74B05 (classical linear elasticity), 74A30 (nonsimple materials), 74A20 (theory of constitutive functions), 74B20 (nonlinear elasticity)

## 1 Introduction

This paper is motivated by our endeavour to better understand isotropy conditions in higher gradient elasticity theories. Therefore, we make use of the old idea expressed by Truesdell [54, Lecture 6] albeit including higher gradient continua.1 To this end we reconsider frame-indifference and isotropy first in nonlinear elasticity and then in classical linear elasticity theories. These investigations are extended to linear higher gradient elasticity with a focus on the indeterminate couple stress theory. We do not discuss general material symmetry conditions a la Noll’s group theoretic framework, as e.g. presented in [11, 50].

In nonlinear elasticity, isotropy requirements are fundamentally different from frame-indifference requirements. Indeed, isotropy rotates the referential coordinate system, while frame-indifference rotates the spatial coordinate system. Both spaces, referential and spatial, are clearly independent of each other and the corresponding rotations of frames are not connected to each other.

In this paper, however, we will later apply the same rotation to the referential and spatial frame simultaneously. It is clear that this is not equivalent to frame-indifference or isotropy as we show with simple explicit examples. What, then, is the use of such a specific transformation (which we will call the -transformation in Section 4). Indeed, once we consider reduced energy expressions which already encode frame-indifference, then applying the -transformation and requiring form-invariance of the energy under this transformation is equivalent to isotropy. Surprisingly, the -transformation remains operative in exactly the same way when applying it to the geometrically linear context. It is this insight that we follow when discussing isotropy conditions for higher gradient continua. We do not believe that our development leads to new results for the representation of isotropic formulations,2 but we believe that there is a conceptual gain of understanding when working directly with transformation connected to finite rotations. In this respect we remind the reader that for linearized frame-indifference it would be sufficient to work with the Lie-algebra of skew-symmetric rotations instead of the orthogonal group. This is, however, not possible with respect to isotropy considerations.

Further, we give a contribution to the discussion in [33], where a fundamental aspect of strain gradient elasticity as proposed by Mindlin [37] is doubted. There we read: ”However, the approach described by equation (1) includes a serious flaw:3 Couple stresses are components of a tensor which has a rank higher than two. In conclusion, the torque arising from an asymmetric stress tensor cannot consequently be compensated by the torque associated with a tensor of higher rank. In fact, there are different transformation rules for tensors of different rank during rotation of a coordinate system. Therefore, couple stress theory does not consequently fulfill the requirement of invariance with respect to rotation of the coordinate system. In spite of this deficiency, strain gradient theory still contains important aspects which are corrected.” Motivated by this statement we show to the contrary that all proper invariance requirements are satisfied.

The paper is now structured as follows. After notational agreements we recall the gradient of continuum rotation, which is the curvature measure in linear couple stress-elasticity. Identities from the scalar triple product built a basis to study transformation rules for this curvature measure later in the text. In Section 2 we start our investigation in the context of nonlinear hyperelasticity by discussing frame-indifference and isotropy for first and second gradient continua. Then we consider the simultaneous transformation of the deformation to new spatial and referential coordinates. The consequence of this transformation in linearized elasticity follows in Section 3. Further, we prepare for higher gradient elasticity by giving details on transformation rules and apply them to the linear momentum balance equations of several models in Section 4. We then define form-invariance in higher gradient elasticity in Section 5 and specify our result for several couple stress theories in Section 6. Finally, we conclude and give an outlook.

### 1.1 Notational agreements and preliminary results

By we denote the set of real second order tensors, written with capital letters. Vectors in are denoted by small letters. Additionally, tensors in are necessary for our discussion, where significant symbols like , , and will be used. The components of vectors and tensors are given according to orthogonal unit vectors , which may be rotated by . Throughout this paper Latin subscripts specify the direction of components in index notation and take the values . For repeating subscripts Einstein’s summation convention applies.

We adopt the usual abbreviations of Lie-algebra theory, i.e., is the Lie-algebra of skew symmetric tensors and is the Lie-algebra of traceless tensors. For all we set , and the deviatoric part and we have the orthogonal Cartan-decomposition of the Lie-algebra

 gl(3) ={sl(3)∩Sym(3)}⊕so(3)⊕R⋅\mathbbm1,X=devsymX+skewX+13tr(X)\mathbbm1, (1.1)

simply allowing to split every second order tensor uniquely into its trace-free symmetric part, skew-symmetric part and spherical part, respectively. For

 ¯¯¯¯A\colonequals⎛⎜⎝0−a3a2a30−a1−a2a10⎞⎟⎠∈so(3),¯¯¯¯Av=a×v,∀v∈R3,a=axl[¯¯¯¯A],¯¯¯¯A=anti[a], (1.2)

the functions and are given by

 axl[¯¯¯¯A]\colonequals−12¯¯¯¯Aijϵijkek,anti[a]:=−ϵijkakei⊗ej=¯¯¯¯Aijei⊗ej, (1.3)

with for even permutation, for odd permutation, and else. Note that the skew-symmetric part of a tensor can be written by the combination of permutations via

 skewX=12ϵnijϵnabXabei⊗ej. (1.4)

From eq.(1.4) and eq.(1.3) one obtains

 axl[skewX]=−12ϵijk12ϵnijδnkϵnabXabek=−12Xabϵabkek. (1.5)

In index notation the typical conventions for differential operations are implied such as comma followed by a subscript to denote the partial derivative with respect to the corresponding coordinate. The gradient of a scalar field and the gradient of a vector field are given by

Similarly, we also define the gradient of a tensor field :

For vectors we let

 ⟨a,b⟩R3=aibi∈R,a×b=aibjϵijkek∈R3, (1.9)

denote the inner and outer product on with associated vector norm . The standard Euclidean inner product on is given by

 ⟨X,Y⟩R3×3=XijYij∈R, (1.10)

and thus the Frobenius tensor norm is . Similarly, on we consider the inner product

 ⟨m,n⟩R3×3×3=\emphmijk\emphnijk∈R, (1.11)

and the tensor norm . The identity tensor on will be denoted by , so that and . Further, we will make repeated use of the identity . The divergence of a vector field reads:

Note that we do not introduce symbolic notation for the double contraction or more complicated contractions for higher order tensors, since this may be confusing and also limited for certain cases. Instead, we use generally index notation for contractions being not defined by the scalar product. Thus, the divergence of a second order tensor is given by

Similarly, the divergence of a third order tensor reads

 DIVxm\colonequals\emphmijk,kei⊗ej∈R3×3. (1.14)

The curl of a vector is given by

 curlxv\colonequals−va,bϵabiei∈R3. (1.15)

Similarly, the Curl of a second order tensor field is defined by

 CurlxX\colonequals−Xia,bϵabjei⊗ej∈R3×3. (1.16)

Using eq.(1.5) the curl of a vector can be written in terms of a gradient via

In this work we consider a body which occupies a bounded open set of the three-dimensional Euclidian space and assume that its boundary is a piecewise smooth surface. An elastic material fills the domain and we refer the motion of the body to the displacement field shifting any point of the reference configuration to the actual configuration .

Since the gradient of a scalar is curl-free

the definition in eq.(1.16) is such that the gradient of a vector field is also curl-free:

However, the skew-symmetric part of the gradient of a vector field is generally not curl-free:

In the indeterminate couple stress model the gradient of the continuum rotation defines the curvature measure

From eq.(1.17) and (1.1) it follows that

Further, it has been shown in [20] (see also [48, p. 27]) that can be also written in terms of

The representations of with the skew-symmetric gradient in eq.(1.22) on the one hand, and the symmetric gradient in eq.(1.23) on the other hand seems amazing. For the proof of relation (1.23) we use eq.(1.1), (1.1), and to obtain the above statement:

Note that the trace of is zero:

### 1.2 Identities from the Levi-Civita tensor

In section 1.1 we have used the baseless permutation to define some functions. Therefore, the base of the functions argument yields the base of the function automatically. On the one hand, this is convenient but on the other hand not sufficiently precise to investigate some transformation rules to be considered in section 4. Thus, we make use of the third order Levi-Civita tensor

 ϵ\colonequals⟨ei,ej×ek⟩ei⊗ej⊗ek=ϵijkei⊗ej⊗ek∈R3×3×3. (1.26)

The components of the Levi-Civita tensor are the scalar triple product of an orthogonal unit base, which may be rotated. Therefore, let be a constant rotation tensor with and , mapping the orthogonal referential system of Euclidean vectors to a rotated system via

 di=Qei,Q=di⊗ei,Qia=⟨di,ea⟩=⟨ea,di⟩,i=1,2,3,a=1,2,3. (1.27)

Note that the components are defined by the commutative inner product. However, the rotation tensor is generally not symmetric: . Since the referential basis vectors are considered to be orthogonal and of unit length the rotated basis vectors are orthogonal and of unit length as well:

 ⟨di,dj⟩=⟨Qiaea,Qjb eb⟩=QiaQjbδab=QiaQja=QiaQTaj=δij. (1.28)

Thus, the Levi-Civita tensor can be written in terms of the rotated base with components

 ϵ♯ijk=⟨di,dj×dk⟩=⟨Qiaea,Qjbeb×Qkcec⟩=QiaQjbQkc⟨ei,ej×ek⟩=QiaQjbQkcϵabc. (1.29)

Since the scalar triple product does not depend on the direction of the orthogonal unit base, the components of the Levi-Civita tensor are independent concerning the direction of the orthogonal unit base4 in . Thus, the Levi-Civita tensor is an isotropic tensor of order 3 with respect to proper orthogonal transformations, which is reading in eq.(1.29), and yielding the identity

 ϵijk=QiaQjbQkcϵabc. (1.30)

Since the symbolic notation of tensor products is only established in we prescind from defining a new kind of simple contraction. This would be necessary to switch from index to symbolic notation in eq.(1.30), where each rotation tensor has one simple contraction with the corresponding index of the permutation.5 Since the rotation in eq.(1.30) is arbitrary, the identity must also hold for the transposed of , which is also in . Thus, a similar identity to eq.(1.30) is easily deduced:

 ϵijk=QaiQbjQckϵabc. (1.31)

Another identity can be found via simple contraction of and the Levi-Civita tensor, which is considered in the basis on the left hand side and by the basis on the right hand side:

 (1.32)

Note that eq.(1.32) links a linear to a quadratic term of .

## 2 Objectivity and isotropy in nonlinear elasticity

### 2.1 Objectivity and isotropy in nonlinear elasticity - the local case

In geometrically nonlinear hyperelasticity we know that frame-indifference is left-invariance of the energy under SO(3)-action and isotropy is right-invariance under SO(3), i.e.6 for the energy density we have

 frame−indifference: W(Q1F)=W(F)∀Q1∈SO(3), (2.1) isotropy: W(FQ2)=W(F)∀Q2∈SO(3). (2.2)

Therefore, by specifying we obtain also the necessary invariance condition

 W(QFQT)=W(F)∀Q∈SO(3). (2.3)

Note that condition (2.3) does not imply objectivity, as can be seen from considering the energy expressions

 W(F) =∥F−\mathbbm1∥2≠∥QF−\mathbbm1∥2=W(QF), W(F) =∥sym(F−\mathbbm1)∥2≠∥sym(QF−\mathbbm1)∥2=W(QF), (2.4)

satisfying (2.3) but not being frame-indifferent [19]. With the same example one sees that condition (2.3) does not imply isotropy. Therefore, applying the transformation (i.e. the -transformation defined in eq.(2.9)) has no intrinsic meaning in nonlinear elasticity theory as such.

It is clear that every frame-indifferent elastic energy can be expressed in the right Cauchy-Green tensor in the sense that there is a function such that

 W(F)=Ψ(FTF). (2.5)

Of course, any of the form (2.5) is automatically frame-indifferent. Applying the isotropy condition (2.2) to the representation in eq. (2.5) we must have

 W(FQ)=W(F)⇔Ψ(QTFTFQ)=W(FQ)=W(F)=Ψ(FTF)∀Q∈SO(3). (2.6)

Therefore, for reduced energy expressions (2.5) the isotropy requirement can be equivalently stated as

 Ψ(QCQT)=Ψ(C)∀Q∈SO(3). (2.7)

In eq.(2.7) we see that isotropy is invariance of the function under simultaneous spatial and referential rotation of the coordinate system with the same rotation by interpreting the Cauchy-Green tensor as a linear mapping , which transforms under such a change of coordinate system as

 C→QCQT. (2.8)

We may now define a transformation of the deformation to new spatial and referential coordinates via7

 φ♯(ξ)\colonequalsQφ(QTξ)=Qφ(x)withξ=Qx,x=QTξ. (2.9)

Then8

and

Gathering our findings so far we can state:

 The nonlinear hyperelastic formulation is frame-indifferent and isotropic if and only if there exists Ψ:Sym+(3)→R such that W(F)=Ψ(FTF) and W(Gradξ[φ♯(ξ)])form−invariance=W(Gradx[φ(x)])[i.e.Ψ(QCQT)=Ψ(C)∀Q∈SO(3)]. (2.12)

In nonlinear elasticity, invariance of the formulation under eq.(2.9) is a consequence of objectivity (left -invariance of the energy). Therefore, the transformation in eq.(2.9) may be used to probe objectivity and isotropy of the formulation, while it is not equivalent to both. If we assume already the reduced representation in , then rotational invariance of the formulation under eq.(2.9) is equivalent to isotropy, see also [36, p. 220] .

The well-known representation theorems imply that any satisfying the isotropy condition and the classical format of material frame-indifference,

 W(QF)=W(F)=Ψ(C)∀Q∈SO(3), (2.13)

must be expressible in terms of the principal invariants of , i.e.,

 W(F)=Ψ(C)=ˆΨ(I1(C),I2(C),I3(C)), (2.14)

where

 I1(C)\colonequalstr(C),I2(C)\colonequalstr(CofC)I3(C)\colonequalsdetC. (2.15)

Incidentally, is not only invariant under compatible changes of the reference configuration with rigid rotations (constant rotations), but also under inhomogeneous rotation fields , that is,

 W(FQ(x))=W(F)∀Q(x)∈SO(3), (2.16)

since

 Ik((FQ(x))TFQ(x))=Ik(Q(x)TFTFQ(x))=Ik(FTF),k=1,2,3. (2.17)

Therefore, the principal invariants are unaffected by inhomogeneous rotations. However, we must stress that isotropy per se is not defined as invariance under right multiplication with . The difference between form-invariance under compatible transformations with rigid rotations (isotropy) and right-invariance under inhomogeneous rotation fields will only become visible in higher gradient elasticity treated in the next section. In any first gradient theory both requirements coincide. Note that likewise, (2.16) is invariant under left multiplication with inhomogeneous rotation fields. We define for further use

 \bf left-local SO(3)-invariance:W(Q(x)F(x))=W(F(x))∀Q(x)∈SO(3)\lx@parboxnewline\lx@parboxnewline\bf right% -local SO(3)-invariance:W(F(x)Q(x))=W(F(x))∀Q(x)∈SO(3). (2.18)

Similarly, we define for constant rotations

 \bf left-global SO(3)-invariance:W(¯¯¯¯QF(x))=W(F(x))∀¯¯¯¯Q∈SO(3)\bf objectivity\lx@parboxnewline\lx@parboxnewline\bf right-global SO(3)-invariance:W(F(x)¯¯¯¯Q)=W(F(x))∀¯¯¯¯Q∈SO(3)\bf isotropy. (2.19)

With these definitions we have shown that for both, local and global, left and right SO(3) invariance are satisfied. Left-global SO(3)-invariance is identical to the Cosserat’s invariance under ”action euclidienne” [8].

### 2.2 Isotropy in second gradient nonlinear elasticity

In this subsection we would like to extend the ”local” picture of the previous subsection to second gradient materials. Using the Noether theorem and Lie-point symmetries, the explicit expressions of the isotropy condition in linear gradient elasticity of grade-2 and in linear gradient elasticity of grade-3 have been derived by Lazar and Anastassiadis [32, eq.(4.76)] and Agiasofitou and Lazar [2, eqs.(57) and (58)], respectively, as consequence of global rotational invariance.

For the sake of clarity we leave first objectivity aside and discuss only isotropy. Moreover, we make the following simplifying assumptions. We consider a homogeneous material given as a finite-sized ball and we want to formalize the statement that first rotating the ball and then to apply the loads leaves invariant the response in the sense that this is indistinguishable from not rotating the ball, see Fig. 1. This is in accordance with the statement in Truesdell & Noll [55, p. 78]: ”For an isotropic material in an undistorted state, a physical test cannot detect whether or not the material has been rotated arbitrarily before the test ist made.”

Let the elastic energy of the body depend also on second gradients of the deformation, i.e. we consider therefore,

We first transform eq.(2.20) to new coordinates via introducing an orientation preserving diffeomorphism ,

 x =ζ(ξ),ξ=ζ−1(x),x=ζ(ζ−1(x)), (2.21)

see also [40, 51]. For further reference, we compute some relations connected to the transformation (2.21). Taking the first and second derivative with respect to in (2.21) and using (2.21) we obtain

and

With help of eq.(2.22) the latter implies

Connected to the coordinate transformation (2.21) we consider the deformation expressed in these new coordinates via setting

 φ♭(ξ)\colonequalsφ(ζ(ξ)),φ♭(ζ−1(x))=φ(x). (2.25)

The standard chain rule induces for the first and second derivative of with respect to the following relations