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Constitutive Laws: Principle of Material Frame-Indifference

Introduction

The principle of Material Frame-Indifference is an important concept in continuum mechanics. Simply put, the concept is the idea that constitutive laws describing the behaviour of a material (for example the relationship between stress and strain) should be independent of the frame of reference or the motion of the observer. For example, Young’s modulus of a material should be the same to an observer whether the observer is standing or moving. In this section we will first introduce the idea of the motion of the observer. Then, the associated changes in the expressions of deformation and stress measures will be investigated.

Motion of the Observer or Change of Frame of Reference

Assuming an embedding of a three dimensional object in $\mathbb{R}^3$ as described in the description of motion section, we will assume that the observer in the spacial configuration is moving with a rigid body motion described by a rotation matrix $Q(t)$ and a translation $c(t)$ where $t$ is time. Let $B=\{e_1,e_2,e_3\}$ be an arbitrary fixed basis set in the spatial configuration and $B'=\{e'_1,e'_2,e'_3\}$ be the spatial basis set that is associated with the motion of the observer. If $x$ and $x'$ are the position vectors with respect to $B$ and $B'$ respectively, then the relationship between their components is described as:

$x'(t)=Q(t)x(t)+c'(t)$

Where $x'=\{x'_1,x'_2,x'_3\}\in\mathbb{R}^3$ with $B'$ as the basis set while $x=\{x_1,x_2,x_3\}\in\mathbb{R}^3$ with $B$ as the basis set. In this case, $Q_{ij}=e'_i\cdot e_j$ . In the above equation, $Q$ is dependent on time through the vectors $e'_i$ while the vectors $e_i$ are assumed fixed. It is important to note that the results in this section would not change if the basis vectors in $B$ are considered fixed or not as long as the quantities $Q(t)$ and $c'(t)$ provide the relative measures between $B'$ and $B$ . It is also important to note that if the dependence on time is omitted, then, the above relationship can simply be regarded as a “passive” change of frame of reference.

With a change of frame of reference (whether that change is dependent on time or not), a scalar field is called objective if the field values are independent of the change of the frame of reference. On the other hand, vector and tensor fields are called objective if their description changes similar to how vector and tensor quantities change with a simple change of coordinates as shown in the change of basis and higher order tensors sections. In the following sections, how components of various kinematic and kinetic quantities change with a change of observer will be presented.

Measures of Deformation

Consider a continuum object whose reference configuration is represented by the set $\Omega_0\subset \mathbb{R}^3$ with basis set $B_r=\{E_1,E_2,E_3\}$ , while its spacial configuration is represented by the set $\Omega \subset \mathbb{R}^3$ with the fixed basis set $B=\{e_1,e_2,e_3\}$ . Let $B'=\{e'_1,e'_2,e'_3\}$ be the basis set associated with a moving spacial observer. Let $x$ and $x'$ be the position vectors with respect to $B$ and $B'$ respectively while $X$ be the position vector in the reference configuration. An important assumption is that both observers agree on the description of the reference configuration, i.e., the reference configuration is not affected by the change of the observer. In other words, at a particular time instant, the two observers agree on the components $X$ of the reference configuration and at that instant the basis sets $B_r$ , $B$ , and $B'$ are assumed to coincide and so does the origin of these systems. This assumption is needed to be able to calculate displacements since the operation $x-X$ is only defined within a particular vector space. For a more general description, a reader is referred to the article by Liu and Sampio which provides a thorough analysis of these concepts. In the following, a deformation measure $D$ is associated with the basis sets $B$ and $B_r$ while a deformation measure $D'$ is associated with the basis sets $B'$ and $B_r$ . Notice is given to those quantities that do not change or that transform similar to a “simple” change of basis as described as in the change of basis section.

Deformation Gradient

In this setting the deformation gradient maps tangent vectors from the reference configuration to tangent vectors in the deformed (spacial) configuration.

$F=\frac{\partial x}{\partial X}$

On the other hand we have

$F'=\frac{\partial x'}{\partial X}=\frac{\partial x'}{\partial x}\frac{\partial x}{\partial X}=QF$

suggesting that $F$ transforms similar to how vectors transform under a simple change of basis. Many authors regard $F$ to be objective even though it does not abide by the definition of objectivity given above as it is considered a two point tensor. Also, in this relationship, $X$ is assumed to be the reference configuration as realized by both observers!

The determinant of $F$ , namely $J$ is an objective scalar field since:

$J'=\det{F'}=\det{(QF)}=\det{F}=J$

If $F=RU$ and $F'=R'U'$ are the right polar decompositions as observed in $B$ and $B'$ , then, utilizing the uniqueness of the decomposition, expressions for the transformation of $R$ and $U$ can be found as follows:

$F'=R'U'=QRU\Rightarrow R'=QR \mbox{ and }U'=U$

Similarly, if $F=VR$ and $F'=V'R'$ are the left polar decompositions as observed in $B$ and $B'$ , then:

$F'=V'R'=QVR\Rightarrow V'=QVQ^T \mbox{ and }R'=QR$

It is important to note that $U$ is independent of the observer. $U$ can be regarded as a linear transformation that acts on the reference configuration vectors with the resulting vectors still in the reference configuration and thus, the motion of the observer does not affect it. On the other hand, $V$ transforms similar to how a second order tensor transforms under a simple change of basis, i.e., $V$ is an objective tensor.

The Right and Left Cauchy Deformation Tensors

The right Cauchy deformation tensor is unaltered by a change of observer:

$C'=F'^TF'=F^TQ^TQF=F^TF=C$

$C$ is considered a field described in the reference configuration, therefore, it is unaffected by the rotation of the observer. The left Cauchy deformation tensor, however, is a spacial field and thus changes according to a simple change of observer:

$B'=F'F'^T=QFF^TQ^T=QBQ^T$

I.e., $B'$ is considered an objective tensor.

Velocity

Naturally, the spacial velocity field will be dependent on the relative velocity of the observers. The spacial velocity field is a map of the velocity of each material point in the deformed configuration and is given by:

$v=\frac{\partial x}{\partial t}=\dot{x}$

On the other hand, a moving observer would see the velocity as:

$v'=\frac{\partial x'}{\partial t}=\dot{x}'=\dot {Q}x+Q \dot {x}+\dot {c}'=\dot {Q}x+Q v+\dot {c}'$

Velocity Gradient

Similarly, the expressions for the velocity gradient will be dependent on the relative velocity of the observers:

$L'=\frac{\partial v'}{\partial x'}=\frac{\partial v'}{\partial x}\frac{\partial x}{\partial x'}=\left(\dot{Q}+QL\right)Q^T=\dot{Q}Q^T+QLQ^T$

Since $Q$ is a rotation matrix, the following is always true:

$QQ^T=I\Rightarrow \dot{Q}Q^T+Q\dot{Q}^T=0$

This relationship can be used to show that the symmetric part of $L$ (The stretch tensor) transforms similar to how second order tensors transform with a simple change of basis:

$D'=\frac{1}{2}\left(L+L^T\right)=\frac{1}{2}\left(\dot{Q}Q^T+Q\dot{Q}^T\right)+QDQ^T=QDQ^T$

I.e., $D$ is an objective tensor. The transformation of the skewsymmetric part (spin tensor), however, contains expressions with the relative rotational velocity between the observers:

$W'=\frac{1}{2}\left(\dot{Q}Q^T-Q\dot{Q}^T\right)+QWQ^T$

Displacement

The displacement as seen by the moving observer has the form:

$u'=x'-X=Qx+c'-X$

Implied in this formula is that the position vector $X$ has the same description when the reference configuration is described using $B$ or $B'$ . In other words, $B$ , $B'$ , and $B_r$ coincide when it comes to the reference configuration along with the origin of these systems.

Displacement gradient

Both the spacial and the referential displacement gradients are not dependent on the relative velocity between the observers. The spacial displacement gradient:

$\nabla u =\frac{\partial u}{\partial x} =I-F^{-1}$

On the other hand we have:

$\nabla u'=\frac{\partial u'}{\partial x'}=\frac{\partial u'}{\partial x}\frac{\partial x}{\partial x'}=(Q-F^{-1})Q^T=I-F^{-1}Q^T=\nabla u Q^T+\left(I-Q^T\right)$

Similarly, the referential displacement gradient is given by:

$\nabla_r u =\frac{\partial u}{\partial X} =F-I$

Where the subscript $r$ refers to differentiating with respect to the coordinates in the reference configuration. On the other hand we have:

$\nabla_r u'=\frac{\partial u'}{\partial X}=\frac{\partial u'}{\partial x}\frac{\partial x}{\partial X}=(Q-F^{-1})F=QF-I=Q\nabla_r u +(Q-I)$

One would expect that the gradients of $u$ would transform similar to a simple change of coordinates. This would have been the case if the reference configuration description changes with the change of the observer such that the relationship between $X$ and $X'$ is similar to the relationship between $x$ and $x'$ . In that case, $F'=QFQ^T$ and the gradients of $u$ would have transformed as expected.

Small Strain Matrix

The small strain matrix is the symmetric part of the referential displacement gradient:

$\varepsilon=\frac{\nabla_r u+\nabla_r u ^T}{2}=\frac{F+F^T-2I}{2}$

It transforms as follows:

$\varepsilon'=\frac{QF+F^TQ^T-2I}{2}$

Similar to the gradients of displacements, one would expect that the small strain matrix would transform such that $\varepsilon'=Q\varepsilon Q^T$ . This would have been the case if the reference configuration description changes with the change of the observer such that the relationship between $X$ and $X'$ is similar to the relationship between $x$ and $x'$ . In that case, $F'=QFQ^T$ and the small strain matrix would have transformed as expected.

The Green Strain Tensor

It is straightforward to show that the Green strain tensor is independent of the observer:

$\varepsilon'_{\mbox{Green}}=\frac{1}{2}\left(C'-I\right)=\frac{1}{2}\left(C-I\right)=\varepsilon_{\mbox{Green}}$

The Green strain tensor is a field in the reference configuration coordinate system and therefore it is independent of the observer.

Measures of Stress

A fundamental assumption that is needed for this section is how forces transform under the change of observer. A force vector $f$ in spacial coordinates is assumed to adopt the form of a simple change of basis when viewed by different observers such that:

$f'=Qf$

Under this assumption, the expressions for the transformation of the the different measures of stress can be studied.

Cauchy Stress Tensor

The symmetric Cauchy stress tensor transforms area vectors to force vectors as follows:

$t_n=\sigma^Tn=\sigma n$

Under a change of observer, the traction vector $t_n$ and the area vector $n$ transform according to simple change of basis, therefore:

$t'_n=\sigma' n'=\sigma' Qn=Qt_n=Q\sigma n\Rightarrow \sigma'=Q\sigma Q^T$

I.e., the Cauchy stress tensor transforms according to simple change of basis, i.e., $\sigma$ is objective.

First Piola Kirchhoff Stress Tensor

The first Piola Kirchhof stress tensor transforms as follows:

$P'=\det(F')\sigma'^TF'^{-T}=\det(F) Q \sigma Q^T QF^{-T}=\det(F) Q \sigma F^{-T}=QP$

I.e., it transforms similar to how $F$ transforms.

Second Piola Kirchhoff Stress Tensor

The second Piola Kirchhoff stress tensor is independent of the observer:

$S'=F'^{-1}P'=F^{-1}Q^TQP=F^{-1}P=S$

$S$ is a field in the reference configuration and therefore is independent of the observer.

Strain Energy

The strain energy rates are scalar fields that are objective. The spacial rate of change of the internal energy per unit volume is given by:

$\dot{\overline{U'}}=\mathrm{Trace}(\sigma' D')=\mathrm{Trace}(Q\sigma Q^TQ DQ^T)=\mathrm{Trace}(\sigma D)=\dot{\overline{U}}$

The rate of change of the internal energy per unit volume of the reference configuration is given by:

$\dot{W'}=\mathrm{Trace}(P'^T F')=\mathrm{Trace}(P^TQ^TQF)=\mathrm{Trace}(P^TF)=\dot{W}$

As the rates are objective, their time integration would be objective as well.

Principle of Material Frame-Indifference

The principle of material frame-indifference provides restrictions on the possible constitutive relationships that describe the material behaviour. In the following section, the restrictions imposed due to the principle on materials whose constitutive laws are dependent on $F$ will be presented.

Hyperelastic Materials

Hyperelastic materials are those materials whose constitutive relationship between the stresses and the deformation is provided by an explicit strain energy density function $W$ . In its simplest form, $W$ is a function of the deformation gradient $F$ . To preserve the objectivity of $W$ , the principle of material frame-indifference implies the following:

$W'=W\Rightarrow W'(F')=W'(QF)=W(F)$

Since $Q$ is arbitrary, we have:

$W'(QF)=W'(R^TRU)=W'(U)$

I.e., to satisfy the principle of material frame-indifference, $W$ is restricted to be a function of the right stretch tensor $U$ .

Elastic Materials

Given an elastic material in which the Cauchy stress matrix is written as a function of $F$ , i.e., $\sigma:\mathbb{M}^3_+\rightarrow \mathbb{M}^3$ , where $\mathbb{M}^3_+$ is the set of matrices with positive determinant (without loss of generality, the dependence on the location within the material is dropped), the principle of material frame-indifference implies the following.

$\sigma'(F')=\sigma'(QF)=Q\sigma(F)Q^T$

In other words, the stresses observed by the moving observer, namely $\sigma'(F')$ are equal to those observed by the fixed observer after a simple change of basis. The arbitrariness of $Q$ imply the following:

$\sigma'(RU)=R\sigma(U)R^T$

Therefore, the principle of material frame-invariance restricts the possible constitutive relationships for $\sigma$ to abide by the above equation.
Similarly, the principle of material frame-invariance implies the following:

$P'(RU)=RP(U)$

and

$S'(RU)=S(U)$

One can show that for hyperelastic materials, the restriction on the arguments of $W$ ensures the above restrictions on the arguments of the stress tensors (See exercise 7 here).
View Mathematica Code:

cc = {-Sqrt[5], 0}
e1 = {2/Sqrt[5], 1/Sqrt[5]}
e2 = {-1/Sqrt[5], 2/Sqrt[5]}
Q = {e1, e2}
x = {2, 1}
Q.x + cc