2.7 Transformation of kinematic and kinetic variables under changes of reference frame

 

In this section we discuss how the vectors and tensors we use in solid mechancis change when we change our coordinate system and the reference configuration that is used to describe shape changes.   This seems a rather abstract and obscure topic for practical purposes.  It becomes important,  when we construct constitutive equations that relate deformation measures (usually one set of tensors) to stresses (another set of tensors).   The constitutive equatinons must behave correctly in all possible choices of coordinate system.

 

This is a difficult concept that requires a more abstract understanding of the mathematical machinery that we use to describe the behavior of deformable solids.   We need to start by understanding what is meant by a ‘reference frame,’ and then work out how our vectors and tensors transform between two particular choices of reference frame.  These relations will be used to ensure that stress-strain relations satisfy the principle of material frame indifference.

 

We start by introducing the concept of the Newtonian ‘inertial frame.’   This is particular choice of coordinate system that describes position in space, in which particle motion and forces (defined in classical mechanics by comparing them to gravity) are observed experimentally to obey Newton’s three laws of motion.   We describe the position of each point in this frame by a triad of real numbers r, and choose to define the distance between any two points as the dot product $d=\sqrt{\left(r_1-r_2\right)\cdot \left(r_1-r_2\right)}$.   Mathematically, this defines a Euclidean vector space.   Of course the choice of three real numbers r characterizing position is not unique, but we can generate all possible inertial frames by a length preserving mapping of the form

$\widehat{r}=v_{0}t+Q(r-r_0)$

Here $\widehat{r}$ is a new triad of real numbers; $v_0$ is a constant vector (another triad of real numbers), t is time, $r_0$ is an arbitrary point in space (expressed in the first choice of frame), and $Q$ is a time independent proper orthogonal tensor ( $Q{Q}^T=I$ and $\det (Q)=1$ ).  It is worth pausing to interpret this equation carefully $–$ it looks like a deformation being applied to a solid, but it actually represents a change of coordinates $–$ we are assigning a different set of numbers to each point in space.

 

To complicate things further, in classical mechanics we also frequently make use of frames that are not inertial.   You have done this when analyzing particle motion using cylindrical polar or spherical-polar basis vectors, for example.  The family of all possible non-inertial frames is generated by a time dependent mapping

$r^{*}=r_0^{*}(t)+Q(t)(r-r_0)$

Here, $r_0^{*}$ and $Q$ are (as before) a vector and a proper orthoganal tensor, but they are now time dependent.   Points in physical space are still identified by a triad of real numbers $r^{*}$, and the distance between two points is still defined by the same dot product $d=\sqrt{r^{*}\cdot r^{*}}$, but the real numbers associated with each point in the Newtonian frame now vary with time.   We can attach a physical significance to this mathematical transformation by noticing that this is how an observer who is rotating and translating relative to the Newtonian inertial frame might choose to describe the world.  

 

These mathematical transformations define what we mean by a ‘change of frame’ or a ‘change of observer’ in classical mechanics.   Any frame is a Euclidean vector space, so we can construct any arbitrary vector in these spaces as a linear combination of three non-parallel position vectors.   We can define a vector or tensor in one frame, and then map it to the other.  What this means in practice is that if we measure some vector quantity, we find a way to quantify its magnitude and direction by three real numbers (and their units!). For tensors we use 9 numbers. But this is meaningful only if we also specify the reference frame we are using for our vector, because the same vector will (usually, but not always) appear as a different set of numbers in the other frame.   How these two sets of numbers are related depends on how the vector is defined $–$ so as we shall see shortly the change of frame is slightly more complicated than a simple change of coordinate system.   

 

For the purposes of discussing the concept of ‘material frame indifference’ we are interested in how vector and tensor quantities transform under a particular change of observer and reference configuration.  The reason for this is discussed in more detail in Section 3.1.  Our first observer chooses to use the inertial frame (one arbitrary choice from the family of possible inertial frames).   Our second observer chooses to use a frame that rotates and translates with respect to the inertial frame.

 

We also assume that both observers use the same time independent coordinates in their separate frames to specify the position of a material particle in the reference configuration for the solid.   The two reference configurations then have the same shape (because both coordinate systems are Euclidean vector spaces with identical definitions of ‘distance’).  Both observers also see a stationary reference configuration in their own frame. But if one observer could see the reference configuration that is used by the other, it would appear to be rotating and translating with respect to their frame (a rigid rotation).     This transformation of observers and reference configuration is illustrated in the figure.   

 

This is perhaps the most difficult idea in this discussion, because we are accustomed to using some physical shape occupied by the solid (usually its unloaded configuration) as the reference configuration.    But this is not necessary.   The reference configuration is a purely mathematical way of assigning a coordinate X to each material particle in a solid.   Physical positions of these material particles in the inertial frame (i.e. what we actually observe) are specified by the deformation mapping $y(X,t)$.   We can describe exactly the same motion of the solid, including its initial position and orientation, using any Euclidean space for X $–$ switching from one to the other will just change the mathematical functions of time and X that appear in our formulas for y.  In the same way, any equations relating stress to strain measure must predict the same behavior in the actual inertial frame regardless of the choice of X.

 

With this background, we will now derive how the vectors and tensors we use in continuum mechanics transform as a result of this simultaneous change in coordinates and reference configuration (we refer to this as a ‘change of reference frame’).  In the discussion to follow, we will use a bold faced symbol with no superscript to denote a vector (3 real numbers) in the inertial frame: position of a point in space is $r$; the position vector of a material particle is $y$, Cauchy stress (9 numbers $–$ with the necessary symmetry!) as $\sigma$, and so on.    Vectors and tensors in the rotating observer’s frame will be denoted with a starred superscript $r^{*},y^{*},{\sigma }^{*}$, and so on.  These symbols just represent a second set of 3 or 9 numbers.

 

· Scalar quantities, such as density or temperature are always invariant $–$ they have the same value in both the inertial frame and in the observer’s frame.

 

· Vector quantities that describe geometry or particle motion in the real world are always defined by measurements conducted in the inertial frame.  We specify the magnitude and direction of these vectors by representing them as a suitable combination of position vectors (a basis) in the inertial frame.  They can be regarded as connecting two fixed points in the inertial frame, and must transform with the line connecting these two points under a change of reference frame.  For example, a normal vector to a deformed surface, body force, velocity, acceleration vectors must transform as

$n^{*}=Qn{b}^{*}=Qb{v}^{*}=Qv{a}^{*}=Qa$

Vectors that transform in this way are said to be frame indifferent, or objective. This is a rather unfortunate terminology because it sounds like the coordinates themselves don’t change.  It really means the opposite $–$ the coordinates must change so that all observers must describe in a consistent way some vector valued quantity that is measured in the inertial frame (which in a Newtonian universe is the only frame in which physical laws hold). 

 

· Similarly, tensor quantities that map a frame indifferent vector onto another frame indifferent vector are similarly said to be frame indifferent, or objective. Examples include the stretch rate tensor (which relates the relative velocity of two ends of an infinitesimal material fiber in the spatial configuration); or Cauchy stress (which maps the normal to a surface in the spatial configuration to the physical traction vector.  As we shall see with specific examples listed below frame indifferent tensors must transform as

${\sigma }^{*}=Q\sigma Q^T{D}^{*}=QD{Q}^T$

 

· The special vectors and tensors that happen to be frame indifferent have the following feature.   Let $\{e_1,e_2,e_3\}$ be an inertial basis (three linearly independent vectors in the inertial frame).   In the observer’s reference frame these vectors transform to a time-dependent triad $\{e_1^{*},e_2^{*},e_3^{*}\}=\{Q{e}_1,Q{e}_2,Q{e}_3\}$. Now, we can compute components of a frame indifferent vector or tensor in either basis

$\begin{array}{l}{b}_i^{*}=e_i^{*}\cdot b^{*}=Q{e}_i\cdot Qb=e_i\cdot Q^{T}Qb=e_i\cdot b=b_i\\ {\sigma }_{ij}^{*}=e_i^{*}{\sigma }^{*}{e}_j^{*}=Q{e}_{i}Q\sigma Q^{T}Q{e}_j=e_i\sigma e_j={\sigma }_{ij}\end{array}$

Thus, the components of a frame indifferent vector or tensor in a basis that rotates with the inertial frame are independent of the observer.  This is why they are called ‘frame indifferent.’

 

· Not all vectors and tensors are frame indifferent.   For example, the position vector of material particles in the reference configuration satisfies (by definition): $X^{*}=X$. Vectors of this form are said to be invariant $–$ they have the same coordinates in all reference frames. This is a matter of choice.   It may seem a bit odd, because we usually use the initial configuration of a solid as the reference, which suggests that the reference configuration has to occupy a fixed region in the inertial frame.   This is not the case, however.   The reference configuration is a purely mathematical entity that we use to quantify changes in length and relative orientation of lines that connect material particles in a solid.   We can do this by assigning each material particle a position in any Euclidean vector space with the same metric as the inertial frame.    We can, for example, specify the reference configuration as follows: (i) choose the initial configuration of the solid; (ii) measure the position X of each material particle in the inertial frame (this is a set of real numbers); and (iii) take the reference configuration to be the region defined by this set of real numbers in the rotating observer’s frame.   This appears to rotate the reference configuration, but of course it does not change its shape.  The behavior of the solid obviously has to be unchanged a shape preserving change of reference configuration.   Of course, we would be unlikely to use the reference configuration in the rotating observer’s frame in any actual calculation $–$ it makes much more sense to use the initial configuration in the inertial frame as reference, and we have assumed that this is the case in the whole of the rest of this text.    We are abandoning this assumption here because it tells us something interesting about constitutive equations, as discussed further in Chapter 3.

 

· This choice of reference configuration means that any tensor that maps a vector from the reference frame onto another vector in the reference frame is also invariant.  So for example we can show that the Lagrange strain tensor and the right stretch tensor satisfy

$E^{*}=E{U}^{*}=U$

 

 

 

With this preamble, we now proceed to derive how various quantities used in continuum mechanics will transform with this simultaneous change in reference configuration and reference frame.

 

· The deformation mapping $y(X,t)$ that describes motion of each material particle in the reference configuraiton in space transforms as

$y^{*}(X,t)=r_0^{*}(t)+Q(t)\left(y(X,t)-r_0\right)=r_0^{*}(t)+Q(t)\left(y(X^{*},t)-r_0\right)$

(recall that $X=X^{*}$ )

 

· Take the time derivatives to see that velocity and acceleration vectors satisfy

$\begin{array}{l}{v}^{*}=Qv=Q{\displaystyle \frac{dy}{dt}=Q{\displaystyle \frac{d}{dt}\left\{Q^T(y^{*}-y_0^{*}(t)\right\}={\displaystyle \frac{d{y}^{*}}{dt}-{\displaystyle \frac{d{y}_0^{*}}{dt}-\Omega (y^{*}-y_0^{*}(t))}}}}\\ a^{*}=Qa=Q{\displaystyle \frac{d^{2}y}{d{t}^2}=Q{\displaystyle \frac{d^2}{d{t}^2}\left\{Q^T(y^{*}-y_0^{*}(t))\right\}}}\\ ={\displaystyle \frac{d^2{y}^{*}}{d{t}^2}-{\displaystyle \frac{d^2{y}_0^{*}}{d{t}^2}+\left({\Omega }^2-{\displaystyle \frac{d\Omega }{dt}}\right)(y^{*}-y_0^{*}(t))-2\Omega ({\displaystyle \frac{d{y}^{*}}{dt}-{\displaystyle \frac{d{y}_0^{*}(t)}{dt})}}}}\end{array}$

where

$\Omega ={\displaystyle \frac{dQ}{dt}{Q}^T=-Q{\displaystyle \frac{d{Q}^T}{dt}}}$

is the spin tensor associated with the rotating observer frame. The additional terms in the acceleration involving $\Omega$ represent the centripetal and coriolis accelerations associated with this rotation.  You will sometimes see statements in the literature that ‘acceleration vectors are not frame indifferent.’   Actually, they are frame indifferent (according to the definition used here) because they transform to the new frame as $a^{*}=Qa$. What these statements are really saying is that in a non-inertial frame $a^{*}\ne d^2{y}^{*}/d{t}^2$.  There is no difficulty analyzing motion in the inertial frame using a second non-inertial rotating frame, provided we use the correct formulas relating $a^{*},y^{*},\Omega$.   We do this without a second’s thought when we analyze particle motion using cylindrical polar coordinates in our intro mechanics courses.

 

· The displacement of a particle from its position in the reference configuration to its position in the deformed solid is $u=y(X,t)-X$ in the inertial frame.   This transforms to

$u^{*}(X,t)=r_0^{*}(t)+Q(t)\left(y(X,t)-r_0\right)-X=r_0^{*}(t)+Q(t)\left(y(X^{*},t)-r_0\right)-X^{*}$

 in the rotating observer frame.  But this measure of displacement is not a very useful quantity (it is clearly not frame indifferent). It is better to measure the displacement of a material particle from its initial position.  This is

$u(X)=y(X,t)-y(X,0).$

This displacement vector evidently transforms as $u^{*}=Qu$ and is frame indifferent.

 

· Differentiate the deformation mapping and recall that $X=X^{*}$ to see that the deformation gradient transforms as

$F^{*}={\displaystyle \frac{\partial y^{*}}{\partial X^{*}}=Q{\displaystyle \frac{\partial y}{\partial X}=QF}}$

 

· The Jacobian of the deformation gradient is frame invariant $J^{*}=J$.

 

 

· Use the definitions to see that the right Cauchy Green strain  Lagrange strain, and the right stretch tensor are related by

$C^{*}=F^{*T}{F}^{*}=F^T{Q}^{T}QF=C{E}^{*}=E{U}^{*}=U$

 

· The left Cauchy Green strain, Eulerian strain, left stretch tensor are frame indifferent

$B^{*}=F^{*}{F}^{*T}=QF{F}^T{Q}^T=QC{Q}^T{V}^{*}=QV{Q}^T$

 

· The velocity gradient, stretch rate, and spin tensors transform as

$\begin{array}{l}{L}^{*}={\dot{F}}^{*}{F}^{*-1}=\left(\dot{Q}F+Q\dot{F}\right)F^{-1}{Q}^T=QL{Q}^T+\Omega \\ D^{*}=(L^{*}+L^{*T})/2=QD{Q}^T\\ W^{*}=(L^{*}-L^{*T})/2=QW{Q}^T+\Omega \end{array}$

where we have noted that $\Omega =-{\Omega }^T$

 

· The infinitesimal strain tensor becomes a rather ambiguous deformation measure with our new choice of reference configuration.  If we use the formal definition $\epsilon =sym(\partial u/\partial X)$ this quantity transforms as ${\epsilon }^{*}=Q\epsilon$.  But this strain is not ‘infinitesimal’ in any sense, and may not be zero in the undeformed solid.  In any practical application of the infinitesimal strain tensor we take the undeformed solid as reference.   It is better to define the infintesimal strain as the symmetric part of the derivative of the displacement vector with respect to position in the undeformed solid, i.e.

$\epsilon =sym\left({\displaystyle \frac{\partial \left(y(t)-y(0)\right)}{\partial y(0)}}\right)=sym\left({\displaystyle \frac{\partial y(t)}{\partial X}{\displaystyle \frac{\partial X}{\partial y(0)}-I}}\right)={\displaystyle \frac{1}{2}\left(F(t)F{(0)}^{-1}+F{(0)}^{-T}F{(t)}^T\right)-I}$

This quantity transforms as

${\epsilon }^{*}=sym(F^{*}(t)F{(0)}^{*-1})-I=sym\left\{QF(t){[QF(0)]}^{-1}-I\right\}=Qsym\left\{(F(t)F{(0)}^{-1}-I\right\}{Q}^T=Q\epsilon Q^T$

With the new definition it is therefore a frame indifferent tensor. 

 

· The Cauchy stress is frame indifferent ${\sigma }^{*}=Q\sigma Q^T$. You can see this from the formal definition $–$ the tensor maps a vector in the inertial frame (the normal to an internal surface) to another vector in the inertial frame (the traction acting on the internal surface).  Or if you find that unconvincing, you can use the fact that the virtual stress power $\sigma :D$ (a scalar) must be invariant under a frame change.  This is left as an exercise.

 

· The material stress is frame invariant ${\Sigma }^{*}=\Sigma$ (can you show this?)

 

· The nominal stress transforms as

$S^{*}=J{(QF)}^{-1}Q\sigma Q^T=J{F}^{-1}\sigma Q^T=S{Q}^T$

(note that this transformation rule will differ if the nominal stress is defined as the transpose of the measure used here…)

 

· The rate of change of Cauchy stress transforms as

${\displaystyle \frac{d{\sigma }^{*}}{dt}={\displaystyle \frac{d}{dt}\left(Q\sigma Q^T\right)={\displaystyle \frac{dQ}{dt}\sigma Q^T+Q{\displaystyle \frac{d\sigma }{dt}{Q}^T+Q\sigma {{\displaystyle \frac{dQ}{dt}}}^T=Q{\displaystyle \frac{d\sigma }{dt}{Q}^T+\Omega {\sigma }^{*}-{\sigma }^{*}\Omega }}}}}$

 

· Since the rate of change of Cauchy stress is not an objective tensor we sometimes construct constitutive equations using alternative measures of stress rate that are objective.   One such example is the Jaumann stress rate

$\stackrel{\nabla }{\sigma }={\displaystyle \frac{d\sigma }{dt}-W\sigma +\sigma W}$

where W is the spin tensor.   This quantity transforms as

$\begin{array}{l}\stackrel{\nabla }{{\sigma }^{*}}={\displaystyle \frac{d{\sigma }^{*}}{dt}-W^{*}{\sigma }^{*}+{\sigma }^{*}{W}^{*}}\\ ={\displaystyle \frac{d}{dt}\left(Q\sigma Q^T\right)-\left(QW{Q}^T+\Omega \right){\sigma }^{*}+{\sigma }^{*}\left(QW{Q}^T+\Omega \right)}\\ =Q{\displaystyle \frac{d\sigma }{dt}{Q}^T-QW\sigma Q^T+Q\sigma W{Q}^T=Q\stackrel{\nabla }{\sigma }{Q}^T}\end{array}$

This quantity is frame indifferent.   There are many other examples of frame indifferent stress rates.

 

· We occasionally encounter higher-order tensors in constitutive equations.   An example is the tensor of elastic moduli in linear elasticity, which maps the infinitesimal strain tensor onto Cauchy stress as

${\sigma }_{ij}=C_{ijkl}{\epsilon }_{kl}$

This maps a frame indifferent tensor (provided we use the frame indifferent definition of infinitesimal strain) onto another frame indifferent tensor, and so must be a frame indifferent tensor itself.   It is easy to show that it transforms as

$C_{ijkl}^{*}=Q_{im}{Q}_{jn}{C}_{mnpq}{Q}_{kp}{Q}_{lq}\equiv C^{*}=QQC{Q}^T{Q}^T$

Transformation laws for other high order tensors may differ, but will follow as a consequence of their formal definition.

 

By this time you are probably asking yourself why anyone could possibly care about all this.   This is a fair question $–$ we don’t use these relations much in practice, but they are useful when we define constitutive laws for a material. This issue is discussed in more detail in Section 3.1