11institutetext: G. Leborgne 22institutetext: ISIMA, University of Clermont
Auvergne, 63178 Aubière Cedex, France
Tel.: +33 (0)4 73 40 50 23,
22email<EMAIL_ADDRESS>
# Virtual Power Principle: A Lie Covariant Approach. Applications to Non-
Linear Elasticity, Turbulence, Visco-elasticity

Gilles Leborgne

###### Abstract

A covariant formulation of the virtual power principle based on Lie
derivatives is proposed. The Lie covariant approach does not require an inner
product and the Cauchy deformation tensor to start, but, at first order in a
Galilean Euclidean setting, gives the usual linear results classically
obtained with the Cauchy deformation tensor. The Lie approach may also enable
to differentiate a fluid from a solid from an analytical point of view, and
leads to propose a model for hysteresis. In the non-linear first order case we
get covariant models for visco-elasticity, non linear fluids and non linear
elasticity which differ from usual models. In the second order case, enriched
modelizations are obtained.

###### Keywords:

Virtual power principle Lie derivative Visco-elasticity Non Newtonian fluids
Non linear elasticity.

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. 3 Classical virtual power principle
  4. 4 Virtual power principle with Lie derivatives
  5. 5 Lie virtual power of internal forces: Linear first order
    1. 5.1 First Order Formulation
    2. 5.2 Galilean setting
    3. 5.3 Galilean Euclidean setting
  6. 6 Stokes Fluid
  7. 7 Linear elasticity
  8. 8 Fluids vs solids, and hysteresis
  9. 9 About hyper-elasticity
  10. 10 Lie virtual power of internal forces: Non linear first order
  11. 11 Non Newtonian fluids 1
  12. 12 Non linear elasticity 1
  13. 13 Visco-elastic fluids 1
  14. 14 Lie virtual power of internal forces: Second order
  15. 15 Non Newtonian fluids 2
  16. 16 Elasticity 2
  17. 17 Visco-elasticity 2
  18. A Tensors and objective contractions
  19. B Push-forward and pull-back
  20. C Lie derivatives: Introduction and interpretation
  21. D Lie derivative versus deformation tensor
  22. E Change of Riesz representation vector
  23. F Incompatibilities with Riesz representation vectors
  24. G Velocity addition formula
  25. H Covariant and isometric objectivities
  26. I About elasticity and classical formulations

##  1 Introduction

The starting point is the classical virtual power principle as stated by
Germain [4]. This principle is based on the “frame invariance principle”, cf.
Truesdell and Noll [9] p. 43, which is given under the “isometric objectivity”
hypothesis: “Change of frame are related by rigid transformations combined
with a time-shift” (p. 41), and “the units of length and time are kept fixed”
(p. 42).

However Marsden and Hughes [6] p. 22 indicate that the considered objectivity
should be covariant: “The use of geometry in attempting to isolate the basic
principles that are covariant—that is, that make intrinsic tensorial sense
independent of a preferred coordinate system—automatically clears up several
basic issues. For example, balance of linear momentum does note make tensorial
sense as it stands. However, one can make covariant sense out of balance
energy principles with no reference to rigid body motions. That is, the Noll
and Green–Rivlin–Naghdi program can be done covariantly, although this is not
obvious given the existing literature.”

But unlike Marsden and Hughes who use a metric, and the Cauchy deformation
tensor $C$ seen as the pull-back of the metric, in this manuscript there is no
use of any metric a priori: It is the objective covariant derivative of Lie
that is used to start. (For the Lie derivative and its interpretation which
motivates this manuscript, see appendix § C; And for a comparison with the
Cauchy deformation tensor, see appendix § D).

The presentation is here limited to the affine space ${\mathbb{R}^{n}}$ to
simplify the writings. After recalling the notations and the classical setting
in § 2 and 3, the Lie derivative approach is introduced in § 4. The first
order, linear, is given in § 5, and with a Galilean Euclidean setting the
classical formulation is recovered. Applications to Newtonian fluids and
elastic solids is proposed in § 6 and § 7. And for solids the Lie approach
immediately gives a linear model for the elastic stress tensor: To be compared
with the classical approach where the linear model is obtained by
linearization of the (quadratic type) Cauchy deformation tensor $C=F^{T}.F$,
see appendix § I. Moreover the Lie approach may give a way to differentiate
fluids from solids, and thus, in the first order case give a hysteresis model,
see § 8. And the Lie approach may also provide a simple characterization of
hyper-elasticity, see § 9. Then the non-linear first order Lie virtual power
principle is introduced in § 10. It produces models for non linear first order
fluids (possibly turbulence), see § 11, non-linear first order elasticity, see
§ 12, and Maxwell type visco-elastic model, see § 13. Then the second order
Lie virtual power principle is introduced in § 14, to get enriched models for
non linear fluids, non linear elasticity, as well as second order models with
some elasticity and some viscosity. And a quite long appendix is provided
since, to the knowledge of the author, there is no obvious (short) reference
in the existing literature concerning covariant objectivity and Lie derivative
for classical mechanics.

##  2 Notation

We use $\displaystyle{\;\mathop{:=}\;}$ to mean “defined by”. Time and space
are decoupled (classical mechanic). The geometric affine space is
${\mathbb{R}^{n}}$, $n=1$, $2$ or $3$. The associated vector space is also
written ${\mathbb{R}^{n}}$ (context removes ambiguities), and is equipped with
its usual topology. And ${\mathbb{R}^{n*}}={\cal
L}({\mathbb{R}^{n}};\mathbb{R})$ is its dual space (the space of linear forms
on ${\mathbb{R}^{n}}$). A basis $({\vec{e}}_{i})_{i=1,...,n}$ is simply
denoted $({\vec{e}}_{i})$. An observer defines a referential ${\cal
R}=(O,({\vec{e}}_{i}))$ (an origin and a basis, an origin of time and a
timescale being implicit). Let ${t_{1}}<{t_{2}}$. The observer locates the
particles of an object ${\it O\\!b\\!j}$ with the mapping, called the motion
of ${\it O\\!b\\!j}$ in ${\cal R}$,

${\mathaccent 869{\Phi}}:\left\\{\hbox{}\,\vbox{\openup
3.0pt\halign{\strut\hfil$\displaystyle{#}$ &
$\displaystyle{{}#}$\hfil\cr[{t_{1}},{t_{2}}]\times{\it
O\\!b\\!j}&\rightarrow{\mathbb{R}^{n}}\cr(t,{P_{\it
O\\!b\\!j}})&\rightarrow{p_{t}}={\mathaccent 869{\Phi}}(t,{P_{\it
O\\!b\\!j}})=O+{\sum_{i=1}^{n}}x^{i}{\vec{e}}_{i},\crcr}}\,\right.$ (1)

where ${p_{t}}$ is to position of ${P_{\it O\\!b\\!j}}$ at $t$ in ${\cal R}$.
For $t$ fixed, let ${\mathaccent 869{\Phi}}_{t}({P_{\it
O\\!b\\!j}}):={\mathaccent 869{\Phi}}(t,{P_{\it O\\!b\\!j}})$. The
configuration of ${\it O\\!b\\!j}$ at $t$ is
${\Omega_{t}}{\;\mathop{:=}\;}{\mathaccent 869{\Phi}}_{t}({\it O\\!b\\!j})$.
And let $\hbox{\large$\cal
C$}:=\bigcup_{t\in[{t_{1}},{t_{2}}]}(\\{t\\}\times\Omega_{t})$, a subset in
the standard Newtonian spacetime.

At $t$, the geometric space ${\mathbb{R}^{n}}$ is written
${\mathbb{R}^{n}_{t}}$ (on a differential manifold the tangent bundle
$T{\Omega_{t}}$ is considered). Assuming ${\mathaccent 869{\Phi}}$ is $C^{1}$
in time, the Eulerian velocity field $\mathaccent
869{\vec{v}}:(t,{p_{t}})\in\hbox{\large$\cal
C$}\rightarrow((t,{p_{t}}),{\vec{v}}(t,{p_{t}}))\in\hbox{\large$\cal
C$}\times{\mathbb{R}^{n}}$ is defined by

${\vec{v}}(t,{p_{t}}){\;\mathop{:=}\;}\lim_{h\rightarrow 0}{{\mathaccent
869{\Phi}}(t{+}h,{P_{\it O\\!b\\!j}})-{\mathaccent 869{\Phi}}(t,{P_{\it
O\\!b\\!j}})\over h}={\partial{\mathaccent 869{\Phi}}\over\partial
t}(t,{P_{\it O\\!b\\!j}})\in{\mathbb{R}^{n}_{t}}.$ (2)

If no confusion arises then $\mathaccent 869{\vec{v}}(t,{p_{t}})$ is
abbreviated as ${\vec{v}}(t,{p_{t}})$ (and ${\vec{v}}(t,{p_{t}})$ is drawn at
$(t,{p_{t}})$). Assuming ${\mathaccent 869{\Phi}}$ is $C^{2}$ in time, the
Eulerian acceleration field is defined with ${p_{t}}={\mathaccent
869{\Phi}}(t,{P_{\it O\\!b\\!j}})$ by
${\vec{\gamma}}(t,{p_{t}})={\partial^{2}{\mathaccent 869{\Phi}}\over\partial
t^{2}}(t,{P_{\it O\\!b\\!j}})$.

For an $C^{1}$ Eulerian function ${\cal E}$ (defined on $\cal C$), the space
differential $d{\cal E}(t,{p_{t}})$ of ${\cal E}$ at
$(t,{p_{t}})\in\\{t\\}\times\Omega_{t}$ is the differential $d{\cal
E}_{t}({p_{t}})$ of ${\cal E}_{t}$ at ${p_{t}}$. (In differential geometry the
tangent map $T{\cal E}_{t}$ of ${\cal E}_{t}$ is used, where $T{\cal
E}_{t}({p_{t}}):=({p_{t}},d{\cal E}_{t}({p_{t}}))$, that is $T{\cal E}_{t}$ is
the “full notation” of $d{\cal E}_{t}$.) And its material derivative ${D{\cal
E}\over Dt}$ is its derivative along a trajectory, that is, is the time
derivative of the function $t\rightarrow{\cal E}(t,{\mathaccent
869{\Phi}}(t,{P_{\it O\\!b\\!j}}))$ at ${p_{t}}={\mathaccent
869{\Phi}}(t,{P_{\it O\\!b\\!j}})$; So, ${D{\cal E}\over
Dt}(t,{p_{t}})={\partial{\cal E}\over\partial t}(t,{p_{t}})+d{\cal
E}(t,{p_{t}}).{\vec{v}}(t,{p_{t}})$. E.g., ${\vec{\gamma}}={D{\vec{v}}\over
Dt}$.

Let ${t_{0}}\in]{t_{1}},{t_{2}}[$ be an initial time of computation as set by
an observer. The notations of Marsden and Hughes [6] are used: Capital letters
in ${\Omega_{t_{0}}}$ and tiny letters in ${\Omega_{t}}$. And with
$t\in[{t_{1}},{t_{2}}]$, let
${\Phi^{t_{0}}_{t}}:{\Omega_{t_{0}}}\rightarrow{\Omega_{t}}$ be defined by
${\Phi^{t_{0}}_{t}}:={\mathaccent 869{\Phi}}_{t}\circ({\mathaccent
869{\Phi}}_{t_{0}})^{-1}$, that is

$\hbox{if}\quad p_{t_{0}}={P}={\mathaccent 869{\Phi}}({t_{0}},{P_{\it
O\\!b\\!j}})\quad\hbox{then}\quad{p_{t}}={\Phi^{t_{0}}_{t}}({P}){\;\mathop{:=}\;}{\mathaccent
869{\Phi}}(t,{P_{\it O\\!b\\!j}}).$ (3)

And the motion relative to the configuration ${\Omega_{t_{0}}}$ is
${\Phi^{t_{0}}}:[{t_{1}},{t_{2}}]\times{\Omega_{t_{0}}}\rightarrow{\mathbb{R}^{n}}$
defined by

$(p_{t}=)\quad{\Phi^{t_{0}}}(t,{P}){\;\mathop{:=}\;}{\Phi^{t_{0}}_{t}}({P}).$
(4)

So, ${\Phi^{t_{0}}}(t,{P})={\mathaccent 869{\Phi}}(t,{P_{\it
O\\!b\\!j}})={p_{t}}$ when ${P}={\mathaccent 869{\Phi}}({t_{0}},{P_{\it
O\\!b\\!j}})$.

The Lagrangian velocity
${\vec{V}^{t_{0}}}:[{t_{1}},{t_{2}}]\times{\Omega_{t_{0}}}\rightarrow{\mathbb{R}^{n}_{t}}$
relative to ${t_{0}}$ is defined by
${\vec{V}^{t_{0}}}(t,{P}):={\partial{\Phi^{t_{0}}}\over\partial t}(t,{P})$.
Let ${\vec{V}^{t_{0}}_{t}}({P}):={\vec{V}^{t_{0}}}(t,{P})$. (The mapping
${\vec{V}^{t_{0}}_{t}}:{\Omega_{t_{0}}}\rightarrow{\mathbb{R}^{n}_{t}}$ does
not define a “vector field” but a two points (${P}$ and ${p_{t}}$) mapping,
see Marsden and Hughes [6] (the vector ${\vec{V}^{t_{0}}}(t,{P})$ is drawn at
$(t,{p_{t}})$ since ${\vec{V}^{t_{0}}}(t,{P})={\vec{v}}(t,{p_{t}})$ is the
tangent vector at ${p_{t}}$ along the trajectory $t\rightarrow{\mathaccent
869{\Phi}}(t,{P_{\it O\\!b\\!j}})={\Phi^{t_{0}}}(t,{P})$).

${\Phi^{t_{0}}_{t}}$ will be assumed to be a $C^{2}$ diffeomorphism, and its
differential
${F^{t_{0}}_{t}}:=d{\Phi^{t_{0}}_{t}}:{\Omega_{t_{0}}}\rightarrow{\cal
L}({\mathbb{R}^{n}_{t_{0}}};{\mathbb{R}^{n}_{t}})$ is named the (covariant)
deformation gradient between ${t_{0}}$ and $t$. (${F^{t_{0}}_{t}}$ does not
define a tensor but a two points (${P}$ and ${p_{t}}$) mapping, see [6].) Let
${F^{t_{0}}}$ be defined by
${F^{t_{0}}}(t,{P}){\;\mathop{:=}\;}{F^{t_{0}}_{t}}({P})$.

Let $L(E;F)$ be the set of continuous linear mappings between two vector
spaces $E$ and $F$, and let $(\cdot,\cdot)_{G}$ (resp. $(\cdot,\cdot)_{g}$) be
an inner product in ${\mathbb{R}^{n}_{t_{0}}}$ (resp. in
${\mathbb{R}^{n}_{t}}$). The transpose $({F^{t_{0}}_{t}}({P}))^{T}\in{\cal
L}({\mathbb{R}^{n}_{t}};{\mathbb{R}^{n}_{t_{0}}})$, relative to
$(\cdot,\cdot)_{G}$ and $(\cdot,\cdot)_{g}$ is characterized by
$(({F^{t_{0}}_{t}}({P}))^{T}.{\vec{w}}_{p_{t}},{\vec{W}}_{P})_{G}=({\vec{w}}_{p_{t}},{F^{t_{0}}_{t}}({P}).{\vec{W}}_{P})_{g}$
for all ${\vec{W}}_{P}\in{\mathbb{R}^{n}_{t_{0}}}$ and
${\vec{w}}_{p_{t}}\in{\mathbb{R}^{n}_{t}}$, see [6]. And the deformation
tensor relative to ${t_{0}}$, $t$, $(\cdot,\cdot)_{G}$ and $(\cdot,\cdot)_{g}$
is
${C^{t_{0}}_{t}}:=({F^{t_{0}}_{t}})^{T}\circ{F^{t_{0}}_{t}}:{\Omega_{t_{0}}}\rightarrow{\cal
L}({\mathbb{R}^{n}_{t_{0}}};{\mathbb{R}^{n}_{t_{0}}})$, written $C=F^{T}.F$.

###### Remark 1

To be able to impose “the same Euclidean basis at two distinct times ${t_{0}}$
and $t$” (and on two different tangent spaces) Marsden and Hughes [6] p. 57
use the shifter: In ${\mathbb{R}^{n}}$ it is the mapping
${S^{t_{0}}_{t}}:({P},{\vec{W}}_{P})\in{\Omega_{t_{0}}}\times{\mathbb{R}^{n}_{t_{0}}}\rightarrow{S^{t_{0}}_{t}}({P},{\vec{W}}_{P})=({p_{t}},{\vec{w}}_{p_{t}})\in{\Omega_{t}}\times{\mathbb{R}^{n}_{t}}$
where ${p_{t}}={\Phi^{t_{0}}_{t}}({P})$ and
${\vec{w}}_{p_{t}}{\;\mathop{:=}\;}{\vec{W}}_{P}$ (parallel displacement in
${\mathbb{R}^{n}}$). In particular, using “the same Euclidean basis at
${t_{0}}$ and $t$” means using a Euclidean basis $({\vec{E}}_{i})$ in
${\mathbb{R}^{n}_{t_{0}}}$ and the basis $({\vec{e}}_{i})$ in
${\mathbb{R}^{n}_{t}}$ given by
$({p_{t}},{\vec{e}}_{i})={S^{t_{0}}_{t}}({P},{\vec{E}}_{i})$, so
${\vec{e}}_{i}={\vec{E}}_{i}$.

Representation. Let ${\mathbb{R}^{n}_{t}}^{*}={\cal
L}({\mathbb{R}^{n}_{t}};\mathbb{R})$ be the space of linear functions on
${\mathbb{R}^{n}_{t}}$. If $({\vec{e}}_{i})$ is a basis in
${\mathbb{R}^{n}_{t}}$, its dual basis $(e^{i})$ is the basis of
${\mathbb{R}^{n}_{t}}^{*}$ made of the linear functions $e^{i}$ characterized
by $e^{i}({\vec{e}}_{j})=\delta^{i}_{j}$ (Kronecker symbol). The $e^{i}$ being
linear, $e^{i}({\vec{v}})$ is written $e^{i}.{\vec{v}}$ for any
${\vec{v}}\in{\mathbb{R}^{n}_{t}}$. A basis in ${\mathbb{R}^{n}_{t}}$ will be
supposed Cartesian to simplify. At ${t_{0}}$ a basis in
${\mathbb{R}^{n}_{t_{0}}}$ will be denoted $({\vec{E}}_{I})$ and its dual
basis $(E^{I})$. At $t$ in a referential ${\cal R}=(o_{t},({\vec{e}}_{i}))$, a
point ${p_{t}}={\Phi^{t_{0}}_{t}}(P)$ is located as the bipoint vector
$\overrightarrow{o_{t}{p_{t}}}=\overrightarrow{o_{t}{\Phi^{t_{0}}_{t}}({P})}={\sum_{i=1}^{n}}\Phi^{i}_{t}({P}){\vec{e}}_{i}$
in ${\mathbb{R}^{n}_{t}}$, and the deformation gradient is denoted

$d{\Phi^{t_{0}}_{t}}({P})={F^{t_{0}}_{t}}({P})={\sum_{i,J=1}^{n}}F^{i}_{J}(t,{P})\,{\vec{e}}_{i}\otimes
E^{J},\quad[F]_{|{\vec{E}},{\vec{e}}}=[F^{i}_{J}].$ (5)

So, $F^{i}_{J}(t,{P})=d\Phi^{i}(t,{P}).{\vec{E}}_{J}$, and with Cartesian
bases $F^{i}_{J}(t,{P})={\partial\Phi^{i}\over\partial X^{J}}(t,{P})$. And
$[F^{i}_{J}(t,{P})]=[{F^{t_{0}}_{t}}({P})]_{|{\vec{E}},{\vec{e}}}=[d{\Phi^{t_{0}}_{t}}({P})]_{|{\vec{E}},{\vec{e}}}$
is the Jacobian matrix of ${\Phi^{t_{0}}_{t}}$ at ${P}$ relatively to the
bases $({\vec{E}}_{I})$ and $({\vec{e}}_{i})$. Since ${\Phi^{t_{0}}_{t}}$ is
supposed to be a diffeomorphism we have
$d{\Phi^{t_{0}}_{t}}(P)^{-1}=(d{\Phi^{t_{0}}_{t}})^{-1}({p_{t}})$ when
${p_{t}}={\Phi^{t_{0}}_{t}}(P)$, and we will denote
${H^{t_{0}}_{t}}({p_{t}}):=(d{\Phi^{t_{0}}_{t}})^{-1}({p_{t}})$. And with the
above bases:

${H^{t_{0}}_{t}}({p_{t}})={H^{t_{0}}}(t,{p_{t}})=\sum_{I,j=1}^{n}H^{I}_{j}(t,{p_{t}}){\vec{E}}_{I}\otimes
e^{j},\quad[H]_{|{\vec{e}},{\vec{E}}}=[H^{I}_{j}].$ (6)

##  3 Classical virtual power principle

We use Germain’s setting, see [4], with no volumic double forces to simplify.
Let $(\cdot,\cdot)_{g}$ be a Euclidean inner product in
${\mathbb{R}^{n}_{t}}$. Let ${\cal V}$ be a vector space of “admissible
velocity fields” (sufficiently regular vector fields for the mathematical
expressions to be meaningful). The virtual power principle connects, at all
times $t$, three linear functionals ${\cal V}\rightarrow\mathbb{R}$: The
virtual power of external forces

${{\cal
P}_{e}}({\vec{v}})=\int_{{\Omega_{t}}}({\vec{f}},{\vec{v}})_{g}\,d\Omega+\int_{{\partial{\Omega_{t}}}}({\vec{g}},{\vec{v}})_{g}\,d\sigma$
(7)

where ${\vec{f}}$ and ${\vec{g}}$ are given vector fields, the virtual power
of mass-acceleration

${{\cal
P}_{a}}({\vec{v}}){\;\mathop{:=}\;}\int_{{\Omega_{t}}}\rho({\vec{\gamma}},{\vec{v}})_{g}\,d\Omega$
(8)

where $\rho$ is the mass density and ${\vec{\gamma}}={D{\vec{w}}\over Dt}$ is
the eulerian acceleration undergone by a material with velocity ${\vec{w}}$,
and the virtual power of internal forces

${{\cal P}_{i}}({\vec{v}})=\int_{{\Omega_{t}}}{p_{\rm
int}}({\vec{v}})\,d\Omega$ (9)

where ${p_{\rm int}}$ is a linear form of ${\vec{v}}$ and its differentials.
And the principle is: For any ${\vec{v}}\in{\cal V}$,

${{\cal P}_{i}}({\vec{v}})+{{\cal P}_{e}}({\vec{v}})={{\cal
P}_{a}}({\vec{v}}).$ (10)

##  4 Virtual power principle with Lie derivatives

The virtual power of external forces (7) and of mass-acceleration (8) are
used; And the “virtual power of pressure”

${{\cal P}_{\it pres}}({\vec{v}})=\int_{\Omega_{t}}p\;{\rm
div}{\vec{v}}\;d{\Omega_{t}}$ (11)

is introduced, where $p$ is a differentiable function, and ${\rm
div}{\vec{v}}$ is the divergence of the velocity field ${\vec{v}}$. (The
virtual power of pressure will enable a simple formulation for the virtual
power of internal forces as is classically done for Maxwell visco-elastic
material, see § 13. Also see § I.)

And the virtual power of internal forces (9) will be used with ${p_{\rm int}}$
(non linear in general) given with Lie derivatives as described in § 5 and
following: This is the main purpose of this manuscript.

Then the principle of virtual power reads: for any admissible vector field
${\vec{v}}$,

${{\cal P}_{i}}({\vec{v}})+{\cal P}_{pres}({\vec{v}})+{{\cal
P}_{e}}({\vec{v}})={{\cal P}_{a}}({\vec{v}}).$ (12)

##  5 Lie virtual power of internal forces: Linear first order

(The Lie derivatives in classical mechanics are described in § C.)

###  5.1 First Order Formulation

First-order conjecture: 1- A material, occupying at $t$ a domain
${\Omega_{t}}$ in ${\mathbb{R}^{n}}$, can be characterized by $3n$ vector
fields ${{\vec{a}}_{00j}},{{\vec{a}}_{01j}},{{\vec{a}}_{10j}}$ and/or $3n$
one-forms ${\alpha_{00}^{j}},{\alpha_{01}^{j}},{\alpha_{10}^{j}}$. 2- At $t$,
in a referential ${\cal R}=(O,({\vec{e}}_{i}))$ and with $(e^{i})$ the dual
basis of $({\vec{e}}_{i})$, the measured density of power of internal forces
depends on an admissible virtual velocity field ${\vec{v}}$ acting on the
material, and reads

${p_{\rm int}}_{1}({\vec{v}})={\sum_{j=1}^{n}}e^{j}.{{\vec{a}}_{00j}}+{\cal
L}_{\vec{v}}e^{j}.{{\vec{a}}_{01j}}+e^{j}.{\cal
L}_{\vec{v}}{{\vec{a}}_{10j}}+{\alpha_{00}^{j}}.{\vec{e}}_{j}+{\alpha_{01}^{j}}.{\cal
L}_{\vec{v}}{\vec{e}}_{j}+{\cal L}_{\vec{v}}{\alpha_{10}^{j}}.{\vec{e}}_{j},$
(13)

where ${\cal L}_{\vec{v}}{\vec{a}}={\partial{\vec{a}}\over\partial
t}+d{\vec{a}}.{\vec{v}}-d{\vec{v}}.{\vec{a}}$ is the Lie derivative of a
vector field ${\vec{a}}$, see (63), and ${\cal
L}_{\vec{v}}\alpha={\partial\alpha\over\partial
t}+d\alpha.{\vec{v}}+\alpha.d{\vec{v}}$ is the Lie derivative of a one-form
$\alpha$, see (64). (For special materials the number of vector fields and/or
one-forms can be chosen to be greater than $n$ the dimension of
${\mathbb{R}^{n}}$.)

###  5.2 Galilean setting

In a Galilean referential and with a Cartesian basis $({\vec{e}}_{i})$, we
have ${\cal L}_{\vec{v}}{\vec{e}}_{j}=-d{\vec{v}}.{\vec{e}}_{j}$ and ${\cal
L}_{\vec{v}}e^{j}=e^{j}.d{\vec{v}}$, and (13) reduces to ${p_{\rm
int}}_{1}({\vec{v}})={\sum_{j=1}^{n}}e^{j}.{{\vec{a}}_{00j}}+e^{j}.d{\vec{v}}.{{\vec{a}}_{01j}}+e^{j}.({\partial{{\vec{a}}_{10j}}\over\partial
t}+d{{\vec{a}}_{10j}}.{\vec{v}}-d{\vec{v}}.{{\vec{a}}_{10j}})+{\alpha_{00}^{j}}.{\vec{e}}_{j}-{\alpha_{01}^{j}}.d{\vec{v}}.{\vec{e}}_{j}+({\partial{\alpha_{01}^{j}}\over\partial
t}+d{\alpha_{01}^{j}}.{\vec{v}}+{\alpha_{01}^{j}}.d{\vec{v}}).{\vec{e}}_{j}$.
And the virtual power of the internal forces is assumed to vanish whenever
$d{\vec{v}}=0$ (Galilean referential). Thus we are left with ${p_{\rm
int}}_{1}({\vec{v}})={\sum_{j=1}^{n}}e^{j}.d{\vec{v}}.{{\vec{a}}_{01j}}-e^{j}.d{\vec{v}}.{{\vec{a}}_{10j}}-{\alpha_{01}^{j}}.d{\vec{v}}.{\vec{e}}_{j}+{\alpha_{01}^{j}}.d{\vec{v}}.{\vec{e}}_{j}$,
that is,

${p_{\rm
int}}_{1}({\vec{v}})=-{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}},\quad-{{\underline{\underline{\tau}}}_{1}}={\sum_{j=1}^{n}}{{\vec{a}}_{1j}}\otimes
e^{j}+{\vec{e}}_{j}\otimes{\alpha_{1}^{j}},\quad\left\\{\hbox{}\,\vbox{\openup
3.0pt\halign{\strut\hfil$\displaystyle{#}$ &
$\displaystyle{{}#}$\hfil\cr&{{\vec{a}}_{1j}}={{\vec{a}}_{01j}}-{{\vec{a}}_{10j}},\cr&{\alpha_{1}^{j}}={\alpha_{01}^{j}}-{\alpha_{01}^{j}},\crcr}}\,\right.$
(14)

where $\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}$ is the double objective
contraction between the ${1\choose 1}$ tensors
${\underline{\underline{\tau}}}_{1}$ and $d{\vec{v}}$, see (46).

###  5.3 Galilean Euclidean setting

In the previous Galilean setting, the basis $({\vec{e}}_{i})$ is chosen to be
Euclidean. Let $(\cdot,\cdot)_{g}$ be the associated inner product. Then the
transposed $d{\vec{v}}^{T}$ of $d{\vec{v}}$ is defined (by
$(d{\vec{v}}^{T}.{\vec{u}},{\vec{w}})_{g}=(d{\vec{v}}.{\vec{w}},{\vec{u}})_{g}$
for any ${\vec{u}},{\vec{w}}\in{\mathbb{R}^{n}_{t}}$). The internal power
being independent of a rigid motion, i.e independent of
${d{\vec{v}}-d{\vec{v}}^{T}\over 2}$, (14) gives

${p_{\rm
int}}_{1}({\vec{v}})=-{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}{d{\vec{v}}+d{\vec{v}}^{T}\over
2}=-{\underline{\underline{\sigma}}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}{d{\vec{v}}+d{\vec{v}}^{T}\over
2}=-{\underline{\underline{\sigma}}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}},\quad{\underline{\underline{\sigma}}}={{{\underline{\underline{\tau}}}_{1}}+{{\underline{\underline{\tau}}}_{1}}^{T}\over
2},$ (15)

which is a classical expression.

##  6 Stokes Fluid

The “pressure forces” being taken into account in the virtual power of
pressure, see (11), we look at the virtual internal power of “viscous forces”.
Consider a fluid animated with a movement ${\mathaccent 869{\Phi}}$, see (1),
let $p(\tau)={\mathaccent 869{\Phi}}(\tau,{P_{\it O\\!b\\!j}})$, and let
${\vec{w}}(\tau,p(\tau))={\partial{\mathaccent
869{\Phi}}\over\partial\tau}(\tau,{P_{\it O\\!b\\!j}})$ be its Eulerian
velocity, see (2). Then, with $t\in]t_{1},t_{2}[$ and $\tau$ in the vicinity
of $t$, consider the associated movement $\Phi^{t}_{\tau}$, see (3), and its
deformation gradient ${F^{t}_{\tau}}=d\Phi^{t}_{\tau}$ between $t$ and $\tau$.
Since
${\partial\Phi^{t}\over\partial\tau}(\tau,{p_{t}})={\vec{w}}(t,\Phi^{t}(\tau,{p_{t}}))$
we have ${\partial
F^{t}\over\partial\tau}(\tau,{p_{t}})=d{\vec{w}}(t,p(\tau)).F^{t}(\tau,{p_{t}})$.

Eventuality 1. Vector fields ${{\vec{a}}_{1j}}$ characterize the fluid, and
the virtual power of internal forces results from their transport by the flow.
Then (14) gives (Galilean setting)

${p_{\rm int}}_{1}({\vec{v}})={\sum_{j=1}^{n}}e^{j}.{\cal
L}_{\vec{v}}{{\vec{a}}_{1j}}=-{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}},\quad{{\underline{\underline{\tau}}}_{1}}=-{\sum_{j=1}^{n}}{{\vec{a}}_{1j}}\otimes
e^{j}.$ (16)

Then consider that a Newtonian fluid (which is isotropic) is characterized by
the $n$ vector fields given by
${{\vec{a}}_{1j}}(\tau,{p_{\tau}}):=-2\mu\,{\partial
F^{t}\over\partial\tau}(\tau,{p_{t}}).{\vec{e}}_{j}$ when $\tau=t$ (no
memory), that is,

${{\vec{a}}_{1j}}(t,{p_{t}})=-2\mu\,d{\vec{w}}(t,{p_{t}}).{\vec{e}}_{j}.$ (17)

(The matrix
$[{{\underline{\underline{\tau}}}_{1}}]_{|{\vec{e}}}=2\mu\,[d{\vec{w}}]_{|{\vec{e}}}$
is made of the columns
$-[{{\vec{a}}_{1j}}]_{|{\vec{e}}}=2\mu\,[{\partial{\vec{w}}\over\partial
x^{j}}]_{|{\vec{e}}}$.) Thus, in a Galilean Euclidean setting, we get
$[{\underline{\underline{\sigma}}}]_{|{\vec{e}}}=2\mu{[d{\vec{w}}]_{|{\vec{e}}}+[d{\vec{w}}]_{|{\vec{e}}}^{T}\over
2}$, see (15), which is the classical model.

E.g. if ${{\vec{a}}_{1j}}={{\vec{a}}_{10j}}$, that is, if ${p_{\rm
int}}_{1}({\vec{v}})={\sum_{j=1}^{n}}e^{j}.{\cal
L}_{\vec{v}}{{\vec{a}}_{10j}}$, see (13) and (14), then the power is measured
by the observer with the projections $e^{j}$. (And if
${{\vec{a}}_{1j}}={{\vec{a}}_{01j}}$, that is if ${p_{\rm
int}}_{1}({\vec{v}})={\sum_{j=1}^{n}}{{\vec{a}}_{01j}}.{\cal
L}_{\vec{v}}e^{j}$, then the power is measured with the $e^{j}$ immersed in
the flow: will be used with the non linear approach.)

Eventuality 2. One-forms characterize the fluid, and their transport by a flow
${\vec{v}}$ give the internal power. In particular in a Galilean Euclidean
setting with a Euclidean basis $({\vec{e}}_{i})$ in ${\mathbb{R}^{n}_{t}}$,
(14) gives ${p_{\rm
int}}_{1}({\vec{v}})=-{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}$ where
${{\underline{\underline{\tau}}}_{1}}=-{\sum_{i=1}^{n}}{\vec{e}}_{i}\otimes{\alpha_{1}^{i}}$.
Here ${\alpha_{1}^{i}}=-2\mu\,e^{j}.d{\vec{w}}=-2\mu\,dw^{i}$, and
$[{{\underline{\underline{\tau}}}_{1}}]_{|{\vec{e}}}$ is the matrix made of
the lines $-[{\alpha_{10}^{i}}]_{|{\vec{e}}}=2\mu[dw^{i}]_{|{\vec{e}}}$.

##  7 Linear elasticity

Consider an elastic solid animated with a movement ${\mathaccent 869{\Phi}}$,
see (1), let ${t_{0}}\in]t_{1},t_{2}[$, let ${\Phi^{t_{0}}}$ be the associated
movement, see (4) and (3), and let ${F^{t_{0}}_{t}}=d{\Phi^{t_{0}}_{t}}$ be
the covariant deformation gradient between ${t_{0}}$ and $t$.

Eventuality 1. One-forms characterize the elastic material (Germain’s point of
view see [3]), and their transport by the flow gives the internal power. Then
(14) gives (Galilean setting)

${p_{\rm int}}_{1}({\vec{v}})={\sum_{i=1}^{n}}{\cal
L}_{\vec{v}}{\alpha_{1}^{i}}.{\vec{e}}_{i}=-{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}},\quad{{\underline{\underline{\tau}}}_{1}}=-{\sum_{i=1}^{n}}{\vec{e}}_{i}\otimes{\alpha_{1}^{i}}.$
(18)

Then consider that an isotropic elastic solid is characterized by $n$ one-
forms ${\alpha_{1}^{i}}$ on ${\Omega_{t}}$ that are push-forwards of one-forms
${\alpha^{i}_{1{t_{0}}}}$ on ${\Omega_{t_{0}}}$, see (56), that is, with
${p_{t}}={\Phi^{t_{0}}_{t}}({P})$ and
${H^{t_{0}}_{t}}({p_{t}})=({F^{t_{0}}_{t}}({P}))^{-1}$,

${\alpha_{1}^{i}}(t,{p_{t}})={\alpha^{i}_{1{t_{0}}}}({P}).{H^{t_{0}}_{t}}({p_{t}}).$
(19)

Then
${{\underline{\underline{\tau}}}_{1}}(t,{p_{t}})=-{\sum_{i=1}^{n}}({\vec{e}}_{i}\otimes{\alpha^{i}_{1{t_{0}}}}({P})).{H^{t_{0}}}(t,{p_{t}})$;
And with $({\vec{E}}_{i})$ a Cartesian basis in ${\mathbb{R}^{n}_{t_{0}}}$,
choose ${\alpha^{i}_{1{t_{0}}}}=E^{i}$ (isotropic material), so that, with
(6),
${{\underline{\underline{\tau}}}_{1}}(t,{p_{t}})=-{\sum_{i=1}^{n}}({\vec{e}}_{i}\otimes
E^{i}).{H^{t_{0}}}(t,{p_{t}})=-\sum_{i,k,j=1}^{n}H^{k}_{j}(t,{p_{t}})({\vec{e}}_{i}\otimes
E^{i}).({\vec{E}}_{k}\otimes
e^{j})=-\sum_{i,j=1}^{n}H^{i}_{j}(t,{p_{t}}){\vec{e}}_{i}\otimes e^{j}$, that
is,

${{\underline{\underline{\tau}}}_{1}}(t,{p_{t}})=-\sum_{i,j=1}^{n}H^{i}_{j}(t,{p_{t}}){\vec{e}}_{i}\otimes
e^{j},\quad\hbox{and}\quad[{{\underline{\underline{\tau}}}_{1}}]_{|{\vec{e}}}=-2{\tilde{\mu}}[{H^{t_{0}}}]_{|{\vec{e}},{\vec{E}}}.$
(20)

So, in a Galilean Euclidean setting, with $({\vec{E}}_{i})$ a Euclidean basis,
we get
$[{\underline{\underline{\sigma}}}]_{|{\vec{e}}}=-2{\tilde{\mu}}{[{H^{t_{0}}}]_{|{\vec{e}},{\vec{E}}}+[{H^{t_{0}}}]_{|{\vec{e}},{\vec{E}}}^{T}\over
2}$, see (15), classical expression for small displacements, except for the
trace part, see § I.

E.g. if ${\alpha_{1}^{j}}={\alpha_{10}^{j}}$, that is if ${p_{\rm
int}}_{1}({\vec{v}})={\sum_{j=1}^{n}}{\cal
L}_{\vec{v}}{\alpha_{10}^{j}}.{\vec{e}}_{j}$, then the power is measured by
the observer with the Euclidean projections along the ${\vec{e}}_{j}$. (And if
${\alpha_{1}^{j}}={\alpha_{01}^{j}}$, that is if ${p_{\rm
int}}_{1}({\vec{v}})={\sum_{j=1}^{n}}{\alpha_{01}^{j}}.{\cal
L}_{\vec{v}}{\vec{e}}_{j}$, then the power is measured with the
${\vec{e}}_{j}$ immersed in the flow; It will be used for visco-elasticity,
see (31).)

Eventuality 2. Vector fields characterize the solid, and their transports give
the internal power. In particular in a Galilean Euclidean setting with a
Euclidean basis $({\vec{e}}_{i})$ in ${\mathbb{R}^{n}_{t}}$, (14) gives

${p_{\rm
int}}_{1}({\vec{v}})=-{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}},\quad{{\underline{\underline{\tau}}}_{1}}=-{\sum_{j=1}^{n}}{{\vec{a}}_{1j}}\otimes
e^{j}.$ (21)

Then suppose that the ${{\vec{a}}_{1j}}$ are the push-forwards of vector
fields ${{\vec{a}}_{j{t_{0}}}}({P})$ see (54), that is, with
${p_{t}}={\Phi^{t_{0}}_{t}}({P})$,
${{\vec{a}}_{1j}}(t,{p_{t}})={F^{t_{0}}_{t}}({P}).{{\vec{a}}_{j{t_{0}}}}({P})$.
Then
${{\underline{\underline{\tau}}}_{1}}(t,{p_{t}})=-{F^{t_{0}}}(t,{P}).{\sum_{j=1}^{n}}{{\vec{a}}_{j{t_{0}}}}({P})\otimes
e^{j}$. With a Euclidean base $({\vec{E}}_{i})$ in ${\mathbb{R}^{n}_{t_{0}}}$,
then for isotropic homogeneous elasticity we may choose
${{\vec{a}}_{j{t_{0}}}}=-2{\tilde{\mu}}{\vec{E}}_{j}$, then, see (5):

${{\underline{\underline{\tau}}}_{1}}(t,{p_{t}})={F^{t_{0}}}(t,{P}).{\sum_{j=1}^{n}}{\vec{E}}_{j}\otimes
e^{j},\quad\hbox{and}\quad[{{\underline{\underline{\tau}}}_{1}}]_{|{\vec{e}}}=2{\tilde{\mu}}[{F^{t_{0}}}]_{|{\vec{E}},{\vec{e}}}.$
(22)

Thus, in a Galilean Euclidean setting, with $({\vec{E}}_{i})$ a Euclidean
basis, we get
$[{\underline{\underline{\sigma}}}]=2{\tilde{\mu}}{[{F^{t_{0}}}]+[{F^{t_{0}}}]^{T}\over
2}$, classical expression for small displacements, except for the trace part,
see § I.

##  8 Fluids vs solids, and hysteresis

For a Stokes fluid, a modelization with vector fields can be considered, see
(16). And for an elastic solid, a modelization with one-forms can be
considered, see (18). Thus fluids would be differentiated from solids in a
non-algebraic way. And “mixed linear materials” could be considered with

${p_{\rm int}}_{1}({\vec{v}})=c_{1}{\sum_{j=1}^{n}}e^{j}.{\cal
L}_{\vec{v}}{{\vec{a}}_{10j}}+c_{2}{\sum_{i=1}^{n}}{\cal
L}_{\vec{v}}{\alpha_{10}^{i}}.{\vec{e}}_{i}=-c_{1}\,{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}-c_{2}\,{\underline{\underline{\kappa}}}_{1}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}},$ (23)

the last equality in a Cartesian setting where
${{\underline{\underline{\tau}}}_{1}}=-{\sum_{j=1}^{n}}{{\vec{a}}_{10j}}\otimes
e^{j}$ characterizes the fluid part,
${\underline{\underline{\kappa}}}_{1}=-{\sum_{i=1}^{n}}{\vec{e}}_{i}\otimes{\alpha_{10}^{i}}$
characterizes the solid part, and $c_{1},c_{2}\in\mathbb{R}$. This could model
simple visco-elasticity or hysteresis.

##  9 About hyper-elasticity

Hyper-elasticity aims to find a “stored energy function” see Marsden and
Hughes [6] (“énergie volumique des déformations élastiques” see Germain [5]).
The classical starting point is a Euclidean setting and the Piola–Kirchhoff
tensor ${P\\!\\!K}=J{\underline{\underline{\sigma}}}.F^{-T}$, simplified
notation of
${P\\!\\!K}^{t_{0}}_{t}({P})=J^{t_{0}}_{t}({P}){\underline{\underline{\sigma}}}(t,{p_{t}}).({F^{t_{0}}_{t}}({P}))^{-T}$,
where ${p_{t}}={\Phi^{t_{0}}_{t}}({P})$, $J=\det(F)$, $F^{-T}$ is the inverse
of the transpose, ${\underline{\underline{\sigma}}}$ is the Cauchy stress
tensor at $(t,{p_{t}})$ built from the Cauchy stress vector at $(t,{p_{t}})$
and the unit orthonormal vector ${\vec{n}}(t,{p_{t}})$ to
${\partial{\Omega_{t}}}$ (depends on the metric). Thus ${P\\!\\!K}$ measures
the force “per unit of undistorted area” (by change of variables in the
integrals). And the material is said to be hyper-elastic if
${P\\!\\!K}^{t_{0}}(t,{P})={\widehat{P\\!\\!K}}^{t_{0}}(t,{F^{t_{0}}}(t,{P}))$
and if there exists a function ${\cal W}^{t_{0}}$ s.t., with (5),
$[{\widehat{P\\!\\!K}}^{t_{0}}]=[{\partial{\cal W}^{t_{0}}\over\partial
F^{i}_{J}}]$. This derivation in terms of the components of the two-point
tensor $F$ is quite intriguing (see [6]).

With the Lie derivative approach, and the introduction of the virtual power of
pressure, see (11), the "hyper-elastic potential" for isotropic homogeneous
elasticity can be considered to be simply ${{\cal
W}^{t_{0}}_{t}}:=2{\tilde{\mu}}{\Phi^{t_{0}}_{t}}$: Then $d{{\cal
W}^{t_{0}}_{t}}=2{\tilde{\mu}}{F^{t_{0}}_{t}}$, and then
$[{{\underline{\underline{\tau}}}_{1}}(t,{p_{t}})]_{|{\vec{e}}}=[d{{\cal
W}^{t_{0}}}(t,{P})]_{|{\vec{E}},{\vec{e}}}=2{\tilde{\mu}}[{F^{t_{0}}}(t,{P})]_{|{\vec{E}},{\vec{e}}}$,
see (22). But the preference may go to

${{\cal
W}^{t_{0}}_{t}}:=-2{\tilde{\mu}}({\Phi^{t_{0}}_{t}})^{-1}:{\Omega_{t}}\rightarrow{\Omega_{t_{0}}}$
(24)

so that $d{{\cal
W}^{t_{0}}_{t}}({p_{t}})=-2{\tilde{\mu}}{H^{t_{0}}_{t}}({p_{t}})$, and
$[{{\underline{\underline{\tau}}}_{1}}(t,.)]_{|{\vec{e}}}=[d{{\cal
W}^{t_{0}}_{t}}]_{|{\vec{e}},{\vec{E}}}=-2{\tilde{\mu}}[{H^{t_{0}}_{t}}]_{|{\vec{e}},{\vec{E}}}$,
see (20). Here ${{\underline{\underline{\tau}}}_{1}}(t,.)$ and $d{{\cal
W}^{t_{0}}_{t}}$, in (24), are defined at ${p_{t}}$, and ${{\cal
W}^{t_{0}}_{t}}$ refers to the past (values at ${t_{0}}$ from ${\Omega_{t}}$)
which is the usual approach of Galilean or general relativity.

##  10 Lie virtual power of internal forces: Non linear first order

(13) is enriched: 1’- The material can be characterized by the preceding
fields and by $n$ vector fields ${{\vec{a}}_{11j}}$ on ${\Omega_{t}}$ and/or
$n$ one-forms ${\alpha_{11}^{j}}$ on ${\Omega_{t}}$. 2- For a velocity field
${\vec{v}}$, at $t$, we get the density of the power of internal forces (first
order nonlinear)

${p_{\rm int}}_{1n}({\vec{v}})={p_{\rm
int}}_{1}({\vec{v}})+{\sum_{j=1}^{n}}{\cal L}_{\vec{v}}{\alpha_{11}^{j}}.{\cal
L}_{\vec{v}}{\vec{e}}_{j}+{\cal L}_{\vec{v}}e^{j}.{\cal
L}_{\vec{v}}{{\vec{a}}_{11j}}.$ (25)

In a Galilean setting, the assumption of zero internal power if $d{\vec{v}}=0$
gives with (14):

${p_{\rm
int}}_{1n}({\vec{v}})=-{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}+({\partial{\alpha_{11}^{j}}\over\partial
t}+d{\alpha_{11}^{j}}.{\vec{v}}+{\alpha_{11}^{j}}.d{\vec{v}}).(-d{\vec{v}}.{\vec{e}}_{j})+e^{j}.d{\vec{v}}.({\partial{{\vec{a}}_{11j}}\over\partial
t}+d{{\vec{a}}_{11j}}.{\vec{v}}-d{\vec{v}}.{{\vec{a}}_{11j}}).$

In (25), in addition to ${p_{\rm
int}}_{1}({\vec{v}})=-{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}$, it appears:

the unsteady linear term $-{\partial{\alpha_{11}^{j}}\over\partial
t}.d{\vec{v}}.{\vec{e}}_{j}+e^{j}.d{\vec{v}}.{\partial{{\vec{a}}_{11j}}\over\partial
t}$,

the non-linear term
$-(d{\alpha_{11}^{j}}.{\vec{v}}).d{\vec{v}}.{\vec{e}}_{j}+e^{j}.d{\vec{v}}.(d{{\vec{a}}_{11j}}.{\vec{v}})$,
non-linear in ${\vec{v}}$ and $d{\vec{v}}$,

and the non-linear term
$-{\alpha_{11}^{j}}.d{\vec{v}}.d{\vec{v}}.{\vec{e}}_{j}-e^{j}.d{\vec{v}}.d{\vec{v}}.{{\vec{a}}_{11j}}$,
non-linear in $d{\vec{v}}.d{\vec{v}}$.

##  11 Non Newtonian fluids 1

$2n$ vector fields ${{\vec{a}}_{1j}}$ and ${{\vec{a}}_{11j}}$ characterize a
non linear first order fluid (non Newtonian, turbulence type model?). Then,
with (16) and
${{\underline{\underline{\tau}}}_{1}}=-{\sum_{j=1}^{n}}{{\vec{a}}_{1j}}\otimes
e^{j}$,

${p_{\rm
int}}_{1n}({\vec{v}})=-{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}+{\sum_{j=1}^{n}}({\partial{{\vec{a}}_{11j}}\over\partial
t}\otimes e^{j}+(d{{\vec{a}}_{11j}}.{\vec{v}})\otimes
e^{j})\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}-({{\vec{a}}_{11j}}\otimes
e^{j})\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}(d{\vec{v}}.d{\vec{v}}).$
(26)

And (for matrix computation) with
${{\underline{\underline{\tau}}}_{11}}=-{\sum_{j=1}^{n}}{{\vec{a}}_{11j}}\otimes
e^{j}$ we get

$\displaystyle{{p_{\rm int}}_{1n}({\vec{v}})=}$ | $\displaystyle{{}-{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern 1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}-({\partial{{\underline{\underline{\tau}}}_{11}}\over\partial t}+d{{\underline{\underline{\tau}}}_{11}}.{\vec{v}}-d{\vec{v}}.{{\underline{\underline{\tau}}}_{11}})\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern 1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}.}$
---|---
(27)

(The expression in parentheses, that is
${\partial{{\underline{\underline{\tau}}}_{11}}\over\partial
t}+d{{\underline{\underline{\tau}}}_{11}}.{\vec{v}}-d{\vec{v}}.{{\underline{\underline{\tau}}}_{11}}$,
is not a Lie derivative of ${{\underline{\underline{\tau}}}_{11}}$ see (65):
Here it is the Lie derivative of the vector fields ${{\vec{a}}_{11j}}$ that
are considered.)

##  12 Non linear elasticity 1

$2n$ one-forms ${\alpha_{1}^{j}}$ and ${\alpha_{11}^{j}}$ characterize non
linear first order elasticity. Then, with (18) and
${{\underline{\underline{\tau}}}_{1}}=-{\sum_{j=1}^{n}}{\vec{e}}_{j}\otimes{\alpha_{1}^{j}}$,

${p_{\rm
int}}_{1n}({\vec{v}})=-{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}-{\sum_{j=1}^{n}}({\vec{e}}_{j}\otimes({\partial{\alpha_{11}^{j}}\over\partial
t}+d{\alpha_{11}^{j}}.{\vec{v}}+{\alpha_{11}^{j}}.d{\vec{v}}))\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}.$ (28)

And (for matrix computation), with
${{\underline{\underline{\tau}}}_{11}}={\sum_{j=1}^{n}}{\vec{e}}_{j}\otimes{\alpha_{11}^{j}}$
we get

$\displaystyle{{p_{\rm int}}_{1n}({\vec{v}})=}$ | $\displaystyle{{}-{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern 1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}-({\partial{{\underline{\underline{\tau}}}_{11}}\over\partial t}+d{{\underline{\underline{\tau}}}_{11}}.{\vec{v}}+{{\underline{\underline{\tau}}}_{11}}.d{\vec{v}})\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern 1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}.}$
---|---
(29)

(The expression in parentheses, that is
${\partial{{\underline{\underline{\tau}}}_{11}}\over\partial
t}+d{{\underline{\underline{\tau}}}_{11}}.{\vec{v}}+{{\underline{\underline{\tau}}}_{11}}.d{\vec{v}}$,
is not a Lie derivative of ${{\underline{\underline{\tau}}}_{11}}$ see (65):
Here it is the Lie derivative of the one-forms ${\alpha_{11}^{i}}$ that are
considered.)

##  13 Visco-elastic fluids 1

Galilean Euclidean setting. The classical model for a visco-elastic fluids of
Maxwell reads

${\underline{\underline{\tau}}}+\lambda{\cal
L}_{\vec{v}}{\underline{\underline{\tau}}}=\mu{d{\vec{v}}+d{\vec{v}}^{T}\over
2},$ (30)

where ${\underline{\underline{\tau}}}$ is a stress tensor, ${\vec{v}}$ the
velocity field of the material, $\lambda$ an elasticity constant, $\mu$ a
viscosity constant, and ${\cal L}_{\vec{v}}{\underline{\underline{\tau}}}$ a
Lie derivative. And the pressure $p$ is classically introduced with (11). In
(30), the derivative of Lie ${\cal L}_{\vec{v}}{\underline{\underline{\tau}}}$
is considered with the velocity ${\vec{v}}$ of the material (not a virtual
velocity), and can the Jaumann or the upper-convected or the lower-convected
Lie derivative, see (65); Or can be a linear combination of such tensors to
improve the numerical results (which otherwise are not convincing), even if
such a linear combination is absurd as far as objectivity is concerned
(addition of vector fields and one-forms).

A possible covariant model could be, with ${\vec{v}}$ a virtual velocity,

${p_{\rm int}}_{1n}({\vec{v}})={\sum_{j=1}^{n}}e^{j}.{\cal
L}_{\vec{v}}{{\vec{a}}_{1j}}+{\sum_{j=1}^{n}}{\cal
L}_{\vec{v}}{\alpha_{11}^{j}}.{\cal
L}_{\vec{v}}{\vec{e}}_{j}=-{{\underline{\underline{\tau}}}_{1}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}-({\partial{{\underline{\underline{\tau}}}_{11}}\over\partial
t}+d{\underline{\underline{\tau}}}.{\vec{v}}+{{\underline{\underline{\tau}}}_{11}}.d{\vec{v}})\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}},$ (31)

obtained by using the Lie derivatives of vector fields for the fluid part and
the Lie derivative of one forms for the elastic part, see (16) and (18), the
last equality being given in a Galilean setting with
${{\underline{\underline{\tau}}}_{1}}=-{\sum_{j=1}^{n}}{{\vec{a}}_{1j}}\otimes
e^{j}$, see (16), and
${{\underline{\underline{\tau}}}_{11}}={\sum_{j=1}^{n}}{\vec{e}}_{j}\otimes{\alpha_{11}^{i}}$,
see the last term in (29).

##  14 Lie virtual power of internal forces: Second order

(25) is enriched with vector fields ${\vec{a}}_{xyj}$ and one-forms
$\alpha^{j}_{xy}$ and the second ordre Lie derivatives

${\alpha_{02}^{j}}.{\cal L}_{\vec{v}}({\cal L}_{\vec{v}}{\vec{e}}_{j})$,
${\cal L}_{\vec{v}}({\cal L}_{\vec{v}}{\alpha_{20}^{j}}).{\vec{e}}_{j}$,
${\cal L}_{\vec{v}}{\alpha_{12}^{j}}.{\cal L}_{\vec{v}}({\cal
L}_{\vec{v}}{\vec{e}}_{j})$, ${\cal L}_{\vec{v}}({\cal
L}_{\vec{v}}{\alpha_{21}^{j}}).{\cal L}_{\vec{v}}{\vec{e}}_{j}$, ${\cal
L}_{\vec{v}}({\cal L}_{\vec{v}}{\alpha_{22}^{j}}).{\cal L}_{\vec{v}}({\cal
L}_{\vec{v}}{\vec{e}}_{j})$, and

$e^{j}.{\cal L}_{\vec{v}}({\cal L}_{\vec{v}}{{\vec{a}}_{02j}})$, ${\cal
L}_{\vec{v}}({\cal L}_{\vec{v}}e^{j}).{{\vec{a}}_{20j}}$, ${\cal
L}_{\vec{v}}e^{j}.{\cal L}_{\vec{v}}({\cal L}_{\vec{v}}{{\vec{a}}_{12j}})$,
${\cal L}_{\vec{v}}({\cal L}_{\vec{v}}e^{j}).{\cal
L}_{\vec{v}}{{\vec{a}}_{21j}}$, ${\cal L}_{\vec{v}}({\cal
L}_{\vec{v}}e^{j}).{\cal L}_{\vec{v}}({\cal L}_{\vec{v}}{{\vec{a}}_{22j}})$.
And, with (25), the measured power density of the internal forces reads:

$\displaystyle{{p_{\rm int}}_{2}({\vec{v}})=}$ | $\displaystyle{{}{p_{\rm int}}_{1n}({\vec{v}})+\sum\hbox{the above second order terms}.}$
---|---
(32)

The second order Lie derivative of a vector field ${\vec{w}}$ is ${\cal
L}_{\vec{v}}({\cal L}_{\vec{v}}{\vec{w}})={\partial^{2}{\vec{w}}\over\partial
t^{2}}+2d{\partial{\vec{w}}\over\partial
t}.{\vec{v}}-2d{\vec{v}}.{\partial{\vec{w}}\over\partial
t}+d{\vec{w}}.{\partial{\vec{v}}\over\partial t}-{\partial
d{\vec{v}}\over\partial
t}.{\vec{w}}+(d^{2}{\vec{w}}.{\vec{v}}).{\vec{v}}+d{\vec{w}}.d{\vec{v}}.{\vec{v}}-2d{\vec{v}}.d{\vec{w}}.{\vec{v}}-(d^{2}{\vec{v}}.{\vec{v}}).{\vec{w}}+d{\vec{v}}.d{\vec{v}}.{\vec{w}}$,
computed with (63). And the second order Lie derivative of a one-form $\alpha$
is ${\cal L}_{\vec{v}}({\cal
L}_{\vec{v}}\alpha)={\partial^{2}\alpha\over\partial
t^{2}}+2d{\partial\alpha\over\partial
t}.{\vec{v}}+2{\partial\alpha\over\partial
t}.d{\vec{v}}+d\alpha.{\partial{\vec{v}}\over\partial t}+\alpha.{\partial
d{\vec{v}}\over\partial
t}+d^{2}\alpha({\vec{v}},{\vec{v}})+d\alpha.(d{\vec{v}}.{\vec{v}})+2(d\alpha.{\vec{v}}).d{\vec{v}}+\alpha.(d^{2}{\vec{v}}.{\vec{v}})+(\alpha.d{\vec{v}}).d{\vec{v}}$,
computed with (56).

(Generalization: Higher order Lie derivative can also be considered.)

##  15 Non Newtonian fluids 2

E.g. (26) is enriched to get e.g. (“one pure second order fluids”)

${p_{\rm int}}_{2}({\vec{v}})={\sum_{j=1}^{n}}e^{j}.{\cal
L}_{\vec{v}}{{\vec{a}}_{1j}}+{\cal L}_{\vec{v}}e^{j}.{\cal
L}_{\vec{v}}{{\vec{a}}_{11j}}+e^{j}.{\cal L}_{\vec{v}}({\cal
L}_{\vec{v}}{{\vec{a}}_{20j}}).$ (33)

(And for a “pure fluid” any other Lie derivative of vector fields can be
added.)

Galilean setting: The internal power vanishing if $d{\vec{v}}=0$ and
${\partial{\vec{v}}\over\partial t}=0$, we get

$\displaystyle{{p_{\rm int}}_{2}({\vec{v}})=}$ | $\displaystyle{{}{p_{\rm int}}_{1n}({\vec{v}})-{\sum_{j=1}^{n}}2e^{j}.d{\vec{v}}.{\partial{{\vec{a}}_{20j}}\over\partial t}+e^{j}.d{{\vec{a}}_{20j}}.{\partial{\vec{v}}\over\partial t}-e^{j}.{\partial d{\vec{v}}\over\partial t}.{{\vec{a}}_{20j}}}$
---|---
|
$\displaystyle{{}+e^{j}.d{{\vec{a}}_{20j}}.d{\vec{v}}.{\vec{v}}-2e^{j}.d{\vec{v}}.d{{\vec{a}}_{20j}}.{\vec{v}}-e^{j}.(d^{2}{\vec{v}}.{\vec{v}}).{{\vec{a}}_{20j}}+e^{j}.d{\vec{v}}.d{\vec{v}}.{{\vec{a}}_{20j}},}$
(34)

which is a “non linear second gradient” (non Newtonian) expression. With
${\underline{\underline{\tau}}}={\sum_{j=1}^{n}}{{\vec{a}}_{1j}}\otimes e^{j}$
and
${\underline{\underline{\kappa}}}_{2}={\sum_{j=1}^{n}}{{\vec{a}}_{20j}}\otimes
e^{j}$ we obtain (for matrix computation with the generic notation ${DT\over
Dt}={\partial T\over\partial t}+dT.{\vec{v}}$):

${p_{\rm int}}_{2}({\vec{v}})={p_{\rm
int}}_{1n}({\vec{v}})-\Bigl{(}{\underline{\underline{\kappa}}}_{2}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}({\partial
d{\vec{v}}\over\partial
t}+d^{2}{\vec{v}}.{\vec{v}}-d{\vec{v}}.d{\vec{v}})+d{\underline{\underline{\kappa}}}_{2}.{D{\vec{v}}\over
Dt}+2{D{\underline{\underline{\kappa}}}_{2}\over
Dt}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern 1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}\Bigr{)}.$ (35)

##  16 Elasticity 2

E.g. (28) is enriched to get e.g. (“a pure second order elastic material”)

${p_{\rm int}}_{2}({\vec{v}})={\sum_{i=1}^{n}}{\cal
L}_{\vec{v}}{\alpha_{1}^{i}}.{\vec{e}}_{i}+{\cal
L}_{\vec{v}}{\alpha_{11}^{j}}.{\cal L}_{\vec{v}}{\vec{e}}_{j}+{\cal
L}_{\vec{v}}({\cal L}_{\vec{v}}{\alpha_{20}^{i}}).{\vec{e}}_{i}.$ (36)

(And for a “pure elastic material” any other Lie derivative of one-forms can
be added.)

Galilean setting: The internal power vanishing if $d{\vec{v}}=0$ and
${\partial{\vec{v}}\over\partial t}=0$, we get

$\displaystyle{{p_{\rm int}}_{2}({\vec{v}})=}$ | $\displaystyle{{}{p_{\rm int}}_{1n}({\vec{v}})+{\sum_{j=1}^{n}}2{\partial{\alpha_{20}^{i}}\over\partial t}.d{\vec{v}}.{\vec{e}}_{i}+d{\alpha_{20}^{i}}.{\partial{\vec{v}}\over\partial t}.{\vec{e}}_{i}+{\alpha_{20}^{i}}.{\partial d{\vec{v}}\over\partial t}.{\vec{e}}_{i}}$
---|---
|
$\displaystyle{{}+d{\alpha_{20}^{i}}.(d{\vec{v}}.{\vec{v}}).{\vec{e}}_{i}+2(d{\alpha_{20}^{i}}.{\vec{v}}).d{\vec{v}}.{\vec{e}}_{i}+{\alpha_{20}^{i}}.(d^{2}{\vec{v}}.{\vec{v}}).{\vec{e}}_{i}+({\alpha_{20}^{i}}.d{\vec{v}}).d{\vec{v}}.{\vec{e}}_{i},}$
(37)

which is a “non linear second gradient” expression. With
${\underline{\underline{\kappa}}}_{2}=-{\sum_{i=1}^{n}}{\vec{e}}_{i}\otimes{\alpha_{20}^{i}}$,
we get (for matrix computation):

${p_{\rm int}}_{2}({\vec{v}})={p_{\rm
int}}_{1n}({\vec{v}})-\Bigl{(}{\underline{\underline{\kappa}}}_{2}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}({\partial
d{\vec{v}}\over\partial
t}+d^{2}{\vec{v}}.{\vec{v}}+d{\vec{v}}.d{\vec{v}})+d{\underline{\underline{\kappa}}}_{2}.{D{\vec{v}}\over
Dt}+2{D{\underline{\underline{\kappa}}}_{2}\over
Dt}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern 1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}\Bigr{)}.$ (38)

With the Eulerian acceleration ${\vec{\gamma}}={D{\vec{v}}\over
Dt}={\partial{\vec{v}}\over\partial t}+d{\vec{v}}.{\vec{v}}$, that gives
$d{\vec{\gamma}}={\partial(d{\vec{v}})\over\partial
t}+d^{2}{\vec{v}}.{\vec{v}}+d{\vec{v}}.d{\vec{v}}$, we have

${p_{\rm int}}_{2}({\vec{v}})={p_{\rm
int}}_{1n}({\vec{v}})-\Bigl{(}{\underline{\underline{\kappa}}}_{2}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{\gamma}}+d{\underline{\underline{\kappa}}}_{2}.{\vec{\gamma}}+2{D{\underline{\underline{\kappa}}}_{2}\over
Dt}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern 1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}\Bigr{)}.$ (39)

(For comparison purposes, the second-order Taylor development of the Cauchy
tensor $C=F^{T}.F$ in the neighborhood of ${t_{0}}$ reads
$C^{t_{0}}({t_{0}}{+}h,{P})=(I+h\,(d{\vec{v}}+d{\vec{v}}^{T})+{h^{2}\over
2}(2d{\vec{v}}^{T}.d{\vec{v}}+d{\vec{\gamma}}+d{\vec{\gamma}}^{T}))({t_{0}},{P})+o(h^{2})$.)

##  17 Visco-elasticity 2

Second-order visco-elastic materials (not purely fluid or elastic) can be
obtained with other Lie combinations.

##  Appendix A Tensors and objective contractions

Let $E$ be a finite dimensional (to simplify) real vector space, and let
${E^{*}}:={\cal L}(E;\mathbb{R})$ be the dual space of $E$ (the space of
linear functions). An element $\ell\in{E^{*}}$ being linear, $\ell({\vec{v}})$
is written $\ell.{\vec{v}}$, for all ${\vec{v}}\in E$. Let ${E^{**}}:={\cal
L}({E^{*}};\mathbb{R})$ be the bidual of $E$, and let ${\cal
J}_{1}:{\vec{v}}\in E\rightarrow v={\cal J}_{1}({\vec{v}})\in{E^{**}}$ be
defined by $v(\ell){\;\mathop{:=}\;}\ell.{\vec{v}}$ for all $\ell\in E^{*}$.
Then ${\cal J}_{1}$ is a natural canonical isomorphism, see e.g. Spivak [8],
where canonical means that the definition only uses the constant $1$ which is
the identity element in multiplication and is the “simplest possible” (quite
blurry), and natural means that ${\cal J}_{1}$ is independent of the observer.
Thus $E$ and ${E^{**}}$ are identified, and $v={\cal J}_{1}({\vec{v}})$ is
written $v={\vec{v}}$. And, by linearity of $v$, $v(\ell)$ is written $v.\ell$
or ${\vec{v}}.\ell$. (There is no natural isomorphism between $E$ and $E^{*}$,
see e.g. [8]: An isomorphism exists but depends on an observer, e.g. depends
on a basis, or e.g. depends on an inner product see (67) and (69)).

Let $E$ and $F$ be two finite dimensional real spaces, and $\dim E=n$ and
$\dim F=m$. Let ${\cal L}(E;F)$ be the set of linear mappings from $E$ to $F$.
Let $({\vec{a}}_{i})$ be a basis in $E$. Then a linear map $L\in{\cal L}(E;F)$
is characterized by the vectors $L.{\vec{a}}_{j}$. Let $({\vec{b}}_{i})$ be a
basis in $F$, and let $(b^{i})$ be its dual basis. And let ${L^{i}_{\;j}}$ be
the components of $L.{\vec{a}}_{j}$ in the basis $({\vec{b}}_{i})$, that is,

$L.{\vec{a}}_{j}={\sum_{i=1}^{m}}{\sum_{j=1}^{n}}{L^{i}_{\;j}}{\vec{b}}_{i},\quad\hbox{so}\quad{L^{i}_{\;j}}=b^{i}.(L.{\vec{a}}_{j}).$
(40)

And we write

$L={\sum_{i=1}^{m}}{\sum_{j=1}^{n}}{L^{i}_{\;j}}{\vec{b}}_{i}\otimes a^{j}.$
(41)

This is coherent with the usual tensorial product: Let ${\cal
L}({F^{*}},E;\mathbb{R})$ the space of bilinear forms on ${F^{*}}\times E$. If
$({\vec{v}}_{F},\ell_{E})\in F\times{E^{*}}$, then their tensorial product
${\vec{v}}_{F}\otimes\ell_{E}$ is the bilinear form in ${\cal
L}({F^{*}},E;\mathbb{R})$ defined by
$({\vec{v}}_{F}\otimes\ell_{E})(\ell_{F},{\vec{v}}_{E}):={\vec{v}}_{F}(\ell_{F})\,\ell_{E}({\vec{v}}_{E})=\ell_{F}({\vec{v}}_{F})\,\ell_{E}({\vec{v}}_{E})$,
thanks to ${\cal J}_{1}$, that is
$({\vec{v}}_{F}\otimes\ell_{E})(\ell_{F},{\vec{v}}_{E}):=(\ell_{F}.{\vec{v}}_{F})\,(\ell_{E}.{\vec{v}}_{E})$.
And we have the natural canonical isomorphism ${\cal J}_{2}:{\mathaccent
869{L}}\in{\cal L}(F^{*},E;\mathbb{R})\rightarrow L={\cal J}_{2}({\mathaccent
869{L}})\in{\cal L}(E;F)$ defined by, see [8],

$\forall(\ell_{F},{\vec{u}})\in F^{*}\times
E,\quad\ell_{F}.(L.{\vec{u}}):={\mathaccent 869{L}}(\ell_{F},{\vec{u}}).$ (42)

Thus if ${\mathaccent
869{L}}={\sum_{i=1}^{m}}{\sum_{j=1}^{n}}{L^{i}_{\;j}}{\vec{b}}_{i}\otimes
a^{j}$, then ${L^{i}_{\;j}}={\mathaccent 869{L}}(b^{i},{\vec{a}}_{j})$, then
${L^{i}_{\;j}}=b^{i}.L.{\vec{a}}_{j}$ where $L={\cal J}_{2}^{-1}({\mathaccent
869{L}})$, which gives (40). And $[{L^{i}_{\;j}}]$ is the matrix
$[L]_{|{\vec{a}},{\vec{b}}}$ of $L\in{\cal L}(E;F)$ as well as the matrix
$[{\mathaccent 869{L}}]_{|{\vec{a}},{\vec{b}}}$ of ${\mathaccent
869{L}}\in{\cal L}(F^{*},E;\mathbb{R})$ when $L={\cal J}_{2}^{-1}({\mathaccent
869{L}})$, relatively to the bases $({\vec{a}}_{i})$ and $({\vec{b}}_{i})$.
And in any case, the Einstein convention is satisfied. In this manuscript this
is applied e.g. with the deformation gradient
$L={F^{t_{0}}_{t}}(P)=d{\Phi^{t_{0}}_{t}}(P):{\mathbb{R}^{n}_{t_{0}}}\rightarrow{\mathbb{R}^{n}_{t}}$
represented with bases in (5), thanks to ${\cal J}_{2}$.

And, with ${\cal J}_{2}$, we have then defined the contraction of a bilinear
mapping ${\mathaccent 869{L}}\in{\cal L}(F^{*},E;\mathbb{R})$ with a vector
${\vec{u}}\in E$: The definition being ${\mathaccent
869{L}}.{\vec{u}}:=L.{\vec{u}}$ when $L={\cal J}_{2}^{-1}({\mathaccent
869{L}})$.

Let $r,s\in\mathbb{N}$ s.t. $r+s\geq 1$. The set ${\cal L}^{r}_{s}(E):={\cal
L}(\underbrace{E^{*},...,E^{*}}_{r\rm\;times},\underbrace{E,...,E}_{s\rm\;times};\mathbb{R})$
of $\mathbb{R}$-multilinear forms is called the set of uniform tensors of type
${r\choose s}$ on $E$. And ${\cal L}^{0}_{0}(E):=\mathbb{R}$. An elementary
uniform tensor in ${\cal L}^{r}_{s}(E)$ is a tensor $u_{1}\otimes...\otimes
u_{r}\otimes\ell_{1}\otimes...\otimes\ell_{s}$, where $u_{i}\in{E^{**}}$ and
$\ell_{i}\in{E^{*}}$, which value on
$(m_{1},...,m_{r},{\vec{v}}_{1},...,{\vec{v}}_{s})\in({E^{*}})^{r}\times
E^{s}$ is
$(u_{1}.m_{1})...(u_{r}.m_{r})(\ell_{1}.{\vec{v}}_{1})...(\ell_{s}.{\vec{v}}_{s})$.
And with $u_{i}={\cal J}_{1}({\vec{u}}_{i})$ this tensor is written
${\vec{u}}_{1}\otimes...\otimes{\vec{u}}_{r}\otimes\ell_{1}\otimes...\otimes\ell_{s}$.

If ${\mathaccent 869{L}}_{1}\in{\cal L}^{r_{1}}_{s_{1}}(E)$ and ${\mathaccent
869{L}}_{2}\in{\cal L}^{r_{2}}_{s_{2}}(E)$ then their tensor product is the
tensor ${\mathaccent 869{L}}_{1}\otimes{\mathaccent 869{L}}_{2}\in{\cal
L}^{r_{1}+r_{2}}_{s_{1}+s_{2}}(E)$ defined by:

$({\mathaccent 869{L}}_{1}\otimes{\mathaccent
869{L}}_{2})(\ell_{1,1},...,\ell_{2,1},...,{\vec{u}}_{1,1},...,{\vec{u}}_{2,1},...):={\mathaccent
869{L}}_{1}(\ell_{1,1},...,{\vec{u}}_{1,1},...){\mathaccent
869{L}}_{2}(\ell_{2,1},...,{\vec{u}}_{2,1},...).$ (43)

If ${\mathaccent 869{L}}_{1}\in{\cal L}^{r_{1}}_{s_{1}}(E)$, ${\mathaccent
869{L}}_{2}\in{\cal L}^{r_{2}}_{s_{2}}(E)$, $\ell\in{\cal
L}^{0}_{1}(E)={E^{*}}$, $u\in{\cal L}^{1}_{0}(E)={E^{**}}$, ${\vec{u}}={\cal
J}^{-1}(u)\in E$, and $u$ written ${\vec{u}}$, the objective tensor
contraction of ${\mathaccent 869{L}}_{1}\otimes\ell\in{\cal
L}^{r_{1}}_{s_{1}+1}(E)$ with ${\vec{u}}\otimes{\mathaccent
869{L}}_{2}\in{\cal L}^{r_{2}+1}_{s_{2}}(E)$ is defined by

$({\mathaccent 869{L}}_{1}\otimes\ell).({\vec{u}}\otimes{\mathaccent
869{L}}_{2}):=(\ell.{\vec{u}})\,{\mathaccent 869{L}}_{1}\otimes{\mathaccent
869{L}}_{2}\in{\cal L}^{r_{1}+r_{2}}_{s_{1}+s_{2}}(E).$ (44)

And the objective tensor contraction of ${\mathaccent
869{L}}_{1}\otimes{\vec{u}}$ with $\ell\otimes{\mathaccent 869{L}}_{2}$ is
defined by $({\mathaccent
869{L}}_{1}\otimes{\vec{u}}).(\ell\otimes{\mathaccent
869{L}}_{2})=({\vec{u}}.\ell)\,{\mathaccent 869{L}}_{1}\otimes{\mathaccent
869{L}}_{2}=(\ell.{\vec{u}})\,{\mathaccent 869{L}}_{1}\otimes{\mathaccent
869{L}}_{2}$.

The objective double tensor contraction
$\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern 1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}$, for compatible tensors, results
from the simple contraction applied twice. E.g. the double objective tensor
contraction of ${\mathaccent
869{L}}_{1}\otimes\ell_{1,1}\otimes{\vec{u}}_{1,2}$ and
$\ell_{2,1}\otimes{\vec{u}}_{2,2}\otimes{\mathaccent 869{L}}_{2}$ is defined
by

$\displaystyle{({\mathaccent 869{L}}_{1}\otimes\ell_{1,1}\otimes{\vec{u}}_{1,2})\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern 1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}(\ell_{2,1}\otimes{\vec{u}}_{2,2}\otimes{\mathaccent 869{L}}_{2})=}$ | $\displaystyle{{}({\vec{u}}_{1,2}.\ell_{2,1})(\ell_{1,1}.{\vec{u}}_{2,2})\,{\mathaccent 869{L}}_{1}\otimes{\mathaccent 869{L}}_{2}.}$
---|---
(45)

Representation with a basis $({\vec{e}}_{i})$ of $E$. E.g. with $S,T\in{\cal
L}^{1}_{1}(E)$, $S={\sum_{j=1}^{n}}S^{i}_{j}{\vec{e}}_{i}\otimes e^{j}$ and
$T={\sum_{j=1}^{n}}T^{i}_{j}{\vec{e}}_{i}\otimes e^{j}$, we get

$S.T={\sum_{i,j,k=1}^{n}}S^{i}_{k}T^{k}_{j}{\vec{e}}_{i}\otimes
e^{j},\quad\hbox{and}\quad
S\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern 1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}T={\sum_{i,j=1}^{n}}S^{i}_{j}T^{j}_{i},$
(46)

that is $S\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}T={\rm Tr}(S.T)$ (the
trace of $S.T$ considered to be an endomorphism thanks to ${\cal J}_{2}$),
which value is independent of the chosen basis (Einstein convention is
satisfied). Thus, if ${\vec{u}}\in E$, $\ell\in{E^{*}}$, $T\in{\cal
L}^{1}_{1}(E)$, then
$({\vec{u}}\otimes\ell)\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}T=\ell.T.{\vec{u}}$; With
a basis $({\vec{e}}_{i})$, and ${\vec{u}}={\sum_{i=1}^{n}}u^{i}{\vec{e}}_{i}$,
$\ell={\sum_{i=1}^{n}}\ell_{i}e^{i}$,
$T={\sum_{j=1}^{n}}T^{i}_{j}{\vec{e}}_{i}\otimes e^{j}$, we have
$({\vec{u}}\otimes\ell)\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}T=\ell.T.{\vec{u}}={\sum_{i,j,k=1}^{n}}\ell_{i}T^{i}_{j}u^{j}$.

Let $A=[A^{i}_{j}]$ and $B=[B^{i}_{j}]$ be two square $n*n$ matrices. The
double matrix contraction of $A$ and $B$ is defined by:

$A:B{\;\mathop{:=}\;}{\sum_{i,j=1}^{n}}A^{i}_{j}B^{i}_{j}.$ (47)

This is not an objective contraction (the Einstein convention is not
satisfied). E.g. $A=[S]_{|{\vec{e}}}$ and $B=[T]_{|{\vec{e}}}$ give
$[S]_{|{\vec{e}}}:[T]_{|{\vec{e}}}=A:B$, value that depends on the choice of
the basis. To be compared with
$S\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern 1.0pt\vbox
to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}T=A^{T}:B$, see (46).

Let ${\cal E}$ be an affine space and $E$ be the associated vector space. Let
${\Omega_{\cal E}}$ be an open set in ${\cal E}$. Let ${\cal F}({\Omega_{\cal
E}};\mathbb{R})$ be the set of real valued functions on ${\Omega_{\cal E}}$. A
tensor of type ${r\choose s}$ on ${\Omega_{\cal E}}$, see Abraham and Marsden
[1], is a mapping $\mathaccent 869{T}:{p}\in{\Omega_{\cal
E}}\rightarrow\mathaccent 869{T}({p})=({p},T({p}))\in{\Omega_{\cal
E}}\times{\cal L}^{r}_{s}({\Omega_{\cal E}})$ that is ${\cal F}({\Omega_{\cal
E}};\mathbb{R})$-multilinear; That is, for all $f\in{\cal F}({\Omega_{\cal
E}};\mathbb{R})$ and all $z_{1},z_{2}$ vector fields or one forms on
${\Omega_{\cal E}}$ where appropriate, we have

$T(...,fz_{1}+z_{2},...)=f\,T(...,z_{1},...)+T(...,z_{2},...).$ (48)

(That is, for all ${p}\in{\Omega_{\cal E}}$,
$T({p})(...,f({p})z_{1}({p})+z_{2}({p}),...)=f({p})\,T({p})(...,z_{1}({p}),...)+T({p})(...,z_{2}({p}),...)$.)
If there is no ambiguity then $\mathaccent 869{T}$ is simply written $T$. The
set of tensors of type ${r\choose s}$ on ${\Omega_{\cal E}}$ is denoted
$T^{r}_{s}({\Omega_{\cal E}})$. And $T^{0}_{0}({\Omega_{\cal E}}):={\cal
F}({\Omega_{\cal E}};\mathbb{R})$. Example: $T^{0}_{1}({\Omega_{\cal E}})$ is
identified with the set $\Omega^{1}({\Omega_{\cal E}})$ of one-forms on
${\Omega_{\cal E}}$, and $T^{1}_{0}({\Omega_{\cal E}})$ is identified with the
set $\Gamma({\Omega_{\cal E}})$ of vector fields on ${\Omega_{\cal E}}$ thanks
to ${\cal J}_{1}$. Counter example: A derivation operator $\nabla$ is
$\mathbb{R}$-linear, but is not ${\cal F}({\Omega_{\cal
E}};\mathbb{R})$-linear because (48) is not satisfied since
$\nabla(fz_{1})=(\nabla f)\,z_{1}+f\,(\nabla z_{1})\neq f\,\nabla z_{1}$ if
$f$ is not constant ($\nabla$ is not a tensor but is a spray, see [1]).

Note that the initial concept is “vector fields”; Then “one-forms” are
introduced, which are functions of vector fields; And then “tensors” are
introduced, which are functions of one forms and velocity fields.

For a complete mathematical introduction (with differential geometry) see e.g.
Spivak [8], Abraham and Marsden [1], Arnold [2], Marsden and Hughes [6].

##  Appendix B Push-forward and pull-back

Push-forwards and pull-backs enable to define, among others, the velocity
addition formula, objectivity, and Lie derivatives. We follow Abraham and
Marsden [1], here in a simplified affine framework.

Let ${\cal E}$ and ${\cal F}$ be affine sets, and let $E$ and $F$ be the
associated vector space supposed to be normed, let ${\Omega_{\cal E}}$ and
${\Omega_{\cal F}}$ be open sets in ${\cal E}$ and ${\cal F}$, and let
$\Psi:{\Omega_{\cal E}}\rightarrow{\Omega_{\cal F}}$ be a diffeomorphism. In
this manuscript $E$ and $F$ are finite dimensional, and $\Psi$ will be either
a motion ${\Phi^{t_{0}}_{t}}:{\Omega_{t_{0}}}\rightarrow{\Omega_{t}}$, see
(3), or a translator ${\Theta_{t}}:{{\cal R}_{\\!B}}\rightarrow{{\cal
R}_{\\!A}}$, see (74).

The push-forward of a function ${f_{\cal E}}:{\Omega_{\cal
E}}\rightarrow\mathbb{R}$ by $\Psi$ is the function $\Psi_{*}{f_{\cal
E}}={f_{{\cal E}*}}:{\Omega_{\cal F}}\rightarrow\mathbb{R}$ defined by
${f_{{\cal E}*}}:={f_{\cal E}}\circ(\Psi)^{-1}$, so, with ${p_{\cal
F}}=\Psi({p_{\cal E}})$,

${f_{{\cal E}*}}({p_{\cal F}}):={f_{\cal E}}({p_{\cal E}}).$ (49)

The pull-back of a function ${f_{\cal F}}:{\Omega_{\cal
F}}\rightarrow\mathbb{R}$ by $\Psi$ is the function $\Psi^{*}{f_{\cal
F}}=(\Psi^{-1})_{*}{f_{\cal F}}={f^{*}_{\cal F}}$, that is ${f^{*}_{\cal
F}}={f_{\cal F}}\circ\Psi$, so ${f^{*}_{\cal F}}({p_{\cal E}})={f_{\cal
F}}({p_{\cal F}})$ when ${p_{\cal F}}=\Psi({p_{\cal E}})$.

Let

${c_{\cal E}}:\left\\{\hbox{}\,\vbox{\openup
3.0pt\halign{\strut\hfil$\displaystyle{#}$ &
$\displaystyle{{}#}$\hfil\cr]s_{1},s_{2}[&\rightarrow{\Omega_{\cal E}}\cr
s&\rightarrow{p_{\cal E}}={c_{\cal E}}(s),\crcr}}\,\right.$ (50)

be a regular curve in ${\Omega_{\cal E}}$. The push-forward of ${c_{\cal E}}$
by $\Psi$ is the curve $\Psi_{*}{c_{\cal E}}={c_{{\cal E}*}}$ defined by
${c_{{\cal E}*}}:=\Psi\circ{c_{\cal E}}$, that is,

${c_{{\cal E}*}}:\left\\{\hbox{}\,\vbox{\openup
3.0pt\halign{\strut\hfil$\displaystyle{#}$ &
$\displaystyle{{}#}$\hfil\cr]s_{1},s_{2}[&\rightarrow{\Omega_{\cal F}}\cr
s&\rightarrow{p_{\cal F}}={c_{{\cal E}*}}(s):=\Psi({c_{\cal
E}}(s))\quad(=\Psi({p_{\cal E}})).\crcr}}\,\right.$ (51)

Let ${c_{\cal F}}:s\in]s_{1},s_{2}[\rightarrow{c_{\cal F}}(s)\in{\Omega_{\cal
F}}$ be a regular curve. The pull-back of ${c_{\cal F}}$ by $\Psi$ is the
curve $\Psi^{*}{c_{\cal F}}={c^{*}_{\cal F}}:=(\Psi^{-1})_{*}{c_{\cal F}}$,
that is ${c^{*}_{\cal F}}=\Psi^{-1}\circ{c_{\cal F}}$.

We use the definition of a vector field given by the tangent vectors to a
curve (see e.g. [1] for equivalent definitions). For a regular curve ${c_{\cal
E}}$ as in (50), its tangent vector at ${p_{\cal E}}={c_{\cal E}}(s)$ is

${{\vec{w}}_{\cal E}}({p_{\cal E}}){\;\mathop{:=}\;}{c_{\cal E}}^{\prime}(s).$
(52)

That defines the vector field ${{\vec{w}}_{\cal E}}$ on ${\rm Im}({c_{\cal
E}})$. The push-forward of ${{\vec{w}}_{\cal E}}$ by $\Psi$ is the vector
field $\Psi_{*}{{\vec{w}}_{\cal E}}={{\vec{w}}_{{\cal E}*}}$ on ${\rm
Im}({c_{{\cal E}*}})$ made of the tangent vectors to the push-forward curve
${c_{{\cal E}*}}$, see (51), that is, ${{\vec{w}}_{{\cal E}*}}({p_{\cal
F}}):={c_{{\cal E}*}{}^{\prime}}(s)$ when ${p_{\cal F}}={c_{{\cal E}*}}(s)$.
So, since (51) gives ${c_{{\cal E}*}{}^{\prime}}(s)=d\Psi({c_{\cal
E}}(s)).{c_{\cal E}}{}^{\prime}(s)$, with ${p_{\cal F}}=\Psi({p_{\cal E}})$ we
have

${{\vec{w}}_{{\cal E}*}}({p_{\cal F}})=d\Psi({p_{\cal E}}).{{\vec{w}}_{\cal
E}}({p_{\cal E}}).$ (53)

(That is, ${{\vec{w}}_{{\cal E}*}}=(d\Psi.{{\vec{w}}_{\cal
E}})\circ\Psi^{-1}$.) And a family of regular curves in ${\Omega_{\cal E}}$
gives a vector field ${{\vec{w}}_{\cal E}}$ on ${\Omega_{\cal E}}$ and its
push-forward ${{\vec{w}}_{{\cal E}*}}$ on ${\Omega_{\cal F}}$. The pull-back
of a vector field ${{\vec{w}}_{\cal F}}:{\Omega_{\cal F}}\rightarrow F$ by
$\Psi$ is the vector field $\Psi^{*}{{\vec{w}}_{\cal F}}={{\vec{w}}^{*}_{\cal
F}}:=(\Psi^{-1})_{*}{{\vec{w}}_{\cal F}}$, that is, with ${p_{\cal
F}}=\Psi({p_{\cal E}})$,

${{\vec{w}}^{*}_{\cal F}}({p_{\cal E}}):=d\Psi({p_{\cal
E}})^{-1}.{{\vec{w}}_{\cal F}}({p_{\cal F}}).$ (54)

The push-forward of a one-form ${\alpha_{\cal E}}\in{E^{*}}$ by $\Psi$ is the
one-form $\Psi^{*}{\alpha_{\cal E}}={\alpha_{{\cal E}*}}$ defined with
${f_{\cal E}}={\alpha_{\cal E}}.{{\vec{w}}_{\cal E}}$, (49) and (53); So, with
${p_{\cal F}}=\Psi({p_{\cal E}})$,

${\alpha_{{\cal E}*}}({p_{\cal F}}):={\alpha_{\cal E}}({p_{\cal
E}}).d\Psi({p_{\cal E}})^{-1}.$ (55)

And the pull-back of a one-form ${\alpha_{\cal F}}\in{F^{*}}$ by $\Psi$ is the
one-form $\Psi^{*}{\alpha_{\cal F}}=(\Psi^{-1})_{*}{\alpha_{\cal
F}}={\alpha^{*}_{\cal F}}$, that is, with ${p_{\cal F}}=\Psi({p_{\cal E}})$,

${\alpha^{*}_{\cal F}}({p_{\cal E}}):={\alpha_{\cal F}}({p_{\cal
F}}).d\Psi({p_{\cal E}}).$ (56)

This pull-back can also be defined thanks to the natural canonical isomorphism
$L\in{\cal L}(E;F)\rightarrow L^{*}\in{\cal L}({F^{*}};{E^{*}})$ defined by
$L^{*}(\ell_{F}).{\vec{u}}_{\cal E}=\ell_{F}.(L.{\vec{u}}_{\cal E})$; So
$L^{*}(\ell_{F})=\ell_{F}.L$; And $L^{*}(\ell_{F})\in{E^{*}}$ is denoted
$\ell_{F}^{*}$ and named the pull-back of $\ell_{F}$ by $L$. So that
$\ell_{F}^{*}=\ell_{F}.L$. And (56) is obtained with $\ell_{F}={\alpha_{\cal
F}}({p_{\cal F}})$ and $L=d\Psi({p_{\cal E}})$, and
$\ell_{F}^{*}={\alpha_{\cal F}}({p_{\cal F}})^{*}$ is denoted
${\alpha^{*}_{\cal F}}({p_{\cal E}})$.

Let $T_{\cal E}$ be a ${r\choose s}$ tensor on ${\Omega_{\cal E}}$, with
$r,s\geq 1$. Its push-forward by $\Psi$ is the ${r\choose s}$ tensor
$\Psi_{*}T_{\cal E}=T_{{}_{\cal E}*}$ on ${\Omega_{\cal F}}$ defined by

$T_{{}_{\cal E}*}(\alpha_{1{\cal F}},...,\alpha_{r{\cal F}},{\vec{w}}_{1{\cal
F}},...,{\vec{w}}_{s{\cal F}})=T_{\cal E}(\alpha_{1{\cal
F}}^{*},...,\alpha_{r{\cal F}}^{*},{\vec{w}}_{1{\cal
F}}^{*},...,{\vec{w}}_{s{\cal F}}^{*}).$ (57)

Let $T_{\cal F}$ be a ${r\choose s}$ tensor on ${\Omega_{\cal F}}$, with
$r,s\geq 1$. Its pull-back by $\Psi$ is the ${r\choose s}$ tensor $T_{\cal
F}^{*}$ on ${\Omega_{\cal E}}$ defined by

$T_{\cal F}^{*}(\alpha_{1{\cal E}},...,\alpha_{r{\cal E}},{\vec{w}}_{1{\cal
E}},...,{\vec{w}}_{s{\cal E}})=T_{\cal F}(\alpha_{1{\cal
E}*},...,\alpha_{r{\cal E}*},{\vec{w}}_{1{\cal E}*},...,{\vec{w}}_{s{\cal
E}*}).$ (58)

##  Appendix C Lie derivatives: Introduction and interpretation

(Classical mechanics.) Let ${\mathaccent 869{\Phi}}$ be a motion, see (1),
${\vec{v}}$ be the associated Eulerian velocity, see (2),
$t,\tau\in\mathbb{R}$, ${\Phi^{t}_{\tau}}$ the associated motion, see (4) and
(3), and ${F^{t}_{\tau}}=d{\Phi^{t}_{\tau}}$. With reference to § B, let
$\Psi={\Phi^{t}_{\tau}}$, ${\Omega_{\cal E}}={\Omega_{t}}$, ${\Omega_{\cal
F}}={\Omega_{\tau}}$, ${p_{\cal E}}={p_{t}}$ and ${p_{\cal
F}}={{p}_{\tau}}={\Phi^{t}_{\tau}}({p_{t}})$. If
$f_{\tau}:{\Omega_{\tau}}\rightarrow\mathbb{R}$ is a function then its pull-
back $({\Phi^{t}_{\tau}})^{*}f_{\tau}={f^{t*}_{\tau}}$ is given by
${f^{t*}_{\tau}}({p_{t}})=f_{\tau}({{p}_{\tau}})$. If ${\vec{w}_{\tau}}$ is a
vector field in ${\Omega_{\tau}}$ then its pull-back
$({\Phi^{t}_{\tau}})^{*}{\vec{w}_{\tau}}={\vec{w}^{t*}_{\tau}}$ is given by
${\vec{w}^{t*}_{\tau}}({p_{t}})={F^{t}_{\tau}}({p_{t}})^{-1}.{\vec{w}_{\tau}}({{p}_{\tau}})$.
If ${\alpha_{\tau}}$ is one-form in ${\Omega_{\tau}}$ then its pull-back
$({\Phi^{t}_{\tau}})^{*}{\alpha_{\tau}}={\alpha^{t*}_{\tau}}$ is given by
${\alpha^{t*}_{\tau}}({p_{t}})={\alpha_{\tau}}({{p}_{\tau}}).{F^{t}_{\tau}}({p_{t}})$.
And we define $f^{*}_{\tau}(t,{p_{t}}):={f^{t*}_{\tau}}({p_{t}})$,
${\vec{w}}^{*}_{\tau}(t,{p_{t}}):={\vec{w}^{t*}_{\tau}}({p_{t}})$, and
$\alpha^{*}_{\tau}(t,{p_{t}}):={\alpha^{t*}_{\tau}}({p_{t}})$.

The Lie derivative of a real-valued function along the flow of velocity
${\vec{v}}$ (or along the motion ${\mathaccent 869{\Phi}}$) at $(t,{p_{t}})$
is defined by

${\cal L}_{\vec{v}}f(t,{p_{t}}){\;\mathop{:=}\;}\lim_{\tau\rightarrow
t}{f^{*}_{\tau}(t,{p_{t}})-f(t,{p_{t}})\over\tau-t}.$ (59)

Thus, with ${{p}_{\tau}}={\Phi^{t}_{\tau}}({p_{t}})$,

${\cal L}_{\vec{v}}f(t,{p_{t}})=\lim_{\tau\rightarrow
t}{f(\tau,{{p}_{\tau}})-f(t,{p_{t}})\over\tau-t}={Df\over Dt}(t,{p_{t}}).$
(60)

Interpretation: An observer does not have the gift of spatial and/or temporal
ubiquity, thus cannot trivially compare values at a two distinct points or
instants: He needs time (from $t$ to $\tau$) and a displacement (from
${p_{t}}$ to ${{p}_{\tau}}$) to compare $f(\tau,{{p}_{\tau}})$ and
$f(t,{p_{t}})$ (to compute the difference
$f(\tau,{{p}_{\tau}})-f(t,{p_{t}})$). So (60) is a computation consequence of
(59).

The Lie derivative ${\cal L}_{\vec{v}}{\vec{w}}$ of a vector field ${\vec{w}}$
along the flow of velocity ${\vec{v}}$ is defined by

${\cal L}_{\vec{v}}{\vec{w}}(t,{p_{t}}){\;\mathop{:=}\;}\lim_{\tau\rightarrow
t}{{\vec{w}}^{*}_{\tau}(t,{p_{t}})-{\vec{w}}(t,{p_{t}})\over\tau-t},$ (61)

and the difference ${\vec{w}}^{t*}(t,{p_{t}})-{\vec{w}}(t,{p_{t}})$ is
computed at a single time $t$ and at a single point ${p_{t}}$ (does not
require ubiquity). In ${\mathbb{R}^{n}}$, ${\cal L}_{\vec{v}}{\vec{w}}$ is
equivalently defined by

${\cal L}_{\vec{v}}{\vec{w}}(t,{p_{t}}){\;\mathop{:=}\;}\lim_{\tau\rightarrow
t}{{\vec{w}}(\tau,{{p}_{\tau}})-{\vec{w}}^{t}_{*}(\tau,{{p}_{\tau}})\over\tau-t}.$
(62)

Interpretation: E.g. in ${\mathbb{R}^{n}}$ with (62): at $\tau$ and
${{p}_{\tau}}$, the numerator
${\vec{w}}(\tau,{{p}_{\tau}})-{\vec{w}}^{t}_{*}(\tau,{{p}_{\tau}})$ compares
the true value ${\vec{w}}(\tau,{{p}_{\tau}})$ of ${\vec{w}}$ at
$(\tau,{{p}_{\tau}})$ with the value ${\vec{w}}^{t}_{*}(\tau,{{p}_{\tau}})$
which corresponds to “the vector that would have let itself transported by the
motion” (the push-forward). So the Lie derivative ${\cal
L}_{\vec{v}}{\vec{w}}(t,{p_{t}})$, which is the limit of the rate
${{\vec{w}}(\tau,{{p}_{\tau}})-{\vec{w}}^{t}_{*}(\tau,{{p}_{\tau}})\over\tau-t}$,
measure the “resistance” of ${\vec{w}}$ “submitted to the motion ${\mathaccent
869{\Phi}}$” (“submitted to the flow”). The Lie Virtual Power Principle
proposed in this manuscript is based on this property. (For comparison with
the classical approach see § D.)

And (61) gives:

${\cal L}_{\vec{v}}{\vec{w}}={D{\vec{w}}\over
Dt}-d{\vec{v}}.{\vec{w}}\quad(={\partial{\vec{w}}\over\partial
t}+d{\vec{w}}.{\vec{v}}-d{\vec{v}}.{\vec{w}}).$ (63)

Indeed with ${{p}_{\tau}}={\Phi^{t}_{\tau}}({p_{t}})$ and
${\vec{g}}:\tau\rightarrow{\vec{g}}(\tau)=d{\Phi^{t}_{\tau}}({p_{t}})^{-1}.{\vec{w}}(\tau,{{p}_{\tau}})$
we have ${\cal L}_{\vec{v}}{\vec{w}}(t,{p_{t}})={\vec{g}}\,^{\prime}(t)$, see
(61); And
$d\Phi^{t}(\tau,{p_{t}}).{\vec{g}}(\tau)={\vec{w}}(\tau,{{p}_{\tau}})$ leads
to
$d{\partial\Phi^{t}\over\partial\tau}(\tau,{p_{t}}).{\vec{g}}(\tau)+d\Phi^{t}(\tau,{p_{t}}).{\vec{g}}\,^{\prime}(\tau)={D{\vec{w}}\over
D\tau}(\tau,{{p}_{\tau}})$, therefore with $\tau=t$ we get
$d{\vec{v}}(t,{p_{t}}).{\vec{w}}(t,{p_{t}})+{\vec{g}}\,^{\prime}(t)={D{\vec{w}}\over
Dt}(t,{p_{t}})$, thus (63). Note that, for a vector field ${\vec{w}}$
submitted to the flow, the term $-d{\vec{v}}.{\vec{w}}$ in ${\cal
L}_{\vec{v}}{\vec{w}}$ takes into account the influence of non uniform flows
${\vec{v}}$ (flows s.t. $d{\vec{v}}\neq 0$) on ${\vec{w}}$.

The Lie derivative ${\cal L}_{\vec{v}}\alpha$ of a one-form $\alpha$ along the
flow of velocity ${\vec{v}}$ is the one-form deduced for example from (63) and
(60) with the derivation property ${\cal L}_{\vec{v}}(\alpha.{\vec{w}})={\cal
L}_{\vec{v}}\alpha.{\vec{w}}+\alpha.{\cal L}_{\vec{v}}{\vec{w}}$. Thus

${\cal L}_{\vec{v}}\alpha={D\alpha\over
Dt}+\alpha.d{\vec{v}}\quad(={\partial\alpha\over\partial
t}+d\alpha.{\vec{v}}+\alpha.d{\vec{v}}).$ (64)

The same result is obtained with the mathematical definition ${\cal
L}_{\vec{v}}\alpha(t,{p_{t}}):=\lim_{\tau\rightarrow
t}{\alpha^{*}_{\tau}(t,{p_{t}})-\alpha(t,{p_{t}})\over\tau-t}$.

And for tensors of order $2$,
${\underline{\underline{\kappa_{m}}}}\in{T^{1}_{1}({\Omega_{t}})}$ (mixed),
${\underline{\underline{\kappa_{u}}}}\in{T^{2}_{0}({\Omega_{t}})}$ (up) and
${\underline{\underline{\kappa_{d}}}}\in{T^{0}_{2}({\Omega_{t}})}$ (down), the
generic derivation property ${\cal L}_{\vec{v}}(a\otimes b)=({\cal
L}_{\vec{v}}a)\otimes b+a\otimes{\cal L}_{\vec{v}}b$ (or definition with pull-
backs) gives

| $\displaystyle{{}{\cal
L}_{\vec{v}}{\underline{\underline{\kappa_{m}}}}={D{\underline{\underline{\kappa_{m}}}}\over
Dt}+{\underline{\underline{\kappa_{m}}}}.d{\vec{v}}-d{\vec{v}}.{\underline{\underline{\kappa_{m}}}}\;\in{T^{1}_{1}({\Omega_{t}})}\quad\hbox{(Jaumann)},}$
---|---
| $\displaystyle{{}{\cal
L}_{\vec{v}}{\underline{\underline{\kappa_{u}}}}={D{\underline{\underline{\kappa_{u}}}}\over
Dt}-{\underline{\underline{\kappa_{u}}}}.(d{\vec{v}})^{*}-d{\vec{v}}.{\underline{\underline{\kappa_{u}}}}\;\in{T^{2}_{0}({\Omega_{t}})}\quad\hbox{(Maxwell
upper-convected)},}$
| $\displaystyle{{}{\cal
L}_{\vec{v}}{\underline{\underline{\kappa_{d}}}}={D{\underline{\underline{\kappa_{d}}}}\over
Dt}+{\underline{\underline{\kappa_{d}}}}.d{\vec{v}}+(d{\vec{v}})^{*}.{\underline{\underline{\kappa_{d}}}}\;\in{T^{0}_{2}({\Omega_{t}})}\quad\hbox{(Maxwell
lower-convected)},}$
(65)

where, for ${\underline{\underline{\sigma_{m}}}}\in{T^{1}_{1}({\Omega_{t}})}$,
the adjoint tensor
${{\underline{\underline{\sigma_{m}}}}}^{*}\in{T^{1}_{1}({\Omega_{t}})}^{*}$
is defined by
${{\underline{\underline{\sigma_{m}}}}}^{*}({\vec{u}},\ell)={\underline{\underline{\sigma_{m}}}}(\ell,{\vec{u}})$,
and ${D{\underline{\underline{\kappa}}}\over
Dt}={\partial{\underline{\underline{\kappa_{d}}}}\over\partial
t}+d{\underline{\underline{\kappa}}}.{\vec{v}}$ (generic notation).

Although an Eulerian velocity field ${\vec{v}}$ is not objective, see § G and
H, we have: If $T$ is an objective tensor, then its Lie derivative ${\cal
L}_{\vec{v}}T$ is an objective tensor; see e.g. Marsden and Hughes [6] p.101.
(But partial derivatives and material derivative are not objective in
general.) Here objectivity refers to “covariant objectivity”, see (78).

##  Appendix D Lie derivative versus deformation tensor

The deformation tensor $C=F^{T}.F$ between ${t_{0}}$ and $t$ is used to
measure the relative deformation between two vectors thanks to the use of two
Euclidean inner products, $(\cdot,\cdot)_{G}$ at ${t_{0}}$ and
$(\cdot,\cdot)_{g}$ at $t$ (see e.g. Marsden and Hughes [6]): With
${P}\in{\Omega_{t_{0}}}$, ${p_{t}}={\Phi^{t_{0}}_{t}}({P})$,
${F^{t_{0}}_{t}}({P}):=d{\Phi^{t_{0}}_{t}}({P})$,
${C^{t_{0}}_{t}}({P})=({F^{t_{0}}_{t}}({P}))^{T}.{F^{t_{0}}_{t}}({P})$,
${\vec{W}}_{i}({P})\in{\mathbb{R}^{n}_{t_{0}}}$ and
${\vec{w}}_{i*}({p_{t}})={F^{t_{0}}_{t}}({P}).{\vec{W}}_{i}({P})$ the push-
forward of ${\vec{W}}_{i}$ by ${\Phi^{t_{0}}_{t}}$ see (53), we have :

$\displaystyle{({\vec{w}}_{1*}({p_{t}}),{\vec{w}}_{2*}({p_{t}}))_{g}=}$ | $\displaystyle{{}({C^{t_{0}}_{t}}({P}).{\vec{W}}_{1}({P}),{\vec{W}}_{2}({P}))_{G}.}$
---|---
(66)

This value is compared with $({\vec{W}}_{1}({P}),{\vec{W}}_{2}({P}))_{G}$ in
classical mechanics. N.B.: The deformation tensor $C$ compares two vectors
that have let themselves deformed by the flow, see (66), since
${\vec{w}}_{1*}$ and ${\vec{w}}_{2*}$ are the push-forwards by the flow.

While the Lie derivative of a vector field ${\vec{w}}$ measures the resistance
of a single vector field ${\vec{w}}$ submitted to a flow, see interpretation
of (62), and does not require a priori the use of inner products (Euclidean or
not) since there is no comparison between two vectors.

##  Appendix E Change of Riesz representation vector

The Riesz representation theorem is often implicitly used in classical
mechanics under isometric objectivity hypotheses (see § H). Unfortunately this
causes problems since covariance cannot be confused with contravariance. E.g.,
Misner, Thorne, Wheeler [7] box 2.1: “Without it [the distinction between
covariance and contravariance], one cannot know whether a vector is meant or
the very different geometric object that is a 1-form.”

Let $E$ be a normed vector space and $E^{*}$ the set of linear continuous
functions $E\rightarrow\mathbb{R}$. There is no natural canonical isomorphism
between $E$ and its dual ${E^{*}}$, see e.g. Spivak [8], but if an observer
introduces an inner product then an isomorphism (depending on the observer) is
obtained:

######  Theorem E.1 (Riesz representation theorem)

If $(\cdot,\cdot)_{g}$ is an inner product in $E$ so that $E$ is a Hilbert
space, then

$\forall\ell\in{E^{*}},\quad\exists!{\vec{\ell}}_{g}\in
E,\quad\forall{\vec{w}}\in
E,\quad\ell.{\vec{w}}=({\vec{\ell}}_{g},{\vec{w}})_{g}.$ (67)

Moreover $||{\vec{\ell}}_{g}||_{g}=||\ell||_{E^{*}}$. And the vector
${\vec{\ell}}_{g}$ is called the Riesz representative vector of the linear
form $\ell$ relatively to the inner product $(\cdot,\cdot)_{g}$, or the
$(\cdot,\cdot)_{g}$-Riesz representative vector of $\ell$.

Proof: If $\ell=0$ then ${\vec{\ell}}_{g}=0$. If $\ell\neq 0$, choose a
${\vec{v}}\notin{\rm Ker}\ell=\\{{\vec{v}}\in
E:\ell({\vec{v}})=0\\}=\ell^{-1}(\\{0\\})$ (closed sub-space since $\ell$ is
continuous). Let ${\vec{v}}_{0}$ be the $(\cdot,\cdot)_{g}$-orthogonal
projection of ${\vec{v}}$ on ${\rm Ker}\ell$, and let
${\vec{n}}={{\vec{v}}-{\vec{v}}_{0}\over||{\vec{v}}-{\vec{v}}_{0}||_{g}}$ (a
$||.||_{g}$-unitary vector normal to ${\rm Ker}\ell$). Then take
${\vec{\ell}}_{g}=\ell({\vec{n}}){\vec{n}}$ to get (67) and
$||{\vec{\ell}}_{g}||_{g}=\sup_{||{\vec{w}}||_{E}=1}|\ell.{\vec{w}}|$
($||\ell||_{E^{*}}$) thanks to the Cauchy–Schwarz inequality. And uniqueness
is trivial.

######  Corollary 1 (Change of Riesz representation vector)

Let $\ell\in{E^{*}}$. Let $(\cdot,\cdot)_{a}$ and $(\cdot,\cdot)_{b}$ be two
inner products, and let ${\vec{\ell}}_{a}$ and ${\vec{\ell}}_{b}$ be the
$(\cdot,\cdot)_{a}$ and $(\cdot,\cdot)_{b}$-Riesz representative of $\ell$.
Then

$\forall{\vec{w}}\in
E,\quad({\vec{\ell}}_{a},{\vec{w}})_{a}=({\vec{\ell}}_{b},{\vec{w}})_{b}.$
(68)

Proof: For all ${\vec{w}}$, (67) gives
$\ell.{\vec{w}}=({\vec{\ell}}_{a},{\vec{w}})_{a}$ and
$\ell.{\vec{w}}=({\vec{\ell}}_{b},{\vec{w}})_{b}$, thus (68).

###### Example 1

Let $(\cdot,\cdot)_{a}$ be the Euclidean inner product built with the English
foot (ft) and $(\cdot,\cdot)_{b}$ be the Euclidean inner product built with
the meter (m). We have $1$ ft $=\mu$ m with $\mu=0.3048$. And
$(\cdot,\cdot)_{b}=\mu^{2}\,(\cdot,\cdot)_{a}$ (and $||.||_{b}=\mu||.||_{a}$),
thus (68) gives
$({\vec{\ell}}_{a},{\vec{w}})_{a}=\mu^{2}({\vec{\ell}}_{b},{\vec{w}})_{a}$ for
all ${\vec{w}}$, therefore:

${\vec{\ell}}_{a}=\mu^{2}{\vec{\ell}}_{b}$ (69)

and the vector ${\vec{\ell}}_{a}$ is $\mu^{2}$ times smaller (more than ten
times smaller) than the vector ${\vec{\ell}}_{b}$. So the Riesz representation
vector depends on the choice of the inner product (an inner product is a
measuring tool, and a change of tool changes the result).

##  Appendix F Incompatibilities with Riesz representation vectors

We have just seen that the Riesz representation vector depends on the
observer, see e.g. (69). But we also have e.g.:

1- Incompatibility with push-forwards. Let $\alpha$ be in
$({\mathbb{R}^{n}_{t_{0}}})^{*}$ and let
${\alpha_{*}}=\alpha.{F^{t_{0}}_{t}}^{-1}\in({\mathbb{R}^{n}_{t}})^{*}$ be its
push-forward by ${\Phi^{t_{0}}_{t}}$, see (55). Let $(\cdot,\cdot)_{G}$ and
$(\cdot,\cdot)_{g}$ be inner products in ${\mathbb{R}^{n}_{t_{0}}}$ and
${\mathbb{R}^{n}_{t}}$. Let ${\vec{\alpha}}_{G}\in{\mathbb{R}^{n}_{t_{0}}}$
and ${{\vec{\alpha}}_{*g}}\in{\mathbb{R}^{n}_{t}}$ be the $(\cdot,\cdot)_{G}$
and $(\cdot,\cdot)_{g}$-Riesz representation vectors of $\alpha$ and
${\alpha_{*}}$, see (67). And let ${\vec{\alpha}}_{G*}\in{\mathbb{R}^{n}_{t}}$
be the push-forward of ${\vec{\alpha}}_{G}$ by ${\Phi^{t_{0}}_{t}}$, see (53).
So, with $p={\Phi^{t_{0}}_{t}}(P)$ and ${\vec{w}}_{p}\in{\mathbb{R}^{n}_{t}}$,
we have
$({{\vec{\alpha}}_{*g}}(p),{\vec{w}}_{p})_{g}={\alpha_{*}}(p).{\vec{w}}_{p}=(\alpha(P).{F^{t_{0}}_{t}}(P)^{-1}).{\vec{w}}_{p}=\alpha(P).({F^{t_{0}}_{t}}(P)^{-1}.{\vec{w}}_{p})=({\vec{\alpha}}_{G}(P),{F^{t_{0}}_{t}}(P)^{-1}.{\vec{w}}_{p})_{G}=({F^{t_{0}}_{t}}(P)^{-T}.{\vec{\alpha}}_{G}(P),{\vec{w}}_{p})_{g}$,
true for all ${\vec{w}}_{p}$, thus

${{\vec{\alpha}}_{*g}}(p)={F^{t_{0}}_{t}}(P)^{-T}.{\vec{\alpha}}_{G}(p).$ (70)

So ${{\vec{\alpha}}_{*g}}$ is not the push-forward of ${\vec{\alpha}}_{G}$,
that is
${{\vec{\alpha}}_{*g}}(p)\neq{F^{t_{0}}_{t}}(P).{\vec{\alpha}}_{G}(p)$, unless
${\Phi^{t_{0}}_{t}}$ is the motion of a solid. Thus the Riesz representation
vectors should not be used if push-forwards of one-forms are needed.

2- Incompatibility with Lie derivative. Let $\beta$ be in
$({\mathbb{R}^{n}_{t}})^{*}$. Let $(\cdot,\cdot)_{g}$ be an inner product in
${\mathbb{R}^{n}_{t}}$, and let ${\vec{\beta}_{g}}$ be the
$(\cdot,\cdot)_{g}$-Riesz representation vector of $\beta$, that is
$\beta.{\vec{w}}=({\vec{\beta}_{g}},{\vec{w}})_{g}$ for any ${\vec{w}}$. Then
we have ${\cal L}_{\vec{v}}\beta.{\vec{w}}\neq({\cal
L}_{\vec{v}}{\vec{\beta}_{g}},{\vec{w}})_{g}$, unless $\Phi$ is the motion of
a solid, that is we have ${D\beta\over
Dt}.{\vec{w}}+\beta.d{\vec{v}}.{\vec{w}}\neq({D{\vec{\beta}_{g}}\over
Dt},{\vec{w}})_{g}-({\vec{\beta}_{g}},d{\vec{v}}.{\vec{w}})_{g}$ in general.
We have:

${\cal L}_{\vec{v}}\beta.{\vec{w}}=({\cal
L}_{\vec{v}}{\vec{\beta}_{g}},{\vec{w}})_{g}+({\vec{\beta}_{g}},(d{\vec{v}}+d{\vec{v}}^{T}).{\vec{w}})_{g}$
(71)

since ${D\beta\over
Dt}.{\vec{w}}+\beta.d{\vec{v}}.{\vec{w}}=({D{\vec{\beta}_{g}}\over
Dt},{\vec{w}})_{g}+({\vec{\beta}_{g}},d{\vec{v}}.{\vec{w}})_{g}$. Thus the
Riesz representation vectors cannot be used if Lie derivative of one-forms are
needed.

##  Appendix G Velocity addition formula

(This § is needed to define objectivity, see the next §.) Classical mechanics
setting. The observers use the same time scale. An observer $A$ defines a
referential ${{\cal R}_{\\!A}}=({O_{\\!A}},({\vec{A}}_{i}))$ and an observer
$B$ defines a referential ${{\cal R}_{\\!B}}=({O_{\\!B}},({\vec{B}}_{i}))$,
with $({\vec{A}}_{i})$ and $({\vec{B}}_{i})$ Cartesian bases. They observe an
object ${\it O\\!b\\!j}$ in a time interval $[t_{1},t_{2}]$. Let
${{\mathaccent 869{\Phi}}_{A}}:[{t_{1}},{t_{2}}]\times{\it
O\\!b\\!j}\rightarrow{{\cal R}_{\\!A}}$ and ${{\mathaccent
869{\Phi}}_{\\!B}}:[{t_{1}},{t_{2}}]\times{\it O\\!b\\!j}\rightarrow{{\cal
R}_{\\!B}}$ be the motions of ${\it O\\!b\\!j}$ as described by $A$ and $B$,
and let ${\vec{v}}_{A}$ and ${\vec{v}}_{B}$ be the associated Eulerian
velocity fields, see (1) and (2), that is,

$\left\\{\hbox{}\,\vbox{\openup 3.0pt\halign{\strut\hfil$\displaystyle{#}$ &
$\displaystyle{{}#}$\hfil\cr&{{p}_{\\!At}}={{\mathaccent
869{\Phi}}_{A}}(t,{P_{\it
O\\!b\\!j}})={O_{\\!A}}+{\sum_{i=1}^{n}}x_{At}^{i}\vec{A}_{i},\cr&{\vec{v}}_{A}(t,{{p}_{\\!At}})={\partial{{\mathaccent
869{\Phi}}_{A}}\over\partial t}(t,{P_{\it
O\\!b\\!j}}),\crcr}}\,\right.\quad\left\\{\hbox{}\,\vbox{\openup
3.0pt\halign{\strut\hfil$\displaystyle{#}$ &
$\displaystyle{{}#}$\hfil\cr&{{p}_{\\!Bt}}={{\mathaccent
869{\Phi}}_{\\!B}}(t,{P_{\it
O\\!b\\!j}})={O_{\\!B}}+{\sum_{i=1}^{n}}x_{Bt}^{i}\vec{B}_{i},\cr&{\vec{v}}_{B}(t,{{p}_{\\!Bt}})={\partial{{\mathaccent
869{\Phi}}_{\\!B}}\over\partial t}(t,{P_{\it O\\!b\\!j}}).\crcr}}\,\right.$
(72)

Let ${{\it O\\!b\\!j}_{\\!B}}$ be the object used by $B$ to define his
referential (e.g. ${{\it O\\!b\\!j}_{\\!B}}=$ Earth). Let ${{\mathaccent
869{\Psi}}_{\\!A}}:[{t_{1}},{t_{2}}]\times{{\it
O\\!b\\!j}_{\\!B}}\rightarrow{{\cal R}_{\\!A}}$ and ${{\mathaccent
869{\Psi}}_{\\!B}}:{{\it O\\!b\\!j}_{\\!B}}\rightarrow{{\cal R}_{\\!B}}$
(static) be the motions of ${{\it O\\!b\\!j}_{\\!B}}$ described by $A$ and
$B$, and let ${\vec{w}}_{A}$ and ${\vec{w}}_{B}$ be the associated Eulerian
velocity fields, see (1) and (2), that is,

$\left\\{\hbox{}\,\vbox{\openup 3.0pt\halign{\strut\hfil$\displaystyle{#}$ &
$\displaystyle{{}#}$\hfil\cr&{q_{\\!At}}={{\mathaccent
869{\Psi}}_{\\!A}}(t,{Q_{Obj\\!B}})={O_{\\!A}}+{\sum_{i=1}^{n}}y_{At}^{i}\vec{A}_{i},\cr&{\vec{w}}_{A}(t,{q_{\\!At}})={\partial{{\mathaccent
869{\Psi}}_{\\!A}}\over\partial
t}(t,{Q_{Obj\\!B}}),\crcr}}\,\right.\quad\left\\{\hbox{}\,\vbox{\openup
3.0pt\halign{\strut\hfil$\displaystyle{#}$ &
$\displaystyle{{}#}$\hfil\cr&{q_{\\!B}}={{\mathaccent
869{\Psi}}_{\\!B}}({Q_{Obj\\!B}})={O_{\\!B}}+{\sum_{i=1}^{n}}y_{B}^{i}\vec{B}_{i},\cr&{\vec{w}}_{B}(t,{q_{\\!B}})=\vec{0}.\crcr}}\,\right.$
(73)

For $t$ fixed, let ${{\mathaccent 869{\Phi}}_{\\!At}}({P_{\it
O\\!b\\!j}}):={{\mathaccent 869{\Phi}}_{A}}(t,{P_{\it O\\!b\\!j}})$ and
${{\mathaccent 869{\Phi}}_{\\!Bt}}({P_{\it O\\!b\\!j}}):={{\mathaccent
869{\Phi}}_{\\!B}}(t,{P_{\it O\\!b\\!j}})$, and ${{\mathaccent
869{\Psi}}_{\\!At}}({P_{\it O\\!b\\!j}}):={{\mathaccent
869{\Psi}}_{\\!A}}(t,{P_{\it O\\!b\\!j}})$.

The mapping ${{\mathaccent 869{\Phi}}_{A}}$, ${{\mathaccent
869{\Psi}}_{\\!A}}$ in ${{\cal R}_{\\!A}}$ and ${{\mathaccent
869{\Phi}}_{\\!B}}$, ${{\mathaccent 869{\Psi}}_{\\!B}}$ in ${{\cal R}_{\\!B}}$
are motions: They are defined by one observer in his referential. The
translation mapping at $t$ from $B$ to $A$ (the translator) is the “inter-
referential” diffeomorphism ${\Theta_{t}}:={{\mathaccent
869{\Psi}}_{\\!At}}\circ{{\mathaccent 869{\Psi}}_{\\!B}}^{-1}$, and we denote

${\Theta_{t}}:\left\\{\hbox{}\,\vbox{\openup
3.0pt\halign{\strut\hfil$\displaystyle{#}$ &
$\displaystyle{{}#}$\hfil\cr{{\cal R}_{\\!B}}&\rightarrow{{\cal
R}_{\\!A}}\cr{q_{\\!B}}={O_{\\!B}}+{\sum_{i=1}^{n}}y_{B}^{i}\vec{B}_{i}&\rightarrow{q_{\\!At}}={\Theta_{t}}({q_{\\!B}})={O_{\\!A}}+{\sum_{i=1}^{n}}y_{At}^{i}\vec{A}_{i}\crcr}}\,\right\\}.$
(74)

So, ${{\mathaccent
869{\Psi}}_{\\!At}}({Q_{Obj\\!B}})={\Theta_{t}}({{\mathaccent
869{\Psi}}_{\\!B}}({Q_{Obj\\!B}}))$ for all ${Q_{Obj\\!B}}\in{{\it
O\\!b\\!j}_{\\!B}}$: If a particle ${Q_{Obj\\!B}}$ of ${{\it
O\\!b\\!j}_{\\!B}}$ is located at ${q_{\\!B}}={{\mathaccent
869{\Psi}}_{\\!B}}({Q_{Obj\\!B}})\in{{\cal R}_{\\!B}}$ by the observer $B$,
then the observer $A$ locates ${Q_{Obj\\!B}}$ at $t$ at ${{\mathaccent
869{\Psi}}_{\\!At}}({Q_{Obj\\!B}})={q_{\\!At}}={\Theta_{t}}({q_{\\!B}})\in{{\cal
R}_{\\!A}}$. E.g. ${\Theta_{t}}({O_{\\!B}})$ is “the position of ${O_{\\!B}}$
in ${{\cal R}_{\\!A}}$ at $t$”; And thus the push-forward
$d{\Theta_{t}}({O_{\\!B}}).{\vec{B}}_{i}$ of ${\vec{B}}_{i}$ by ${\Theta_{t}}$
is “the basis vector ${\vec{B}}_{i}$ as seen by $A$ at
${\Theta_{t}}({O_{\\!B}})$” (see § B with the diffeomorphism
$\Psi={\Theta_{t}}$ and ${c_{\cal E}}=$ the $i$-th coordinate line in ${{\cal
R}_{\\!B}}$).

With (74) define $\Theta:[{t_{1}},{t_{2}}]\times{{\cal
R}_{\\!B}}\rightarrow{{\cal R}_{\\!A}}$ by
$\Theta(t,{q_{\\!B}}){\;\mathop{:=}\;}{\Theta_{t}}({q_{\\!B}})$. And define
the “$\Theta$-Eulerian velocity” at ${q_{\\!At}}\in{{\cal R}_{\\!A}}$ by:

${\vec{w}}_{\Theta}(t,{q_{\\!At}})={\partial\Theta\over\partial
t}(t,{q_{\\!B}})\quad\hbox{if}\quad{q_{\\!At}}={\Theta_{t}}({q_{\\!B}}).$ (75)

Note that ${\vec{w}}_{\Theta}$ looks like a Eulerian velocity but is not since
$\Theta$ is not a motion (${\Theta_{t}}$ is an inter-referential mapping).

Interpretation of ${\vec{w}}_{\Theta}$: With ${{\mathaccent
869{\Psi}}_{\\!A}}(t,{Q_{Obj\\!B}})=\Theta(t,{{\mathaccent
869{\Psi}}_{\\!B}}({Q_{Obj\\!B}}))$ and ${q_{\\!B}}={{\mathaccent
869{\Psi}}_{\\!B}}({Q_{Obj\\!B}})$ we get ${\partial{{\mathaccent
869{\Psi}}_{\\!A}}\over\partial
t}(t,{Q_{Obj\\!B}})={\partial\Theta\over\partial t}(t,{q_{\\!B}})$, thus, with
(73) and (75) we have

${\vec{w}}_{A}(t,{q_{\\!At}})={\vec{w}}_{\Theta}(t,{q_{\\!At}}),$ (76)

equality in ${{\cal R}_{\\!A}}$; And the “$\Theta$-Eulerian velocity”
${\vec{w}}_{\Theta}(t,{q_{\\!At}})$ is the Eulerian velocity
${\vec{w}}_{A}(t,{q_{\\!At}})$ of the particle ${Q_{Obj\\!B}}\in{{\it
O\\!b\\!j}_{\\!B}}$ that is at $t$ at ${q_{\\!At}}={{\mathaccent
869{\Psi}}_{\\!At}}({Q_{Obj\\!B}})={\Theta_{t}}({q_{\\!B}})\in{{\cal
R}_{\\!A}}$.

Velocity addition formula. We also have ${{\mathaccent
869{\Phi}}_{\\!At}}={\Theta_{t}}\circ{{\mathaccent 869{\Phi}}_{\\!Bt}}$, that
is, with (72), ${{p}_{\\!At}}={\Theta_{t}}({{p}_{\\!Bt}})$ (inter-referential
relations between positions at $t$). So ${{\mathaccent
869{\Phi}}_{A}}(t,{P_{\it O\\!b\\!j}})=\Theta(t,{{\mathaccent
869{\Phi}}_{\\!B}}(t,{P_{\it O\\!b\\!j}}))$, and with
${{p}_{\\!Bt}}={{\mathaccent 869{\Phi}}_{\\!B}}(t,{P_{\it O\\!b\\!j}})$ we get
${\partial{{\mathaccent 869{\Phi}}_{A}}\over\partial t}(t,{P_{\it
O\\!b\\!j}})={\partial\Theta\over\partial
t}(t,{{p}_{\\!Bt}})+d\Theta(t,{{p}_{\\!Bt}}).{\partial{{\mathaccent
869{\Phi}}_{\\!B}}\over\partial t}(t,{P_{\it O\\!b\\!j}})$, that is,
${\vec{v}}_{A}(t,{{p}_{\\!At}})={\vec{w}}_{\Theta}(t,{{p}_{\\!At}})+d\Theta(t,{{p}_{\\!Bt}}).{\vec{v}}_{B}(t,{{p}_{\\!Bt}})$,
that is with (76),

${{\vec{v}}_{\\!At}}({{p}_{\\!At}})={{\vec{v}}_{Bt*}}({{p}_{\\!At}})+{{\vec{w}}_{\\!At}}({{p}_{\\!At}})\quad\hbox{where}\quad{{\vec{v}}_{Bt*}}({{p}_{\\!At}}){\;\mathop{:=}\;}d{\Theta_{t}}({{p}_{\\!Bt}}).{{\vec{v}}_{\\!Bt}}({{p}_{\\!Bt}}).$
(77)

This formula is the velocity addition formula. Interpretation: At $t$, if $A$
is the “absolute observer” and $B$ is the “relative observer” then the formula
reads: “$({{\vec{v}}_{\\!At}}$ the absolute velocity) = (${{\vec{v}}_{Bt*}}$
the relative velocity translated for $A$) \+ (${{\vec{w}}_{\\!At}}$ the
velocity of the coordinate system of $B$ in ${{\cal R}_{\\!A}}$)”, equality in
${{\cal R}_{\\!A}}$.

##  Appendix H Covariant and isometric objectivities

(Mainly from Marsden and Hughes [6].) Setting of the previous §.

###### Definition 1

Let ${{\vec{u}}_{\\!Bt}}$ be a vector field in ${{\cal R}_{\\!B}}$ at $t$ as
described by the observer $B$. Let ${{\vec{u}}_{\\!Bt*}}$ be its translation
for $A$, that is, the push-forward of ${{\vec{u}}_{\\!Bt}}$ by ${\Theta_{t}}$
(so
${{\vec{u}}_{\\!Bt*}}({{p}_{\\!At}})=d{\Theta_{t}}({{p}_{\\!Bt}}).{{\vec{u}}_{\\!Bt}}({{p}_{\\!Bt}})$
when ${{p}_{\\!At}}={\Theta_{t}}({{p}_{\\!Bt}})$, see (53)). Then
${{\vec{u}}_{\\!Bt*}}$ is called the objective transform of
${{\vec{u}}_{\\!Bt}}$ by ${\Theta_{t}}$. More generally, at $t$ the objective
transform of a tensor $T_{Bt}$ in ${{\cal R}_{\\!B}}$ is its push-forward
$T_{Bt*}$ by ${\Theta_{t}}$.

###### Definition 2

Consider “a quantity” that can be described as a vector field by any observer,
quantity written ${\vec{u}}$. At $t$, $A$ describes ${\vec{u}}$ as the vector
field ${{\vec{u}}_{\\!At}}$ in ${{\cal R}_{\\!A}}$, and $B$ describes
${\vec{u}}$ as the vector field ${{\vec{u}}_{\\!Bt}}$ in ${{\cal R}_{\\!B}}$.
Then ${\vec{u}}$ is (covariant) objective if and only if, for all $t$ and all
observers $A$ and $B$, ${{\vec{u}}_{\\!At}}$ is the objective transform of
${{\vec{u}}_{\\!Bt}}$ by ${\Theta_{t}}$, that is,

${{\vec{u}}_{\\!At}}={{\vec{u}}_{\\!Bt*}}.$ (78)

In other words, ${\vec{u}}$ is objective iff, for all $t$ and all observers
$A$ and $B$, whenever ${q_{\\!At}}={\Theta_{t}}({q_{\\!B}})$ we have
${{\vec{u}}_{\\!At}}({q_{\\!At}})=d{\Theta_{t}}({q_{\\!B}}).{{\vec{u}}_{\\!Bt}}({q_{\\!B}})$.
Similarly, a tensor $T$ is objective iff $T_{At}=T_{Bt*}$ for all $t$ and all
observers.

Counter-example: a velocity field ${\vec{v}}$ is not objective, see (77):
${{\vec{w}}_{\\!At}}\neq 0$ in general. So the objective vector fields in the
definition 2 refer, e.g., to the “force fields” ${\vec{f}}$ of Newton’s
fundamental law $\sum{\vec{f}}=m{\vec{\gamma}}$ where
${\vec{\gamma}}={D{\vec{v}}\over Dt}$ (acceleration).

###### Definition 3

To work in an objective covariant framework means to consider any
diffeomorphism ${\Theta_{t}}$ (any translator).

###### Definition 4

To work in an “objective isometric” framework means to only consider observers
$A$ and $B$ using a unique metric $(\cdot,\cdot)_{g}$ (isometric setting),
that is, to only consider the ${\Theta_{t}}$ that are isometries relatively to
one chosen inner product $(\cdot,\cdot)_{g}$ (and
$({\Theta_{t}}.{\vec{u}}_{1},{\Theta_{t}}.{\vec{u}}_{2})_{g}=({\vec{u}}_{1},{\vec{u}}_{2})_{g}$
for any ${\vec{u}}_{1},{\vec{u}}_{2}$). (And in classical mechanics the chosen
metric $(\cdot,\cdot)_{g}$ is often assumed to a Euclidean metric.)

E.g. an English observer using feet and a french observer using meters have no
choice but to use covariant objectivity if they want to communicate (they use
different metrics). And in general relativity “objective isometry” is
meaningless.

##  Appendix I About elasticity and classical formulations

The appendix follows Germain [5], apart from the remarks where ${\rm Tr}(F)$
is replaced by $\det(F)$.

$\bullet$ In ${\mathbb{R}^{n}_{t_{0}}}$. The abbreviated notations $C=F^{T}.F$
(deformation tensor) and $E={C-I\over 2}$ (Green-Lagrange tensor) stand for
the endomorphisms in ${\mathbb{R}^{n}_{t_{0}}}$ relative to ${t_{0}}$, $t$ and
${P}\in{\Omega_{t_{0}}}$ defined by
${C^{t_{0}}_{t}}({P})={F^{t_{0}}_{t}}({P})^{T}.{F^{t_{0}}_{t}}({P})$ and
${E^{t_{0}}_{t}}({P})={{C^{t_{0}}_{t}}({P})-I_{t_{0}}\over 2}$ where
$I_{t_{0}}$ is the identity in ${\mathbb{R}^{n}_{t_{0}}}$. The classical
isotropic homogeneous elasticity is e.g. stated as,

${\underline{\underline{\sigma}}}=\lambda{\rm Tr}(E)\,I+2\mu E,$ (79)

abbreviated notation for the endomorphism
${{\underline{\underline{\sigma}}}^{t_{0}}_{t}}({P})=\lambda{\rm
Tr}({E^{t_{0}}_{t}}({P}))\,I+2\mu{E^{t_{0}}_{t}}({P})$ in
${\mathbb{R}^{n}_{t_{0}}}$.

Small deformations: The linearization $E={C-I\over 2}\simeq{F+F^{T}\over 2}-I$
gives (linear elasticity)

${\underline{\underline{\sigma}}}\simeq\lambda{\rm
Tr}({\underline{\underline{\varepsilon}}})\,I+2\mu{\underline{\underline{\varepsilon}}},\quad{\underline{\underline{\varepsilon}}}={F+F^{T}\over
2}-I.$ (80)

But ${\underline{\underline{\varepsilon}}}$ has no functional meaning because
${F^{t_{0}}_{t}}(P):{\mathbb{R}^{n}_{t_{0}}}\rightarrow{\mathbb{R}^{n}_{t}}$,
${F^{t_{0}}_{t}}(P)^{T}:{\mathbb{R}^{n}_{t}}\rightarrow{\mathbb{R}^{n}_{t_{0}}}$,
and $I=I_{t_{0}}$ stays in ${\mathbb{R}^{n}_{t_{0}}}$, so that the sum
${{F^{t_{0}}_{t}}+{F^{t_{0}}_{t}}^{T}\over 2}-I_{t_{0}}$ is not a function
(and ${\rm Tr}(\varepsilon)$ has no functional meaning either), unless the
shifter is introduced, see remark 1, and/or (80) is considered in the matrix
sense after having chosen a unique Euclidean basis at ${t_{0}}$ and $t$.

Remark relative to the introduction of the virtual power of pressure (11):
Relatively to Euclidean bases, the volume change at ${P}\in{\Omega_{t_{0}}}$
is $J=\det(F)=(\det(C))^{1\over 2}$, where $J$, $F$ and $C$ stands for
${J^{t_{0}}_{t}}({P})$, ${F^{t_{0}}_{t}}({P})$ and ${C^{t_{0}}_{t}}({P})$. And
for small displacements ${\rm Tr}(C-I)\simeq\det(C)-1$: Indeed with an
orthonormal basis of diagonalization of $C$ we have $[C]={\rm
diag}(1{+}\varepsilon_{1},...,1{+}\varepsilon_{n})$, and then ${\rm
Tr}(C)=n+\sum_{i}\varepsilon_{i}$ and
$\det(C)=1+\sum_{i}\varepsilon_{i}+o(\varepsilon)$ in the neighborhood of
$\varepsilon=0$ where $\varepsilon=\max(|\varepsilon_{i}|)$. So ${\rm
Tr}(E)={{\rm Tr}(C)-n\over 2}\simeq{\det(C)-1\over 2}$. Then, for small
displacements, (79) can be replaced by

${\underline{\underline{\sigma}}}={\lambda\over 2}(\det(C)-1)\,I+\mu(C-I).$
(81)

And $\det(C)=\det(F^{T}.F)=\det(F)^{2}$, and
$\det(F)^{2}-1=(\det(F)-I)(\det(F)+I)\simeq 2(\det(F)-I)$ for small
displacements. Then a pressure term can be considered with (11).

$\bullet$ In ${\mathbb{R}^{n}_{t}}$. Instead of the deformation tensor
$C=F^{T}.F$ we may prefer to use the Finger tensor
${\underline{\underline{b}}}=F.F^{T}$, and more precisely its inverse
${\underline{\underline{b}}}^{-1}=H^{T}.H$ where $H=F^{-1}$. Unabbreviated
notation: At $t$ and ${p_{t}}={\Phi^{t_{0}}_{t}}({P})$,
${H^{t_{0}}_{t}}({p_{t}})={F^{t_{0}}_{t}}({P})^{-1}$ and
$({{\underline{\underline{b}}}^{t_{0}}_{t}})^{-1}({p_{t}})={H^{t_{0}}_{t}}({p_{t}})^{T}.{H^{t_{0}}_{t}}({p_{t}})$.
And instead of the Green–Lagrange tensor, we may use the Euler–Almansi tensor
defined by

${\underline{\underline{a}}}={I_{t}-{\underline{\underline{b}}}^{-1}\over
2}={I_{t}-H^{T}.H\over 2},$ (82)

simplified notation of the endomorphism
${{\underline{\underline{a}}}^{t_{0}}_{t}}({p_{t}})={I_{t}-({{\underline{\underline{b}}}^{t_{0}}_{t}})^{-1}({p_{t}})\over
2}$ in ${\mathbb{R}^{n}_{t}}$. The classical elasticity is then e.g. stated
as, with $\lambda,\mu\in\mathbb{R}$,

${\underline{\underline{\sigma}}}=\lambda{\rm
Tr}({\underline{\underline{a}}})\,I+2\mu{\underline{\underline{a}}},$ (83)

to compare with (79). And here
${\underline{\underline{\sigma}}}\mathop{\;\raise-1.55pt\hbox{\large$0$}\mkern-9.5mu\mbox{\kern
1.0pt\vbox to4.30554pt{\hbox{.}\vss\vss\hbox{.}}}\;}d{\vec{v}}$ is meaningful
(double objective contraction in ${\mathbb{R}^{n}_{t}}$). For small
displacements, and for matrix computation, (82) yields
$[{\underline{\underline{a}}}]\simeq[I]-{[H]+[H]^{T}\over 2}$ (linearization),
and (83) reads (linear elasticity)

$[{\underline{\underline{\sigma}}}]=\lambda{\rm
Tr}([{\underline{\underline{a}}}])\,I+2\mu[{\underline{\underline{a}}}]\quad\hbox{with}\quad[{\underline{\underline{a}}}]=[I]-{[H]+[H]^{T}\over
2}].$ (84)

Since $[{F^{t_{0}}}(t,{P})]-[I]=[I]-[{H^{t_{0}}}(t,{p}(t))]+o(t{-}{t_{0}})$ if
${p}(t)={\Phi^{t_{0}}}(t,{P})$, we get back to (80). And, as for (81), instead
of (83) we can consider, for small displacements,

${\underline{\underline{\sigma}}}={\lambda\over
2}(1-\det({\underline{\underline{b}}}^{-1}))\,I+\mu(I-{\underline{\underline{b}}}^{-1}).$
(85)

(Or with $(1-\det(H))$ instead of ${\rm Tr}(I-H)$.) And a pressure term can be
considered with (11).

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