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chapterthird.tex
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\en{\chapter{Nonlinear elastic momentless continuum}}
\ru{\chapter{Нелинейно\hbox{-\hspace{-0.2ex}}упругая безмоментная среда}}
\thispagestyle{empty}
\label{chapter:nonlinearcontinuum}
\en{\section{Continuum and the two approaches to describe it}}
\ru{\section{\ruContinuum{} и два подхода к его описанию}}
\label{section:continuum}
\dropcap{\en{A}\ru{С}}{\en{ccording}\ru{огласно}}
\en{to~}\href{https://en.wikipedia.org/wiki/Atomic_theory}{\en{the~atomic theory}\ru{атомной теории}},
\en{a~substance}\ru{вещество}
\en{is composed}\ru{состоит}
\en{of discrete particles}\ru{из дискретных частиц}\:---
\en{atoms}\ru{атомов}.
\en{Therefore}\ru{Поэтому}
\en{a~model}\ru{модель}
\en{of a~system of~particles}\ru{системы частиц}
\en{with }\ru{с~}\en{masses}\ru{массами}~${m_k}$
\en{and}\ru{и}~\en{location vectors}\ru{векторами положения}~${\currentlocationvector_k (t)}$
\en{may seem suitable}\ru{может показаться подходящей}
\en{yet despite}\ru{даже несмотр\'{я} на}
\en{an~unimaginable}\ru{невообразимое}
\en{number of degrees of freedom}\ru{число степеней свободы},
\en{because}\ru{так как}
\en{amounts of~memory}\ru{объёмы памяти}
\en{and}\ru{и}~\en{the~speed}\ru{быстрота}
\en{of~modern computers}\ru{современных компьютеров}
\en{are characterized}\ru{характеризуются}
\en{also}\ru{тоже}
\en{by }\en{astronomical numbers}\ru{астрономическими числами}.
\en{But anyway}\ru{Но~всё~же},
\en{maybe}\ru{может быть}\ru{,}
\en{it’s worth choosing}\ru{ст\'{о}ит выбрать}
\en{a~}\en{fundamentally}\ru{фундаментально}
\en{and}\ru{и}~\en{qualitatively}\ru{качественно}
\en{different model}\ru{иную модель}\:---
\en{a~model}\ru{модель}
\en{of the~material continuum}\ru{материального \rucontinuum{}а},
\en{or}\ru{или}
\en{the~continuous medium}\ru{сплошной среды},
\en{where}\ru{где}
\en{the~mass}\ru{масса}
\en{is distributed}\ru{распределена}
\en{continuously}\ru{континуально~(непрерывно)}
\en{within a~volume}\ru{в~объёме},
\en{and}\ru{и}
\en{the~finite volume}\ru{конечный объём}~$\mathcal{V}$
\en{contains}\ru{содержит}
\en{the~mass}\ru{массу}
\nopagebreak\vspace{-0.2em}
\begin{equation}\label{themassofthevolume}
m = \hspace{-0.5ex} \scalebox{.9}{$\displaystyle\integral\displaylimits_{\mathcal{V}}$} \hspace{-0.2ex} \rho \hspace{.25ex} d\mathcal{V}
, \hspace{.5em}
dm = \rho \hspace{.2ex} d \mathcal{V} ,
\end{equation}
\vspace{-0.25em}\noindent
\en{here}\ru{здесь}
${\rho\hspace{.1ex}}$\ru{\:---}\en{ is}
\en{the~volume(tric) mass density}\ru{объёмная плотность массы}
\en{and}\ru{и}~${d \mathcal{V}}$\ru{\:---}\en{ is}
\en{the~infinitesimal volume element}\ru{бесконечно\-малый элемент объёма}.
\en{A~real matter}\ru{Реальная материя}
\en{is modeled}\ru{моделируется}
\en{as}\ru{как}
\en{a~continuum}\ru{\rucontinuum},
\en{which can be thought of as}\ru{который может быть мыслим как}
\en{an~infinite set}\ru{бесконечное множество}
\en{of~vanishingly small}\ru{исчезающе м\'{а}лых}
\en{particles}\ru{частиц},
\en{joined together}\ru{соединённых вместе}.
\en{A~space of material points}\ru{Пространство материальных точек}\en{ is}\ru{\:---}
\en{only the first and simple}\ru{лишь первая и~простая}
\en{idea}\ru{идея}
\en{of a~continuous distribution}\ru{непрерывного распределения}
\en{of~mass}\ru{массы}.
\en{More complex models are possible too}\ru{Возможны более сложные модели}, \en{where}\ru{где} \en{particles}\ru{частицы} \en{have}\ru{имеют} \en{more degrees of freedom}\ru{больше степеней свободы}: \en{not only of~translation}\ru{не~только трансляции}, \en{but also}\ru{но также} \en{of~rotation}\ru{поворота}, \en{of~internal deformation}\ru{внутренней деформации}\en{,} \en{and others}\ru{и~другие}.
\en{Knowing that}\ru{Зная, что} \en{such models}\ru{такие модели} \en{are attracting}\ru{привлекают} \en{more and more interest}\ru{всё б\'{о}льший интерес}, \en{in this chapter}\ru{в~этой главе} \en{we will consider}\ru{мы рассмотрим} \en{the classical concept}\ru{классический концепт} \en{of a~continuous medium}\ru{о~сплошной среде} \en{as}\ru{как} \inquotesx{\en{made}\ru{сделанной} \en{of}\ru{из}~\en{simple points}\ru{простых точек}}[.]
\en{At~every moment of~time}\ru{В~каждый момент времени}~$t$\en{,}
\en{a~deformable continuum}\ru{деформируемый \rucontinuum}
\en{occupies}\ru{занимает}
\en{a~certain volume}\ru{некий объём}~${\mathcal{V}\hspace{.1ex}}$
\en{of the space}\ru{пространства}.
\en{This volume}\ru{Этот объём}
\en{moves}\ru{движется}
\en{and}\ru{и}~\en{deforms}\ru{деформируется},
\en{but}\ru{но}
\en{the~set of particles}\ru{набор частиц}
\en{inside this volume}\ru{внутри этого объёма}
\en{is constant}\ru{постоянен}.
\en{It is}\ru{Это}
\en{the~balance of~mass}\ru{баланс массы}
(\inquotes{\en{matter is neither created nor annihilated}\ru{материя не~создаётся и~не~аннигилируется}})
\nopagebreak\vspace{-0.2em}
\begin{equation}\label{thebalanceofmass}
dm \hspace{-0.15ex}
= \hspace{-0.2ex} \rho \hspace{.2ex} d\mathcal{V} \hspace{-0.2ex}
= \hspace{-0.2ex} \rho\hspace{.1ex}' \hspace{-0.2ex} d\mathcal{V}\hspace{.2ex}' \hspace{-0.5ex}
= \hspace{-0.2ex} \mathcircabove{\rho} \hspace{.22ex} d\mathcircabove{\mathcal{V}} ,
\hspace{.6em}
%
m = \hspace{-0.4ex} \scalebox{0.9}{$\displaystyle\integral\displaylimits_{\mathcal{V}}$} \hspace{-0.2ex} \rho \hspace{.2ex} d\mathcal{V} \hspace{-0.1ex}
= \hspace{-0.4ex} \scalebox{0.9}{$\displaystyle\integral\displaylimits_{\mathcal{V}\hspace{.12ex}'}$} \hspace{-0.2ex} \rho\hspace{.1ex}' \hspace{-0.2ex} d\mathcal{V}\hspace{.2ex}' \hspace{-0.4ex}
= \hspace{-0.4ex} \scalebox{0.9}{$\displaystyle\integral\displaylimits_{\mathcircabove{\mathcal{V}}}$} \hspace{-0.2ex} \mathcircabove{\rho} \hspace{.22ex} d\mathcircabove{\mathcal{V}} .
\vspace{-0.5em}\end{equation}
\en{Introducing}\ru{Вводя} \en{some}\ru{какие\hbox{-}либо} \en{variable parameters}\ru{переменные параметры}~${q^{\hspace{.1ex}i}\hspace{-0.3ex}}$\:--- \en{the curvilinear coordinates}\ru{криволинейные координаты}, \en{we have a~relation}\ru{имеем отношение} \en{for locations of particles}\ru{для положений частиц}
\nopagebreak\vspace{-0.2em}\begin{equation}\label{particlelocationbycoordinatesandtime}
\currentlocationvector \hspace{-0.4ex} = \hspace{-0.3ex} \currentlocationvector \hspace{.1ex} (q^{\hspace{.1ex}i} \hspace{-0.33ex}, t)
\hspace{.1ex} .
\end{equation}
...
\begin{otherlanguage}{russian}
% the material (or Lagrangian) description
\emph{\en{Material}\ru{Материальное} \en{description}\ru{описание}}
\textcolor{magenta}{\en{at the~initial moment}\ru{в~начальный момент}, \en{in~the~so\hbox{-}called}\ru{в~так называемой} \en{initial}\ru{начальной}~(\en{original}\ru{оригинальной}, \en{\sout{reference}}\ru{\sout{отсчётной}}, \inquotes{\en{material}\ru{материальной}}) \en{configuration}\ru{конфигурации}}
\en{at some initial moment}\ru{в~какой\hbox{-}то начальный момент} ${t \narroweq 0}$
\inquotes{запоминается} начальная~(\inquotes{материальная}) конфигурация\:---
locations in space of~particles at some arbitrarily chosen \inquotes{initial} moment ${t \narroweq 0}$
\nopagebreak\vspace{-0.1em}\begin{equation*}
\initiallocationvector \hspace{.1ex} (q^{\hspace{.1ex}i}) \hspace{-0.2ex}
\equiv
\locationvector \hspace{.1ex} (q^{\hspace{.1ex}i} \hspace{-0.33ex}, 0)
\end{equation*}
\en{Morphism}\ru{Морфизм}~(\en{function}\ru{функция})~${\initiallocationvector \hspace{-0.4ex} = \hspace{-0.3ex} \initiallocationvector(q^{i})}$
isomorphism (bijective mapping) (invertible one\hbox{-}to\hbox{-}one relation) (взаимно однозначное)
%%\en{material}\ru{материальное}~(\en{or}\ru{или} \inquotes{\en{Lagrangian}\ru{лагранжево}})
Subsequent locations in space of particles are then dependent variables\:--- functions of time and of the~initial~(material, \inquotes{Lagrangian}) \textcolor{magenta}{coordinates}/location~${\initiallocationvector}$
\nopagebreak\vspace{-1em}\begin{equation*}
\currentlocationvector \hspace{-0.4ex} = \hspace{-0.3ex} \currentlocationvector \hspace{.1ex} (\initiallocationvector, t)
\hspace{.1ex} .
\end{equation*}
\noindent
Для пространственного дифференцирования (постоянных во~времени) отношений \en{like}\ru{типа} ${\varphi \hspace{-0.4ex} = \hspace{-0.4ex} \varphi(\initiallocationvector)}$
вводится локальный касательный базис~${\initiallocationvector_\differentialindex{i}}$ и~взаимный базис~${\smash{\initiallocationvector^i}}$
\nopagebreak\vspace{-0.2em}\begin{equation*}
\initiallocationvector_\differentialindex{i} \hspace{-0.2ex} \equiv \partial_i \initiallocationvector
\hspace{.3em} \bigl( \partial_i \hspace{-0.1ex} \equiv \hspace{-0.1ex} \scalebox{0.8}{$ \displaystyle\frac{\raisemath{-0.2em}{\partial}}{\partial q^i} $} \hspace{.1ex} \bigr) \hspace{-0.1ex}
, \hspace{.5em}
\initiallocationvector^i \hspace{-0.3ex} \dotp \initiallocationvector_\differentialindex{\hspace{-0.1ex}j} \hspace{-0.3ex} = \hspace{-0.1ex} \delta^{i}_{\hspace{-0.2ex}j}
\end{equation*}
\inquotes{материальный} оператор Hamilton’а~${\hspace{-0.25ex}\smash{\boldnablacircled}}$
\nopagebreak\vspace{-0.2em}\begin{equation}
\UnitDyad = \initiallocationvector^i \initiallocationvector_\differentialindex{i} \hspace{-0.1ex} = \initiallocationvector^i \partial_i \initiallocationvector = \hspace{-0.25ex} \boldnablacircled \initiallocationvector
, \hspace{.5em}
\boldnablacircled \equiv \hspace{.1ex} \initiallocationvector^i \partial_i \hspace{.1ex} ,
\end{equation}
\vspace{-0.2em}\noindent
тогда ${d\varphi = d\initiallocationvector \dotp \hspace{-0.2ex} \boldnablacircled \varphi}$.
...
\end{otherlanguage}
% the spatial (or Eulerian) description
% introduced by d’Alembert
\en{But}\ru{Но} \ru{может быть эффективен }\en{yet another approach}\ru{ещё иной подход}\en{ may be effective}\:---
\emph{\en{the~spatial}\ru{пространственное}~(\en{or}\ru{или} \inquotes{\hbox{\en{Eulerian}\ru{эйлерово}}}) \en{description}\ru{опис\'{а}ние}},
\en{when}\ru{когда}
\en{instead of focusing on}\ru{вместо фокусирования на~том,} \en{how particles}\ru{как частицы} \en{of~a~continuum}\ru{\rucontinuum{}а} \en{move}\ru{движутся} \en{from the~initial configuration}\ru{из начальной конфигурации} \en{through space and~time}\ru{в~пространстве и~времени},
\en{processes}\ru{процессы} \en{are~considered}\ru{рассматриваются} \en{at fixed points in~space}\ru{в~неподвижных точках пространства} \en{as~time progresses}\ru{с~течением времени}.
\en{With relations}\ru{C~отношениями} \en{like}\ru{типа} ${\rho \hspace{-0.3ex} = \hspace{-0.44ex} \rho \hspace{.16ex} (\hspace{-0.1ex}\currentlocationvector, t)}$,
\en{we track}\ru{мы следим} \en{what’s happening}\ru{за про\-ис\-хо\-дя\-щим} \en{exactly in this place}\ru{именно в~этом месте}.
\en{Various particles}\ru{Разные частицы}, \en{continuously}\ru{непрерывно} \en{leaving and coming here}\ru{уходящие и~приходящие сюда}, \en{do not confuse us}\ru{не смущают нас}.
...
\en{the~balance of~mass}\ru{баланс массы} \en{in~spatial description}\ru{в~пространственном описании}
(\en{the~continuity equation}\ru{уравнение непрерывности~(сплошности, неразрывности)} \en{for}\ru{для} \en{mass}\ru{массы})
...
Jaumann derivative (\inquotes{\textcolor{magenta}{corotational} time derivative}) was first introduced by \href{https://en.wikipedia.org/wiki/Gustav_Jaumann}{Gustav Jaumann}\footnote{%
\bookauthor{Gustav Jaumann}. \href{http://www.physikdidaktik.uni-karlsruhe.de/download/jaumann_1911.pdf}{Geschlossenes System phy\-si\-ka\-li\-scher und che\-mi\-scher Differential\-gesetze (I.\;Mit\-teilung)~//~Sitzungs\-berichte der~Kaiser\-lichen Akademie der~Wissen\-schaften in~Wien, Mathematisch\hbox{-}natur\-wissen\-schaftliche Klasse, Abteilung~IIa, Band~CXX, 1911. Seiten 385\hbox{--}530.}}
\nopagebreak\begin{tcolorbox}[breakable, enhanced, colback = orange!8, before upper={\parindent3.2ex}, parbox = false]
\small%
\setlength{\abovedisplayskip}{2pt}\setlength{\belowdisplayskip}{2pt}%
Es sei ${\frac{\partial}{\partial t}}$ der~Operator der~\loosetexttr[80]{lokalen Fluxion}, d.\:i. der~partiellen Fluxion in einem gegen das~Koordinatensystem ruhenden Punkte des~Raumes. Ferner sei ${\frac{d}{dt}}$ der~Operator der~\loosetexttr[80]{totalen Fluxion}, welcher definiert wird durch
\nopagebreak\vspace{.1em}\begin{equation*}
\begin{array}{r@{\hspace{.66ex}}c@{\hspace{.75ex}}l}
\scalebox{0.92}[0.92]{$\displaystyle\frac{\raisemath{-0.125em}{da}}{dt}$} & = & \scalebox{0.92}[0.92]{$\displaystyle\frac{\raisemath{-0.125em}{\partial a}}{\partial t}$} + \bm{v} \dotp \hspace{-0.2ex} \boldnabla a
\hspace{.1ex} ,
\\[.8em]
\scalebox{0.92}[0.92]{$\displaystyle\frac{\raisemath{-0.125em}{d \bm{a}}}{dt}$} & \stackrel{3}{=} & \scalebox{0.92}[0.92]{$\displaystyle\frac{\raisemath{-0.125em}{\partial \bm{a}}}{\partial t}$} + \bm{v} \dotp \hspace{-0.2ex} \boldnabla\bm{;} \hspace{-0.26ex} \bm{a} - \hspace{.1ex} \smalldisplaystyleonehalf \bigl( \hspace{.1ex} \operatorname{rot}\hspace{.1ex} \bm{v} \hspace{.1ex} \bigr) \hspace{-0.4ex} \times \hspace{-0.2ex} \bm{a}
\hspace{.1ex} ,
\\[.8em]
\scalebox{0.92}[0.92]{$\displaystyle\frac{\raisemath{-0.125em}{d \mathboldalpha}}{dt}$} & \stackrel{9}{=} & \scalebox{0.92}[0.92]{$\displaystyle\frac{\raisemath{-0.125em}{\partial \mathboldalpha}}{\partial t}$} + \bm{v} \dotp \hspace{-0.2ex} \boldnabla\bm{;} \hspace{-0.28ex} \mathboldalpha - \hspace{.1ex} \smalldisplaystyleonehalf \bigl( \hspace{.1ex} \operatorname{rot}\hspace{.1ex} \bm{v} \hspace{-0.2ex} \times \hspace{-0.2ex} \mathboldalpha - \mathboldalpha \hspace{-0.2ex} \times \hspace{-0.2ex} \operatorname{rot}\hspace{.1ex} \bm{v} \hspace{.1ex} \bigr)
\hspace{-0.1ex} .
\end{array}
\end{equation*}
Endlich verwenden wir die~\loosetexttr[80]{körperliche Fluxion} eines Skalars:
\nopagebreak\vspace{-0.1em}\begin{equation*}
\scalebox{0.92}[0.92]{$\displaystyle\frac{\raisemath{-0.125em}{\delta}}{\delta t}$} \hspace{.15ex} a
= \scalebox{0.92}[0.92]{$\displaystyle\frac{\raisemath{-0.125em}{\partial}}{\partial t}$} \hspace{.15ex} a + \operatorname{div}\hspace{.1ex} a \bm{v}
= \scalebox{0.92}[0.92]{$\displaystyle\frac{\raisemath{-0.125em}{d}}{dt}$} \hspace{.15ex} a + a \operatorname{div}\hspace{.1ex} \bm{v}
\hspace{.1ex} .
\end{equation*}
\par
\end{tcolorbox}
\begin{otherlanguage}{russian}
körperliche\:--- bodily/телесная, material/вещественная(материальная), physical/физическая
${
\boldnabla \hspace{-0.1ex} \dotp \hspace{-0.15ex} \bigl( a \bm{v} \bigr) \hspace{-0.2ex} = \hspace{.1ex} a \hspace{.2ex} \boldnabla \hspace{-0.2ex} \dotp \hspace{-0.1ex} \bm{v} + \bm{v} \dotp \hspace{-0.2ex} \boldnabla a
}$
...
Пусть~${\upupsilon(\initiallocationvector, t)}$\:--- какое\hbox{-}либо поле \textcolor{magenta}{(?? только в~материальном описании от~${\initiallocationvector}$ ??)}.
Найдём скорость изменения интеграла по~объёму
\end{otherlanguage}
\nopagebreak\vspace{-0.25em}\begin{equation*}
\Upsilon
\equiv \hspace{-0.4ex} \scalebox{0.9}{$\displaystyle\integral\displaylimits_{\mathcal{V}}$} \hspace{-0.1ex} \rho \hspace{.19ex} \upupsilon \hspace{.11ex} d\mathcal{V}
\end{equation*}
\vspace{-0.4em}\noindent
(\inquotes{$\upupsilon$ \en{is}\ru{есть} $\Upsilon$ \en{per mass unit}\ru{на~единицу массы}}).
\en{Seemingly difficult}\ru{Кажущееся сложным} \en{calculation of}\ru{вычисление}~${\mathdotabove{\Upsilon}}$ (\en{since}\ru{ведь} $\mathcal{V}$ \en{is deforming}\ru{деформируется}) \en{is actually quite simple}\ru{на~с\'{а}мом деле весьма простое} \en{together}\ru{вместе} \en{with}\ru{с}~\eqref{thebalanceofmass}:
\nopagebreak\vspace{-0.1em}\begin{equation}\label{rateofvolumeintegralchange}
\Upsilon = \hspace{-0.4ex} \scalebox{0.9}{$\displaystyle\integral\displaylimits_{\mathcircabove{\mathcal{V}}}$} \hspace{-0.1ex} \mathcircabove{\rho} \hspace{.19ex} \upupsilon \hspace{.11ex} d\mathcircabove{\mathcal{V}}
\;\Rightarrow\;
\mathdotabove{\Upsilon}
= \hspace{-0.4ex} \scalebox{0.9}{$\displaystyle\integral\displaylimits_{\mathcircabove{\mathcal{V}}}$} \hspace{-0.1ex} \mathcircabove{\rho} \hspace{.19ex} \mathdotabove{\upupsilon} \hspace{.11ex} d\mathcircabove{\mathcal{V}}
= \hspace{-0.4ex} \scalebox{0.9}{$\displaystyle\integral\displaylimits_{\mathcal{V}}$} \hspace{-0.1ex} \rho \hspace{.19ex} \mathdotabove{\upupsilon} \hspace{.11ex} d\mathcal{V} .
\end{equation}
\nopagebreak\begin{equation*}
\Psi = \hspace{-0.25ex}\scalebox{1.4}{$\integral$}_{\hspace{-0.5ex}\raisemath{-0.05em}{\mathcal{V}}} \hspace{.3ex} \rho \hspace{.2ex} \psi \hspace{.2ex} d\mathcal{V} \hspace{-0.1ex}
= \hspace{-0.2ex} \scalebox{1.4}{$\integral$}_{\hspace{-0.5ex}\raisemath{-0.05em}{\mathcal{V}\hspace{.12ex}'}} \hspace{.2ex} \rho \hspace{.1ex}' \psi \hspace{.2ex} d\mathcal{V}\hspace{.2ex}'
\hspace{.2ex} \Rightarrow \hspace{.33ex}
\mathdotabove{\Psi} = \hspace{-0.25ex}\scalebox{1.4}{$\integral$}_{\hspace{-0.5ex}\raisemath{-0.05em}{\mathcal{V}}} \hspace{.3ex} \rho \hspace{.2ex} \mathdotabove{\psi} \hspace{.2ex} d\mathcal{V} \hspace{-0.1ex}
= \hspace{-0.2ex} \scalebox{1.4}{$\integral$}_{\hspace{-0.5ex}\raisemath{-0.05em}{\mathcal{V}\hspace{.12ex}'}} \hspace{.2ex} \rho \hspace{.1ex}' \mathdotabove{\psi} \hspace{.2ex} d\mathcal{V}\hspace{.2ex}'
%%\hspace{-0.5ex} ,
\end{equation*}
...
\en{It is not worth it}\ru{Не~ст\'{о}ит} \en{to contrapose}\ru{противопоставлять} \en{the material}\ru{материальное} \en{and}\ru{и}~\en{the spatial}\ru{пространственное} \en{descriptions}\ru{описания}.
\en{In this book}\ru{В~этой книге} \en{both are used}\ru{используются оба}, \en{depending on the situation}\ru{в~зависимости от~ситуации}.
\newpage
\en{\section{Motion gradient}}
\ru{\section{Градиент движения}}
\label{section:motiongradient}
\en{Having}\ru{Имея}
\en{the motion function}\ru{функцию движения}~%
${\currentlocationvector \narroweq \hspace{-0.1ex} \currentlocationvector(q^{\hspace{.1ex}i} \hspace{-0.3ex}, t)}$,
${\initiallocationvector(q^{\hspace{.1ex}i}) \equiv \currentlocationvector(q^{\hspace{.1ex}i} \hspace{-0.3ex}, 0)}$,
\en{the }\ru{операторы }\inquotes{\en{nabla}\ru{набла}}\en{ operators}
${\boldnabla \equiv \currentlocationvector^{i} \partial_i}$,
${\boldnablacircled \equiv \initiallocationvector^{i} \partial_i}$
\en{and}\ru{и}
\en{looking}\ru{гл\'{я}дя}
\en{at differential relations}\ru{на дифференциальные отношения}
\en{for}\ru{для}
\en{a~certain}\ru{какого\hbox{-}либо}
\en{infinitesimal vector}\ru{бесконечномалого вектора}
\en{in two configurations}\ru{в~двух конфигурациях},
\en{the~current}\ru{текущей}
\en{with}\ru{с}~${d\currentlocationvector}$
\en{and}\ru{и}~\en{the~initial}\ru{начальной}
\en{with}\ru{с}~${d\initiallocationvector}$
\nopagebreak\en{\vspace{1.3em}}\ru{\vspace{2.1em}}
\begin{equation}
\begin{array}{c}
d\currentlocationvector = d\initiallocationvector \dotp \hspace{-0.2ex} \tikzmark{beginFtransposed} \boldnablacircled \currentlocationvector \tikzmark{endFtransposed} = \hspace{-0.2ex} \tikzmark{beginMotionGradient} \boldnablacircled \currentlocationvector^{\T} \tikzmark{endMotionGradient} \hspace{-0.44ex} \dotp d\initiallocationvector
\\[.2em]
%
d\initiallocationvector = d\currentlocationvector \dotp \hspace{-0.2ex} \tikzmark{beginFtransposedinverse} \boldnabla \initiallocationvector \tikzmark{endFtransposedinverse} = \hspace{-0.2ex} \tikzmark{beginFinverse} \boldnabla \initiallocationvector^{\T} \hspace{-0.4ex} \tikzmark{endFinverse} \dotp d\currentlocationvector
\end{array}
\end{equation}%
\AddOverBrace[line width=.75pt][0,0.6ex][yshift=.1em]%
{beginFtransposed}{endFtransposed}{${\begin{array}{c}
\hspace{.12em} \scalebox{0.85}{$\bm{F}^{\hspace{.1ex}\T}$}
\\[-0.33em]
\scriptstyle \initiallocationvector^{i} \hspace{-0.2ex} \currentlocationvector_\differentialindex{i}
\\[-0.36em]
\end{array}}$}
\AddOverBrace[line width=.75pt][0,0.6ex][yshift=.1em]%
{beginMotionGradient}{endMotionGradient}{${\begin{array}{c}
\scalebox{0.85}{$\bm{F}$}
\\[-0.33em]
\scriptstyle \currentlocationvector_\differentialindex{i} \initiallocationvector^{i}
\\[-0.36em]
\end{array}}$}
\AddUnderBrace[line width=.75pt][0,0.2ex][yshift=.2em]%
{beginFtransposedinverse}{endFtransposedinverse}{${\begin{array}{c}
\scriptstyle \currentlocationvector^{i} \initiallocationvector_\differentialindex{i}
\\[-0.1em]
\scalebox{0.85}{$\bm{F}^{\hspace{.1ex}\expminusT}$}
\end{array}}$}
\AddUnderBrace[line width=.75pt][0,0.2ex][yshift=.2em]%
{beginFinverse}{endFinverse}{${\begin{array}{c}
\scriptstyle \initiallocationvector_\differentialindex{i} \currentlocationvector^{i}
\\[-0.1em]
\scalebox{0.85}{$\bm{F}^{\hspace{.1ex}\expminusone}$}
\end{array}}$}
\vspace{1.5em}
\noindent
\en{here comes to~mind}\ru{приходит на~ум} \en{to introduce}\ru{ввести} \en{the }\inquotes{\en{motion gradient}\ru{градиент движения}}\footnote{%
\en{Tensor}\ru{Тензору}~$\bm{F}$ \en{doesn’t well suit}\ru{не~вполне подходит} \en{its}\ru{его} \en{more popular}\ru{более популярное} \en{name}\ru{название} \inquotesx{\en{deformation gradient}\ru{градиент деформации}}[,] \en{because}\ru{поскольку} \en{this tensor}\ru{этот тензор} \en{describes}\ru{описывает} \en{not only the~deformation itself}\ru{не~только сам\'{у} деформацию}, \en{but also the rotation of a~body as a~whole without de\-for\-ma\-tion}\ru{но~и поворот тела как~целого без деформации}.
}%
\hbox{\hspace{-0.6ex},} \en{picking}\ru{взяв} one of these tensor multipliers for it:
${\bm{F} \equiv \hspace{-0.2ex} \boldnablacircled \currentlocationvector^{\T} \hspace{-0.33ex} = \currentlocationvector_\differentialindex{i} \initiallocationvector^{i} \hspace{-0.4ex}}$.
\en{Why this one}\ru{Почему именно этот}?
\en{The~reason}\ru{Причина}
\en{to choose}\ru{выбрать}~${\hspace{-0.25ex}\boldnablacircled \currentlocationvector^{\T}\hspace{-0.2ex}}$\ru{\:---}\en{ is}
\en{another expression}\ru{другое выражение}
\en{for the~differential}\ru{для дифференциала}
\begin{gather*}
\begin{array}{c@{\hspace{2em}}c}
d\currentlocationvector = \scalebox{0.9}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial \currentlocationvector}}{\partial \initiallocationvector} $} \dotp d\initiallocationvector
&
\bm{F} \hspace{-0.1ex} = \scalebox{0.9}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial \currentlocationvector}}{\partial \initiallocationvector} $}
\end{array}
\\
%
\begin{array}{c@{\hspace{2em}}c}
d\initiallocationvector = \scalebox{0.9}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial \initiallocationvector}}{\partial \currentlocationvector} $} \dotp d\currentlocationvector
&
\bm{F}^{\expminusone} \hspace{-0.3ex} = \scalebox{0.9}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial \initiallocationvector}}{\partial \currentlocationvector} $}
\end{array}
\end{gather*}
\begin{equation*}
\scalebox{.9}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial \bm{\zeta}}}{\partial \initiallocationvector} $} = \partial_i \bm{\zeta} \hspace{.1ex} \initiallocationvector^i
\hspace{2em}
\scalebox{.9}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial \bm{\zeta}}}{\partial \currentlocationvector} $} = \partial_i \bm{\zeta} \currentlocationvector^{i}
\end{equation*}
....
\nopagebreak\vspace{-0.2em}\begin{equation*}
\UnitDyad
= \tikzmark{unitTensorAsOriginalDerivativeBegin} \hspace{-0.25ex} \boldnablacircled \initiallocationvector \tikzmark{unitTensorAsOriginalDerivativeEnd}
= \tikzmark{unitTensorAsCurrentDerivativeBegin} \hspace{-0.25ex} \boldnabla \currentlocationvector \tikzmark{unitTensorAsCurrentDerivativeEnd}
\end{equation*}%
\AddUnderBrace[line width=.75pt][0.1ex,0.2ex]%
{unitTensorAsOriginalDerivativeBegin}{unitTensorAsOriginalDerivativeEnd}%
{${ \scalebox{0.8}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial \initiallocationvector}}{\partial \initiallocationvector} $} }$}%
\AddUnderBrace[line width=.75pt][0.1ex,0.2ex]%
{unitTensorAsCurrentDerivativeBegin}{unitTensorAsCurrentDerivativeEnd}%
{${ \scalebox{0.8}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial \currentlocationvector}}{\partial \currentlocationvector} $} }$}
...
\en{For cartesian coordinates}\ru{Для декартовых координат} \en{with orthonormal basis}\ru{с~ортонормальным базисом} ${\bm{e}_i \hspace{-0.16ex} = \boldconstant}$
\nopagebreak\vspace{-0.2em}\begin{equation*}
\currentlocationvector = \hspace{-0.15ex} x_{i}(t) \hspace{.2ex} \bm{e}_i
\hspace{.1ex} , \:\;
\initiallocationvector = \hspace{-0.15ex} x_{i}(0) \hspace{.2ex} \bm{e}_i \hspace{-0.16ex} = \mathcircabove{x}_i \hspace{.1ex} \bm{e}_i
\hspace{.1ex} , \:\:
\mathcircabove{x}_i \hspace{-0.15ex} \equiv x_{i}(0)
\hspace{.1ex} ,
\end{equation*}
\nopagebreak\vspace{-0.25em}\begin{equation*}
\boldnablacircled \hspace{-0.1ex}
= \bm{e}_i \hspace{.2ex} \scalebox{0.9}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial}}{\partial \mathcircabove{x}_i} $} \hspace{-0.1ex}
= \bm{e}_i \hspace{.15ex} \mathcircabove{\partial}_i
\hspace{.1ex} , \:\:
%
\boldnabla \hspace{-0.1ex}
= \bm{e}_i \hspace{.2ex} \scalebox{0.9}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial}}{\partial x_i} $} \hspace{-0.1ex}
= \bm{e}_i \hspace{.15ex} \partial_i
\hspace{.1ex} ,
\end{equation*}
\nopagebreak\vspace{-0.4em}\begin{equation*}
\begin{array}{c}
\boldnablacircled \currentlocationvector
= \bm{e}_i \hspace{.2ex} \scalebox{.9}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial \currentlocationvector}}{\partial \mathcircabove{x}_i} $} \hspace{-0.1ex}
= \bm{e}_i \hspace{.2ex} \scalebox{.9}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial \hspace{.1ex} ( \hspace{-0.1ex} x_{\hspace{-0.1ex}j} \hspace{.2ex} \bm{e}_j )}}{\partial \mathcircabove{x}_i} $}
= \hspace{.2ex} \scalebox{.9}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial x_{\hspace{-0.1ex}j}}}{\partial \mathcircabove{x}_i} $} \hspace{.25ex} \bm{e}_i \bm{e}_{\hspace{-0.1ex}j} \hspace{-0.2ex}
= \mathcircabove{\partial}_i \hspace{.1ex} x_{\hspace{-0.1ex}j} \hspace{.1ex} \bm{e}_i \bm{e}_{\hspace{-0.1ex}j}
\hspace{.1ex} ,
\\[.66em]
%
\boldnabla \initiallocationvector
= \bm{e}_i \hspace{.2ex} \scalebox{.9}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial \hspace{.1ex} \initiallocationvector}}{\partial x_{i}} $} \hspace{-0.1ex}
= \hspace{.2ex} \scalebox{.9}{$ \displaystyle \frac{\raisemath{-0.2em}{\partial \mathcircabove{x}_{\hspace{-0.1ex}j}}}{\partial x_{i}} $} \hspace{.25ex} \bm{e}_i \bm{e}_{\hspace{-0.1ex}j} \hspace{-0.2ex}
= \partial_i \hspace{.1ex} \mathcircabove{x}_{\hspace{-0.1ex}j} \hspace{.1ex} \bm{e}_i \bm{e}_{\hspace{-0.1ex}j}
\end{array}
\end{equation*}
...
\en{By the polar decomposition theorem}\ru{По теореме о~полярном разложении}~(\chapterdotsectionref{chapter:mathapparatus}{section:polardecomposition}), \en{the motion gradient}\ru{градиент движения} \en{decomposes into}\ru{разлож\'{и}м на} \en{the rotation tensor}\ru{тензор поворота}~$\rotationtensor$ \en{and}\ru{и}~\en{the symmetric}\ru{симметричные} \en{positive}\ru{положительные} \en{stretch tensors}\ru{тензоры искажений}~${\bm{U}\hspace{-0.25ex}}$ \en{and}\ru{и}~${\bm{V}\hspace{-0.1ex}}$:
\nopagebreak\vspace{-0.1em}\begin{equation*}
\bm{F} \hspace{-0.1ex} = \rotationtensor \dotp \hspace{.25ex} \bm{U} \hspace{-0.2ex} = \bm{V} \hspace{-0.3ex} \dotp \hspace{.15ex} \rotationtensor
\end{equation*}
...
\en{When}\ru{Когда} \en{there's no rotation}\ru{поворота нет}~(${\rotationtensor = \hspace{-0.1ex} \UnitDyad \hspace{.1ex}}$), \en{then}\ru{тогда} ${\bm{F} \hspace{-0.1ex} = \hspace{.1ex} \bm{U} \hspace{-0.25ex} = \bm{V}\hspace{-0.3ex}}$.
...
\en{\section{Measures (tensors) of deformation}}
\ru{\section{Меры (тензоры) деформации}}
\label{section:deformationtensors}
\en{And this}\ru{А это}\en{ is}\ru{\:---} \en{where}\ru{где} \ru{возникает }\en{the extra complexity}\ru{сверх сложность}\en{ arose}.
Although, the multivariance is often seen as a big gift.
\en{Motion gradient}\ru{Градиент движения}~$\bm{F}$ \en{characterizes}\ru{характеризует}
\en{both the~deformation of a~body}\ru{и~деформацию тела,}
\en{and the~rotation of a~body as a~whole}\ru{и~поворот тела как~целого}.
%
\en{The deformation-only tensors}\ru{Тензорами лишь-деформации}
\en{are}\ru{являются}
\en{the stretch tensors}\ru{тензоры искажений}~${\bm{U}\hspace{-0.25ex}}$
\en{and}\ru{и}~${\bm{V}\hspace{-0.1ex}}$
\en{from the polar decomposition}\ru{из полярного разложения}
${\bm{F} \hspace{-0.1ex} = \rotationtensor \dotp \hspace{.25ex} \bm{U} \hspace{-0.2ex} = \bm{V} \hspace{-0.3ex} \dotp \hspace{.15ex} \rotationtensor}$,
\en{as well as}\ru{так~же как и}~\en{another tensors}\ru{другие тензоры},
\en{originating}\ru{происходящие}
\en{from}\ru{от}~${\bm{U}\hspace{-0.25ex}}$
\ru{или\,(и)}\en{or\,(and)}~${\bm{V}\hspace{-0.3ex}}$.
\en{The widely used ones are}\ru{Широко используются}
\en{the~}\inquotes{\en{squares}\ru{квадраты}}
\en{of~}${\bm{U}\hspace{-0.25ex}}$
\en{and}\ru{и}~${\bm{V}\hspace{-0.1ex}}$
\nopagebreak\vspace{-0.1em}\begin{equation}\label{deformationtensors.nonlinear}
\begin{array}{c}
\hspace*{-2.33em} \bigl( \hspace{.1ex} \bm{U}^{\hspace{.1ex}2} \hspace{-0.3ex} = \hspace{.22ex} \bigr) \hspace{.8ex}
\bm{U} \hspace{-0.3ex} \dotp \hspace{.1ex} \bm{U} \hspace{-0.2ex}
= \bm{F}^{\hspace{.1ex}\T} \hspace{-0.4ex} \dotp \bm{F}
\equiv \bm{G}
\hspace{.1ex} ,
\\[.1em]
%
\hspace*{-2.33em} \bigl( \hspace{.1ex} \bm{V}^{\hspace{.04ex}2} \hspace{-0.3ex} = \hspace{.22ex} \bigr) \hspace{.8ex}
\bm{V} \hspace{-0.3ex} \dotp \bm{V} \hspace{-0.25ex}
= \bm{F} \dotp \bm{F}^{\hspace{.1ex}\T} \hspace{-0.36ex}
\equiv \mathboldPhi
\hspace{.1ex} .
\end{array}
\end{equation}
\vspace{-0.25em}\noindent
\en{These are}\ru{Это}
\en{the }\ru{тензор деформации }Green’\en{s}\ru{а}\en{ deformation tensor}
(\en{or}\ru{или}
\en{the right}\ru{правый}
\ru{тензор }Cauchy--Green\ru{’а}\en{ tensor})~$\bm{G}$
\en{and}\ru{и}
\en{the }\ru{тензор деформации }Finger’\en{s}\ru{а}\en{ deformation tensor}
(\en{or}\ru{или}
\en{the left}\ru{левый}
\ru{тензор }Cauchy--Green\ru{’а}\en{ tensor})~$\mathboldPhi$.
%
\en{They have}\ru{У~них есть}
\en{the~convenient link}\ru{удобная связь}
\en{with}\ru{с}~\en{the motion gradient}\ru{градиентом движения}~$\bm{F}$,
\en{without}\ru{без}
\en{calculating square roots}\ru{вычисления квадратных корней}
(\en{as}\ru{как}
\en{it’s needed}\ru{это нужно}
\en{for}\ru{для}~${\bm{U}\hspace{-0.25ex}}$
\en{and}\ru{и}~${\bm{V}\hspace{-0.25ex}}$%
).
%
\en{That’s}\ru{Таков\'{а}}
\en{the big reason}\ru{больш\'{а}я причина}\ru{,}
\en{why}\ru{почему}
\en{tensors}\ru{тензоры}~$\bm{G}$
\en{and}\ru{и}~$\mathboldPhi$
\en{are so widely used}\ru{так широк\'{о} используются}.
\en{Tensor}\ru{Тензор}~$\bm{G}$
\en{was first used}\ru{впервые использовал}
\en{by }George Green\hspace{-0.1ex}%
\footnote{%
\href{https://en.wikipedia.org/wiki/George_Green_(mathematician)}{\bookauthor{Green, George}}.
\href{https://hdl.handle.net/2027/mdp.39015027059651?urlappend=\%3Bseq=133}{(1839) On the~propagation of~light in crystallized media. \emph{Transactions of the~Cambridge Philosophical Society.} 1842, vol.\:7, part~II, pages 121\hbox{--}140.}
}\hspace{-0.5ex}.
\en{An~inversion}\ru{Обращение}
\en{of~}$\mathboldPhi$
\en{and}\ru{и}~$\bm{G}$
\en{gives}\ru{даёт} \en{the two more}\ru{ещё два} \en{deformation tensors}\ru{тензора деформации}
\nopagebreak\vspace{-0.2em}\begin{equation}\label{moredeformationtensors.nonlinear}
\begin{array}{c}
\bm{V}^{\expminustwo} \hspace{-0.25ex}
= \mathboldPhi^{\expminusone} \hspace{-0.2ex}
= \hspace{-0.1ex} \left( \bm{F} \dotp \bm{F}^{\hspace{.1ex}\T} \hspace{.1ex} \right)^{\hspace{-0.33ex}\expminusone} \hspace{-0.4ex}
= \bm{F}^{\expminusT} \hspace{-0.3ex} \dotp \bm{F}^{\expminusone} \hspace{-0.25ex}
\equiv {^2\hspace{-0.2ex}\bm{c}}
\hspace{.2ex} ,
\\
%
\bm{U}^{\expminustwo} \hspace{-0.25ex}
= \bm{G}^{\hspace{.12ex}\expminusone} \hspace{-0.2ex}
= \hspace{-0.1ex} \left( \bm{F}^{\hspace{.1ex}\T} \hspace{-0.4ex} \dotp \bm{F} \hspace{.2ex} \right)^{\hspace{-0.33ex}\expminusone} \hspace{-0.4ex}
= \bm{F}^{\expminusone} \hspace{-0.3ex} \dotp \bm{F}^{\expminusT} \hspace{-0.3ex}
\equiv {^2\hspace{-0.4ex}\bm{f}}
\hspace{-0.1ex} ,
\end{array}
\end{equation}
\vspace{-0.2em}\noindent
\en{each of~which}\ru{каждый из~которых} \en{is sometimes called}\ru{иногда называется} \ru{тензором }\en{the~}\href{https://en.wikipedia.org/wiki/Gabrio_Piola}{Piola}\en{ tensor} \en{or}\ru{или} \ru{тензором }\en{the~}\href{https://en.wikipedia.org/wiki/Josef_Finger}{Finger}\ru{’а}\en{ tensor}.
\en{The~inverse}\ru{Обратный} \en{of the left}\ru{к~левому} \ru{тензору }Cauchy--Green\ru{’а}\en{ tensor}~${\hspace{-0.1ex}\mathboldPhi\hspace{.1ex}}$\en{ is}\ru{\:---} \ru{тензор деформации }\en{the }Cauchy\en{ deformation tensor}~${\hspace{-0.2ex}{^2}\hspace{-0.2ex}\bm{c}}$.
\en{The components}\ru{Компоненты}
\en{of these tensors}\ru{этих тензоров}\en{ are}
\nopagebreak\vspace{-0.1em}
\begin{equation*}\label{componentsofdeformationtensors}
\begin{array}{r@{\hspace{.5em}}l}
\bm{G} = \initiallocationvector^i \hspace{-0.25ex} \currentlocationvector_\differentialindex{i} \hspace{-0.1ex} \dotp \currentlocationvector_\differentialindex{\hspace{-0.1ex}j} \initiallocationvector^j \hspace{-0.25ex}
= G_{\hspace{-0.15ex}i\hspace{-0.1ex}j} \hspace{.1ex} \initiallocationvector^i \initiallocationvector^j
\hspace{-0.3ex} , &
G_{\hspace{-0.15ex}i\hspace{-0.1ex}j} \hspace{-0.2ex} \equiv
\currentlocationvector_\differentialindex{i} \hspace{-0.1ex} \dotp \currentlocationvector_\differentialindex{\hspace{-0.1ex}j}
\hspace{.1ex} ,
\\[.25em]
%
{{^2}\hspace{-0.4ex}\bm{f}} \hspace{-0.2ex} = \initiallocationvector_\differentialindex{i} \currentlocationvector^{i} \hspace{-0.2ex} \dotp \hspace{-0.15ex} \currentlocationvector^j \hspace{-0.1ex} \initiallocationvector_\differentialindex{\hspace{-0.1ex}j} \hspace{-0.2ex}
= G^{\hspace{.1ex}i\hspace{-0.1ex}j} \initiallocationvector_\differentialindex{i} \initiallocationvector_\differentialindex{\hspace{-0.1ex}j} \hspace{-0.2ex}
\hspace{.2ex} , &
G^{\hspace{.1ex}i\hspace{-0.1ex}j} \hspace{-0.2ex} \equiv
\currentlocationvector^{i} \hspace{-0.2ex} \dotp \hspace{-0.15ex} \currentlocationvector^j
\hspace{-0.3ex} ,
\\[.25em]
%
{{^2}\hspace{-0.2ex}\bm{c}} = \hspace{-0.1ex} \currentlocationvector^{i} \initiallocationvector_\differentialindex{i} \hspace{-0.1ex} \dotp \hspace{.1ex} \initiallocationvector_\differentialindex{\hspace{-0.1ex}j} \currentlocationvector^{j} \hspace{-0.25ex}
= \textsl{g}_{i\hspace{-0.1ex}j} \hspace{.1ex} \currentlocationvector^{i} \hspace{-0.2ex} \currentlocationvector^{j}
\hspace{-0.3ex} , &
\textsl{g}_{i\hspace{-0.1ex}j} \hspace{-0.15ex} \equiv
\initiallocationvector_\differentialindex{i} \hspace{-0.1ex} \dotp \hspace{.1ex} \initiallocationvector_\differentialindex{\hspace{-0.1ex}j}
\hspace{.1ex} ,
\\[.25em]
%
\mathboldPhi = \hspace{-0.1ex} \currentlocationvector_\differentialindex{i} \hspace{.1ex} \initiallocationvector^i \hspace{-0.15ex} \dotp \hspace{.1ex} \initiallocationvector^j \hspace{-0.25ex} \currentlocationvector_\differentialindex{\hspace{-0.1ex}j} \hspace{-0.2ex}
= \textsl{g}^{\hspace{.2ex}i\hspace{-0.1ex}j} \currentlocationvector_\differentialindex{i} \currentlocationvector_\differentialindex{\hspace{-0.1ex}j}
\hspace{.1ex} , &
\textsl{g}^{\hspace{.2ex}i\hspace{-0.1ex}j} \hspace{-0.15ex} \equiv
\initiallocationvector^{\hspace{.1ex}i} \hspace{-0.15ex} \dotp \hspace{.1ex} \initiallocationvector^{\hspace{.1ex}j}
\hspace{.1ex} ,
\end{array}
\end{equation*}
\vspace{-0.2em}\noindent
\en{and }\ru{и~}\en{they}\ru{они}
\en{coincide}\ru{совпадают}
\en{with the components}\ru{с~компонентами}
\en{of the unit}\ru{единичного}~(\en{metric}\ru{метрического})
\en{tensor}\ru{тензора}
\nopagebreak\vspace{-0.4em}
\begin{multline*}
\shoveleft{
\hspace{3em} \UnitDyad
= \hspace{-0.2ex}
\currentlocationvector_\differentialindex{i} \hspace{-0.1ex} \currentlocationvector^{i}
\hspace{-0.2ex} =
G_{\hspace{-0.15ex}i\hspace{-0.1ex}j} \currentlocationvector^{i} \hspace{-0.2ex} \currentlocationvector^j
\hspace{-0.25ex} = \hspace{-0.2ex}
\currentlocationvector^{i} \hspace{-0.2ex} \currentlocationvector_\differentialindex{i}
\hspace{-0.2ex} =
G^{\hspace{.1ex}i\hspace{-0.1ex}j} \hspace{-0.2ex} \currentlocationvector_\differentialindex{i} \currentlocationvector_\differentialindex{\hspace{-0.1ex}j}
\hfill }
\\[-0.2em]
%
= \initiallocationvector^i \initiallocationvector_\differentialindex{i} \hspace{-0.2ex}
= \textsl{g}^{\hspace{.2ex}i\hspace{-0.1ex}j} \initiallocationvector_\differentialindex{i} \initiallocationvector_\differentialindex{\hspace{-0.1ex}j} \hspace{-0.2ex}
= \initiallocationvector_\differentialindex{i} \initiallocationvector^i \hspace{-0.25ex}
= \textsl{g}_{i\hspace{-0.1ex}j} \hspace{.1ex} \initiallocationvector^i \initiallocationvector^j
\hspace{-0.2ex} ,
\end{multline*}
\vspace{-0.2em}\noindent
\en{but}\ru{но}
\en{the components’ bases}\ru{базисы компонент}
\en{are different}\ru{разные}.
%
\en{Using}\ru{Пользуясь}
\en{only}\ru{только}
\en{the~index notation}\ru{индексной записью},
\en{it’s easy to get confused}\ru{легко запутаться}
\en{due to the~differences}\ru{из\hbox{-}за различий}
\en{between}\ru{между}
\en{the~unit}\ru{единичным}
\en{tensor}\ru{тензором}~$\UnitDyad$
\en{and the~strain tensors}\ru{и~тензорами деформации}
$\bm{G}$,
$\mathboldPhi$,
${{^2}\hspace{-0.4ex}\bm{f}\hspace{-0.1ex}}$,
${{^2}\hspace{-0.2ex}\bm{c}}$.
%
\en{The direct indexless notation}\ru{Прямая безиндексная запись}
\en{has}\ru{имеет}
\ru{тут }\en{the obvious}\ru{явное}
\en{advantage}\ru{преимущество}\en{ here}.
\en{As was mentioned}\ru{Как упоминалось}
\en{in}\ru{в}~\chapterdotsectionref{chapter:mathapparatus}{section:polardecomposition},
\en{the~invariants}\ru{инварианты}
\en{of~the~stretch tensors}\ru{тензоров искажений}
$\bm{U}\hspace{-0.1em}$
\en{and}\ru{и}~$\bm{V}\hspace{-0.1em}$
\en{are the~same}\ru{одинаковые}.
%
\en{If}\ru{Если}
${w_{i}}$~\en{are}\ru{это}
\en{the~three eigenvalues}\ru{три собственных значения}
\en{of~}${\bm{U}\hspace{-0.1em}}$
\en{and}\ru{и}~${\bm{V}\hspace{-0.2em}}$,
\en{that is}\ru{то есть}
\en{the~roots}\ru{корни}
\en{of the characteristic equations}\ru{характеристических уравнений}
\en{for these tensors}\ru{для этих тензоров},
\en{then}\ru{то}
\en{here are}\ru{вот}
\en{their}\ru{их}
\en{invariants}\ru{инварианты}:
\nopagebreak\vspace{-0.3em}
\begin{gather*}
\anyfirstinvariantof{\bm{U}}
= \anyfirstinvariantof{\bm{V}}
= \trace{\bm{U}} \hspace{-0.25ex}
= \trace{\hspace{-0.2ex}\bm{V}} \hspace{-0.25ex}
= \textstyle\sum \hspace{-0.2ex} U_{\hspace{-0.2ex}j\hspace{-0.2ex}j}
= \textstyle\sum \hspace{-0.2ex} V_{\hspace{-0.1ex}j\hspace{-0.2ex}j}
= \textstyle\sum \hspace{-0.2ex} w_{i}
\hspace{.1ex} ,
\\
%
\anysecondinvariantof{\bm{U}} \hspace{-0.2ex}
= \anysecondinvariantof{\bm{V}} \hspace{-0.2ex}
= \vphantom{\textstyle\sum}
w_1 w_2 \hspace{-0.2ex}
+ w_1 w_3 \hspace{-0.2ex}
+ w_2 \hspace{.15ex} w_3
\hspace{.1ex} ,
\\
%
\anythirdinvariantof{\bm{U}} \hspace{-0.2ex}
= \anythirdinvariantof{\bm{V}} \hspace{-0.2ex}
= \vphantom{\textstyle\sum}
w_1 w_2 \hspace{.15ex} w_3
\hspace{.1ex} .
\end{gather*}
\en{The invariants}\ru{Инварианты}
\en{of~}$\bm{G}$
\en{and}\ru{и}~$\mathboldPhi$
\en{coincide too}\ru{тоже совпадают}:
\nopagebreak
\begin{equation*}
\anyfirstinvariantof{\bm{G}}
\hspace{-0.2ex}
= \anyfirstinvariantof{\mathboldPhi}
\hspace{.1ex} , \dots
\end{equation*}
\en{Without a~deformation}\ru{Без деформации}
\nopagebreak\vspace{-0.2em}\begin{equation*}
\bm{F} = \bm{U} \hspace{-0.3ex} = \bm{V} \hspace{-0.3ex} = \bm{G} = \mathboldPhi = {^2}\hspace{-0.4ex}\bm{f} = {^2}\hspace{-0.2ex}\bm{c} = \UnitDyad
\hspace{.1ex}
,
\end{equation*}
\vspace{-0.2em}\noindent
\en{thus}\ru{поэтому}
\en{as characteristics of~deformation}\ru{как характеристики деформации}
\en{it’s worth taking}\ru{ст\'{о}ит взять}
\en{the differences}\ru{разности}
\en{like}\ru{типа}
${\bm{U} \hspace{-0.2ex} - \UnitDyad}$,
${\bm{U} \hspace{-0.3ex} \dotp \bm{U} \hspace{-0.2ex} - \UnitDyad}$, \dots
...
\subsection*{The right Cauchy\hbox{--}Green deformation tensor}
George Green discovered a deformation tensor known as the right Cauchy\hbox{--}Green deformation tensor or Green’s deformation tensor
\nopagebreak\begin{equation*}
\bm{G}
= \bm{F}^{\hspace{.1ex}\T} \hspace{-0.5ex} \dotp \bm{F}
= \bm{U}^{2}
\hspace{1em} \text{\en{or}\ru{или}} \hspace{1em}
G_{i\hspace{-0.1ex}j} \hspace{-0.2ex}
= F_{k' i} \hspace{.25ex} F_{k' \hspace{-0.15ex}j} \hspace{-0.15ex}
= \frac{\partial \hspace{-0.1ex} x_{\hspace{-0.1ex}k'}}{\partial \mathcircabove{x}_{i}} \hspace{.15ex} \frac{\partial \hspace{-0.1ex} x_{\hspace{-0.1ex}k'}}{\partial \mathcircabove{x}_{\hspace{-0.2ex}j}}
\hspace{.1ex} .
\end{equation*}
This tensor \textcolor{magenta}{gives the~\inquotes{square} of local change in distances} due to deformation:
${\displaystyle d\currentlocationvector \dotp d\currentlocationvector = d\initiallocationvector \dotp \bm{G} \dotp d\initiallocationvector}$
The most popular invariants of~${\bm{G}}$ are
%%used in expressions for the potential energy of elastic deformations of an~isotropic body.
\[
\begin{array}{r@{\hspace{.25em}}c@{\hspace{.33em}}l}
\anyfirstinvariantof{\bm{G}} & \equiv &
\trace{\bm{G}}
= G_{ii} \hspace{-0.2ex} = \gamma_{1}^{2} + \gamma_{2}^{2} + \gamma_{3}^{2}
\\[.25em]
%
\anysecondinvariantof{\bm{G}} & \equiv &
\smalldisplaystyleonehalf \bigl( G_{\hspace{-0.2ex}j\hspace{-0.1ex}j}^{\hspace{.25ex}2} \hspace{-0.1ex} - G_{ik} G_{ki} \hspace{.1ex} \bigr) \hspace{-0.25ex}
= \gamma_{1}^{2}\gamma_{2}^{2} + \gamma_{2}^{2}\gamma_{3}^{2} + \gamma_{3}^{2}\gamma_{1}^{2}
\\[.4em]
%
\anythirdinvariantof{\bm{G}} & \equiv &
\determinant \hspace{.1ex} \bm{G}
= \gamma_{1}^{2}\gamma_{2}^{2}\gamma_{3}^{2}
\end{array}
\]
where ${\gamma_{i}\hspace{-0.2ex}}$ are stretch ratios for unit fibers that are initially oriented along directions of eigenvectors of the right stretch tensor~${\bm{U}\hspace{-0.2ex}}$.
\subsection*{The inverse of Green’s deformation tensor}
Sometimes called the Finger tensor or the Piola tensor, the~inverse of the right Cauchy\hbox{--}Green deformation tensor
\nopagebreak\vspace{-0.25em}\begin{equation*}
{^2\hspace{-0.4ex}\bm{f}}
= \bm{G}^{\expminusone} \hspace{-0.25ex}
= \bm{F}^{\expminusone} \hspace{-0.4ex} \dotp \bm{F}^{\expminusT}
\hspace{1em} \text{\en{or}\ru{или}} \hspace{1em}
f_{i\hspace{-0.1ex}j} \hspace{-0.2ex} = \frac{\partial \mathcircabove{x}_{i}}{\partial x_{\hspace{-0.1ex}k'}} \hspace{.15ex} \frac{\partial \mathcircabove{x}_{\hspace{-0.2ex}j}}{\partial x_{\hspace{-0.1ex}k'}}
\end{equation*}
\subsection*{The left Cauchy\hbox{--}Green or Finger deformation tensor}
Swapping multipliers in the formula for the right Green–Cauchy deformation tensor leads to the left Cauchy\hbox{--}Green deformation tensor, defined as
\nopagebreak\vspace{-0.2em}\begin{equation*}
\mathboldPhi
= \bm{F} \dotp \bm{F}^{\hspace{.1ex}\T} \hspace{-0.4ex}
= \bm{V}^{2}
\hspace{1em} \text{\en{or}\ru{или}} \hspace{1em}
\Phi_{i\hspace{-0.1ex}j} \hspace{-0.2ex}
= \frac{\partial x_{i}}{\partial \mathcircabove{x}_{k}} \hspace{.15ex} \frac{\partial x_{\hspace{-0.15ex}j}}{\partial \mathcircabove{x}_{k}}
\end{equation*}
The left Cauchy\hbox{--}Green deformation tensor is often called the Finger’s deformation tensor, named after Josef Finger (1894).
Invariants of ${\mathboldPhi}$ are also used in expressions for strain energy density functions.
The conventional invariants are defined as
\nopagebreak\begin{equation*}
\begin{aligned}
I_{1} & \equiv \Phi_{ii} = \lambda_{1}^{2} + \lambda_{2}^{2} + \lambda_{3}^{2}
\\
%
I_{2} & \equiv \tfrac{1}{2} \bigl( \Phi_{ii}^{2} - \Phi_{jk}\Phi_{kj} \bigr) = \lambda_{1}^{2}\lambda_{2}^{2} + \lambda_{2}^{2}\lambda_{3}^{2} + \lambda_{3}^{2}\lambda_{1}^{2}
\\
%
I_{3} & \equiv \det \mathboldPhi = J^{2} \hspace{-0.4ex} = \lambda_{1}^{2}\lambda_{2}^{2}\lambda_{3}^{2}
\end{aligned}
\end{equation*}
\vspace{-0.2em}\noindent
(${J \equiv \det{\bm{F}}}$\en{ is}\ru{\:---} \en{the Jacobian}\ru{якобиан}, \en{determinant of the motion gradient}\ru{определитель градиента движения})
\subsection*{The Cauchy deformation tensor}
The Cauchy deformation tensor is defined as the~inverse of the left Cauchy\hbox{--}Green deformation tensor
\nopagebreak\vspace{-0.4em}\begin{equation*}
{^2\hspace{-0.2ex}\bm{c}} = \mathboldPhi^{\expminusone} \hspace{-0.25ex}
= \bm{F}^{\expminusT} \hspace{-0.4ex} \dotp \bm{F}^{\expminusone}
\hspace{1em} \text{\en{or}\ru{или}} \hspace{1em}
c_{i\hspace{-0.1ex}j} \hspace{-0.2ex}
= \frac{\partial \mathcircabove{x}_{k}}{\partial \hspace{-0.1ex} x_{i}} \hspace{.15ex} \frac{\partial \mathcircabove{x}_{k}}{\partial \hspace{-0.1ex} x_{\hspace{-0.15ex}j}}
\end{equation*}
${\displaystyle d\initiallocationvector \dotp d\initiallocationvector = d\currentlocationvector \dotp {^2\hspace{-0.2ex}\bm{c}} \dotp d\currentlocationvector}$
This tensor is also called the Piola tensor or the Finger tensor in rheology and fluid dynamics literature.
\subsection*{Finite strain tensors}
The concept of \emph{strain} is used to evaluate how much a~given displacement differs locally from a~body displacement as a~whole (a~\inquotes{rigid body displacement}). One of such strains for large deformations is the \emph{Green strain tensor} (\emph{Green\hbox{--}Lagrangian strain tensor}, \emph{Green\hbox{--}Saint\hbox{-\hspace{-0.2ex}}Venant strain tensor}), defined as
\nopagebreak\begin{equation*}
\displaystyle \bm{C} = \smalldisplaystyleonehalf \bigl( \bm{G} - \UnitDyad \hspace{.1ex} \bigr)
\hspace{1em} \text{\en{or}\ru{или}} \hspace{1em}
C_{i\hspace{-0.1ex}j} \hspace{-0.2ex} = \onehalf \Bigl( \frac{\partial x_{k'}}{\partial \mathcircabove{x}_{i}} \hspace{.15ex} \frac{\partial x_{k'}}{\partial \mathcircabove{x}_{\hspace{-0.2ex}j}} - \delta_{i\hspace{-0.1ex}j} \Bigr)
\end{equation*}
\noindent
or as the function of the displacement gradient tensor
\nopagebreak\begin{equation*}
\displaystyle \bm{C} = \smalldisplaystyleonehalf \hspace{-0.3ex} \left( \hspace{-0.1ex}
\boldnablacircled\bm{u}
+ \hspace{-0.1ex} \boldnablacircled\bm{u}^{\hspace{-0.1ex}\T} \hspace{-0.3ex}
+ \hspace{-0.1ex} \boldnablacircled\bm{u} \dotp \hspace{-0.1ex} \boldnablacircled\bm{u}^{\hspace{-0.1ex}\T}
\right)
\end{equation*}
\noindent
in cartesian coordinates
\nopagebreak\begin{equation*}
\displaystyle C_{i\hspace{-0.1ex}j} \hspace{-0.2ex} = \onehalf \hspace{-0.25ex} \left(
\frac{\partial u_{\hspace{-0.1ex}j}}{\partial \mathcircabove{x}_{i}}
+ \frac{\partial u_{i}}{\partial \mathcircabove{x}_{\hspace{-0.2ex}j}}
+ \frac{\partial u_{k}}{\partial \mathcircabove{x}_{i}} \frac{\partial u_{k}}{\partial \mathcircabove{x}_{\hspace{-0.2ex}j}}
\right)
\end{equation*}
The Green strain tensor measures how much $\bm{G}$ differs from~$\UnitDyad$.
The \emph{Almansi\hbox{--}Hamel strain tensor}, referenced to the deformed configuration (\inquotes{Eulerian description}), is defined as
\nopagebreak\vspace{-0.5em}\begin{equation*}
{^2\hspace{-0.2ex}\bm{a}} = \smalldisplaystyleonehalf \bigl( \UnitDyad - \hspace{-0.15ex} {^2\hspace{-0.2ex}\bm{c}} \hspace{.3ex} \bigr) \hspace{-0.3ex}
= \smalldisplaystyleonehalf \bigl( \UnitDyad - \mathboldPhi^{\expminusone} \hspace{.2ex} \bigr)
\hspace{1em} \text{\en{or}\ru{или}} \hspace{1em}
a_{i\hspace{-0.1ex}j} \hspace{-0.2ex}
= \onehalf \hspace{-0.25ex} \left( \hspace{-0.4ex} \delta _{i\hspace{-0.1ex}j} - \frac{\partial \mathcircabove{x}_{k}}{\partial \hspace{-0.1ex} x_{i}} \hspace{.15ex} \frac{\partial \mathcircabove{x}_{k}}{\partial \hspace{-0.1ex} x_{\hspace{-0.15ex}j}} \right)
\end{equation*}
\vspace{-0.4em}\noindent
or as function of the displacement gradient
\nopagebreak\begin{equation*}
{^2}\hspace{-0.2ex}\bm{a} = \smalldisplaystyleonehalf \bigl(
\boldnabla\bm{u}^{\hspace{-0.1ex}\T} \hspace{-0.3ex}
+ \hspace{-0.1ex} \boldnabla\bm{u}
- \hspace{-0.1ex} \boldnabla\bm{u} \dotp \hspace{-0.1ex} \boldnabla\bm{u}^{\hspace{-0.1ex}\T}
\bigr)
\end{equation*}
\nopagebreak\vspace{-0.2em}\begin{equation*}
\displaystyle a_{i\hspace{-0.1ex}j} \hspace{-0.2ex} = \onehalf \hspace{-0.25ex} \left(
\frac{\partial u_{i}}{\partial x_{\hspace{-0.15ex}j}}
+ \frac{\partial u_{\hspace{-0.1ex}j}}{\partial \hspace{-0.1ex} x_{i}}
- \frac{\partial u_{k}}{\partial x_{i}} \frac{\partial u_{k}}{\partial x_{\hspace{-0.15ex}j}}
\right)
\end{equation*}
\subsection*{Seth\hbox{--}Hill family of abstract strain tensors}
B. R. Seth was the first to show that the Green and Almansi strain tensors are special cases of a more abstract measure of deformation.
The idea was further expanded upon by Rodney Hill in~1968 \textcolor{red}{(publication??)}.
The Seth\hbox{--}Hill family of strain measures (also called Doyle\hbox{--}Ericksen tensors) is expressed as
\nopagebreak\vspace{-0.1em}\begin{equation*}
\displaystyle \bm{D}_{(m)} \hspace{-0.2ex}
= \frac{\raisebox{-0.2em}{1}}{2m} \left( \hspace{.1ex} \bm{U}^{2m} \hspace{-0.4ex} - \UnitDyad \hspace{.2ex} \right)
= \frac{\raisebox{-0.2em}{1}}{2m} \left( \bm{G}^{m} \hspace{-0.4ex} - \UnitDyad \hspace{.1ex} \right) \end{equation*}
\vspace{.1em} \noindent \en{For various}\ru{Для разных}~$m$
\en{it gives}\ru{это даёт}
\nopagebreak\begin{equation*}
\begin{array}{r@{\hspace{0.1em}}ll}
\bm{D}_{(1)} & = \smalldisplaystyleonehalf \hspace{-0.25ex} \left( \bm{U}^{2} \hspace{-0.25ex} - \UnitDyad \right) = \smalldisplaystyleonehalf (\bm{G} - \UnitDyad) & \text{\scalebox{0.9}{Green strain tensor}}
\\[.4em]
\bm{D}_{(\nicefrac{1}{2})} & = \bm{U} \hspace{-0.15ex} - \UnitDyad = \bm{G}^{\hspace{.1ex}\nicefrac{1}{2}} \hspace{-0.25ex} - \UnitDyad & \text{\scalebox{0.9}{Biot strain tensor}}
\\[.4em]
\bm{D}_{(0)} & = \ln \bm{U} = \smalldisplaystyleonehalf \ln \bm{G} & \text{\scalebox{0.9}{logarithmic strain, Hencky strain}}
\\[.4em]
\bm{D}_{(-\hspace{-0.1ex}1)} & = \smalldisplaystyleonehalf \hspace{-0.25ex} \left( \hspace{-0.1ex} \UnitDyad - \bm{U}^{-2} \hspace{.1ex} \right) & \text{\scalebox{0.9}{Almansi strain}}
\end{array}
\end{equation*}
The second\hbox{-}order approximation of these tensors is
\[ \bm{D}_{(m)} \hspace{-0.2ex} =
\infinitesimaldeformation
+ \smalldisplaystyleonehalf \hspace{.1ex} \boldnabla\bm{u} \dotp \hspace{-0.1ex} \boldnabla\bm{u}^{\hspace{-0.1ex}\T} \hspace{-0.3ex}
- (1 - m) \hspace{.2ex} \infinitesimaldeformation \dotp \infinitesimaldeformation \]
\vspace{-0.25em}\noindent
where ${\infinitesimaldeformation \equiv \hspace{-0.2ex} \boldnabla {\bm{u}}^{\hspace{.1ex}\mathsf{S}}}$ is the infinitesimal deformation tensor.
Many other different definitions of measures~$\bm{D}$ are possible, provided that they satisfy these conditions:
\begin{itemize}
\item $\bm{D}$ vanishes for any movement of a~body as a~rigid whole
\item dependence of~$\bm{D}$ on displacement gradient tensor~${\nabla \bm{u}}$ is continuous, continuously differentiable and monotonic
\item it’s desired that $\bm{D}$ reduces to the infinitesimal linear deformation tensor~${\infinitesimaldeformation}$ when ${\boldnabla \bm{u} \to 0}$
\end{itemize}
\noindent For example, tensors from the set
\[ \displaystyle \bm{D}^{(n)} \hspace{-0.32ex} = \left( {\bm{U}}^{n} \hspace{-0.4ex} - {\bm{U}}^{-n} \right) \hspace{-0.4ex} / \hspace{.25ex} 2n \]
aren’t from the Seth\hbox{--}Hill family, but for any~$n$ they have the same 2nd\hbox{-}order approximation as Seth\hbox{--}Hill measures with~${m=0}$.
\vspace{.4em} \noindent \hfill \textboldoblique{Wikipedia, the free encyclopedia}\:--- \href{https://en.wikipedia.org/wiki/Finite_strain_theory}{Finite strain theory}
...
\subsection*{\en{Logarithmic strain, Hencky’s strain}\ru{Логарифмическая деформация, деформация Hencky}}
\href{https://en.wikipedia.org/wiki/Heinrich_Hencky}{%
\bookauthor{Heinrich Hencky}%
}.
Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen.
Zeitschrift für technische Physik, Vol.\:9~(1928),
Seiten~215\hbox{--}220.
....
\en{\section{Velocity field}}
\ru{\section{Поле скоростей}}
\label{section:velocityfield}
\en{This topic}\ru{Эта тема} \en{is discussed}\ru{обсуждается} \en{in nearly any}\ru{в~почти любой} \en{book}\ru{книге} \en{about~continuum mechanics}\ru{о~механике сплошной среды}, \en{however}\ru{однако} \en{for}\ru{для} \en{solid elastic continua}\ru{твёрдых упругих сред} \en{it’s not very vital}\ru{она не~столь насущна}.
\en{Among}\ru{Среди} \en{various}\ru{разных} \en{models}\ru{моделей} \en{of~a~material continuum}\ru{материального \rucontinuum{}а}, \en{an elastic solid body}\ru{упругое твёрдое тело} \en{is distinguished}\ru{выделяется} \en{by interesting possibility}\ru{интересной возможностью} \en{of~deriving}\ru{вывода} \en{the~complete set}\ru{полного набора}~(\en{system}\ru{системы}) \en{of~equations}\ru{уравнений} \en{for it}\ru{для него} \en{via the single logically flawless procedure}\ru{единой логически безупречной процедурой}.
\en{But now}\ru{Но пока} \en{we follow the~way}\ru{мы идём путём}, \en{usual}\ru{обычным} \en{for}\ru{для} \en{fluid continuum mechanics}\ru{механики сплошной текучей среды}.
\en{So}\ru{Итак}, \en{there’s}\ru{есть} \en{velocity field}\ru{поле скоростей} \en{in spatial description}\ru{в~пространственном описании}
${\bm{v} \equiv \hspace{-0.1ex} \mathdotabove{\currentlocationvector} = \bm{v}(\currentlocationvector, \hspace{-0.1ex} t)}$.
\en{Decomposition}\ru{Разложение} \en{of~tensor}\ru{тензора} ${\hspace{-0.1ex}\boldnabla \bm{v} \hspace{-0.16ex} = \hspace{-0.2ex} \boldnabla \mathdotabove{\currentlocationvector} = \hspace{-0.16ex} \currentlocationvector^{i} \partial_i \mathdotabove{\currentlocationvector} = \hspace{-0.16ex} \currentlocationvector^{i} \mathdotabove{\currentlocationvector}_\differentialindex{i}}$%
\kern-0.15ex\footnote{For sufficiently smooth functions, partial derivatives always commute, space and time ones too.
Thus
\nopagebreak\vspace{-0.8em}\begin{equation*}
\scalebox{0.92}{$\displaystyle
\frac{\raisemath{-0.2em}{\partial}}{\partial q^i} \frac{\raisemath{-0.2em}{\partial \hspace{.1ex} \currentlocationvector}}{\partial t}
= \frac{\raisemath{-0.2em}{\partial}}{\partial t} \frac{\raisemath{-0.2em}{\partial \hspace{.1ex} \currentlocationvector}}{\partial q^i}
$}
\hspace{.8em} \text{\en{or}\ru{или}} \hspace{.8em}
\partial_i \mathdotabove{\currentlocationvector} = \hspace{-0.1ex} \mathdotabove{\currentlocationvector}_\differentialindex{i}
\end{equation*}
}
\en{into symmetric and skewsymmetric parts}\ru{на симметричную и~кососимметричную части}~(\chapterdotsectionref{chapter:mathapparatus}{section:tensors.symmetric+skewsymmetric})
\nopagebreak\vspace{-0.1em}\begin{equation*}
\boldnabla \mathdotabove{\currentlocationvector}
= \hspace{-0.2ex} \boldnabla \mathdotabove{\currentlocationvector}^{\hspace{.3ex}\mathsf{S}} \hspace{-0.25ex}
- \hspace{.1ex} \smalldisplaystyleonehalf \bigl( \boldnabla \hspace{-0.25ex}\times\hspace{-0.25ex} \mathdotabove{\currentlocationvector} \bigr) \hspace{-0.4ex} \times \hspace{-0.2ex} \UnitDyad