Integral Balance Equations in reference space

2.6. Integral Balance Equations in reference space#

2.6.1. Mass#

Integal balance for a material volume $V_{t}$ reads

\begin{array}{r} \begin{aligned} 0 & = \frac{d}{d t} \int_{V_{t}} ρ (\vec{r}, t) d V = \\ = \frac{d}{d t} \int_{V_{0}} ρ (\vec{r} ({\vec{r}}_{0}, t), t) J ({\vec{r}}_{0}, t) d V_{0} = \\ = \frac{d}{d t} \int_{V_{0}} ρ_{0} ({\vec{r}}_{0}, t) J ({\vec{r}}_{0}, t) d V_{0} = \\ = \int_{V_{0}} \frac{D}{D t} (ρ_{0} ({\vec{r}}_{0}, t) J ({\vec{r}}_{0}, t)) d V_{0} . \end{aligned} \end{array}

Since the domain $V_{0}$ is arbitrary, with some abuse of notation to indicate the density field as $ρ$ , hidihg the dependence on the independet fvariables $ρ_{0} ({\vec{r}}_{0}, t)$ , the differential balance in reference space follows

\frac{D}{D t} (ρ J) = 0

or

ρ J = ρ^{0},

i.e. the product $ρ J$ equals the initial density field $ρ^{0}$ , assuming that the determinant of the transformation is $J^{0} = 1$ , in the reference configuration.

2.6.2. Momentum#

Integral balance for a material volume $V_{t}$ reads

\begin{array}{r} \begin{aligned} \frac{d}{d t} \int_{V_{t}} ρ \vec{v} d V = \int_{V_{t}} ρ \vec{g} + \oint_{\partial V_{t}} \hat{n} \cdot T d S \\ \frac{d}{d t} \int_{V_{0}} ρ J \vec{v} d V_{0} = \int_{V_{0}} ρ J \vec{g} + \oint_{\partial V_{t}} {\hat{n}}_{0} \cdot (J F^{- T} \cdot T) d S_{0} \\ \frac{d}{d t} \int_{V_{0}} ρ^{0} \vec{v} d V_{0} = \int_{V_{0}} ρ^{0} \vec{g} + \oint_{\partial V_{0}} {\hat{n}}_{0} \cdot Σ_{n} d S_{0} \\ \int_{V_{0}} ρ^{0} \frac{D}{D t} \vec{v} d V_{0} = \int_{V_{0}} ρ^{0} \vec{g} + \oint_{V_{0}} \nabla_{0} \cdot Σ_{n} d S_{0} \end{aligned} \end{array}

Since the domain $V_{0}$ is arbitrary, the differential balance in reference space follows

ρ^{0} \frac{D \vec{v}}{D t} = ρ^{0} \vec{g} + \nabla_{0} \cdot Σ_{n}

Example 2.1 (Relation between description in physical and reference space)

\begin{array}{r} \begin{aligned} ρ^{0} \frac{D \vec{v}}{D t} & = ρ^{0} \vec{g} + \nabla_{0} \cdot Σ_{n} \\ J ρ \frac{D \vec{v}}{D t} & = J ρ \vec{g} + \nabla_{0} \cdot Σ_{n} \\ ρ \frac{D \vec{v}}{D t} & = ρ \vec{g} + \frac{1}{J} \nabla_{0} \cdot Σ_{n} \end{aligned} \end{array}

thus,

\begin{array}{r} \begin{aligned} \nabla \cdot T & = \frac{1}{J} \nabla_{0} \cdot Σ_{n} = \\ = \frac{1}{J} \nabla_{0} \cdot (J F^{- T} \cdot T) \end{aligned} \end{array}

todo Prove it with derivation!

2.6.3. Kinetic energy#

\begin{array}{r} \begin{aligned} 0 & = \vec{v} \cdot {ρ^{0} \frac{D \vec{v}}{D t} - ρ^{0} \vec{g} - \nabla_{0} \cdot Σ_{n}} = \\ = ρ^{0} \frac{D}{D t} \frac{| \vec{v} |^{2}}{2} - ρ^{0} \vec{v} \cdot \vec{g} - \vec{v} \cdot \nabla_{0} \cdot Σ_{n} = \\ = ρ^{0} \frac{D}{D t} \frac{| \vec{v} |^{2}}{2} - ρ^{0} \vec{v} \cdot \vec{g} - \nabla_{0} \cdot (\vec{v} \cdot Σ_{n}) + \nabla_{0} \vec{v} : Σ_{n} \end{aligned} \end{array}

v_{i} \partial_{k}^{0} Σ_{k i} = \partial_{k}^{0} (v_{i} Σ_{k i}) - \partial_{k}^{0} v_{i} Σ_{k i}

\begin{array}{r} \begin{aligned} d V = J d V_{0} \\ d r_{i} n_{i} d S = J d r_{k}^{0} n_{k}^{0} d S_{0} \\ d r_{k}^{0} \frac{\partial r_{i}}{\partial r_{k}^{0}} n_{i} d S = d r_{k}^{0} J n_{k}^{0} d S_{0} \end{aligned} \end{array}

\begin{array}{r} \begin{aligned} \frac{\partial r_{i}}{\partial r_{k}^{0}} n_{i} d S & = J n_{k}^{0} d S_{0} \\ \underset{= δ_{i j}}{\underset{⏟}{\frac{\partial r_{k}^{0}}{\partial r_{j}} \frac{\partial r_{i}}{\partial r_{k}^{0}}}} n_{i} d S & = J \frac{\partial r_{k}^{0}}{\partial r_{j}} n_{k}^{0} d S_{0} \\ n_{j} d S & = J \frac{\partial r_{k}^{0}}{\partial r_{j}} n_{k}^{0} d S_{0} \end{aligned} \end{array}

F = {\hat{e}}_{k}^{0} {\hat{e}}_{i}^{0} \frac{\partial r_{i}}{\partial r_{k}^{0}}

F^{- 1} = {\hat{e}}_{j}^{0} {\hat{e}}_{k}^{0} \frac{\partial r_{k}^{0}}{\partial r_{j}}

\begin{array}{r} \begin{aligned} F^{- 1} \cdot F & = ({\hat{e}}_{j}^{0} {\hat{e}}_{k}^{0} \frac{\partial r_{k}^{0}}{\partial r_{j}}) \cdot ({\hat{e}}_{l}^{0} {\hat{e}}_{i}^{0} \frac{\partial r_{i}}{\partial r_{l}^{0}}) = {\hat{e}}_{j}^{0} {\hat{e}}_{i}^{0} \frac{\partial r^{i}}{\partial r_{j}} = {\hat{e}}_{j}^{0} {\hat{e}}_{i}^{0} δ_{i j} \\ F \cdot F^{- 1} & = ({\hat{e}}_{l}^{0} {\hat{e}}_{i}^{0} \frac{\partial r_{i}}{\partial r_{l}^{0}}) \cdot ({\hat{e}}_{j}^{0} {\hat{e}}_{k}^{0} \frac{\partial r_{k}^{0}}{\partial r_{j}}) = {\hat{e}}_{l}^{0} {\hat{e}}_{k}^{0} \frac{\partial r^{k}}{\partial r_{l}} = {\hat{e}}_{l}^{0} {\hat{e}}_{k}^{0} δ_{l k} \end{aligned} \end{array}

\frac{\partial r_{k}^{0}}{\partial r_{i}} \frac{\partial r_{i}}{\partial r_{l}^{0}} = δ_{k l}

Σ := Σ_{n} \cdot F^{- 1} = J F^{- T} \cdot T \cdot F^{- 1}

Σ_{n} = Σ \cdot F

Σ_{i k} = Σ_{n, i j} {(F^{- 1})}_{j k} = Σ_{n, i j} \frac{\partial r_{k}^{0}}{\partial r_{j}}

Σ_{n, i j} = Σ_{i k} \frac{\partial x_{j}}{\partial x_{k}^{0}}

\begin{array}{r} \begin{aligned} \nabla_{0} \vec{v} : Σ_{n} & = \frac{D}{D t} F : Σ_{n} = \\ = \frac{\partial v_{j}}{\partial x_{i}^{0}} Σ_{n, i j} = \\ = \frac{\partial v_{j}}{\partial x_{i}^{0}} Σ_{i k} \frac{\partial x_{j}}{\partial x_{k}^{0}} = \\ = Σ_{i k} \frac{1}{2} (\frac{\partial v_{j}}{\partial x_{i}^{0}} \frac{\partial x_{j}}{\partial x_{k}^{0}} + \frac{\partial v_{j}}{\partial x_{k}^{0}} \frac{\partial x_{j}}{\partial x_{i}^{0}}) \end{aligned} \end{array}

if $Σ$ is symmetric, $Σ_{i k} = Σ_{k i}$ , or with tensor notation

\begin{array}{r} \begin{aligned} \nabla_{0} \vec{v} : Σ_{n} & = \frac{D}{D t} F : Σ_{n} = \\ = \nabla_{0} \vec{v} : (Σ \cdot F) = \\ = Σ : \frac{1}{2} (\frac{D F}{D t} \cdot F^{T} + F \cdot \frac{D F^{T}}{D t}) = \\ = Σ : \frac{D}{D t} E, \end{aligned} \end{array}

having recognized the time derivative (1.6) of the Green-Lagrange tensor (1.5).

Integral of the volume stress in the reference space can be recast as the volume in the physical space

\int_{V_{0}} \nabla_{0} \vec{v} : Σ_{n} d V_{0} = \int_{V_{0}} Σ : \frac{D E}{D t} d V_{0}

Σ_{n, i k} = J \frac{\partial x_{i}^{0}}{\partial x_{k}} T_{j k}

\begin{array}{r} \begin{aligned} \int_{V_{0}} \nabla_{0} \vec{v} : Σ_{n} d V_{0} & = \int_{V_{0}} \frac{\partial v_{k}}{\partial x_{i}^{0}} Σ_{n, i k} d V_{0} = \\ = \int_{V_{0}} \underset{}{\underset{⏟}{\frac{\partial v_{k}}{\partial x_{i}^{0}} (\frac{\partial x_{i}^{0}}{\partial x_{j}}}} T_{k j} \underset{d V}{\underset{⏟}{J) d V_{0}}} = \\ = \int_{V} \frac{\partial v_{k}}{\partial x^{j}} T_{k j} d V = \\ = \int_{V} \frac{1}{2} (\frac{\partial v_{k}}{\partial x^{j}} + \frac{\partial v_{j}}{\partial x^{k}}) T_{k j} d V = \\ = \int_{V} D_{j k} T_{k j} d V = \\ = \int_{V} D : T d V . \end{aligned} \end{array}

2.6.3.1. Variational principles#

Using an arbitrary test function $\vec{w} ({\vec{r}}_{0})$ ,

0 = \vec{w} \cdot {ρ^{0} \frac{D \vec{v}}{D t} - ρ^{0} \vec{g} - \nabla_{0} \cdot Σ_{n}}

and using rule of product

\begin{array}{r} \begin{aligned} w_{i} \frac{\partial Σ_{n, j i}}{\partial x_{j}^{0}} & = \frac{\partial}{\partial x_{j}^{0}} (w_{i} Σ_{n, j i}) - \frac{\partial w_{i}}{\partial x_{j}^{0}} Σ_{n, j i} = \end{aligned} \end{array}

and the second term can be transformed using the relation bewteen normal stress and second Piola-Kirchhoff tensor

\begin{array}{r} \frac{\partial w_{i}}{\partial x_{j}^{0}} Σ_{n, j i} = \frac{\partial w_{i}}{\partial x_{j}^{0}} Σ_{j k} \frac{\partial x_{i}}{\partial x_{k}^{0}} = Σ_{j k} \frac{1}{2} [\frac{\partial x_{i}}{\partial x_{k}^{0}} \frac{\partial w_{i}}{\partial x_{j}^{0}} + \frac{\partial x_{i}}{\partial x_{j}^{0}} \frac{\partial w_{i}}{\partial x_{k}^{0}}] = Σ_{j k} W_{i j} (\vec{w}), \end{array}

having defined the tensor

W (\vec{w}) := \frac{1}{2} [\nabla_{0} \vec{w} \cdot F^{T} + F \cdot \nabla_{0}^{T} \vec{w}],

with the evident analogy with the time derivative of Green-Lagrange strain tensor, namely

ϵ = W (\vec{v}),

being $\vec{v}$ the velocity field. Integrating on the domain $V_{0}$ and using divergence theorem, the problem is written in its weak form

\int_{V_{0}} {ρ^{0} \vec{w} \cdot \frac{D \vec{v}}{D t} + W (\vec{w}) : Σ} = \int_{V_{0}} ρ^{0} \vec{w} \cdot \vec{g} + \oint_{\partial V_{0}} {\hat{n}}_{0} \cdot Σ_{n} \cdot \vec{w},

with the proper boundary conditions and the corresponding conditions on the test function $\vec{w}$ . As an example, if the boundary is composed of two different regions, $S_{D, 0} \cup S_{N, 0} = \partial V_{0}$ , $S_{D} \cap S_{N} = \emptyset$ where either position (called $S_{D}$ from Dirichlet boundary) and stress (called $S_{N}$ from Neumann boundary) are prescribed

\begin{array}{r} \begin{aligned} \vec{r} = {\vec{r}}_{b}, & \vec{w} = \vec{0} & (on S_{D, 0} Dirichlet - essential - boundary) \\ {\hat{n}}_{0} \cdot Σ_{n} = {\vec{t}}_{0, {\hat{n}}_{0}}, & (on S_{N, 0} Neumann - natural - boundary) \end{aligned} \end{array}

the weak form of the equation reads

\int_{V_{0}} {ρ^{0} \vec{w} \cdot \frac{D \vec{v}}{D t} + W (\vec{w}) : Σ} = \int_{V_{0}} ρ^{0} \vec{w} \cdot \vec{g} + \int_{S_{n, 0}} {\hat{n}}_{0} \cdot {\vec{t}}_{{\hat{n}}_{0}}

2.6.4. Total energy#

Using Nanson’s relation $\hat{n} d S = {\hat{n}}_{0} \cdot (J F^{- T}) d S_{0}$ ,

\begin{array}{r} \begin{aligned} \frac{d}{d t} \int_{V_{t}} ρ e^{t} d V & = \int_{V_{t}} ρ \vec{g} \cdot \vec{v} d V + \oint_{\partial V_{t}} {\vec{t}}_{\hat{n}} \cdot \vec{v} d S - \oint_{\partial V_{t}} \hat{n} \cdot \vec{q} + \int_{V_{t}} ρ r d V = \\ = \int_{V_{t}} ρ \vec{g} \cdot \vec{v} d V + \oint_{\partial V_{t}} \hat{n} \cdot T \cdot \vec{v} d S - \oint_{\partial V_{t}} \hat{n} \cdot \vec{q} d S + \int_{V_{t}} ρ r d V \\ = \int_{V_{0}} ρ^{0} \vec{g} \cdot \vec{v} d V_{0} + \oint_{\partial V_{0}} {\hat{n}}_{0} \cdot (J F^{- T} \cdot T) \cdot \vec{v} d S_{0} - \oint_{\partial V_{0}} {\hat{n}}_{0} \cdot (J F^{- T} \cdot \vec{q}) d S_{0} + \int_{V_{0}} ρ^{0} r d V_{0} . \end{aligned} \end{array}

and the differential form reads

ρ^{0} \frac{D e^{t}}{D t} = ρ^{0} \vec{g} \cdot \vec{v} + \nabla_{0} \cdot (J F^{- T} \cdot T \cdot \vec{v}) - \nabla_{0} \cdot (J F^{- T} \cdot \vec{q}) + ρ^{0} r .

or

ρ^{0} \frac{D e^{t}}{D t} = ρ^{0} \vec{g} \cdot \vec{v} + \nabla_{0} \cdot (Σ_{n} \cdot \vec{v}) - \nabla_{0} \cdot {\vec{q}}_{0} + ρ^{0} r .

and dividing by $J$ and using the relation (see below) $\nabla_{0} \cdot (J F^{- T} \cdot \vec{a}) = J \nabla \cdot \vec{a}$ ,

ρ \frac{D e^{t}}{D t} = ρ \vec{g} \cdot \vec{v} + \nabla \cdot (T \cdot \vec{v}) - \nabla \cdot \vec{q} + ρ r .

Comparison with equation in physical space (dividing by $J$ ) suggests the identity

\frac{1}{J} \nabla_{0} \cdot (J F^{- T} \cdot \vec{a}) = \nabla \cdot \vec{a},

and thus

\nabla_{0} \cdot (J F^{- T}) = \vec{0},

since

\nabla_{0} \cdot (J F^{- T} \cdot \vec{a}) = \nabla_{0} \cdot (J F^{- T}) \cdot \vec{a} + J F^{- T} : \nabla_{0} \vec{a} = \nabla_{0} \cdot (J F^{- T}) \cdot \vec{a} + J \nabla \cdot \vec{a}

Proof.

J = | \frac{\partial r_{k}}{\partial r_{i}^{0}} | = ε_{i_{1}, i_{2}, i_{3}} \frac{\partial r_{i_{1}}}{\partial r_{1}^{0}} \frac{\partial r_{i_{2}}}{\partial r_{2}^{0}} \frac{\partial r_{i_{3}}}{\partial r_{3}^{0}}

{(F^{- T})}_{i j} = \frac{\partial r_{i}^{0}}{\partial r_{j}}

\begin{array}{r} \begin{aligned} {\nabla_{0} \cdot (J F^{- T})}_{j} & = \frac{\partial}{\partial r_{i}^{0}} (ε_{i_{1}, i_{2}, i_{3}} \frac{\partial r_{i_{1}}}{\partial r_{1}^{0}} \frac{\partial r_{i_{2}}}{\partial r_{2}^{0}} \frac{\partial r_{i_{3}}}{\partial r_{3}^{0}} \frac{\partial r_{i}^{0}}{\partial r_{j}}) = \\ = ε_{i_{1}, i_{2}, i_{3}} \frac{\partial}{\partial r_{i}^{0}} (\frac{\partial r_{i_{1}}}{\partial r_{1}^{0}}) \frac{\partial r_{i_{2}}}{\partial r_{2}^{0}} \frac{\partial r_{i_{3}}}{\partial r_{3}^{0}} \frac{\partial r_{i}^{0}}{\partial r_{j}} + ε_{i_{1}, i_{2}, i_{3}} \frac{\partial r_{i_{1}}}{\partial r_{1}^{0}} \frac{\partial}{\partial r_{i}^{0}} (\frac{\partial r_{i_{2}}}{\partial r_{2}^{0}}) \frac{\partial r_{i_{3}}}{\partial r_{3}^{0}} \frac{\partial r_{i}^{0}}{\partial r_{j}} + \\ + ε_{i_{1}, i_{2}, i_{3}} \frac{\partial r_{i_{1}}}{\partial r_{1}^{0}} \frac{\partial r_{i_{2}}}{\partial r_{2}^{0}} \frac{\partial}{\partial r_{i}^{0}} (\frac{\partial r_{i_{3}}}{\partial r_{3}^{0}}) \frac{\partial r_{i}^{0}}{\partial r_{j}} + ε_{i_{1}, i_{2}, i_{3}} \frac{\partial r_{i_{1}}}{\partial r_{1}^{0}} \frac{\partial r_{i_{2}}}{\partial r_{2}^{0}} \frac{\partial r_{i_{3}}}{\partial r_{3}^{0}} \frac{\partial}{\partial r_{i}^{0}} (\frac{\partial r_{i}^{0}}{\partial r_{j}}) = \\ = ε_{i_{1}, i_{2}, i_{3}} \underset{= \frac{\partial}{\partial r_{j}}}{\underset{⏟}{\frac{\partial r_{i}^{0}}{\partial r_{j}} \frac{\partial}{\partial r_{i}^{0}}}} (\frac{\partial r_{i_{1}}}{\partial r_{1}^{0}}) \frac{\partial r_{i_{2}}}{\partial r_{2}^{0}} \frac{\partial r_{i_{3}}}{\partial r_{3}^{0}} + ε_{i_{1}, i_{2}, i_{3}} \frac{\partial r_{i_{1}}}{\partial r_{1}^{0}} \underset{= \frac{\partial}{\partial r_{j}}}{\underset{⏟}{\frac{\partial r_{i}^{0}}{\partial r_{j}} \frac{\partial}{\partial r_{i}^{0}}}} (\frac{\partial r_{i_{2}}}{\partial r_{2}^{0}}) \frac{\partial r_{i_{3}}}{\partial r_{3}^{0}} + \\ + ε_{i_{1}, i_{2}, i_{3}} \frac{\partial r_{i_{1}}}{\partial r_{1}^{0}} \frac{\partial r_{i_{2}}}{\partial r_{2}^{0}} \underset{= \frac{\partial}{\partial r_{j}}}{\underset{⏟}{\frac{\partial r_{i}^{0}}{\partial r_{j}} \frac{\partial}{\partial r_{i}^{0}}}} (\frac{\partial r_{i_{3}}}{\partial r_{3}^{0}}) + ε_{i_{1}, i_{2}, i_{3}} \frac{\partial r_{i_{1}}}{\partial r_{1}^{0}} \frac{\partial r_{i_{2}}}{\partial r_{2}^{0}} \frac{\partial r_{i_{3}}}{\partial r_{3}^{0}} \frac{\partial}{\partial r_{j}} \underset{= 3}{\underset{⏟}{(\frac{\partial r_{i}^{0}}{\partial r_{i}^{0}})}} = \\ = 0, \end{aligned} \end{array}