Crack Tip Equation of Motion in Dynamic Gradient Damage Models

Abstract

We propose in this contribution to investigate the link between the dynamic gradient damage model and the classical Griffith’s theory of dynamic fracture during the crack propagation phase. To achieve this main objective, we first rigorously reformulate two-dimensional linear elastic dynamic fracture problems using variational methods and shape derivative techniques. The classical equation of motion governing a smoothly propagating crack tip follows by considering variations of a space-time action integral. We then give a variationally consistent framework of the dynamic gradient damage model. Owing to the analogies between the variational ingredients of these two models and under some basic assumptions concerning the damage band structuration, one obtains a generalized Griffith criterion which governs the crack tip evolution within the non-local damage model. Assuming further that the internal length is small compared to the dimension of the body, the previous criterion leads to the classical Griffith’s law through a separation of scales between the outer linear elastic domain and the inner damage process zone.

Appendices

Appendix A: Calculation of the First-Order Action Variation

We will carefully explore the first-order stability principle (15) by calculating the action variation with respect to arbitrary displacement and crack variations. The following easily established identities

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}l_{t}}\det\nabla\boldsymbol{\phi} \bigl(\mathbf{x}^{*} \bigr) &= \det \nabla\boldsymbol{\phi} \bigl(\mathbf{x}^{*} \bigr)\operatorname {tr} \bigl(\nabla \boldsymbol{\theta}^{*} \bigl(\mathbf{x}^{*} \bigr)\nabla\boldsymbol {\phi} \bigl( \mathbf{x}^{*} \bigr)^{-1} \bigr)=\operatorname{div} \boldsymbol{ \theta}_{t}(\mathbf{x})\det\nabla\boldsymbol{\phi} \bigl(\mathbf{x}^{*} \bigr), \\ \frac{\mathrm{d}}{\mathrm{d}l_{t}}\nabla\boldsymbol{\phi} \bigl(\mathbf{x}^{*} \bigr)^{-1} &= -\nabla\boldsymbol{\phi} \bigl(\mathbf{x}^{*} \bigr)^{-1}\nabla \boldsymbol{\theta}^{*} \bigl(\mathbf{x}^{*} \bigr)\nabla \boldsymbol {\phi} \bigl( \mathbf{x}^{*} \bigr)^{-1}=-\nabla\boldsymbol{ \phi} \bigl(\mathbf{x}^{*} \bigr)^{-1}\nabla\boldsymbol{ \theta}_{t}(\mathbf{x}), \end{aligned}$$

will be used for all subsequent calculations.

The classical wave equation can be obtained by calculating the action variation with zero crack advance $\delta l=0$

$$\begin{aligned} \mathcal{A}' \bigl(\mathbf{u}^{*},l \bigr) \bigl(\mathbf{w}^{*},0 \bigr)&= \int_{I}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma_{0}} \biggl(\mathsf{A} \biggl({ \frac{1}{2}}\nabla \mathbf{u}_{t}^{*}\nabla\boldsymbol{\phi} ^{-1}+{ \frac{1}{2}}\nabla\boldsymbol{\phi}^{-\mathsf {T}} \bigl(\nabla \mathbf{u}_{t}^{*} \bigr)^{\mathsf{T}} \biggr)\\ &\quad{}\cdot \biggl({ \frac{1}{2}}\nabla\mathbf{w}_{t}^{*}\nabla \boldsymbol{ \phi}^{-1}+{\frac{1}{2}}\nabla\boldsymbol{ \phi}^{-\mathsf {T}} \bigl(\nabla\mathbf{w}_{t}^{*} \bigr)^{\mathsf{T}} \biggr)\det\nabla \boldsymbol{\phi} \\ &\quad{}-\rho \bigl(\dot{\mathbf{u}}_{t}^{*}-\dot{l}_{t}\nabla \mathbf{u}_{t}^{*}\nabla\boldsymbol{\phi} ^{-1}\boldsymbol{ \theta}^{*} \bigr)\cdot \bigl(\dot{\mathbf{w}}_{t}^{*}-\dot{l}_{t} \nabla\mathbf{w}_{t}^{*}\nabla\boldsymbol{\phi}^{-1}\boldsymbol{ \theta}^{*} \bigr)\det\nabla\boldsymbol{\phi} \biggr)\, \mathrm {d}\mathbf{x}^{*}\\ &\quad{}- \mathcal{W}_{t}^{*} \bigl(\mathbf{w}^{*}_{t} \bigr), \end{aligned}$$

which gives

$$\begin{aligned} \mathcal{A}' \bigl(\mathbf{u}^{*},l \bigr) \bigl( \mathbf{w}^{*},0 \bigr)&= \int_{I}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma_{t}} \bigl(\boldsymbol{\sigma }_{t}\cdot \boldsymbol{\varepsilon}(\mathbf{w}_{t})+\rho\dot{l}_{t}\dot{ \mathbf{u}}_{t}\cdot\nabla\mathbf{w}_{t}\boldsymbol{ \theta}_{t} \bigr)\,\mathrm{d}\mathbf{x}-\mathcal{W}_{t}( \mathbf{w}_{t}) \\ &\quad{}+\underbrace{ \int_{I}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma _{0}}\rho\frac{\mathrm{d}}{\mathrm{d}t} \bigl( \bigl(\dot{ \mathbf{u}}_{t}^{*}-\nabla\mathbf{u}_{t}^{*}\nabla\boldsymbol{ \phi}^{-1}\,\dot{l}_{t}\boldsymbol{\theta} ^{*} \bigr)\det\nabla \boldsymbol{\phi} \bigr)\cdot\mathbf{w}_{t}^{*}\,\mathrm{d} \mathbf{x}^{*}}_{R}, \end{aligned}$$

(88)

where $\mathbf{w}$ denotes the pushforward of $\mathbf{w}^{*}$ to the current cracked configuration via (6).

To proceed, we observe that the real acceleration $\ddot{\mathbf{u}}_{t}$ can be obtained by differentiating (8)

$$ \ddot{\mathbf{u}}_{t}(\mathbf{x})=-\nabla\dot{ \mathbf{u}}_{t}(\mathbf{x})\dot{l}_{t}\boldsymbol{\theta}^{*} \bigl(\mathbf{x}^{*} \bigr)+\frac{\mathrm{d}}{\mathrm {d}t} \bigl(\dot{\mathbf{u}}_{t}^{*} \bigl(\mathbf{x}^{*} \bigr)-\nabla\mathbf{u}_{t}^{*} \bigl(\mathbf {x}^{*} \bigr) \nabla\boldsymbol{\phi} \bigl(\mathbf{x}^{*} \bigr)^{-1}\dot {l}_{t}\boldsymbol{\theta}^{*} \bigl(\mathbf{x}^{*} \bigr) \bigr), $$

(89)

where $\nabla\dot{\mathbf{u}}_{t}$ is the (Eulerian) velocity gradient. Using (89), the last term above can be written

$$\begin{aligned} R &= \int_{I}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma_{0}} \bigl(\rho \bigl(\ddot {\mathbf{u}}_{t} \circ \boldsymbol{\phi} +(\nabla\dot{\mathbf{u}}_{t}\circ\boldsymbol { \phi}) \dot{l}_{t}\boldsymbol{\theta}^{*} \bigr)\cdot\mathbf{w}_{t}^{*} \det\nabla\boldsymbol{\phi}+\rho\dot{l}_{t} \bigl(\dot{ \mathbf{u}}_{t}^{*}-\nabla\mathbf{u}_{t}^{*}\nabla\boldsymbol{ \phi}^{-1}\,\dot{l}_{t}\boldsymbol{\theta}^{*} \bigr) \\ &\quad{}\cdot \mathbf{w}_{t}^{*}\operatorname{tr} \bigl(\nabla\boldsymbol{ \phi}^{-1}\nabla\boldsymbol{\theta}^{*} \bigr)\det\nabla\boldsymbol {\phi} \bigr)\,\mathrm{d}\mathbf{x}^{*} \\ &= \int_{I}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma_{t}}(\rho\ddot{\mathbf {u}}_{t}\cdot \mathbf{w}_{t}+\rho\dot{l}_{t}\nabla\dot{ \mathbf{u}}_{t}\boldsymbol{\theta}_{t}\cdot \mathbf{w}_{t}+\rho\dot{l}_{t}\dot{\mathbf{u}}_{t} \cdot\mathbf{w}_{t}\operatorname{div}\boldsymbol{\theta}_{t}) \,\mathrm{d}\mathbf{x} \\ &= \int_{I}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma_{t}}(\rho\ddot{\mathbf {u}}_{t}\cdot \mathbf{w}_{t}-\rho\dot{l}_{t}\dot{\mathbf{u}}_{t} \cdot\nabla\mathbf{w}_{t}\boldsymbol{\theta}_{t})\,\mathrm{d} \mathbf{x}, \end{aligned}$$

(90)

where an integration by parts in $\varOmega\setminus\varGamma_{t}$ has been used on establishing the last equality. Regrouping (88) and (90), we obtain thus the spatially weak dynamic equilibrium

$$ \mathcal{A}' \bigl(\mathbf{u}^{*},l \bigr) \bigl( \mathbf{w}^{*},0 \bigr)= \int_{I}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma_{t}} \bigl(\boldsymbol{\sigma }_{t}\cdot \boldsymbol{\varepsilon}(\mathbf{w}_{t})+\rho\ddot{\mathbf{u}}_{t} \cdot\mathbf{w}_{t} \bigr)\,\mathrm{d}\mathbf{x} -\mathcal{W}_{t}( \mathbf{w}_{t}). $$

(91)

An integration by parts then gives the desired wave equation (18) for the displacement.

We then evaluate the action variation with respect to arbitrary crack increment $\delta l$ but zero displacement variation

$$\begin{aligned} &\mathcal{A}' \bigl(\mathbf{u}^{*},l \bigr) ( \mathbf{0}, \delta l) \\ &\quad= \int_{I}G_{\mathrm{c}}\cdot\delta l_{t}\, \mathrm{d}t+ \int_{I}\delta l_{t}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma_{t}} \bigl( \bigl(\psi \bigl(\boldsymbol{\varepsilon}( \mathbf{u}_{t}) \bigr)-\kappa(\dot{\mathbf{u}}_{t}) \bigr) \operatorname{div}\boldsymbol{\theta}_{t}-\boldsymbol{ \sigma}_{t}\cdot(\nabla\mathbf{u}_{t}\nabla\boldsymbol{ \theta}_{t})-\operatorname{div}(\mathbf{f}_{t}\otimes \boldsymbol{\theta}_{t})\cdot\mathbf{u}_{t} \bigr)\, \mathrm{d}\mathbf{x} \\ &\qquad{}-\underbrace{ \int_{I}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma _{0}} \bigl(\rho \bigl(\dot{\mathbf{u}}^{*}_{t}- \dot {l}_{t}\nabla\mathbf{u}^{*}_{t}\nabla\boldsymbol{ \phi}^{-1}\boldsymbol{\theta}^{*} \bigr)\cdot \bigl(-\nabla \mathbf{u}_{t}^{*}\nabla\boldsymbol{\phi}^{-1}\boldsymbol{ \theta}^{*}\cdot\dot{\overline{\delta l}}_{t}+\dot{l}_{t} \nabla \mathbf{u}_{t}^{*}\nabla\boldsymbol{\phi}^{-1}\nabla \boldsymbol{ \theta}^{*}\nabla\boldsymbol{\phi} ^{-1}\boldsymbol {\theta}^{*} \cdot\delta l_{t} \bigr)\det\nabla\boldsymbol{\phi} \bigr)\, \mathrm{d} \mathbf{x}^{*}}_{R}. \end{aligned}$$

(92)

The last term can be written using integration by parts in the time domain

$$\begin{aligned} R =& \int_{I}\delta l_{t}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma_{0}}\rho\frac{\mathrm {d}}{\mathrm{d}t} \bigl( \bigl(\dot{ \mathbf{u}}^{*}_{t}-\dot{l}_{t}\nabla\mathbf{u}^{*}_{t} \nabla\boldsymbol{\phi}^{-1}\boldsymbol{\theta}^{*} \bigr)\cdot \bigl( \nabla\mathbf{u}_{t}^{*}\nabla\boldsymbol{\phi}^{-1}\boldsymbol{ \theta}^{*} \bigr)\det\nabla\boldsymbol{\phi} \bigr)\,\mathrm {d}\mathbf{x}^{*} \\ &{}+ \int_{I}\delta l_{t}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma_{t}}\rho\dot{l}_{t}\dot{\mathbf{u}}_{t} \cdot\nabla\mathbf{u}_{t}\nabla\boldsymbol{\theta}_{t} \boldsymbol{\theta}_{t}\,\mathrm{d}\mathbf{x}, \end{aligned}$$

which gives

$$\begin{aligned} R&= \int_{I}\delta l_{t}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma_{0}} \bigl(\rho \bigl(\ddot {\mathbf{u}}_{t} \circ \boldsymbol{\phi}+(\nabla\dot{\mathbf{u}}_{t}\circ\boldsymbol{ \phi}) \dot{l}_{t}\boldsymbol{\theta}^{*} \bigr)\cdot \bigl(\nabla \mathbf{u}_{t}^{*}\nabla\boldsymbol{\phi}^{-1}\boldsymbol{ \theta}^{*} \bigr)\det\nabla\boldsymbol{\phi} \\ &\quad{}+\rho \bigl(\dot{\mathbf{u}}^{*}_{t}-\dot{l}_{t}\nabla \mathbf{u}^{*}_{t}\nabla\boldsymbol{\phi} ^{-1}\boldsymbol{ \theta}^{*} \bigr)\cdot \bigl(\nabla\dot{\mathbf{u}}^{*}_{t}\nabla \boldsymbol{ \phi}^{-1}\boldsymbol{\theta} ^{*}-\dot{l}_{t}\nabla \mathbf{u}^{*}_{t}\nabla\boldsymbol{\phi}^{-1}\nabla\boldsymbol{ \theta}^{*}\nabla\boldsymbol{\phi}^{-1}\boldsymbol{\theta}^{*} \bigr)\det \nabla\boldsymbol{\phi} \bigr)\, \mathrm{d}\mathbf{x}^{*} \\ &\quad{}+ \int_{I}\delta l_{t}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma_{t}}(\rho\dot{l}_{t}\dot{\mathbf{u}}_{t} \cdot\nabla\mathbf{u}_{t}\boldsymbol{\theta}_{t} \operatorname{div}\boldsymbol{\theta}_{t}+\rho\dot{l}_{t} \dot{\mathbf{u}}_{t}\cdot\nabla\mathbf{u}_{t}\nabla \boldsymbol{\theta}_{t}\boldsymbol{\theta}_{t})\,\mathrm{d} \mathbf{x}. \end{aligned}$$

We obtain thus

$$\begin{aligned} R&= \int_{I}\delta l_{t}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma _{t}}(\rho\ddot{\mathbf{u}}_{t}\cdot\nabla \mathbf{u}_{t}\boldsymbol{\theta}_{t}+\rho \dot{l}_{t}\nabla\dot{\mathbf{u}}_{t}\boldsymbol{ \theta}_{t}\cdot\nabla\mathbf{u}_{t}\boldsymbol{ \theta}_{t}+\rho\dot{l}_{t}\dot{\mathbf{u}}_{t}\cdot \nabla\mathbf{u}_{t}\boldsymbol{\theta}_{t}\operatorname{div} \boldsymbol{\theta}_{t})\,\mathrm{d}\mathbf{x} \\ &\quad{}+ \int_{I}\delta l_{t}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma_{0}}\rho \bigl(\dot{\mathbf {u}}^{*}_{t}-\dot {l}_{t}\nabla\mathbf{u}^{*}_{t}\nabla\boldsymbol{ \phi}^{-1}\boldsymbol{\theta}^{*} \bigr)\cdot \bigl(\nabla\dot{ \mathbf{u}}^{*}_{t}\nabla\boldsymbol{\phi}^{-1}\boldsymbol{ \theta}^{*} \bigr)\det\nabla\boldsymbol{\phi}\,\mathrm{d}\mathbf{x}^{*}. \end{aligned}$$

Differentiating (7) to obtain the material time derivative of the deformation gradient $(\mathrm{d}/\mathrm{d}t)(\nabla\mathbf{u}_{t})$

$$\frac{\mathrm{d}}{\mathrm{d}t} \bigl(\nabla\mathbf{u}_{t}(\mathbf {x}) \bigr)= \nabla \dot{\mathbf{u}}_{t}^{*} \bigl(\mathbf{x}^{*} \bigr)\nabla\boldsymbol {\phi} \bigl(\mathbf{x}^{*} \bigr)^{-1}-\dot{l}_{t}\nabla \mathbf{u}_{t}(\mathbf{x})\nabla\boldsymbol{\theta}_{t}( \mathbf{x}), $$

and with its definition

$$\frac{\mathrm{d}}{\mathrm{d}t} \bigl(\nabla\mathbf{u}_{t}(\mathbf {x}) \bigr)= \nabla \dot{\mathbf{u}}_{t}(\mathbf{x})+\nabla^{2} \mathbf{u}_{t}(\mathbf{x})\dot{l}_{t}\boldsymbol{\theta}^{*} \bigl(\mathbf{x}^{*} \bigr), $$

where $\nabla^{2}\mathbf{u}_{t}$ is the second gradient of the displacement field (a third-order tensor), we obtain

$$\begin{aligned} R&= \int_{I}\delta l_{t}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma_{t}} \bigl(\rho\ddot{\mathbf {u}}_{t}\cdot\nabla \mathbf{u}_{t}\boldsymbol{\theta}_{t}+\rho \dot{l}_{t}\nabla\dot{\mathbf{u}}_{t}\boldsymbol{ \theta}_{t}\cdot\nabla\mathbf{u}_{t}\boldsymbol{ \theta}_{t}+\rho\dot{\mathbf{u}}_{t}\cdot\nabla\dot{ \mathbf{u}}_{t}\boldsymbol{\theta}_{t}+\rho \dot{l}_{t}\dot{\mathbf{u}}_{t}\cdot \bigl(\nabla ^{2}\mathbf{u}_{t}\boldsymbol{\theta}_{t} \bigr) \boldsymbol{\theta}_{t} \\ &\quad{}+\rho\dot{l}_{t}\dot{\mathbf{u}}_{t}\cdot\nabla \mathbf{u}_{t}\nabla\boldsymbol{\theta}_{t}\boldsymbol{\theta} _{t}+\rho\dot{l}_{t}\dot{\mathbf{u}}_{t}\cdot\nabla \mathbf{u}_{t}\boldsymbol{\theta}_{t}\operatorname{div} \boldsymbol{\theta} _{t} \bigr)\,\mathrm{d}\mathbf{x}. \end{aligned}$$

Using an integration by parts in the domain $\varOmega\setminus \varGamma_{t}$ knowing that $\boldsymbol{\theta}_{t}=\mathbf{0}$ on $\partial \varOmega$ and $\boldsymbol{\theta}_{t}\cdot \mathbf{n}=0$ on $\varGamma_{t}$ by definition

$$\begin{aligned} &\int_{\varOmega\setminus\varGamma_{t}}\rho\dot{l}_{t}\dot{\mathbf{u}}_{t} \cdot\nabla\mathbf{u}_{t}\nabla\boldsymbol{\theta}_{t} \boldsymbol{\theta}_{t}\,\mathrm{d}\mathbf{x}\\ &\quad=- \int_{\varOmega \setminus\varGamma_{t}} \bigl(\rho\dot{l}_{t}\dot{ \mathbf{u}}_{t}\cdot\nabla\mathbf{u}_{t}\boldsymbol{ \theta}_{t}\operatorname{div}\boldsymbol{\theta}_{t}+\rho \dot{l}_{t}\nabla\dot{\mathbf{u}}_{t}\boldsymbol{ \theta}_{t}\cdot\nabla\mathbf{u}_{t}\boldsymbol{\theta }_{t}+\rho\dot{l}_{t}\dot{\mathbf{u}}_{t}\cdot \bigl(\nabla^{2}\mathbf{u}_{t}\boldsymbol{ \theta}_{t} \bigr)\boldsymbol{\theta}_{t} \bigr)\,\mathrm{d} \mathbf{x}, \end{aligned}$$

we get finally

$$R= \int_{I}\delta l_{t}\,\mathrm{d}t \int_{\varOmega\setminus\varGamma _{t}}(\rho\ddot{\mathbf{u}}_{t}\cdot\nabla \mathbf{u}_{t}\boldsymbol{\theta}_{t}+\rho\dot{ \mathbf{u}}_{t}\cdot\nabla\dot{\mathbf{u}}_{t}\boldsymbol{ \theta} _{t})\,\mathrm{d}\mathbf{x} $$

which permits with (92) to deduce the desired equations (20) and (21).

Appendix B: Local Energy Balance Condition

In this section we will derive the equivalent local condition of the global energy balance (16), which gives the desired Griffith’s law of motion (22) when combined with the local stability condition (20). The Lagrangian defined in (14) is explicitly dependent on time solely through the external work potential (11). Its total derivative can thus be given by

$$ \frac{\mathrm{d}\mathcal {L}}{\mathrm{d}t}=\frac{\partial\mathcal {L}}{\partial\mathbf{u}_{t}^{*}}\dot{\mathbf{u}}_{t}^{*}+ \frac{\partial\mathcal{L}}{\partial\dot {\mathbf{u}}_{t}^{*}}\ddot{\mathbf{u}}_{t}^{*}+\frac{\partial\mathcal {L}}{\partial l_{t}}\dot {l}_{t}+\frac{\partial\mathcal{L}}{\partial\dot{l}_{t}}\ddot {l}_{t}+\frac {\partial\mathcal{L}}{\partial t}. $$

(93)

Using the weak dynamic equilibrium (91) and the fact that $\dot{\mathbf{u}}_{t}^{*}-\dot{\mathbf{U}}_{t}\in\mathcal {C}_{0}$, we have

$$ \frac{\partial\mathcal{L}}{\partial \mathbf{u}_{t}^{*}} \bigl(\dot{\mathbf{u}}_{t}^{*}-\dot{ \mathbf{U}}_{t} \bigr)-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial \mathcal {L}}{\partial\dot {\mathbf{u}}_{t}^{*}} \bigl(\dot{ \mathbf{u}}_{t}^{*}-\dot{\mathbf{U}}_{t} \bigr)=\mathbf{0}. $$

(94)

Plugging (94) into (93), we obtain

$$ \frac{\mathrm{d}\mathcal{L}}{\mathrm {d}t}=\frac{\mathrm {d}}{\mathrm{d}t} \biggl(\frac{\partial \mathcal{L}}{\partial\dot{\mathbf{u}}_{t}^{*}} \dot{\mathbf{u}}_{t}^{*} \biggr)+\frac {\partial\mathcal{L}}{\partial\mathbf{u}_{t}^{*}}\dot{ \mathbf{U}}_{t}-\frac {\mathrm{d} }{\mathrm{d}t}\frac{\partial\mathcal{L}}{\partial\dot{\mathbf {u}}_{t}^{*}}\dot{ \mathbf{U}}_{t}+\frac{\partial\mathcal{L}}{\partial l_{t}}\dot{l}_{t}+ \frac {\partial \mathcal{L}}{\partial\dot{l}_{t}}\ddot{l}_{t}+\frac{\partial\mathcal {L}}{\partial t}. $$

(95)

With all necessary temporal regularity, we note that the energy balance condition (16) can be equivalently written as

$$ \frac{\mathrm{d}\mathcal{H}}{\mathrm {d}t}=\frac{\mathrm {d}}{\mathrm{d}t} (\mathcal{L}+2\mathcal{K} )=\frac{\mathrm{d}}{\mathrm{d}t} \biggl(\mathcal{L}-\frac {\partial\mathcal{L}}{\partial\dot{\mathbf{u}}_{t}^{*}}\dot{ \mathbf{u}}_{t}^{*}-\frac {\partial\mathcal{L}}{\partial\dot{l}_{t}}\dot{l}_{t} \biggr)= \frac {\partial \mathcal{L}}{\partial\mathbf{u}_{t}^{*}}\dot{\mathbf{U}}_{t}-\frac {\mathrm {d}}{\mathrm{d}t} \frac {\partial\mathcal{L}}{\partial\dot{\mathbf{u}}_{t}^{*}}\dot{\mathbf {U}}_{t}+\frac {\partial\mathcal{L}}{\partial t}. $$

(96)

Comparing (95) and (96), we obtain the desired local energy balance condition

$$\biggl(\frac{\partial\mathcal{L}}{\partial l_{t}}-\frac{\mathrm {d}}{\mathrm{d}t}\frac {\partial\mathcal{L}}{\partial\dot{l}_{t}} \biggr)\cdot \dot{l}_{t}=0. $$