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Thermodynamics of crack nucleation

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Abstract

Crack nucleation issue is addressed in a comprehensive micromechanics-based framework allowing to bridge the 2D model with the more realistic 3D representation of a crack. The sudden and abrupt nature of the nucleation process argues in favour of adiabatic conditions rather than isothermal so that the formulation of the energy balance is formulated in terms of internal energy instead of Helmholtz free energy. The proposed theory provides the mean to evaluate the temperature rise as a function of the created entropy at the microscopic scale and the internal energy crack density at the macroscopic one.

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Notes

  1. The volume \(\varOmega (\ell )\) over which is taken the integral in (5) can be replaced by \(\varOmega _0\) if needed.

Abbreviations

t :

Time

\({\varvec{x}}\) :

Macroscopic position vector

\({\varvec{z}}\) :

Microscopic position vector

f :

Volume fraction of microcracks in the REV

\({d}\) :

Damage parameter

\({\varvec{T}}\) :

Stress vector

\({\varvec{\xi }}\) :

Displacement field

\({\varvec{\sigma }}\) :

Cauchy stress tensor

\({\varvec{\varepsilon }}\) :

Infinitesimal strain tensor

\(\rho \) :

Mass density

\(\ell \) :

Crack length

\(\mathcal {F}(\ell )\) :

2D crack subset

\(\mathcal {F}^\pm (\ell )\) :

Upper and lower lips of the crack

\({\varvec{N}}\) :

Outer unit normal to the crack upper lip

\(\epsilon \) :

Finite thickness of the macrocrack

\(\mathcal {L}_\epsilon \) :

3D geometrical model of a macrocrack

\(\varOmega _0\) :

Whole structure, including the crack

\(\varOmega (\ell )\) :

Complementary subset of the crack in \(\varOmega _0\)

\(\partial \varOmega _0\) :

Boundaries of \(\varOmega _0\)

\(\partial \varOmega ^T\) :

Subset of \(\partial \varOmega _0\) where the stress boundary conditions are defined

\(\partial \varOmega ^\xi \) :

Subset of \(\partial \varOmega _0\) where the displacement boundary conditions are defined

\(\rho {\varvec{F}}\) :

Mass density force

\(\mathcal {C}_F(t)\) :

Scalar time function controlling the time-dependent intensity of the body forces

\(\mathcal {C}_T(t)\) :

Scalar time function controlling the time-dependent intensity of the stress boundary conditions

\(\mathcal {C}_\xi (t)\) :

Scalar time function controlling the time-dependent intensity of the displacement boundary conditions

\({U}_{\mathrm{tot}}\) :

Internal energy of \(\varOmega _0\)

\({U}_{\mathrm{bulk}}\) :

Internal energy of \(\varOmega (\ell )\)

\({U}_{\mathcal {F}}\) :

Internal energy of \(\mathcal {F}\)

\(\varPsi \) :

Helmholtz free energy

\({K}\) :

Kinetic energy

\(\mathcal {P}_\mathrm{e}\) :

Rate of work done by external forces acting on the system in its actual motion

\(\varPhi ({\varvec{\xi }})\) :

Work of the given external forces

\(G_\mathrm{c}\) :

Critical energy release rate

\({u}\) :

Internal energy density

\(\psi \) :

Helmholtz free energy density

\({s}\) :

Entropy density

\(\overset{\circ }{{s}}_{\mathrm{cr}}\mathrm{d}t\) :

Volume entropy created between time t and \(t+\mathrm{d}t\)

\({\overset{\circ }{q}}\) :

Heat input density

r :

Heat supply density

\({\varvec{q}}\) :

Heat flow vector

\({D}\) :

Dissipation per unit volume

\({T}\) :

Temperature

\(T^0\) :

Temperature of reference

\(T^0{c}\) :

Heat capacity density

\(T^0{{\varvec{k}}}\) :

Strain latent heat density

\({\mathbb {C}}_{\mathrm{iso}}\) :

Isothermal elastic tensor

\({\mathbb {C}}_{\mathrm{ad}}\) :

Adiabatic elastic tensor

\(c^{\mathrm{th}}\) :

Thermal diffusivity

\(t^\mathrm{c}\) :

Characteristic time of heat transfer

U :

Macroscopic internal energy density

\(U^{\mathrm{FD}}\) :

Internal energy density of a fully damaged material

\(\tau \) :

Temperature variation with respect to the reference state

\({\varvec{E}}\) :

Average strain of the REV

\({\mathbb {C}}^{\mathrm{hom}}({d})\) :

Macroscopic homogenized elasticity of a microcracked domain

\({\varvec{\varSigma }}\) :

Macroscopic stress

S :

Macroscopic entropy density

\(\mathbb {I}\) :

Symmetrical fourth-order identity tensor

\(\mathbb {A}\) :

Strain concentration tensor

\(g_\mathrm{c}\) :

Material constant characterizing the dissipative process

\(G_\mathrm{c}^{\mathrm{ad}}\) :

Critical energy release rate in adiabatic context

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Acknowledgements

The authors gratefully acknowledge the reviewers for their constructive suggestions which significantly improved the paper. The authors are also grateful for the financial support by ANDRA (Agence Nationale pour la gestion des Déchets Radioactifs).

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Authors and Affiliations

  1. Laboratoire Navier, CNRS UMR8205, Ecole des Ponts Paristech, 6-8 Avenue Blaise Pascal, 77420, Champs-sur-Marne, France

    Thomas Carlioz, Luc Dormieux & Eric Lemarchand

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  1. Thomas Carlioz
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  2. Luc Dormieux
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  3. Eric Lemarchand
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Correspondence to Eric Lemarchand.

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Communicated by Andreas Öchsner.

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Appendix

Appendix

In order to integrate (86) between the sound and the fully damaged states, it is necessary to detail the quantities \({\varvec{K}}({d}){:}\,{\varvec{E}}\) and C. Hence, we will give some precisions about their order of magnitude when dealing with a claystone material. Let us consider the following orders of magnitude for such a material:

$$\begin{aligned} {c} = 1\times 10^{3}\,\hbox { J}\,\hbox {K}^{-2}\,\hbox {m}^{-3},\, \mathbb {X} \sim {\mathbb {S}} = 1\times 10^{-10}\,\hbox {Pa}^{-1},\, {{\varvec{k}}} = 1\times {10}^{5}\,\hbox {Pa}\,\hbox {K}^{-1} \end{aligned}$$
(91)

where the thermal expansion coefficient \(\mathbf {\alpha }\) is around \(1\times {10}^{-5}\,\hbox {K}^{-1}\). In this framework, the following simplification may be used:

$$\begin{aligned} {{\varvec{k}}}{:}\,\mathbb {X}{:}\,{{\varvec{k}}} \ll {c} \end{aligned}$$
(92)

Therefore C can be approximated by c.

Damage takes place when equality in (64) is reached. Considering the internal energy derived in (59), (64) reads:

$$\begin{aligned} -\dfrac{1}{2}{\varvec{E}}{:}\,\dfrac{\partial {\mathbb {C}}_{\mathrm{ad}}}{\partial d}{:}\,{\varvec{E}} +\dfrac{S-{s}^0}{{c}}\dfrac{\partial {\varvec{K}}}{\partial d}{:}\,{\varvec{E}} = g_{\mathrm{c}} \end{aligned}$$
(93)

Provided that the damage parameter range of variation is O(1), the order of magnitude of \(\frac{\partial {\mathbb {C}}_{\mathrm{ad}}}{\partial d}\) and \(\frac{\partial {\varvec{K}}}{\partial d}\) is \(|{{\mathbb {C}}_{\mathrm{ad}}}|\) and \(|{\varvec{K}}|\), respectively. Taking advantage of (91) and (92), the scale analysis of (93) gives:

$$\begin{aligned} |{\varvec{E}}| \sim \dfrac{1}{|{\mathbb {C}}_{\mathrm{ad}}|}\left( \dfrac{\left( S-{s}^0\right) \,|{\varvec{K}}|}{{c}}\right) \quad \Rightarrow \quad \dfrac{|{\varvec{K}}{:}\,{\varvec{E}}|}{\left( S-{s}^0\right) } \sim \dfrac{|{\varvec{K}}|^2}{|{\mathbb {C}}_{\mathrm{ad}}| {c}} \ll 1. \end{aligned}$$
(94)

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Carlioz, T., Dormieux, L. & Lemarchand, E. Thermodynamics of crack nucleation. Continuum Mech. Thermodyn. 32, 1515–1531 (2020). https://doi.org/10.1007/s00161-020-00863-7

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