Minimum theorems in 3D incremental linear elastic fracture mechanics

Abstract

The crack propagation problem for linear elastic fracture mechanics has been studied by several authors exploiting its analogy with standard dissipative systems theory (see e.g. Nguyen in Appl Mech Rev 47, 1994, Stability and nonlinear solid mechanics. Wiley, New York, 2000; Mielke in Handbook of differential equations, evolutionary equations. Elsevier, Amsterdam, 2005; Bourdin et al. in The variational approach to fracture. Springer, Berlin, 2008). In a recent publication (Salvadori and Carini in Int J Solids Struct 48:1362–1369, 2011) minimum theorems were derived in terms of crack tip “quasi static velocity” for two-dimensional fracture mechanics. They were reminiscent of Ceradini’s theorem (Ceradini in Rendiconti Istituto Lombardo di Scienze e Lettere A99, 1965, Meccanica 1:77–82, 1966) in plasticity. Following the cornerstone work of Rice (1989) on weight function theories, Leblond et al. (Leblond in Int J Solids Struct 36:79–103, 1999; Leblond et al. in Int J Solids Struct 36:105–142,1999) proposed asymptotic expansions for stress intensity factors in three dimensions—see also Lazarus (J Mech Phys Solids 59:121–144,2011). As formerly in 2D, expansions can be given a Colonnetti’s decomposition (Colonnetti in Rend Accad Lincei 5, 1918, Quart Appl Math 7:353–362, 1950) interpretation. In view of the expression of the expansions proposed in Leblond (Int J Solids Struct 36:79–103, 1999), Leblond et al. (Int J Solids Struct 36:105–142, 1999) however, symmetry of Ceradini’s theorem operators was not evident and the extension of outcomes proposed in Salvadori and Carini (Int J Solids Struct 48:1362–1369, 2011) not straightforward. Following a different path of reasoning, minimum theorems have been finally derived.

Appendices

Appendix A: Crack propagation requirements

Stable crack propagation is meant to be a sequence of equilibrium states. At each load $k(\tau )$ corresponds a crack configuration l(s, τ) which eventually evolves quasi-statically, keeping the SIFs at the onset of propagation ${\text{K}}^{*} (\tau ){ \in }\partial {\mathbb{E}}$. Assume “time” τ in an interval ]t, t*] in which crack grows steadily with “velocity” $0 < \dot{l}\left( {s, \, \tau } \right) \, < \infty$. One has in view of (8):

$$ \begin{aligned} \varphi (\tau ) - \varphi (t) = & \,A\left( {K_{1}^{*2} (\tau ) - K_{1}^{*2} (t) + K_{2}^{*2} (\tau ) - K_{2}^{*2} (t)} \right) \\ & + B\left( {K_{3}^{*2} (\tau ) - K_{3}^{*2} (t)} \right) = 0 \\ \end{aligned} $$

(47)

with $A = \frac{{1 - v^{2} }}{E}$ and $B = \frac{1 - v}{E}$. If the crack path in the normal plane at any $s \in \mathcal{F}$ is taken to be smooth in the “time” interval ]t, t*], in other words along a curve at least of class C ¹, then the kinking angle $\theta (s,\tau ) = 0$ and ${\mathbf{K}}^{*} (\tau ) = {\mathbf{K}}(\tau )$ for being F = l in expansion (2). If furthermore one selects the local symmetry (Goldstein and Salganik 1974) as a kinking angle criterion, then $K_{2}^{*} = 0$ and therefore:

$$ \begin{aligned} \varphi (\tau ) - \varphi (t) = & A\left[ {K_{1}^{*} (\tau ) + K_{1}^{*} (t)} \right]\left[ {K_{1}^{*} (\tau ) + K_{1}^{*} (t)} \right] \\ & + B\left[ {K_{3}^{*} (\tau ) + K_{3}^{*} (t)} \right]\left[ {K_{3}^{*} (\tau ) + K_{3}^{*} (t)} \right] = 0 \\ \end{aligned} $$

(48)

Using SIFs expansion (18) one has:

$$ \begin{aligned} [K_{1}^{*} (\tau ) - K_{1}^{*} (t)] = & {\mathbf{e}}_{{\mathbf{1}}} \cdot \left\{ {{\mathbf{K}}^{*} (t)\frac{k(\tau )}{k(t)}} \right. \\ & + {\mathbf{K}}^{(1/2)} \frac{k(\tau )}{k(t)}\sqrt {l(s,\tau )} + {\mathbf{K}}_{0}^{(1)} \frac{k(\tau )}{k(t)}l(s,\tau ) \\ & + {\mathbf{K}}_{1}^{(1)} \frac{k(\tau )}{k(t)}(s,\tau ) + {\mathbf{K}}_{nl}^{(1)} \frac{k(\tau )}{k(t)}[l(s^{{\prime }} ,\tau ) - l(s,\tau ))] \\ & - {\mathbf{K}}^{*} (t)\} + o(l) \\ = & {\mathbf{e}}_{{\mathbf{1}}} \cdot \left\{ {{\mathbf{K}}^{*} (t)\frac{\delta k}{k(t)} + {\mathbf{K}}^{(1/2)} \sqrt {l(s,\tau )} + {\mathbf{K}}_{0}^{(1)} l(s,\tau )} \right. \\ & \left. { + {\mathbf{K}}_{1}^{(1)} \frac{\partial l}{\partial s}(s,\tau ) + {\mathbf{K}}_{nl}^{(1)} [l(s^{{\prime }} ,\tau ) - l(s,\tau )]} \right\} + o(\delta k \cdot l) \\ \end{aligned} $$

(49)

where e ₁ denotes the unit vector in direction 1. Analogously:

$$ \begin{aligned} [K_{3}^{*} (\tau ) - K_{3}^{*} (t)] & \\ & = {\mathbf{e}}_{3} \cdot \left\{ {{\mathbf{K}}^{*} (t)\frac{\delta k}{k(t)} + {\mathbf{K}}^{(1/2)} \sqrt {l(s,\tau )} + {\mathbf{K}}_{0}^{(1)} l(s,\tau )} \right. \\ &\;\; \left. { + {\mathbf{K}}_{1}^{(1)} \frac{\partial l}{\partial s}(s,\tau ) + {\mathbf{K}}_{nl}^{(1)} [l(s^{{\prime }} ,\tau ) - l(s,\tau )]} \right\} + o(\delta k \cdot l) \\ \end{aligned} $$

(50)

where e3 denotes the unit vector in direction 3.

By noting that l(s, 0) = 0 by definition of l(s, t) and assuming that it exists a bounded quasi-static velocity $\dot{l}\left( {s, \, t} \right)$ so that $l\left( {s, \, \tau } \right) = \dot{l}\left( {s, \, t} \right)(\tau - t)$ and a load variation velocity so that δk =$\dot{k}$(t)(τ − t) one has for τ → t ⁺:

$$ \begin{aligned} 0 = & 2(A\;K_{1}^{*} (t){\mathbf{e}}_{1} + B\;K_{3}^{*} (t){\mathbf{e}}_{3} ) \\ & \cdot {\mathbf{K}}^{(1/2)} \sqrt {\dot{l}(s,t)} \sqrt {\tau - t} \\ & + 2(A\;K_{1}^{*} (t){\mathbf{e}}_{1} + B\;K_{3}^{*} (t){\mathbf{e}}_{3} ) \\ & \cdot \left\{ {{\mathbf{K}}^{*} \frac{{\dot{k}(t)}}{k(t)} + {\mathbf{K}}_{0}^{(1)} \dot{l}(s,t) + {\mathbf{K}}_{1}^{(1)} \dot{l}^{{\prime }} (s,t)} \right. \\ & \left. { + {\mathbf{K}}_{nl}^{(1)} [\dot{l}(s^{{\prime }} ,t) - \dot{l}(s,t)]} \right\}(\tau - t) + o(\tau - t) \\ \end{aligned} $$

(51)

whence the conditions:

$$ \frac{\partial \varphi }{{\partial {\mathbf{K}}^{*} }} \cdot {\mathbf{K}}^{(1/2)} = 0 $$

(52)

$$ \begin{aligned} & \frac{\partial \varphi }{{\partial {\mathbf{K}}^{*} }} \cdot [{\mathbf{K}}^{*} \frac{{\dot{k}(t)}}{k(t)} + {\mathbf{K}}_{0}^{(1)} \dot{l}(s,t) + {\mathbf{K}}_{1}^{(1)} \dot{l}^{{\prime }} (s,t) \\ & \quad + {\mathbf{K}}_{nl}^{(1)} [\dot{l}(s^{{\prime }} ,t) - \dot{l}(s,t))] = 0 \\ \end{aligned} $$

(53)

at ${\mathbf{K}}_{2}^{*} = 0$

Appendix B: MERR and LS

Maximum energy release rate (MERR) onset of propagation uses as a magnitude ϑ the energy released during crack advance at any point along the crack front. Such a magnitude is related to stress intensity factors after a kink via Irwin’s formula, recently revised at a kink by several authors (Ichikawa and Tanaka (1982); Chambolle et al. (2010)) in two-dimensions, whence the onset of propagation reads

$$ \varphi = \frac{{1 - v^{2} }}{E}(K_{1}^{*2} + K_{2}^{*2} ) + \frac{1 - v}{E}K_{3}^{*2} - G_{C} $$

(54)

It seems extremely desirable, although probably quite involved, an extension of Chambolle et al. (2010) to the three dimensional case. As its formal derivation in the presence of kinking seems not to be available, the validity of Irwin’s formula for 3D, widely accepted in the fracture mechanics community, is here assumed.

As stated already several times in this note, the principle of Local Symmetry (LS) and MERR share the same onset of propagation. They differ on the criteria for kinking angle prediction. The kink angle predicted by the MERR descends from the general form (7). It reads:

$$ \begin{aligned} \frac{\partial \varphi }{\partial \theta } = &\, \frac{{1 - v^{2} }}{E}\left( {2K_{1}^{*} \frac{{\partial K_{1}^{*} }}{\partial \theta } + 2K_{2}^{*} \frac{{\partial K_{2}^{*} }}{\partial \theta }} \right) \\ & + \frac{1 - v}{E}2K_{3}^{*} \frac{{\partial K_{3}^{*} }}{\partial \theta } = 0 \\ \end{aligned} $$

(55)

Matrix ${\mathbb{F}}$ has been defined in terms of the ratio m = θ/π in Leblond (1999), Leblond et al. (1999) as

$$ \begin{aligned} F_{11} (m) = & 4.1m^{20} + 1.6.3m^{18} - 4.059m^{16} + 2.996m^{14} \\ & - 0.0925m^{12} - 2.88312m^{10} + 5.0779m^{8} \\ & + \left( {\frac{{\pi^{2} }}{9} - \frac{{11\pi^{4} }}{72} + \frac{{119\pi^{6} }}{15360}} \right)m^{6} \\ & + \left( {\pi^{2} - \frac{{5\pi^{4} }}{128}} \right)m^{4} - \frac{{3\pi^{2} m^{2} }}{8} + 1 \\ F_{12} (m) = & 4.56m^{19} + 4.21m^{17} - 6.915m^{15} + 4.0216m^{13} \\ & + 1.5793m^{11} - 7.32433m^{9} + 12.313906m^{7} \\ & + \left( { - 2\pi - \frac{{133\pi^{3} }}{180} + \frac{{59\pi^{5} }}{1280}} \right)m^{5} \\ & + \left( {\frac{10\pi }{3} + \frac{{\pi^{3} }}{16}} \right)m^{3} - \frac{3\pi m}{2} \\ F_{21} (m) = & - 1.32m^{19} - 3.95m^{17} + 4.684m^{15} - 2.07m^{13} \\ & - 1.534m^{11} + 4.44112m^{9} - 6.176023m^{7} \\ & + \left( { - \frac{2\pi }{3} + \frac{{13\pi^{3} }}{30} - \frac{{59\pi^{5} }}{3840}} \right)m^{5} \\ & + \left( {\frac{4\pi }{3} + \frac{{\pi^{3} }}{48}} \right)m^{3} + \frac{\pi m}{2} \\ F_{22} (m) = & 12.5m^{20} + 0.25m^{18} - 7.591m^{16} + 7.28m^{14} \\ & - 1.8804m^{12} - 4.78511m^{10} + 10.58254m^{8} \\ & + \left( { - \frac{32}{15} - \frac{{4\pi^{2} }}{9} - \frac{{1159\pi^{4} }}{7200} + \frac{{119\pi^{6} }}{15360}} \right)m^{6} \\ & + \left( {\frac{8}{3} + \frac{{29\pi^{2} }}{18} - \frac{{5\pi^{4} }}{128}} \right)m^{4} - \left( {4 + \frac{{3\pi^{2} }}{8}} \right)m^{2} + 1 \\ F_{33} (m) = & \left( {\frac{1 - m}{1 + m}} \right)^{m/2} \\ F_{13} (m) = & F_{31} (m) = F_{32} (m) = F_{23} (m) = 0 \\ \end{aligned} $$

It is straightforward to show that at K ₁ ≠ 0 Eq. (55) is equivalent to

$$ \begin{aligned} & \left[ {(F_{11} + \alpha_{2} F_{12} )\left( {\frac{{\partial F_{11} }}{\partial \theta } + \alpha_{2} \frac{{\partial F_{12} }}{\partial \theta }} \right)} \right. \\ & \left. { + (F_{21} + \alpha_{2} F_{22} )\left( {\frac{{\partial F_{21} }}{\partial \theta } + \alpha_{2} \frac{{\partial F_{22} }}{\partial \theta }} \right)} \right] \\ & + \frac{1}{1 - v}\alpha_{3}^{2} \frac{{\partial F_{33} }}{\partial \theta } = 0 \\ \end{aligned} $$

(56)

where $\alpha_{2} = K_{2} /K_{1}$ and $\alpha_{3} = K_{3} /K_{1}$. For a given material (i.e. a given Poisson ratio) at any couple α₂, α₃ the corresponding kink angle θ_MERR solves Eq. (56). At K₁ = 0 angle θ_MERR is plot as a function of ratio $\alpha_{32} = K_{3} /K_{2}$ and of Poisson ratio in Fig. 7.

The local symmetry criterion is the only notable exception to the mathematical representation (7). It provides the kink angle $\theta_{LS}$ through the equation $K_{2}^{*} = 0$:

$$ F_{21} + \alpha_{2} F_{22} = 0 $$

(57)

where $\alpha_{2} = K_{2} /K_{1}$ For any α₂ Eq. (57) provides the kink angle $\theta_{LS}$ which turns out to be independent on the mode 3 stress intensity factor.

Whereas thus in 2D the two angles θ_MERR and $\theta_{LS}$ differ from very small amounts, in 3D the scenario changes completely as it can be readily seen in Fig. 8. This fact may allow experimental campaigns of investigation to provide conclusive statements on which criteria better describes crack kinking in brittle materials.