Initiation of a periodic array of cracks in the thermal shock problem: A gradient damage modeling

Abstract

This paper studies the initiation of cracks in the thermal shock problem through the variational analysis of the quasi-static evolution of a gradient damage model. We consider a two-dimensional semi-infinite slab with an imposed temperature drop on its free surface. The damage model is formulated in the framework of the variational theory of rate-independent processes based on the principles of irreversibility, stability and energy balance. In the case of a sufficiently severe shock, we show that damage immediately occurs and that its evolution follows first a fundamental branch without localization. Then it bifurcates into another branch in which damage localization will take place to finally generate cracks. The determination of the time and mode of that bifurcation allows us to explain the periodic distribution of the so-initiated cracks and to calculate the crack spacing in terms of the material and loading parameters. Numerical investigations complete and quantify the analytical results.

Size effect of thermal shock crack patterns in ceramics and numerical predictions

“…Further, Li et al 34 proposed a non-local failure model to simulate the thermal shock crack evolution. On the basis of the variational model to fracture developed by Francfort et al 35,36 , Bourdin 37,38 , Sicsic 39 , and Cyron 40 et al conducted numerical studies for fracture analysis induced by thermal shock. These recent works verified and complemented each other, much promoted the studies on thermal shock cracking phenomenon of ceramics.…”

An overview of the modelling of fracture by gradient damage models

“…As shown in [47], for ∆T < ∆T c the solution is purely elastic with no damage (α = 0 everywhere). For ∆T > ∆T c the solution evolves qualitatively as in Figure 10, with (i) the immediate creation of an x-homogeneous damage band parallel to the exposed surface, (ii) the bifurcation of this solution toward an x-periodic one, which (iii) further develops in a periodic array of crack bands orthogonal to the exposed surface.…”

“…We resume here the results reported in [13,47] on the thermal shock of a brittle slab. For this problem a number of experimental work are available in the literature [6,45,21].…”

“…These bands further propagate with a period doubling phenomenon (iv). A detailed analytical study of the bifurcation phenomenon observed in the phase (ii) is reported in [47]. The three columns in Figure 10 show the phases (ii)-(iv) of the evolution for ∆T /∆T c equal to 2, 4 and 8.…”

“…It constitutes essentially a summary of the works devoted to the variational approach to fracture, starting from [19] and followed by several other papers like [12] where the first numerical tests using Ambrosio-Tortorelli's model are presented, [38,39] where a general method for the construction of gradient damage models is proposed, [42] where a method for the experimental identification of a gradient damage model is proposed, [9] or [40,41] where some general qualitative properties of such models are discussed, [4] where the delicate issue of the asymmetric behavior between tension and compression is addressed, [46] where a fundamental link with Griffith's fracture theory is proved, [47] and [13] where a complete theoretical and numerical analysis of the thermal shock problem is made and compared to experimental results.…”