Civil Engineering Departmental Seminar - Fracture and Complexity: Scale effects on the ductile-to-brittle transition in plain and reinforced materials

Fracture and Complexity: Scale effects on the ductile-to-brittle transition in plain and reinforced materials

Brittle solids often show an unstable structural behaviour represented by a negative slope in the load-displacement softening response. This means that the load must decrease to obtain a stable crack propagation. In extremely brittle cases, crack propagation occurs suddenly with a catastrophic drop in the load carrying capacity, and the load-displacement softening branch assumes a virtual positive slope. If the loading process is controlled by the displacement, the curve presents a discontinuity, and the representative point drops onto the lower branch with negative slope. In this case, both load and displacement must decrease to obtain a controlled crack propagation. Such a phenomenon, the so-called snap-back instability, was deeply investigated with reference to crack growth in quasi-brittle materials (Carpinteri, 1984; 1989). In the framework of Linear Elastic Fracture Mechanics, the cusp catastrophe represents the classical Griffith instability for very brittle systems.

Part I deals with Nonlinear Fracture Mechanics models (in particular, the Cohesive Crack Model to describe strain-localization both in tension and in compression) and their peculiar consequences: fold catastrophes (post-peak strain-softening and snap-through instabilities) or cusp catastrophes (snap-back instabilities) in plain or reinforced structural elements. How can a relatively simple nonlinear constitutive law, which is scale-independent, generate a size-scale dependent ductile-to-brittle transition? Constant reference is made to Dimensional Analysis and to the definition of suitable nondimensional brittleness numbers that govern the transition. These numbers can be defined in different ways, according to the selected theoretical model. The simplest way is that of directly comparing critical LEFM conditions and plastic limit analysis results. This is an equivalent way −although more effective for finite-sized cracked plates− to describe the ductile-to-brittle size-scale transition, if compared to the traditional evaluation of the crack tip plastic-zone extension in an infinite plate. In extremely brittle cases, the plastic zone or process zone tends to disappear and the cusp catastrophe conditions prevail over the strain-softening ones and tend to coincide with the LEFM critical conditions in the case of initially cracked plates.

Part II deals with the unstable mechanical behaviour of fibre-reinforced materials with a linear elastic matrix and the analogy with weakened materials.

Let us consider a tension test specimen where reinforcing fibres are embedded in the matrix, as illustrated in Figure 1a. In addition, let us consider the case of a specimen containing a distribution of collinear micro-cracks, as illustrated in Figure 1b. A load P is applied, opening the faces of an edge crack that propagates through the fibres or the collinear micro-cracks. Propagation will occur alternately within the matrix and through the heterogeneities (Carpinteri and Accornero, 2018; 2019). The loading process is controlled by the monotonically increasing crack length.

In both cases, the structural response presents a discrete number of snap-back instabilities with related peaks and valleys (Fig.1c). After each single peak, the crack starts growing in the matrix. Thus, the descending branches after peaks describe the crack growth between a fibre and the next, or between a micro-crack tip and the next. The crack arrests at the minimum of each valley, which represents the achievement of the next fibre or crack tip (Fig. 1a,b). The analogy between strengthened and weakened zones consists therefore in a multiple snap-back mechanical response, where descending branches of propagating cracks alternate with ascending (linear) branches of arrested cracks.

References

Carpinteri, A. (1984) “Interpretation of the Griffith instability as a bifurcation of the global equilibrium”, Proceedings of the N.A.T.O. Advanced Research Workshop on Application of Fracture Mechanics to Cementitious Composites, 287-316.

Carpinteri, A. (1989) “Cusp catastrophe interpretation of fracture instability”, Journal of the Mechanics and Physics of Solids, 37:567-582.

Carpinteri A., Accornero F. (2018) “Multiple snap-back instabilities in progressive microcracking coalescence”, Engineering Fracture Mechanics, 187:272-281.

Carpinteri A., Accornero F. (2019) “The Bridged Crack model with multiple fibers: Local instabilities, scale effects, plastic shake-down, and hysteresis”, Theoretical and Applied Fracture Mechanics, 104:102351.

Image

Figure 1. Regular distribution of reinforcing fibers in a brittle-matrix specimen with an edge crack (a); Brittle specimen with an edge crack and collinear micro-cracks (b); Load-displacement response diagram (c).