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Computational Technology Reviews
ISSN 2044-8430
Computational Technology Reviews
Volume 6, 2012
Material Length Scales in Fracture Analysis: From Gradient Elasticity to the Theory of Critical Distances
L. Susmel and H. Askes

Department of Civil and Structural Engineering, University of Sheffield, United Kingdom

Full Bibliographic Reference for this paper
L. Susmel, H. Askes, "Material Length Scales in Fracture Analysis: From Gradient Elasticity to the Theory of Critical Distances", Computational Technology Reviews, vol. 6, pp. 63-80, 2012. doi:10.4203/ctr.6.3
Keywords: gradient elasticity, generalised continuum, theory of critical distances, length scale, crack tip, fracture.

Various modelling tools exist for the simulation of the stress and strain fields around cracks. Many of these are based, in some form or another, on fracture mechanics. Much progress has been made in the formulation of linear and nonlinear versions of fracture mechanics. Furthermore, much progress also has been made in the development of novel numerical simulation tools for crack analysis and crack propagation, such as mesh adaptivity, meshless methods or the more recent family of partition-of-unity formulations of the finite element method. However, the combination of sophisticated material modelling with a sophisticated numerical technique often leads to simulation tools that are complicated to use. It is thus of interest to develop simulation technology that allows for a straightforward, simple and effective engineering interpretation of crack stability and crack propagation.

In the area of mechanics of materials, so-called generalised continua have received much attention in recent decades. Nonlocal, strain-gradient, micro-polar or micro-morphic continuum models have been formulated whereby the governing equations are extended with additional terms and, or additional kinematic variables that capture the intrinsic micro-structural behaviour of the material. With such enrichments, typical material behaviour can be described that is governed by an interaction between macro-structural excitation and micro-structural responses, such as dispersive wave propagation or indeed stress concentration around the tips of sharp cracks. For instance, many such generalised continuum models predict strongly localised, yet finite stresses and strains around the tip of a sharp crack – in contrast to the classical theory of elasticity. Of particular interest here is the strain-gradient elasticity theory attributed to Aifantis, whereby the Laplacian of the strains enters the constitutive equations. One of the main advantages of the Aifantis theory is that the strain-gradient dependence can be decoupled from the macro-structural response; in terms of equations this means that the governing equations can be split into two sets of equations. Firstly, the macro-structural response is determined using the standard equations of elasticity. Secondly, these results are used as input for a subsequent stage of analysis whereby the strain-gradient dependence is introduced. With this particular formulation of the Aifantis theory, finite element simulations are simplified enormously.

A second area of interest is that of the fracture and fatigue of materials. It has been noted by many researchers that reliable predictions on crack growth or crack arrest can be made not by evaluating the stresses and strains at the crack tip itself, but instead by assessing the stresses and strains at a finite distance away from the crack tip. This is the so-called theory of critical distances; the distance at which the stresses and strains are evaluated is a material parameter, which has been quantified for a wide range of materials and composites. Interestingly, numerical analysis according to the theory of critical distances follows the same philosophy as that of Aifantis’ theory of strain-gradient elasticity: a two-step approach is used, whereby a standard elastic analysis is followed by a post-processing step that introduces the material length scale and, thus, the micro-structural effects.

The similarities between gradient elasticity and the theory of critical distances are obvious and thus warrant further exploration. In this paper, we will review both theories and establish a formal link between them. This then subsequently can be used to develop robust finite element technology (learning from recent progress in finite element implementations for gradient elasticity) and materials science (learning from various developments in the theory of critical distances, in particular in identifying and quantifying the intrinsic length scales of a range of materials).

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