A number of different cementitious material models were briefly discussed in Sec- tion 2.1.1. In this section the dominant role of micro-cracking in governing the non- linear behaviour of these materials was discussed. This micro-cracking is thought
to occur initially in the matrix material, at matrix-inclusion interfaces, where the highest stresses are reached (Prado & Van Mier 2003).
There are two particular requirements when describing micro-cracks in con- crete. The first is to estimate the effective elastic properties of a micro-cracked media and the second is to determine the micro-crack initiation and evolution law. Micro-cracks are typically idealised as either penny-shaped, slit like or flat ellip- soidal voids (Mura 1987). Complex phenomena such as interacting micro-cracks, micro-cracking recovery and micro-cracking closure effects can also be described at the micro-scale. The directional dependent crack density parameter of Budiansky & O’Connell (1976) can be regarded as a damage variable (Jefferson & Bennett 2007) for brittle materials which indirectly links the penny-shaped crack volume fraction to directional damage degree and micro-crack distribution.
The effective elastic properties of the micro-cracked media are generally eval- uated using one of two approaches. In the direct method the micro-cracks or frac- ture strains are added to the elastic homogenised macro-strain. In the homogenised method, the micro-cracks are included in the homogenisation procedure.
The direct approach to determining the effective elastic properties has been given by many authors (Mura 1987, Nemat-Nasser & Hori 1999, Krajcinovic 2000, Voyiadjis & Kattan 2006). An example of the direct approach specifically used in micromechanical models, is given by the work of Pensée et al. (2002). A full 3D anisotropic damage model for brittle materials was developed using an elastic solid combined with penny-shaped open and closed micro-cracks. The additional fracture strain is provided by adding the contributions from a set of penny-shaped micro-cracks evaluated using Eshelby theory (Nemat-Nasser & Hori 1999, Mura 1987). This fracture strain was added to the strain contribution for the elastic solu- tion. A similar approach was employed by Jefferson & Bennett (2007) and Jefferson & Bennett (2010). Pensee & Kondo (2003) found that strain based formulations, taking account of moderate crack density, are preferred to stress based formula- tions, having non-interacting cracks, for brittle anisotropic damage with unilateral micro-cracking effects.
The alternative approach to determining the effective elastic properties of con- crete is the homogenised method which is typically based on the standard Eshelby homogenisation procedure. A composite with micro-crack inclusions in the matrix are up-scaled (Eshelby 1957, Mura 1987). However, in these basic models, crack interaction is not taken into account. Lee & Ju (2007) analysed the stress within an infinite solid containing both a penny-shaped micro-crack and a spherical inclusion. A two step superposition scheme was used to obtain a stress field over the crack site. This was interpreted in terms of the stress intensity factor for a penny-shaped crack. Both the direct and homogenised approaches are compared by Zhu et al. (2008).
The direct approach used is from Pensée et al. (2002) and the homogenisation schemes include Mori & Tanaka (1973) and Castaneda & Willis (1995). It was found that only Castaneda & Willis (1995) can take account of the spacial distri- bution of frictional micro-cracks (Zhu et al. 2009). This work was developed fur- ther to included tensile and compressive tests (Zhu et al. 2011). Their formulation, that combines micro-cracking and frictional sliding, led to a hybrid mechanistic- phenomenological model. Another example of such an approach was given by Brencich & Gambarotta (2001) where a plane crack model was simplified to an isotropic damage model with frictional sliding for closed cracks under compres- sion. Isotropic versions of this model were validated with experimental data. In later works (Gambarotta 2004) developed this model into an anisotropic version and applied it to a biaxial stress state problem.
Jefferson & Bennett (2007) looked to retain a mechanistic approach throughout their derivation of a micromechanical model for concrete by introducing a rough crack closure model alongside penny-shaped micro-cracks. This contact model was based on the Craft concrete model which simulated rough crack contact behaviour (Jefferson 2003).
Pichler, Hellmich & A. Mang (2007) studied the fracture process zone ahead of a main macro-crack and developed a model where the penny-shaped micro-cracks were treated as void inclusions using the classical Eshelby (1957) procedure. The penny-shaped voids were specialised for sharp open cracks to simplify the approach. This included the void having zero stiffness, a volume fraction linked to the Budi- ansky crack density parameter and where the height of micro-crack tends towards zero. Micromechanics and fracture energy theories were combined successfully al- though they showed that the approach led to an overly brittle post peak solution, thought to be due to the model not accounting for crack arresting due to the pres- ence of aggregate particles or large voids. This initial work was later developed to study the cracking risk of partially saturated porous geomaterials in a thermody- namically based microporoelasticity model (Pichler & Dormieux 2010a). Spherical micropores at the micro-scale were homogenised and up-scaled to form the matrix which included penny-shaped micro-cracks. This was further extended and applied to a drying shrinkage case (Pichler & Dormieux 2010b). One of the difficulties in simulating anisotropic cracking using a void is that, upon first cracking, the once isotropic medium turns into an anisotropic medium and the standard Eshelby ten- sors are no longer applicable.
Having addressed the effective elastic properties of a cracked media, attention is now briefly given to damage initiation and evolution laws. Pioneering work of Kachanov (1982) incorporated the initiation and propagation of micro-cracks in the theoretical framework of fracture mechanics. Many researchers have dedicated time
to dealing with issues of how these cracks are incorporated and homogenised with the elastic components. Alongside each of these developments an initiation crite- rion and continuing yield conditions are been proposed. A few of the significant contributions included in the review above are the energy-based yield criterion with a general elastic predictor and damage corrector scheme (Pensée et al. 2002) and a criterion based on a combination of friction and damage limit states (Gambarotta 2004). However a strain based damage rule with an experimentally derived expo- nential equation has been shown to be particularly effective (Jefferson & Bennett 2007, 2010, Mihai & Jefferson 2011). This damage initiation and evolution is pre- sented in Chapter 3, Section 3.3.
Avoiding the need for a numerical solution to evaluate the Eshelby tensor for a changing generally anisotropic matrix material has great advantages (Desrumaux et al. 2001). It is apparent that a combination of volumetric cracking built into the homogenisation equations, added strains to allow for directional cracking and allowing for progression from micro-cracks to macro-cracks would be very useful for describing a 3D cementitious material. Having a void as inclusions built into the modelling framework provides an opportunity to bring in self-healing through filling of these voids. Utilising these micromechanical frameworks, self-healing mechanisms are discussed in Section 2.4 and in Chapter 6.