Abstract
Besides ductile materials considered hitherto, a variety of brittle materials, like concrete, rocks and ceramics, are widely employed in engineering practice. Their mechanical behavior can not be described by the elastic-plastic damage theory or by the viscoplastic damage theory discussed already. The present chapter is concerned with the damage and the deformation behavior of elastic-brittle materials, and the related continuum damage mechanics theory to describe them. In Section 9.1, to begin with, the microscopic mechanisms of damage and the ensuing mechanical behavior of microcracks in concrete will be discussed. Application of the simplest theory of isotropic damage to the damage process of concrete with unilateral crack effect is descried in Section 9.2.
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Notes
- 1.
Concrete, rocks and ceramics are usually called brittle materials. Observed in more detail, however, these materials may be accompanied by certain amount of dissipation prior to fracture.
Thus, more rigorous definition of the material is sometimes employed for this reason. Namely, a material is defined a brittle material if it has no dissipation before the crack initiation and its fracture does not results in any irreversible strain (e.g., glass and ceramics). On the other hand, a material is said a quasi-brittle material if a certain amount of dissipation occurs before crack initiation although it shows no irreversible strain in its whole process (e.g., concrete).
Hereafter, a term “brittle material” will be used both for the brittle and the quasi-brittle material, unless their distinction is particularly needed.
- 2.
This increase in volume is caused by shear deformation in addition to microcrack initiation due to compressive stress, and is called dilatancy.
- 3.
According to the definition of Eq. (3.26), Gibbs potential Γ is negative in general.
- 4.
According to the definition of Eq. (3.26), Gibbs potential Γ usually has a negative value.
- 5.
Though \({\mathbb{S}}^{D + }\) and \({\mathbb{S}}^{D - }\) of Eq. (9.61) are furnished by Eq. (9.63), the integration is not simple and straightforward. Moreover, for the change in stress sign, or the change of crack state between active and passive state, the constitutive equation of Eq. (9.61) may be discontinuous in some cases (Krajcinovic 1996).
- 6.
The evolution equations of Eqs. (9.63) and (9.64) have been derived on the basis of the experimentally observed features of the microcrack patterns under simple loading histories for \(\sigma _1 > \sigma _2 > \sigma _3\). As will be seen in Section 9.5.4, they describe well the damage development and its effect on the mechanical behavior of material under a uniaxial or a simple proportional loading. However, their applicability to the case of more general non-proportional loading is not obvious.
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Murakami, S. (2012). Elastic-Brittle Damage. In: Continuum Damage Mechanics. Solid Mechanics and Its Applications, vol 185. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2666-6_9
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