# Material Models for Damage and Failure

Computer analysis of concrete structures requires a general and robust material model for distributed cracking and other types of strain-softening damage such as softening plastic-frictional slip. The material model must perform realistically under a wide range of circumstances. The problem ean be approached through two types of models: (1) the continuum approach, in which case the structure is usually solved by finite element discretization (although boundary elements and other methods are possible), and (2) the discrete (or lattice) approach, taking the form of discrete element method or its variants—the random particle model or lattice model. In the former approach, the material is characterized by a general nonlinear triaxial stress-strain relation coupled with a nonlocal formulation. In the latter approach, the material is represented by a lattice of particles and connecting bars for which simple rules of deformation and breakage must be devised.

At present, the continuum approach is more general, more widely applicable to structural analysis under general types of loading. The discrete approach provides some valuable insight into the micromcchanics of failure and the role of heterogeneity, but only when the failure is due principally to tensile cracking and fracture. The computational demands of the discrete approach are still prohibitive for large structures and three-dimensional analysis, and attempts to develop the discrete approach for compression or compression – shear failures have so far been unsuccessful. In this chapter, we will first discuss the continuum approach and later briefly review the lattice approach.

The preceding chapter, dealing with nonlocal formulations, already presented one of two necessary components of the continuum approach to general analysis of softening damage due to microcracking and frictional-plastic slip. The other necessary component–the general triaxial stress-strain relation—will be described in this chapter. We have already touched this subject in chapter 8 while describing the triaxial stress-strain relation for the crack-band model, such as the lixcd-crack and rotating crack models. But these simple models are not sufficiently general to deal with compression splitting and compression shear, cracking combined with plastic-frictional slip and softening slip.

Formulation of a general constitutive relation for such phenomena is a rather difficult problem, to which numerous studies have been devoted during the last two decades. Although many valuable advances have been made, this chapter will present in detail only one approach—the microplane model, which currently appears most realistic, powerful, and versatile. Other approaches, which use classical types of constitutive relations based on the invariants of the stress and strain tensors and include models such as plasticity, continuum damage mechanics, fracturing theory, plastic-fracturing theory, and endochronie theory, will not be treated.

All the constitutive models describing fracture exhibit properties such as post-peak strain softening and deviations from the normality rule (or Drucker’s postulate). As discussed in Chapter 8, these properties, which are inevitable if the constitutive relation should describe cracking, friction, and loss of cohesion realistically, cause well-known mathematical difficulties such as ill-posedness of the boundary value problem, spurious localization instabilities, and spurious mesh sensitivity. To avoid these difficulties, the constitutive relations presented in this chapter must be combined with some kind of localization limiter. The nonlocal approach described in the preceding chapter is an effective method of solving these problems.

It is often thought that the continuum approach cannot be applied to the final stages of failure, in which damage localiz. es into large continuous cracks. However, the continuum approach can provide a relatively good (albeit not perfect) model for the propagation of such cracks. The reasons have already been explained in Chapter 8, in connection with the crack band model, which may be regarded as the simplest version of the nonlocal approach. The width of the localized damage band has, in most cases, negligible influence on the results of structural analysis. A zero width, that is, a distinct crack, and a

finite width (not excessively large, of course) often yields about the same results. Forcing, through the nonlocal concept, the distinct crack to spread over a width of several finite element sizes, or forcing a narrow damage band to be wider than the real width, is usually admissible, provided that the energy dissipation per unit length of advance of the band is adjusted to remain the same. It should be noted that such spreading of damage over a width of several element sizes is also a convenient way to avoid the directional bias of finite element mesh.