Quasibrittle materials, such as concrete, rock, tough 2.1 Isotropic damage ceramics, or ice, are characterized by the develop- The simple isotropic damage model considered here ment of nonlinear fracture process zones, which can is described by the basic equations be…
Quasibrittle materials, such as concrete, rock, tough 2.1 Isotropic damage ceramics, or ice, are characterized by the develop- The simple isotropic damage model considered here ment of nonlinear fracture process zones, which can is described by the basic equations be macroscopically described as regions of highly lo- calized strains. The degradation of strength and stiff- σ = (1−ω)De:ε (1) nessduetotheprogressivegrowthofmicrocracksand their coalescence is convenientlydescribedbymodels ω = g(κ) (2) based on continuumdamage mechanics. If a standard (local) continuum theory is used, the softening part f (ε,κ) ≤ 0, κ˙ ≥ 0, κf˙ (ε,κ) = 0 (3) of the stress-strain law must be properly adjusted ac- cording to the size of the numerically resolved band in which σ is the stress tensor, ε is the strain tensor, of localized strain, which typically corresponds to the D is the elastic stiffness tensor, ω is a scalar dam- size of a finite element, or to a similar discretiza- e tion parameter if another numerical technique is used. age variable, κ is an internal variable that drives dam- This adjustment eliminates the pathological depen- age,andf isthedamageloadingfunction.Thetypical dence of the solution on the finite element mesh and form of the loading function is ensures that the global energy dissipation in the fail- f (ε,κ) = ε˜(ε)−κ (4) ure process is captured correctly.