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Using 5.20 and 5.21, the evolutionary model for damage can be expressed as:

d =1 £0[2exp(-b(£ - £o)) - exp(-2b(£ - £o))] (5.23) If a simple linear softening curve is assumed, Figure 5.8-D can show that:

The fundamental issue of this approach lies in the introduction of a geometrical factor, /ch, in the constitutive model. When the finite element method is used, a so-called mesh-dependent hardening modulus is obtained. This technique was proposed by Pietruszcak and Mroz and was employed by a number of authors. Using equation 5.18 ensures conservation of the energy dissipated by the material. Amongst many strategies used to ensure mesh objectivity, the mesh-dependent hardening technique is the most practical for mass structures such as dams. Condition in equation 5.22 should be interpreted as a localization limiter on the characteristic length of the volume representing the global behaviour of the material. In other words, if lch > fGf, it is not possible to develop strain softening in the volume. For mass concrete, using average values for E = 30000 MPa, Gf = 200 N/m and ft = 2MPa yields a limit for /ch = 3 m. This leads therefore to a reasonable mesh size for dam models. The introduction of such parameter is not for numerical convenience. The characteristic length can be related to the Fracture Process Zone (FPZ) commonly used in fictitious crack model for concrete.

5.6.7 Opening and closing of the crack and initial damage

When the strain is increasing, damage will also increase. During cyclic loading the strains are reversed and unloading will occur. Experimental cyclic loading of concrete in tension shows evidence of permanent strain after unloading. Under compression the material recovers its stiffness. The classical split of the total strain in a recoverable elastic part £e and an inelastic strain £m gives:

£ = ee + e%n = £e + Aemax where £max is the maximum principal strain reached by the material and A is a calibration factor varying from 0 tol. Figure 5.8-E illustrates this criterion. The value A = 0.2 is selected; the unloading-reloading stiffness becomes :

When the principal strain is less than £in the crack is considered closed.

Alternatively, the crack will open when the principal strain is greater than a£

The damage model presented here is based on three parameters: the elastic modulus E, the initial strain threshold £0 and the fracture energyGf. If the element of volume is initially damaged (d = do), the secant modulus and the fracture energy are reduced by (1 — do) so the effective values are (Figure 5.8-F):

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