## E doE Gf doGf

5.6.8 ANALYTICAL PROCEDURES IN A FINITE ELEMENT MODEL

In a finite element model, the four-node isoparametric element is preferred in the implementation of the local approach of fracture-based models and has been used for the implementation of the described constitutive model. The standard local definition of damage is modified such that it refers to the status of the complete element. The average of the strain at the four Gauss points is obtained and the damage is evaluated from the corresponding principal strains of the element. The constitutive matrix [Cd] is updated depending on the opening/closing and the damage state of the element. The stresses at each integration point are then computed using the matrix [Cd] and the individual strains.

The characteristic length of the element is calculated approximately, using the square root of its total area. For an efficient control of the damage propagation in the finite element mesh, some adjustments have to be considered. During the softening regime the stiffness is reduced as a consequence of damage ,equation 5.23. At the unloading stage the secant modulus is calculated using equation 5.24 until closing of the crack. When the crack closes the material recovers its initial properties. In the reloading regime the last damage calculated before unloading is used again.

Finite element implementation of the damage mechanics

2. Initialization I select = 0, Emax = 0

3. Loop over elements e = 1,nel

(a) Compute deformations for each Gauss point: {e}gp = [B]3p{u}e

(b) If the element is already damaged call subroutine UPDATE

(c) If not: (selection of element with largest energy density)

Compute Ee = 2{a}e{e}e if Ee > Emax then Emax = Ee, Iselect = e

(e) Compute the internal force vector re = j'A [B]T{a}edA

f) Assemble the element contribution r ^ re

(a) Correction of stresses due to damage

(b) Update data base of damaged elements

Subroutine UPDATE (Stress and damage update)

1. Set damage parameter d = doid

2. Compute the average strains within the element and the corresponding principal strains: £1, e2 set £ = Max(e1, £2)

3. Check for loading/unloading and closing/opening of crack:

(a) if (£ > £max) the element is in a loading state

Calculate d from equation 5.23 update if em, £max = £

4. Compute[Cd] using equation 5.16

0 0