Issue 43

L.C.H. Ricardo, Frattura ed Integrità Strutturale, 43 (2018) 57-78; DOI: 10.3221/IGF-ESIS.43.04 68 where:         0 0 3 4 max 1 S K C C S (2.33)   max max K S c F (2.34)       max 0 eff K S S c F The constants C 1 to C5 are determined by experimental test under constant amplitude loading. The factor F is the boundary correction factor on stress intensity. The analytical closure model provides no information about the amount of crack growth per cycle. Extending the crack an incremental value at the maximum applied stress simulates crack growth. Modified Dugdale model (crack closing) Various modified Dugdale models were proposed [59-60]. After Elber [45] defined crack closure, the research community developed analytical or numerical models to simulate fatigue crack growth and closure. These models were designed to calculate the growth and closure behavior instead of assuming such behavior as in the empirical models. Seeger [61] and Newman [48] developed two type of models. Seeger modified the Dugdale model and Newman developed the ligament or strip yield model. Later, a large group of similar models were also developed using the Dugdale framework. Budiansky & Hutchinson [62] studied crack closure using an analytical model, while Dill & Saff [30], Fuhring & Seeger [63], and Newman [64] modified the Dugdale model. Some have used analytical functions to model the plastic zone, while others divided the plastic zone into a number of elements. The model by Wang & Blom [65] is a modification of Newman’s model [64] but their model was the first to include weight functions in analyzing another crack configuration. C RACK PROPAGATION BY F INITE E LEMENT M ETHOD lber´s [45] experiments of crack closure with constant amplitude loading proposed the following equation for fatigue crack propagation rates:     ( ) n eff a C K N (3.1) where C and n are constants of the material and  K eff is the effective stress intensity factor range that can be calculated by:     eff eff K S c F (3.2) where: c - half length of the crack, F – boundary correction factor  S eff – effective stress range Fig. 7 shows the center crack panel that was used to evaluate the crack propagation. Fig. 8 shows the panel idealized to finite element method. The panel material was assumed to be elastic perfect plastic with a tensile (and compressive) yield stress,  0 , of 350.0 MPa and modulus of elasticity of 70000 MPa. These properties are for an aluminum alloy. The released nodes range from node A to node F. The accuracy of the calculated crack opening stresses would be affected by the mesh size chosen to model the crack tip region. A finer element mesh size would give more accurate results. Newman [29] evaluated three types of mesh, as shows in Tab. 3. E