PSI - Issue 42

Vítor M. G. Gomes et al. / Procedia Structural Integrity 42 (2022) 1552–1559 V.M.G. Gomes et al. / Structural Integrity Procedia 00 (2019) 000–000

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This research work intends to determine the critical spots and the corresponding fatigue crack initiation for parabolic leaf springs under real operating conditions. The loading conditions are predominantly under bending, however, non-vertical loadings exist during the suspension operationality. The developed model simulates the real fixtures and real loading on leaf springs. The elastoplastic behaviour of 51CrV4 is considered. A combined isotropic and kinematic hardening model approach is considered. The properties of fatigue strain-life and the cyclic hardening model are estimated from mechanical monotonic properties. With respect to the fatigue approach in this context, a critical plane approach is considered. Instead of the classical definitions of determining the orientation of the critical plane, in which the shear stress is calculated for all possible planes, an approach based on the maximum variance method, MVM, is considered. The MVM considered the maxi mum variance of the resolved shear stress as the primary parameter for idenfying the critical plane. This approach is coupled with the finite element method. The advantage of MVM over classical methods is that once the critical plane orientation is identified, the problem is no longer dependent on the length of the input loading history and the history of the shear stresses is normal to this one. This results in a very significant reduction of computation time in fatigue analysis (Susmel (2010)). The results of this numerical study were presented in terms of evaluating the maximum variance of the resolved shear stress on the upper face of the master leaf spring for two di ff erent scenarios of random lateral and vertical loading. The numerical results were compared to typical fracture surfaces for di ff erent potential regions for fatigue failure. Considering that the fatigue crack initiation in random amplitude regime is proportional to the variance of stress signal, then, according to the maximum variance method, the critical point for the crack initiation by fatigue should occur in the direction of the maximum variance of the stress at the critical plane. In the case of metallic materials, initiation occurs along the critical plane where the resolved shear stress variance is maximum. The variance of the resolved shear stress, var  τ q ( t )  is given by the fundamental mathematical framework the MVM (Susmel (2010)): var  τ q ( t )  = d T C v d , (1) where C v is the variance-covariance matrix and d is the vector that takes into account the orientation angles which define the orientation of critical plane: 2. The Maximum-Variance Method based on FEM

     1 2  sin( θ ) sin(2 φ ) cos( α ) + sin( α ) sin(2 θ ) cos 1 2  − sin( θ ) sin(2 φ ) cos( α ) + sin( α ) sin( θ ) sin − 1 2 sin( α ) sin(2 θ ) 1 2 sin( α ) sin(2 φ ) sin(2 θ ) − cos( α ) cos(2 φ ) sin( θ ) sin( α ) cos( φ ) sin(2 θ ) + cos( α ) sin( φ ) cos( θ ) sin( α ) sin( φ ) cos(2 θ ) − cos( α ) cos( φ ) cos( θ ) 2 ( φ )  2 ( φ ) 

    

    V x C x , y C x , z C x , xy C x , xz C x , yz C x , y V y C y , z C y , xy C y , xz C y , yz C x , z C y , z V z C z , xy C z , xz C z , yz C x , xy C y , xy C z , xy V xy C xy , xz C xy , yz C x , xz C y , xz C z , xz C xy , xz V xz C xz , yz C x , yz C y , yz C z , yz C xy , yz C xz , yz V yz

   

C v =

, and d =

with V i = var [ σ i ( t )] with i = x , y , z , xy , xz , yz , and C i , j = covar  σ i ( t ) , σ j ( t )  with i , j = x , y , z , xy , xz , yz . Figure 1 illustrates the definition of the resolved shear stress direction. The numerical optimization algorithm is suggested in (Susmel (2010)). In summary, the terms of variance and covariance are computed firstly, and then, using a multi variable optimization method, the optimal angles, θ p , α p , and φ p are found out. The numerical optimization method used is the Gradient Ascent Method (Snyman (2005)).