Issue 37

J. Kramberger et alii, Frattura ed Integrità Strutturale, 37 (2016) 153-159; DOI: 10.3221/IGF-ESIS.37.21 155 Newton method. The number of Fourier terms, the number of iterations and the incrementation during the cyclic time period can be controlled to improve the accuracy [15]. The damage initiation criterion is a phenomenological model for predicting the onset of damage due to stress reversals and the accumulation of inelastic strain in a low-cycle fatigue analysis. It is characterized by the accumulated inelastic hysteresis energy per cycle, Δw, in a material point when the structure response is stabilized in the cycle. The number of stress cycles N0, corresponding to the damage initiation, is given by [15]:   2 0 1 c N c w (1) where c1 and c2 are material constants. The damage evolution law describes the rate of the material stiffness degradation per cycle once the corresponding damage initiation condition has been reached. For damage in ductile materials, Abaqus/Standard assumes that the degradation of the stiffness can be modelled using a scalar damage variable D where the stress tensor at any given loading cycle during the numerical analysis can be expressed as follows [15]:       1 D (2) where  is the effective (or undamaged) stress tensor that would exist in the material in the absence of damage computed in the current increment. Once the damage criterion is satisfied at the material integration point, the damage state is calculated and updated based on the inelastic hysteresis energy for the stabilized cycle. The rate of damage per cycle is given by:  4 3 Δ c c w dD dN L (3) where c3 and c4 are material constants, and L is the characteristic length referred to the material point and based on the finite element geometry. It is assumed that material loses its load capacity when D = 1. When this condition is satisfied in a certain finite element, such an element can be removed from the mesh. In such a way, the damage (crack) propagation can be monitored in the proposed computational model. C OMPUTATIONAL MODEL general structure of lotus-type porous material is simplified in the computational model and represented by a square with random pore sizes and patterns, as is shown in Fig. 2. The size of the model is 3.3 × 3.3 mm [16]. Five computational models with different pore topologies and the same porosity, equal to 0.234, are generated and examined. The 2D numerical analysis is performed under plane strain loading conditions. All models are discretized with linear plane strain finite elements (CPE4R and CPE3 type of elements). The global size of finite elements 0.03 mm is chosen. The geometric structure, shown in Fig. 2a, is used as a basis for FEM analysis. Fig. 2b shows the applied boundary conditions with bi-axial loading. The load is applied to the top and bottom edge under displacement rate control. The displacement load vary sinusoidal in-phase, in such a way that the global deformation corresponds to values, given by Eq. (4) and (5):       0 sin y t (4)           0 1 sin xy t (5) where ε 0 is a global deformation in y-direction. Selected maximum value of global deformation is ε 0 = 0.1%. The load is applied in the low-cycle fatigue step with a time period of 1 second. In the numerical analyses, the linear kinematic hardening model with linearized monotonic stress-plastic strain behaviour presented in Tab. 1 is used. Furthermore, the Young’s modulus E = 200 GPa and the Poisson’s ratio ν = 0.33 is assumed. A