PSI - Issue 28

R. Moreira et al. / Procedia Structural Integrity 28 (2020) 943–949 R.Moreira et al. / Structural Integrity Procedia 00 (2019) 000–000 3 for compression branch is obtained using the experimental data at points 4, 3, 2 and 1. This procedure is repeated for shear hysteresis loop. The functions � � �� , � and � � �� , � estimate the stresses for the maximum total strains, for positive and negative loading directions, respectively. The functions � � �� , � and � � �� , � estimate the plastic strain increments inherent to the maximum total strains and the functions � � �� , � and � � �� , � estimate the back stresses. Under tension-torsion loading conditions, it is obtained two hysteresis loops, one for the axial loading component and other for the shear one. Therefore, two different hysteresis loops are obtained, which are dependent of each other. This dependence is captured by the strain amplitude ratio ( � � ) given by the shear strain to axial strain ratio. The load level influence on cyclic response is modelled considering the biaxial strain level given by: �� � � � � � � � . 945

Fig. 1. Third degree polynomial interpolation for two hysteresis loops (axial and shear), Anes et al. (2019).

The axial and shear hysteresis loop for a given total strain is given by Eq. (3) for the axial stress loading component and by Eq. (4) for the shear one.

 

3

2

t t t a b c d      

Tension t  

t        3 2 t t t t t

t

(3) Where � � , � � , � � and � � are the polynomial coefficients for the tension branch of the hysteresis loop and � � , � � , � � and ℎ � � are the coefficients for the compression branch. In the same equation � is the axial total strain. (4) Similarly, � � , � � , � � and � � are the polynomial coefficients for the positive direction branch and � � , � � , � � and ℎ � � are the coefficients for the negative direction branch. In the same equation � is the shear total strain. The polynomial of Eq. (3) and (4) has as input values the output values of the functions � � �� , � � to � � �� , � � . These functions have the following shape under multiaxial loading conditions: (5) t Compression t t t t e f g h                3 2 t t t t Direction t t t t a b c d                 3 2 t t t t Direction t t t t e f g h               2 2 3 3 2 2 a b c d e f g h i j                         , , axial n sl i i sl i i sl i i sl i sl i i sl i sl P  