Issue 33

T. Itoh et alii, Frattura ed Integrità Strutturale, 33 (2015) 289-301; DOI: 10.3221/IGF-ESIS.33.33 292 S I 1  ( t )  ( t ) S I ( t ) S I ( t 0 ) S I 2 S I 3 Δ S I 0.5Δ S I S I mean S I max S I min S I max e X e Y e Z e R Figure 3 : Definition of principal range and mean principal value. C ASE STUDIES his section discusses a couple of case studies for  S I and S Imean under proportional and non-proportional loadings under biaxial plane stress condition. The symbol S takes (  ) in the case studies in the followings. Proportional loading Fig. 4 (a) illustrates three proportional strain paths for normal and shear strainings under plane stress condition in the normal strain – shear strain (  –  /  3) plot. The path A-A’ is push-pull loading, the B-B’ combined tension and shear loading and the path C-C’ shear loading. Figs 4 (b)  (d) show the variations of  1 ( t ),  3 ( t ),  I ( t ),  ( t )/2 and  ( t ) and Fig. 4 (e) the strain path in the polar figure presentation. Principal strains and their angle change are obtained in the stages of OA-OB-OC and OA’-OB’-OC’ as follows. Stage OA-OB-OC:   I I I 2 1 2 2 2 3 1 1 3 max ( ) 1 1 1 ( ) ( ) 2 2 ( ) ( ) ( ) ( ) ( ) , ( ) 0 , ( ) 0 t t t t t Since of t t k k A B or C t t                                  (7.a) Stage OA’-OB’-OC’:   I 2 1 2 2 2 3 3 1 3 ( ) ( ) 1 1 1 ( ) 2 2 ( ) ( ) ( ) ( ) ( ) 180 , ( ) 0 t t t t t Since of t t t t                              (7.b)  is the Poisson’s ratio. Note that  /  3=0 in the loading stages of OA and OA’ and  =0 in the loading stages of OC and OC’. The maximum principal strains (  1 ( t )) vary along the solid lines in Fig. 4 (b) in the loading stages of OA, OB and OC, and the minimum principal strains (  3 ( t )) along the dashed lines in the same loading stages. In these cases, the maximum principal strains all have a positive sign, whereas the minimum principal strains a negative sign.  I ( t ) makes the positive triangle waveforms as illustrated in Fig. 4 (c) because  I ( t ) is either  1 ( t ) or  3 ( t ) taking larger absolute strain as shown in Eq. (1). The principal strain,  I ( t ), changes its direction by an angle of 180  (  /2=90  ) at the origin O in the reversed T