PSI- Issue 9

Riccardo Fincato et al. / Procedia Structural Integrity 9 (2018) 126–135 Author name / Structural Integrity Procedia 00 (2018) 000 – 000 Author name / Structural Integrity Procedia 00 (2018) 000 – 000 Author name / Structural Integrity Procedia 00 (2018) 000 – 000 Author name / Structural Integrity Procedia 00 (2018) 000 – 000 2 s    Author name / Structural Integrity Procedia 00 (2018) 000 – 0 0

4 4 4 4 4

1 1 1 1 1 Y Y Y

1 1 1 1 1

   

129

s s s s

D D D D D

H s  H s  H s  H s  H s 

    

                       1  1  1  1  1 1 1 1 1 1 D s                                               2 2 2 2 Y         D s      D s      D s              D s         Y       4 H 4 H 4 H 4 H 4 H H  H  H  H

3 3 3 3 3

(1 ) (1 ) (1 ) (1 ) (1 ) 

    

H H H H H

    

(6) (6) (6) (6) (6)

 1 1 1 1 2 1 1    2 1 1    1 1 H    H if H if H  H if H if H  2 H if     2 2 H if H if H  H if H if H               

          H     H       H       H       H              H H H H H     1 1 1 1

         

1               1           1 1 1

1   1 1   1 1   1 1   1  1 1

         

        

    

      

if H

a) a) a) a) a)

b) b) b) b) b)

Figure 2 a) Effect of the parameter β, b) effect of the parameter α 2 . Figure 2 a) Effect of the parameter β, b) effect of the parameter α 2 . Figure 2 a) Effect of the parameter β, b) effect of the parameter α 2 . Figure 2 a) Effect of the parameter β, b) effect of the parameter α 2 . The term Figure 2 a) Effect of the parameter β, b) effect of the parameter α 2 . 3 H s  was introduced to prevent the damage to accumulate at low values of accumulated plastic H   (see Figure 2) is responsible for regulating the ductile damage evolution under different loading condition, modifying the shape of the fracture envelope. A detailed explanation is offered in Fincato and Tsutsumi (2017a). The introduction of the new variable H   , function of the principal stress and the cumulative plastic strain, requires the calibration of three additional material parameters for a total number of six constants for the description of the damaging behavior of metals. A different approach was presented in Algarni et al. (2015) and in Bai and Wierzbicki (2010) adopting a modified version of the Mohr-Coulomb criterion for the material failure. This criterion has been successfully used for porous and granular materials (soil, rocks, concrete, etc.). However, its ability to take into account the exponential decay of the ductility with the stress triaxiality and the Lode angle make it particularly suitable for the application to metallic materials too. At this point, it is useful to introduce two dimensionless parameters which help to characterize the stress state: the stress triaxiality and the Lode angle parameter.   1.0 6 m Mises          (7) The locus of points formed by each set of stress triaxiality, Lode angle parameter and the equivalent strain at fracture define the failure envelope for the material. In particular, if the Lode angle effect on plasticity is limited and the pressure dependency of the yield surface can be neglected, then the expression of the failure envelope assume a simplified form, function of only four material parameters A , N , c 1 and c 2 (Bai and Wierzbicki, 2010): 3 H s  was introduced to prevent the damage to accumulate at low values of accumulated plastic strain, since no elastic modulus degradation can be detected at this stage. The new variable H   (see Figure 2) is responsible for regulating the ductile damage evolution under different loading condition, modifying the shape of the fracture envelope. A detailed explanation is offered in Fincato and Tsutsumi (2017a). The introduction of the new variable H   , function of the principal stress and the cumulative plastic strain, requires the calibration of three additional material parameters for a total number of six constants for the description of the damaging behavior of metals. 2.2. The modified Mohr- Coulomb’s ductile damage evolution law A different approach was presented in Algarni et al. (2015) and in Bai and Wierzbicki (2010) adopting a modified version of the Mohr-Coulomb criterion for the material failure. This criterion has been successfully used for porous and granular materials (soil, rocks, concrete, etc.). However, its ability to take into account the exponential decay of the ductility with the stress triaxiality and the Lode angle make it particularly suitable for the application to metallic materials too. At this point, it is useful to introduce two dimensionless parameters which help to characterize the stress state: the stress triaxiality and the Lode angle parameter.   1.0 6 m Mises          (7) The locus of points formed by each set of stress triaxiality, Lode angle parameter and the equivalent strain at fracture define the failure envelope for the material. In particular, if the Lode angle effect on plasticity is limited and the pressure dependency of the yield surface can be neglected, then the expression of the failure envelope assume a simplified form, function of only four material parameters A , N , c 1 and c 2 (Bai and Wierzbicki, 2010): 3 H s  was introduced to prevent the damage to accumulate at low values of accumulated plastic strain, since no elastic modulus degradation can be detected at this stage. The new variable H   (see Figure 2) is responsible for regulating the ductile damage evolution under different loading condition, modifying the shape of the fracture envelope. A detailed explanation is offered in Fincato and Tsutsumi (2017a). The introduction of the new variable H   , function of the principal stress and the cumulative plastic strain, requires the calibration of three additional material parameters for a total number of six constants for the description of the damaging behavior of metals. 2.2. The modified Mohr- Coulomb’s ductile damage evolution law A different approach was presented in Algarni et al. (2015) and in Bai and Wierzbicki (2010) adopting a modified version of the Mohr-Coulomb criterion for the material failure. This criterion has been successfully used for porous and granular materials (soil, rocks, concrete, etc.). However, its ability to take into account the exponential decay of the ductility with the stress triaxiality and the Lode angle make it particularly suitable for the application to metallic materials too. At this point, it is useful to introduce two dimensionless parameters which help to characterize the stress state: the stress triaxiality and the Lode angle parameter.   1.0 6 m Mises          (7) The locus of points formed by each set of stress triaxiality, Lode angle parameter and the equivalent strain at fracture define the failure envelope for the material. In particular, if the Lode angle effect on plasticity is limited and the pressure dependency of the yield surface can be neglected, then the expression of the failure envelope assume a simplified form, function of only four material parameters A , N , c 1 and c 2 (Bai and Wierzbicki, 2010): 3 H s  was introduced to prevent the damage to accumulate at low values of accumulated plastic strain, since no elastic modulus degradation can be detected at this stage. The new variable 2.2. The modified Mohr- Coulomb’s ductile damage evolution law 1  strain, since no elastic modulus degradation can be detected at this stage. The new variable 2.2. The modified Mohr- Coulomb’s ductile damage evolution law The term The term The term H   (see Figure 2) is responsible for regulating the ductile damage evolution under different loading condition, modifying the shape of the fracture envelope. A detailed explanation is offered in Fincato and Tsutsumi (2017a). The introduction of the new variable H   , function of the principal stress and the cumulative plastic strain, requires the calibration of three additional material parameters for a total number of six constants for the description of the damaging behavior of metals. A different approach was presente in Algarni et al. (2015) and in Bai and Wierzbicki (2010) adopting a modified version of the Mohr-Coulomb criterion for the material failure. This criterion has been successfully used for porous and granular materials (soil, rocks, concrete, etc.). However, its ability to take into account the exponential decay of the ductility with the stress triaxiality and the Lode angle make it particularly suitable for the application to metallic materials too. At this point, it is useful to introduce two dimensionless parameters which help to characterize the stress state: the stress triaxiality and the Lode angle parameter.   1.0 6 m Mises          (7) The locus of points formed by each set of stress triaxiality, Lode angle parameter and the equivalent strain at fracture define the failure envelope for the material. In particular, if the Lode angle effect on plasticity is limited and the pressure dependency of the yield surface can be neglected, then the expression of the failure envelope assume a simplified form, function of only four material parameters A , N , c 1 and c 2 (Bai and Wierzbicki, 2010): The term 3 H s  was introduc d to prevent the damage to accumulate at low values of accumulated plastic str in, since no elastic modulus degra a ion can be detected at this stage. The new variable H   (see Figure 2) is esponsible for regulating the ductile damage evolution under different loading condition, modifying the shape of the fracture envelope. A detailed explanation is offered in Fincato and Tsutsumi (2017a). The introductio of the new variable H   , function of the principal stress and the cumulative plastic strain, requires the calibration of three additional material parameters for a total number of six constants for the description of the damaging behavior of metals. 2.2. The modified Mohr- Coulomb’s uctile damag evolution law A dif e ent approach was presented in A garni et al. (2015) and in Bai and W erzbicki (2010) ad pting a modified version of the Mohr-Coulomb criterion for the m terial failure. This crit rion has been successfully used for porous and granular materials (soil, rocks, concrete, etc.). However, its ability to take into account the exponential decay of the ductility with the stress triaxial ty and the Lode angle make it pa ticularly suitable for the application to metallic materials too. At this point, it is useful to introduce two dimensionless parameters which help to characterize the stress state: the stress triaxiality and the Lode angle parameter.   1.0 6 m Mises          (7) The locus of points f rmed by each set of stress triaxiality, Lode angle parameter and the equivalent strain at fracture define the failure envelope for the material. In particular, if the Lode angle effect on plastic ty s limited and the pressure dependency of the yield surface can be neglected, then the expression of the failure envelope assume a simplified form, function of only four material parameters A , N , c 1 and c 2 (Bai and Wierzbicki, 2010): 1

                       

                       

                

                

1 1 N N N 1 N N

       

                   

6       6       6                   6     

                                  

c c c c c

2 2 2 2 2

1 1 1 1 1

    

2 2 2 2 2 A A A A A

1 1 1 1 1 sin sin 3 6 3 6 3 6 3 6 3 6 sin sin sin

   

1 1 1 1 1 c                            c c c c

(8) (8) (8) (8) (8)

cos cos cos cos cos

f  f  f  f  

    

1 1 1 1 1

c c c c c

3 3 3 3 3

Using the definition of the equivalent strain to fracture as a function of the stress triaxiality and Lode angle parameter it is possible to express the incremental damage as: Using the definition of the equivalent strain to fracture as a function of the stress triaxiality and Lode angle parameter it is possible to express the incremental damage as: Using the definition of the equivalent strain to fracture as a function of the stress triaxiality and Lode angle parameter it is possible to express the incremental damage as: Using the definition of the equivalent strain to fracture as a function of the stress triaxiality and Lode angle parameter it is possible to express the incremental damage as: 6 f Using the definition of the equivalent strain to fracture as a function of the stress triaxiality and Lode angle