/MAT/LAW116

Block Format Keyword Describes mixed mode, strain rate dependent material model with damage and failure.

This material is only compatible with solid hexahedron elements (/BRICK) and the cohesive solid property (/PROP/TYPE43 (CONNECT)).
Note: Not compatible with any failure model. All damage and failure are defined inside of the material directly.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW116/mat_ID/unit_ID
mat_title
ρi                
EI MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBaaaleaacaWGjbaabeaaaaa@37BB@ EII MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBaaaleaacaWGjbaabeaaaaa@37BB@ Thick Imass Idel Icrit  
GCI_ini MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaadoeadaWgaaWcbaGaamysaiaac+facaWGPbGaamOBaiaadMgaaeqaaaaa@3C37@ GCI_inf MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaadoeadaWgaaWcbaGaamysaiaac+facaWGPbGaamOBaiaadMgaaeqaaaaa@3C37@ ε˙GI MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaiaadaWgaaWcbaGaam4ramaaBaaameaacaWGjbaabeaaaSqabaaaaa@39A4@ fGI MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaadEeadaWgaaWcbaGaamysaaqabaaaaa@38A8@    
GCII_ini MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaadoeadaWgaaWcbaGaamysaiaadMeacaGGFbGaamyAaiaad6gacaWGPbaabeaaaaa@3D05@ GCII_inf MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaadoeadaWgaaWcbaGaamysaiaadMeacaGGFbGaamyAaiaad6gacaWGPbaabeaaaaa@3D05@ ε˙GII MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaiaadaWgaaWcbaGaam4ramaaBaaameaacaWGjbGaamysaaqabaaaleqaaaaa@3A72@ fGII MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaadEeadaWgaaWcbaGaamysaiaadMeaaeqaaaaa@3976@    
σA_I MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadgeacaGGFbGaamysaaqabaaaaa@3A5D@ σB_I MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadgeacaGGFbGaamysaaqabaaaaa@3A5D@ ε˙I MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaiaadaWgaaWcbaGaamysaaqabaaaaa@38A0@ Iorder_I Ifail_I    
σA_II MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadgeacaGGFbGaamysaiaadMeaaeqaaaaa@3B2B@ σB_II MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadgeacaGGFbGaamysaiaadMeaaeqaaaaa@3B2B@ ε˙II MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaiaadaWgaaWcbaGaamysaiaadMeaaeqaaaaa@396E@ Iorder_II Ifail_II    

Definition

Field Contents SI Unit Example
mat_ID Material identifier.

(Integer, maximum 10 digits)

unit_ID Optional unit identifier.

(Integer, maximum 10 digits)

mat_title Material title.

(Character, maximum 100 characters)

ρi Initial density.

(Real)

[kgm3]
EI MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBaaaleaacaWGjbaabeaaaaa@37BB@ Young’s (stiffness) modulus in normal direction per unit length.

(Real)

[Pam] MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaadaWcaaqaaiaadcfacaWGHbaabaGaamyBaaaaaiaawUfacaGLDbaaaaa@3AA3@
EII MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBaaaleaacaWGjbaabeaaaaa@37BB@ Shear (stiffness) modulus in tangent direction per unit length.

Default = EII=EI MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBaaaleaacaWGjbGaamysaaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaadMeaaeqaaaaa@3B5C@ (Real)

[Pam] MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaadaWcaaqaaiaadcfacaWGHbaabaGaamyBaaaaaiaawUfacaGLDbaaaaa@3AA3@
Thick Reference cohesive thickness.

(Real)

[m]
Imass Mass calculation flag.
= 1 (Default)
Element mass is calculated using density and mean area.
= 2
Element mass is calculated using density and volume.

(Integer)

Idel Failure flag indicating the number of integration points to delete the element (between 1 and 4).

Default = 1 (Integer)

Icrit Yield and damage initiation flag.
= 1 (Default)
Based on quadratic nominal stress.
= 2
Based on maximum nominal stress.

(Integer)

GCI_ini MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaadoeadaWgaaWcbaGaamysaiaac+facaWGPbGaamOBaiaadMgaaeqaaaaa@3C37@ Initial critical energy release rate for mode I (normal direction).

(Real)

[J]
GCI_inf MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaadoeadaWgaaWcbaGaamysaiaac+facaWGPbGaamOBaiaadMgaaeqaaaaa@3C37@ Upper bound of critical energy release rate. Indicates the strain rate dependency of GCI MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaadoeadaWgaaWcbaGaamysaaqabaaaaa@3884@ .

Default = 0.0 (Real)

[J]
ε˙GI MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaiaadaWgaaWcbaGaam4ramaaBaaameaacaWGjbaabeaaaSqabaaaaa@39A4@ Reference (lower bound) strain rate for GC strain rate dependency.

Must be defined if GCI_inf>0 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaadoeadaWgaaWcbaGaamysaiaac+facaWGPbGaamOBaiaadAgaaeqaaOGaeyOpa4JaaGimaaaa@3DFF@ .

(Real)

[Hz]
fGI MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaadEeadaWgaaWcbaGaamysaaqabaaaaa@38A8@ Shape factor for energy release rate before failure in mode I.

(Real)

GCII_ini MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaadoeadaWgaaWcbaGaamysaiaadMeacaGGFbGaamyAaiaad6gacaWGPbaabeaaaaa@3D05@ Initial critical energy release rate for mode II (shear).

(Real)

[J]
GCII_inf MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaadoeadaWgaaWcbaGaamysaiaadMeacaGGFbGaamyAaiaad6gacaWGPbaabeaaaaa@3D05@ Upper bound of critical energy release rate. Indicates the strain rate dependency of GCII MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaadoeadaWgaaWcbaGaamysaiaadMeaaeqaaaaa@3952@ .

Default = 0.0 (Real)

[J]
ε˙GII MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaiaadaWgaaWcbaGaam4ramaaBaaameaacaWGjbGaamysaaqabaaaleqaaaaa@3A72@ Reference (lower bound) strain rate for GC strain rate dependency.

Must be defined if GCII_inf>0 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaadoeadaWgaaWcbaGaamysaiaadMeacaGGFbGaamyAaiaad6gacaWGMbaabeaakiabg6da+iaaicdaaaa@3ECD@ .

(Real)

[Hz]
fGII MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaadEeadaWgaaWcbaGaamysaiaadMeaaeqaaaaa@3976@ Shape factor for energy release rate before failure in mode II.

(Real)

σA_I MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadgeacaGGFbGaamysaaqabaaaaa@3A5D@ Static yield stress in mode I.

(Real)

[Pa]
σB_I MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadgeacaGGFbGaamysaaqabaaaaa@3A5D@ Strain rate dependent yield stress term in mode I.

(Real)

[Pa]
ε˙I MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaiaadaWgaaWcbaGaamysaaqabaaaaa@38A0@ Reference (lower bound) strain rate value for yield stress rate dependency in mode I.

Must be defined σB_I>0 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadkeacaGGFbGaamysaaqabaGccqGH+aGpcaaIWaaaaa@3C29@ .

(Real)

[Hz]
Iorder_I Order of yield stress dependency on strain rate in mode I.
= 1 (Default)
Linear logarithmic dependency of strain rate.
= 2
Quadratic logarithmic dependency of strain rate.

(Integer)

 
Ifail_I Failure criteria defined by fGI MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaadEeadaWgaaWcbaGaamysaaqabaaaaa@38A8@ :
= 1 (Default)
Ratio of fracture energy.
= 2
Ratio of fracture displacements.

(Integer)

 
σA_II MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadgeacaGGFbGaamysaiaadMeaaeqaaaaa@3B2B@ Static yield stress in mode II.

(Real)

[Pa]
σB_II MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadgeacaGGFbGaamysaiaadMeaaeqaaaaa@3B2B@ Strain rate dependent yield stress term in mode II.

(Real)

[Pa]
ε˙II MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaiaadaWgaaWcbaGaamysaiaadMeaaeqaaaaa@396E@ Reference (lower bound) strain rate value for yield stress rate dependency in mode II.

Must be defined if σB_II>0 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadkeacaGGFbGaamysaiaadMeaaeqaaOGaeyOpa4JaaGimaaaa@3CF8@ .

(Real)

[Hz]
Iorder_II Order of yield stress dependency of strain rate in mode II.
= 1 (Default)
Linear logarithmic dependency of strain rate.
= 2
Quadratic logarithmic dependency of strain rate.

(Integer)

Ifail_II Failure criteria defined by fGII MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaadEeadaWgaaWcbaGaamysaiaadMeaaeqaaaaa@3976@ :
= 1 (Default)
Ratio of fracture energy.
= 2
Ratio of fracture displacements.

(Integer)

Example

Comments

  1. The elastic stiffness is defined with:


    Figure 1.
    Where,
    GP
    Plastic energy under constant stress
    GC
    Total energy
    i MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGPbaaaa@39A5@ ={I,II}
    The mode I (normal) and mode II (shear)
    The shape of the traction separation law is defined with:
    • Failure criteria defined by ratio of fracture energy (Ifail_i=1)(1)
      0fGi=GCi(ε˙eq)GCi(ε˙eq)<1σ(ε˙eq)22GCi(ε˙eq)Ei<1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6305@
    • Failure criteria defined by ratio fracture displacements (Ifail_i=2)(2)
      0fGi=δi2δi1δifδi1<1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgsMiJkaadAgacaWGhbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaacqaH0oazdaWgaaWcbaGaamyAaiaaikdaaeqaaOGaeyOeI0IaeqiTdq2aaSbaaSqaaiaadMgacaaIXaaabeaaaOqaaiabes7aKnaaBaaaleaacaWGPbGaamOzaaqabaGccqGHsislcqaH0oazdaWgaaWcbaGaamyAaiaaigdaaeqaaaaakiabgYda8iaaigdaaaa@4E30@
  2. The yield stress is defined as:
    • When Iorder_i=1:(3)
      σ(ε˙eq)=σA_i+σB_i.[max(0,ln(ε˙eqε˙i))] MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaeWaaeaacuaH1oqzgaGaamaaBaaaleaacaWGLbGaamyCaaqabaaakiaawIcacaGLPaaacqGH9aqpcqaHdpWCdaWgaaWcbaGaamyqaiaac+facaWGPbaabeaakiabgUcaRiabeo8aZnaaBaaaleaacaWGcbGaai4xaiaadMgaaeqaaOGaaiOlamaadmaabaGaciyBaiaacggacaGG4bWaaeWaaeaacaaIWaGaaiilaiGacYgacaGGUbWaaeWaaeaadaWcaaqaaiqbew7aLzaacaWaaSbaaSqaaiaadwgacaWGXbaabeaaaOqaaiqbew7aLzaacaWaaSbaaSqaaiaadMgaaeqaaaaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@5A93@
    • When Iorder_i=2:(4)
      σ(ε˙eq)=σA_i+σB_i.[max(0,ln(ε˙eqε˙i))]2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaeWaaeaacuaH1oqzgaGaamaaBaaaleaacaWGLbGaamyCaaqabaaakiaawIcacaGLPaaacqGH9aqpcqaHdpWCdaWgaaWcbaGaamyqaiaac+facaWGPbaabeaakiabgUcaRiabeo8aZnaaBaaaleaacaWGcbGaai4xaiaadMgaaeqaaOGaaiOlamaadmaabaGaciyBaiaacggacaGG4bWaaeWaaeaacaaIWaGaaiilaiGacYgacaGGUbWaaeWaaeaadaWcaaqaaiqbew7aLzaacaWaaSbaaSqaaiaadwgacaWGXbaabeaaaOqaaiqbew7aLzaacaWaaSbaaSqaaiaadMgaaeqaaaaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaaGaay5waiaaw2faamaaCaaaleqabaGaaGOmaaaaaaa@5B7C@

      Where, i MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGPbaaaa@39A5@ ={I,II}, the mode I and mode II.

  3. The equivalent strain rate is defined with:(5)
    ε˙eq=Δ˙2I+Δ˙2IIThick MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaiaadaWgaaWcbaGaamyzaiaadghaaeqaaOGaeyypa0ZaaSaaaeaadaGcaaqaaiqbfs5aezaacaWaaWbaaSqabeaacaaIYaaaaOWaaSbaaSqaaiaadMeaaeqaaOGaey4kaSIafuiLdqKbaiaadaahaaWcbeqaaiaaikdaaaGcdaWgaaWcbaGaamysaiaadMeaaeqaaaqabaaakeaacaWGubGaamiAaiaadMgacaWGJbGaam4Aaaaaaaa@47EA@
    Where,
    Δ˙I MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuiLdqKbaiaadaWgaaWcbaGaamysaaqabaaaaa@385F@
    Normal velocity.
    Δ˙II MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuiLdqKbaiaadaWgaaWcbaGaamysaiaadMeaaeqaaaaa@392D@
    Shear velocity.
  4. The rate dependent fracture energies are defined with:(6)
    GCi(ε˙eq)=GCi_ini+(GCi_infGCi_ini).exp(ε˙Giε˙eq) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6316@

    Where, i MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGPbaaaa@39A5@ ={I,II}, the mode I and mode II.

  5. Yield stress and damage law scheme:


    Figure 2.
  6. For yield and damage based on quadratic nominal stress (Icrit=1):
    • Mixed-mode yield initiation displacement is:(7)
      δm1=δI1δII1.1+β2δII12+(β.δI1)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaad2gacaaIXaaabeaakiabg2da9iabes7aKnaaBaaaleaacaWGjbGaaGymaaqabaGccqaH0oazdaWgaaWcbaGaamysaiaadMeacaaIXaaabeaakiaac6cadaGcaaqaamaalaaabaGaaGymaiabgUcaRiabek7aInaaCaaaleqabaGaaGOmaaaaaOqaaiabes7aKnaaDaaaleaacaWGjbGaamysaiaaigdaaeaacaaIYaaaaOGaey4kaSIaaiikaiabek7aIjaac6cacqaH0oazdaWgaaWcbaGaamysaiaaigdaaeqaaOGaaiykamaaCaaaleqabaGaaGOmaaaaaaaabeaaaaa@54E7@
      Where,
      δi1=σiEi MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaadMgacaaIXaaabeaakiabg2da9maalaaabaGaeq4Wdm3aaSbaaSqaaiaadMgaaeqaaaGcbaGaamyramaaBaaaleaacaWGPbaabeaaaaaaaa@3F5B@
      i MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGPbaaaa@39A5@ ={I,II}, the mode I and mode II.
      β=ΔIIΔI MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaeyypa0ZaaSaaaeaacqqHuoardaWgaaWcbaGaamysaiaadMeaaeqaaaGcbaGaeuiLdq0aaSbaaSqaaiaadMeaaeqaaaaaaaa@3E45@
    • Mixed-mode damage initiation is:(8)
      δm2=δI2δII2.1+β2δII22+(β.δI2)2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaad2gacaaIYaaabeaakiabg2da9iabes7aKnaaBaaaleaacaWGjbGaaGOmaaqabaGccqaH0oazdaWgaaWcbaGaamysaiaadMeacaaIYaaabeaakiaac6cadaGcaaqaamaalaaabaGaaGymaiabgUcaRiabek7aInaaCaaaleqabaGaaGOmaaaaaOqaaiabes7aKnaaDaaaleaacaWGjbGaamysaiaaikdaaeaacaaIYaaaaOGaey4kaSIaaiikaiabek7aIjaac6cacqaH0oazdaWgaaWcbaGaamysaiaaikdaaeqaaOGaaiykamaaCaaaleqabaGaaGOmaaaaaaaabeaaaaa@54EC@
      Where,
      δi2=δi1+fGi.GCiσi MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaadMgacaaIYaaabeaakiabg2da9iabes7aKnaaBaaaleaacaWGPbGaaGymaaqabaGccqGHRaWkdaWcaaqaaiaadAgacaWGhbWaaSbaaSqaaiaadMgaaeqaaOGaaiOlaiaadEeacaWGdbWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgaaeqaaaaaaaa@4819@
      i MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaakeaacaWGPbaaaa@39A5@ ={I,II}, the mode I and mode II.
  7. For yield and damage based on quadratic nominal stress (Icrit=2).
    • Mixed-mode yield initiation displacement is:
      If βδII1δI1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaeyizIm6aaSaaaeaacqaH0oazdaWgaaWcbaGaamysaiaadMeacaaIXaaabeaaaOqaaiabes7aKnaaBaaaleaacaWGjbGaaGymaaqabaaaaaaa@40E8@ :(9)
      δm1=δI1.1+β2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaad2gacaaIXaaabeaakiabg2da9iabes7aKnaaBaaaleaacaWGjbGaaGymaaqabaGccaGGUaWaaOaaaeaacaaIXaGaey4kaSIaeqOSdi2aaWbaaSqabeaacaaIYaaaaaqabaaaaa@42D1@
      If β>δII1δI1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaeyOpa4ZaaSaaaeaacqaH0oazdaWgaaWcbaGaamysaiaadMeacaaIXaaabeaaaOqaaiabes7aKnaaBaaaleaacaWGjbGaaGymaaqabaaaaaaa@403B@ :(10)
      δm1=δII1β.1+β2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaad2gacaaIXaaabeaakiabg2da9maalaaabaGaeqiTdq2aaSbaaSqaaiaadMeacaWGjbGaaGymaaqabaaakeaacqaHYoGyaaGaaiOlamaakaaabaGaaGymaiabgUcaRiabek7aInaaCaaaleqabaGaaGOmaaaaaeqaaaaa@4550@
      Where,
      β=ΔIIΔI MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaeyypa0ZaaSaaaeaacqqHuoardaWgaaWcbaGaamysaiaadMeaaeqaaaGcbaGaeuiLdq0aaSbaaSqaaiaadMeaaeqaaaaaaaa@3E45@
      ΔI MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaSbaaSqaaiaadMeaaeqaaaaa@3857@
      Displacement is mode I (normal).
      ΔII MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaSbaaSqaaiaadMeacaWGjbaabeaaaaa@3925@
      Displacement is mode II (shear).
    • Mixed-mode damage initiation is:
      If βδII2δI2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaeyizIm6aaSaaaeaacqaH0oazdaWgaaWcbaGaamysaiaadMeacaaIYaaabeaaaOqaaiabes7aKnaaBaaaleaacaWGjbGaaGOmaaqabaaaaaaa@40EA@ :(11)
      δm2=δI2.1+β2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaad2gacaaIYaaabeaakiabg2da9iabes7aKnaaBaaaleaacaWGjbGaaGOmaaqabaGccaGGUaWaaOaaaeaacaaIXaGaey4kaSIaeqOSdi2aaWbaaSqabeaacaaIYaaaaaqabaaaaa@42D3@
      If β>δII2δI2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaeyOpa4ZaaSaaaeaacqaH0oazdaWgaaWcbaGaamysaiaadMeacaaIYaaabeaaaOqaaiabes7aKnaaBaaaleaacaWGjbGaaGOmaaqabaaaaaaa@403D@ :(12)
      δm2=δII2β.1+β2 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaad2gacaaIYaaabeaakiabg2da9maalaaabaGaeqiTdq2aaSbaaSqaaiaadMeacaWGjbGaaGOmaaqabaaakeaacqaHYoGyaaGaaiOlamaakaaabaGaaGymaiabgUcaRiabek7aInaaCaaaleqabaGaaGOmaaaaaeqaaaaa@4552@
  8. The mixed-mode final damage is (Icrit=1,2):(13)
    δmf=δm1.(δm1δm2)EIGCIIcos2γ+GCI.(2GCII+δm1.(δm1δm2)EIIsin2γ)δm1(EIGCIIcos2γ+EIIGCIsin2γ) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8EE8@
    Where,
    γ=arccos(ΔIΔm) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaeyypa0JaciyyaiaackhacaGGJbGaai4yaiaac+gacaGGZbGaaiikamaalaaabaGaeuiLdq0aaSbaaSqaaiaadMeaaeqaaaGcbaGaeuiLdq0aaSbaaSqaaiaad2gaaeqaaaaakiaacMcaaaa@449A@
    Δm MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaSbaaSqaaiaad2gaaeqaaaaa@387B@
    Displacement is mixed mode.
  9. The plastic strain is defined as:
    • Mode I:(14)
      ΔpI=max(ΔpI(t1),ΔpIδm1cosγ,0) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaamiCamaaBaaaleaacaWGjbaabeaakiabg2da9iGac2gacaGGHbGaaiiEamaabmaabaGaeuiLdqKaamiCamaaBaaaleaacaWGjbaabeaakmaabmaabaGaamiDaiabgkHiTiaaigdaaiaawIcacaGLPaaacaGGSaGaeuiLdqKaamiCamaaBaaaleaacaWGjbaabeaakiabgkHiTiabes7aKnaaBaaaleaacaWGTbGaaGymaaqabaGcciGGJbGaai4BaiaacohacqaHZoWzcaGGSaGaaGimaaGaayjkaiaawMcaaaaa@54AA@

      Where, (t1) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaWG0bGaeyOeI0IaaGymaaGaayjkaiaawMcaaaaa@3A21@ is the value from previous time step.

    • Mode II:

      If (ΔII1ΔpII1(t1))2+(ΔII2ΔpII2(t1))2>δm1 MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6000@

      The plastic strain is computed for each direction 1 and 2 in the shear plane.(15)
      ΔpII1=ΔpII1(t1)+ΔII1ΔII1(t1) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaamiCamaaBaaaleaacaWGjbGaamysaiabgkHiTiaaigdaaeqaaOGaeyypa0JaeuiLdqKaamiCamaaBaaaleaacaWGjbGaamysaiabgkHiTiaaigdaaeqaaOWaaeWaaeaacaWG0bGaeyOeI0IaaGymaaGaayjkaiaawMcaaiabgUcaRiabfs5aenaaBaaaleaacaWGjbGaamysaiabgkHiTiaaigdaaeqaaOGaeyOeI0IaeuiLdq0aaSbaaSqaaiaadMeacaWGjbGaeyOeI0IaaGymaaqabaGcdaqadaqaaiaadshacqGHsislcaaIXaaacaGLOaGaayzkaaaaaa@5689@
      (16)
      ΔpII2=ΔpII2(t1)+ΔII2ΔII2(t1) MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaamiCamaaBaaaleaacaWGjbGaamysaiabgkHiTiaaigdaaeqaaOGaeyypa0JaeuiLdqKaamiCamaaBaaaleaacaWGjbGaamysaiabgkHiTiaaigdaaeqaaOWaaeWaaeaacaWG0bGaeyOeI0IaaGymaaGaayjkaiaawMcaaiabgUcaRiabfs5aenaaBaaaleaacaWGjbGaamysaiabgkHiTiaaigdaaeqaaOGaeyOeI0IaeuiLdq0aaSbaaSqaaiaadMeacaWGjbGaeyOeI0IaaGymaaqabaGcdaqadaqaaiaadshacqGHsislcaaIXaaacaGLOaGaayzkaaaaaa@5689@
  10. Stress value is reduced linearly from damage initiation to final damage ( Δm>δm2 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaSbaaSqaaiaad2gaaeqaaOGaeyOpa4JaeqiTdq2aaSbaaSqaaiaad2gacaaIYaaabeaaaaa@3D0C@ ).(17)
    D=max(Δmδm2δmfδm2,D(t1),0) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2da9iGac2gacaGGHbGaaiiEamaabmaabaWaaSaaaeaacqqHuoardaWgaaWcbaGaamyBaaqabaGccqGHsislcqaH0oazdaWgaaWcbaGaamyBaiaaikdaaeqaaaGcbaGaeqiTdq2aaSbaaSqaaiaad2gacaWGMbaabeaakiabgkHiTiabes7aKnaaBaaaleaacaWGTbGaaGOmaaqabaaaaOGaaiilaiaadseacaGGOaGaamiDaiabgkHiTiaaigdacaGGPaGaaiilaiaaicdaaiaawIcacaGLPaaaaaa@5242@

    The stress reduction is computed in the normal direction as:

    If ΔI>ΔpI MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaSbaaSqaaiaadMeaaeqaaOGaeyOpa4JaeuiLdqKaamiCamaaBaaaleaacaWGjbaabeaaaaa@3CBD@ , then σI=EI(ΔIΔpI) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMeaaeqaaOGaeyypa0JaamyramaaBaaaleaacaWGjbaabeaakmaabmaabaGaeuiLdq0aaSbaaSqaaiaadMeaaeqaaOGaeyOeI0IaeuiLdqKaamiCamaaBaaaleaacaWGjbaabeaaaOGaayjkaiaawMcaaaaa@43D1@ .

    otherwise, σI=EI(1D)(ΔIΔpI) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMeaaeqaaOGaeyypa0JaamyramaaBaaaleaacaWGjbaabeaakmaabmaabaGaaGymaiabgkHiTiaadseaaiaawIcacaGLPaaadaqadaqaaiabfs5aenaaBaaaleaacaWGjbaabeaakiabgkHiTiabfs5aejaadchadaWgaaWcbaGaamysaaqabaaakiaawIcacaGLPaaaaaa@47CB@ .

    For each direction 1 and 2 in the shear plane.(18)
    σII1=EII(1D)(ΔII1ΔpII1) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMeacaWGjbGaeyOeI0IaaGymaaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaadMeacaWGjbaabeaakmaabmaabaGaaGymaiabgkHiTiaadseaaiaawIcacaGLPaaadaqadaqaaiabfs5aenaaBaaaleaacaWGjbGaamysaiabgkHiTiaaigdaaeqaaOGaeyOeI0IaeuiLdqKaamiCamaaBaaaleaacaWGjbGaamysaiabgkHiTiaaigdaaeqaaaGccaGLOaGaayzkaaaaaa@4FFB@
    (19)
    σII2=EII(1D)(ΔII2ΔpII2) MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMeacaWGjbGaeyOeI0IaaGOmaaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaadMeacaWGjbaabeaakmaabmaabaGaaGymaiabgkHiTiaadseaaiaawIcacaGLPaaadaqadaqaaiabfs5aenaaBaaaleaacaWGjbGaamysaiabgkHiTiaaikdaaeqaaOGaeyOeI0IaeuiLdqKaamiCamaaBaaaleaacaWGjbGaamysaiabgkHiTiaaikdaaeqaaaGccaGLOaGaayzkaaaaaa@4FFE@
  11. The connection element is deleted when Δm>δmf MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaSbaaSqaaiaad2gaaeqaaOGaeyOpa4JaeqiTdq2aaSbaaSqaaiaad2gacaWGMbaabeaaaaa@3D3B@ .