U.S. Department of Transportation
Federal Highway Administration
1200 New Jersey Avenue, SE
Washington, DC 20590
202-366-4000


Skip to content
Facebook iconYouTube iconTwitter iconFlickr iconLinkedInInstagram

Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations

Report
This report is an archived publication and may contain dated technical, contact, and link information
Publication Number: FHWA-HRT-04-095
Date: November 2004

Manual for LS-DYNA Soil Material Model 147

PDF Version (1.96 MB)

PDF files can be viewed with the Acrobat® Reader®

APPENDIX A. DETERMINATION OF PLASTICITY GRADIENTS

In the following, the gradient of the yield surface in stress space is determined. The yield function is:

Equation 34. F equals negative P times the sine of phi plus the square root of the following: J subscript 2 times K theta squared, plus the product of a parameter for determining how close the modified surface is fitted to the standard Mohr-Coulomb yield surface, A, squared, times the sine, squared, of theta. Then subtract C times the cosine of theta, and this equals 0. 34

where:

J3= third invariant of the stress deviator

e= material parameter describing the ratio of triaxial extension strength to triaxial compression strength

We need to find equation 34d. A convenient method is:

Equation 35. Partial F divided by Partial sigma equals C subscript 1 times the quotient of Partial P divided by sigma, plus C subscript 2 times the quotient of Partial times the square root of J subscript 2 divided by Partial sigma, plus C subscript 3 times the quotient of Partial J subscript 3 divided by sigma. 35

where:

Here, si is the stress deviator. Now, we just need to determine the coefficients C1, C2, and C3.

Equation 36. C subscript 1 equals negative sine of theta. (36)

 

Equation 37. C subscript 2 equals the quotient of the square root of J subscript 2 times K theta divided by the square root of the following: J subscript 2 times K theta plus A squared plus the sine, squared, of theta. Then multiply that by K theta minus the tangent of 3 theta times the quotient of Partial K theta divided by Partial theta. (37)

 

Equation 38. C subscript 3 equals the quotient of the square root of J subscript 2 times K theta divided by the square root of the following: J subscript 2 times K theta plus A squared plus the sine, squared, of theta. Then multiply that by the quotient of the square root of 3 divided by 2 times the cosine of 3 theta J subscript 2, times the quotient of Partial K theta divided by Partial theta. (38)

 

Equation 39. The quotient of Partial K theta divided by Partial theta equals the quotient of 8 times the sum of 1 minus E squared times the cosine of theta times the sine of theta, divided by the following: 2 times the sum of 1 minus E squared times the cosine of theta, plus the product of negative 1 plus 2 times E times the following raised to the half power: negative 4 times E plus 5 times E squared plus 4 times the sum of 1 minus E squared, times the cosine, squared, of theta. Then take this entire result and subtract the quotient of the square of negative 1 minus 2E, plus 4 times the sum of 1 minus E squared, times the cosine, squared, of theta, times negative 2 times the sum of 1 minus E squared times the sine of theta, minus the quotient of 4 times the sum of negative 1 plus 2 times E, times the sum of 1 minus E squared, times the cosine of theta times the sine of theta, divided by the following, raised to the half power: negative 4 times E plus 5 times E squared plus 4 times the sum of 1 minus E squared times the cosine, squared, of theta. Then divide the entire amount by 2 times the sum of 1 minus E squared times the cosine of theta, plus the product of negative 1 plus 2 times E times the following, squared: negative 4 times E plus 5 times E squared plus 4 times the sum of 1 minus E squared, times the cosine, squared, of theta. (39)

Note that equation 39a is not defined at e = 0.5 and equation 39b.

For the hardening functions, see equations 40 and 41.

Equation 40. Partial F divided by Partial phi equals P times the cosine of phi plus C times the sine of phi plus the quotient of A squared times the cosine of phi times the sine of phi, divided by the square root of the following: J subscript 2 times K theta squared, plus A squared times the sine, squared, of phi. (40)

 

Equation 41. Partial phi divided by Partial E subscript EP equals H times the following: 1 minus the quotient of phi minus phi subscript init divided by N times phi subscript max. (41)

 

Equation 42. H equals Partial F divided by Partial sigma times C times Partial F divided by Partial sigma, which equals 9 times KB subscript 0 squared plus 4 times GB subscript 1 squared times J subscript 2 plus four-thirds times GB subscript 2 squared times J subscript 2 squared plus 12 times GB subscript 1 times B subscript 2 times J subscript 2. (42)

where:

In comparison with Abbo and Sloan:

Equation 43. B subscript 0 equals the sine of theta divided by 3. (43)

 

Equation 44. B subscript 1 equals C subscript 2 divided by 2 times the square root of J subscript 2. (44)

 

Equation 45. B subscript 2 equals C subscript 3. (45)

Therefore,

Equation 46. H equals K times the sine of theta squared, plus GC subscript 2 squared, plus four-thirds times GC subscript 3 squared times J subscript 2 squared plus the quotient of 6 times GC subscript 2 times C subscript 3 times J subscript 3 divided by the square root of J subscript 2. (46)

 

Equation 47. The change in sigma subscript IJ equals the change in Lambda times C subscript IJKL times the quotient of Partial F divided by Partial sigma subscript KL, which equals the change in Lambda times the following: K times the sine of theta times Kronecker delta subscript IJ plus the quotient of GC subscript 2 divided by the square root of J subscript 2, times S subscript IJ plus 2 times GC subscript 3 times T subscript IJ. (47)

where:


Previous | Table of Contents | Next

Federal Highway Administration | 1200 New Jersey Avenue, SE | Washington, DC 20590 | 202-366-4000
Turner-Fairbank Highway Research Center | 6300 Georgetown Pike | McLean, VA | 22101