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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:


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