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Publication Number: FHWA-HRT-04-097
Date: August 2007
Measured Variability Of Southern Yellow Pine - Manual for LS-DYNA Wood Material Model 143
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APPENDIX F. DERIVATION OF LIMITING FUNCTION FOR HARDENING MODEL
The functions that restrict the motion of each yield surface so that they cannot translate outside the ultimate surfaces are labeled G|| and G^ for the parallel and perpendicular modes, respectively. Each function is derived from the yield surface definition and hardening stress update. The limiting function for the parallel modes is derived here.
The desired attributes of the limiting function are G|| = 1 at initial yield and G|| = 0 at ultimate yield. Hardening is modeled in compression, but not shear or tension, so the only stress component with hardening is s11. The initial yield strength in compression is defined as , and the ultimate strength in compression is defined as . The relationship between these strengths is:
where 1 – N|| is the user-supplied reduction factor. For combined stress states, the ultimate yield strength from equation 13 is:
For the case of uniaxial compressive stress, the ultimate yield strength is .
The longitudinal stress update with hardening is:
where a11 is the hardening stress (backstress). At ultimate yield, this relationship becomes:
where is the maximum backstress that can be attained. As previously defined, is the total stress with hardening (at ultimate yield) and is the stress without hardening (at initial yield).
Substitution of equation 198 into equation 195 and rearranging gives:
The above function has the desired attribute in that it equals zero when the stress state lies on the ultimate yield surface. Thus, one defines:
The value of the limiting function is G|| = 1 at initial yield because a11 = 0 at initial yield. The value of the limiting function is G|| = 0 at ultimate yield because from equation 199. Thus, G|| limits the growth of the backstress as the ultimate yield surface is approached.