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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 initial yield strength in compression subscript 11, and the ultimate strength in compression is defined as the ultimate strength in compression subscript 11 superscript Capital F. The relationship between these strengths is:

This equation reads the initial yield strength in compression subscript 11 equals the ultimate strength in compression subscript 11 superscript Capital F times parenthesis 1 minus parallel hardening initiation parameter parenthesis.

where 1 – N|| is the user-supplied reduction factor. For combined stress states, the ultimate yield strength from equation 13 is:

This equation reads ultimate strength in compression subscript 11 superscript Capital F equals the product of parallel wood strength compression times the square-root of parenthesis 1 minus the quotient Trial elastic stress invariant subscript 4 divided by parallel shear strength superscript 2 parenthesis.

For the case of uniaxial compressive stress, the ultimate yield strength is ultimate strength in compression subscript 11 superscript Capital F equals X subscript c .

The longitudinal stress update with hardening is:

This equation reads Orthotropic stress component subscript 11 equals initial yield strength in compression subscript 11 plus backstress tensor subscript 11.

where a11 is the hardening stress (backstress). At ultimate yield, this relationship becomes:

This equation reads ultimate strength in compression subscript 11 superscript Capital F equals yield strength in compression subscript 11 plus backstress tensor subscript 11 superscript max.

where backstress tensor subscript 11 superscript max is the maximum backstress that can be attained. As previously defined, ultimate strength in compression subscript 11 superscript Capital F is the total stress with hardening (at ultimate yield) and yield strength in compression subscript 11 is the stress without hardening (at initial yield).

Substitution of equation 198 into equation 195 and rearranging gives:

This equation reads the difference of 1 minus backstress subscript 11 superscript max divided by the product of parallel hardening initiation parameter times strength in compression subscript 11 superscript Capital F equals zero.

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:

This equation reads Parallel hardening model translational limit function equals 1 minus the quotient of backstress subscript 11 divided by the product of parallel hardening initiation parameter times strength in compression subscript 11 superscript Capital F.

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 ultimate strength in compression subscript 11 superscript Capital F from equation 199. Thus, G|| limits the growth of the backstress as the ultimate yield surface is approached.

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