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Help!!! How to calculate the consistency parameter by consistency condition (potential function) of plasitc flow

hi, there!

I have read a "Manual for LS-DYNA Wood Material Model 143"(website: http://www.tfhrc.gov/safety/pubs/04097/sec150.htm#toc_1_5_1 and http://www.tfhrc.gov/safety/pubs/04097/append_e.htm).

I have two question:

1. how to calculate the consistency parameter (formula (180)) by potential function, I'm quite not understant how the formula (19), (20), (25), and (26) came out. Could anyone explain it to me or tell me what material I should get and read?

This equation reads Strain increment superscript p equals the product of plasticity consistency parameter divided by delta viscid with damage stress tensor evaluated at n.

This equation reads lowercase delta I subscript 1 over lowercase delta parallel lambda equals negative 2 times C subscript 11 times stress invariant subscript 1 over general parallel wood strength superscript 2.

This equation reads lowercase delta I subscript 4 over lowercase delta parallel lambda equals negative 4 shear moduli of an orthotropic material subscript 12 stress invariant subscript 4 over parallel shear strength superscript 2.

This equation reads the quotient of lowercase delta I subscript 2 over lowercase delta perpendicular lambda equals  the quantity C subscript 22 plus C subscript 23 end quantity, times stress invariant subscript 2, times the quantity of negative 4 over general perpendicular wood strength superscript 2 plus 1 over perpendicular shear strength superscript end quantity.

click on the image for Section 508 compliancy text

2. the effective strain-rate increment is given by formula (37), I wonder if the strain-rate increment is equal to strain increment  divided by time increment?

This equation reads Parallel scalar effective strain-rate increment equals the square root of the quantity of strain-rate increments parallel to the grain superscript 2 plus 2 times the strain-rate increments parallel to the grain superscript 2 plus 2 times strain-rate increments parallel to the grain superscript 2, end quantity. 

thank you!

Comments

my question may be simplified to: how to calculate the partial derivative of stress to plastic multiplier (or consistency parameter), eg:

This equation reads lowercase delta I subscript 1 over lowercase delta parallel lambda equals negative 2 times C subscript 11 times stress invariant subscript 1 over general parallel wood strength superscript 2. 

 thank you!!!

 

 

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