I am doing a comparison between different anisotropy models. Now, I am understanding the Barlat 1991 model. I understand the concept but I have some doubts about how to proceed to calculate the parameters. I would be very greatful if you could help me.
1- In the original paper by Barlat says that the a, b, c parameters can be obtained from Newton Raphson based on three uniaxial test in the anisotropy axes. I did a trial considering an effective stress of 350MPa. With 1, 2 and 3 referring to anisotropic axes the uniaxial yield stress in each direction are: s1 = 350, s2 = 350 and s3 = 345. Taking the definition of yield function gives by Barlat in Eq(22d) and using Mathematica with FindRoot to calculate a, b, and c parameters
FindRoot[{phi1 == 350, phi2 == 350,
phi3 == 350}, {{a, 1}, {b, 1}, {c, 1}}]
where, ph1 is the yield function with the only stress different from 0 is s1, phi2 with only s2 and phi3 with only s3.
So, the values of a, b, c are very high a = 250.001, b = 250.001, c = 199.111.
Newton Raphson should be a multivariable approach, right?
Obiously, I am doing something wrong, but I don't know what.
2- In the original paper by Barlat says that the other three parameters (f, g, h) are derived from the three shear yield stress. With the only non zero stress component sxy, for example, I am able to calculate h, but as in the previous point the value is so high.
3- The princiapl deviatoric stresses given by Barlat are comple, but other authors as in the thesis by Boxun WU "Application of Constitutive Equations based on Non-Associated Flow Rules for the Plastic Deformation of Anisotropic Sheet Metals" paragraph 4.2.3, principal deviatoric stress seem to be not complex. Are they equivalent to Barlat ones?
4- Other authors use minimize function to optimize the parameters, Which is the best way to construct the minimization function? I mean, the r values dependent of orientation are calculated by R_angle = -dε2/(dε3) = -dε2/(dε1+ dε2), where dε1 is the derivative of yield function respect s1, dε2 respect s2 and dε3 respect s3. The directional r values is a monster function (with 1, 2, 3 the anisotropic axes).
Anyway, how many terms are recommended to include into minimization function?
is it possible to calculate rb value (r value for equibiaxial stress)? If yes, how is it calculated? Is rb calculated with dε2/dε1 for an orientation of 45º respect rolling direction?
5- In Hill 1948 the width strain increment (the increment of strain at right angles to the direction of loading, alpha) is written as:
dεα+π2=dε1sin2α+dε2cos2α−dγ12sinαcosα,
However, in Barlat 1989 paper this relation is given by:
dεα+π2=dε1sin2α+dε2cos2α−2 dγ12sinαcosα
in the Barlat 1989 this term has a factor of 2 compared with Hill 1948. Why this difference in the shear strain term?
Thanks in advance
Question 1 and 2 fixed. A mistake in the yield function exponent was the cause.
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