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- Conventional formulation for true stress-strain and engineering
stress-strain conversion are:
Strain e=ln(1+E) (1)
Stresss s=S(1+E) (2)
Here e and s are true strain and stress, and E and S are engineering
strain and stress.
The equation for strain conversion is by definition, therefore no
question about it.
The equation for stress conversion is based on the assumption:
a. material is volume preserved, which is true for
metal material under extensive plastic deformation.
b. Stress distribution in the specimen cross section is
uniformly distributed, which is ok in general for standard specimen before
necking occurs.
For common metal material, elastic deformation is
compressible, therefore eq.(2) will have error in elastic and early stage of
plastic deformation because of neglecting elastic volume changes. If one
wants to account for elastic volume change, the formulation is as follows:
s=S(1+E)/(1+(1-2nu)S/Y) (3)
Y is young's modulus and nu is Poisson's ratio. Now you can
see how much you can get from this exact formulation. Compare to the
experimental scatter this is real nothing. This equation is get by adjust
elastic volume strain as epsv=(1-2nu)*S/Y for unaxial tension. for common
metal, this value is really smale, usually less then a few of thousand
percents.
- For finite element analysis with plasticity, what we really need is
stress-plastic strain curve. Although for your convenience, ANSYS let you
input stress-total strain curve, but we internally convert it to
stress-plastic strain curve. The elastic deformation is actually defined by
young's modulus and Poisson's ratio. But one need to make sure the first
slope in stress-strain curve is actually the Young's modulus.
- To get a right stress-strain curve, usually it is more than simple
using equation (1) and (2) to convert the S-E to s-e curve, one need to
adjust the gauge slip, mechine compliance, and may be other factors. Some of
these factors may have been adjusted when you get test data, such as machine
compliance and some system errors. A simple way is to plot the s-e curve in
log-log scale, then find straight line for the Young's modulus, and then
define yield stress. With these two values you can now adjust your s-e
curve. A right Young's modulus is not easy to get although looks simple, it
really needs a very careful conducted test and it is better to have several
tests to make sure the value is right. Also you may need to smooth your
curve and get rid some of data points for the plastic part of curve. At this
point, you can see it is really no point to say that eq. (3) should be used
instead of (2).
- After necking happens it is really hard to get a right true stress
value. You may want to stop your test a few times and measure the necking
radius to get a better average true stress, which of course is not an easy
task. A further step, you may then run finite element analysis at the
loading point where you stop and get the stress distribution, and then try
to get a better value for the true stress. However this is very tricky. I
haven't read the paper refered in this post, so I can't comment on that.
they may have a clever way. fortunately, for practical interest, we usually
do need that part of data.
- For elastic-plastic finite element analysis, conventional
displacement formulation is usually more than enough. Mixed u-p formulation
only helps when nu is very close to 0.5, which is not usual for metal
materials. However, mixed u-p elements are very good for incompressible or
nearly incompressible materials, such as rubber. For elastic-plastic finite
element analysis in case of metal material, mixed u-p is usually much costly
and perform no better than u formulation. Linear elements with B-bar may be
the best choice. By the way, in additional to the existing hyper elements in
ANSYS, in 5.7 we have also introduced mixed u-p formulation for following
elements 182, 183, 85, 186, and 187, in which the Lagrangian multiplier is
introduced for the pressure as independent degree of freedom. These elements
are applicable for both elastic-plastic material as well as rubber like
materials.
- If you wish to define stress-plastic strain curve to avoid double
defining Young's modulus, there is an undocumented command for you in 5.7.
TB,NLISO,mat,NTEMP,NPTS,3.
TBTEMP,t1
TBPT,,x,y
TBPT,,x,y
...
TBTEMP,t2
TBPT,,x,y
TBPT,,x,y
...
Where NTEMP is number of temperature points and NPTS is number of
data points.
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