The question demands numerous implicit assumptions from the responder.  Determination of mechanical properties normally depends on the assumption of uniform loading, and Considere's method can be used in a tensile test, to determine the onset of necking.  Then Bridgman's analysis can be used to determine the true stress during deformation after necking develops.  In the compression test there are complicating factors, such as the greasing of the platens.  How does the questioner compute "true stress" if the stress state and the specimen shape are continually changing?  So is the questioner presenting the unidirectional engineering stress based on the initial specimen cross sectional area?  Is it a cylindrical specimen?  As other responders have suggested, the answer depends on the purpose of the experiment and the criteria for the mechanical properties, as well as the ductility of the metal.  Eisenacher's answer is reasonable with a quibble about the failure being complete over the section.  If the metal is coming apart in shear, it may not be complete across the section.  So what is the criterion for failure?

Thank you very much, I have upload the clear figure, and I want to know the difference of failure initiation strain and failure strain.

From your Figure, the material seems to be ductile. In this case, you correctly pointed out in green the maximum stress (that is the Rm limit).

It depends on your application,  if plastic deformations are accepted or not. If not, the failure stress should be considered Rp0,2, divided by a safety factor (usually greater than 1.1).

As Mr. Viorel Paleu mentioned it is depending on your application and acceptance criteria, if the plastic deformation is not critical then the green line limit is correct. In addition to this you need to check at tensile region of the model in your simulation/lab test model to define the failure criteria/limit.

The question demands numerous implicit assumptions from the responder.  Determination of mechanical properties normally depends on the assumption of uniform loading, and Considere's method can be used in a tensile test, to determine the onset of necking.  Then Bridgman's analysis can be used to determine the true stress during deformation after necking develops.  In the compression test there are complicating factors, such as the greasing of the platens.  How does the questioner compute "true stress" if the stress state and the specimen shape are continually changing?  So is the questioner presenting the unidirectional engineering stress based on the initial specimen cross sectional area?  Is it a cylindrical specimen?  As other responders have suggested, the answer depends on the purpose of the experiment and the criteria for the mechanical properties, as well as the ductility of the metal.  Eisenacher's answer is reasonable with a quibble about the failure being complete over the section.  If the metal is coming apart in shear, it may not be complete across the section.  So what is the criterion for failure?

Thanks for  your answer. The specimen is a cylindrical shape. The aim of the compression test is to determine the plastic flow stress and the failure parameters. So I want to know how to identify the failure initiation strain from the compression test results.

The specimen has failed.  Why not tell me how it failed, the dimensions of the specimen, your method of calculating the true stress-strain curve, the shape it developed before failing, what steel was it (if you know) and any other factor you think relevant, e.g microstructural pre-test condition?  Then we can get an improved answer.

The failure initiation strain will be when the slope of the stress-strain curve changes (stress 1200, strain 0.05), and the strain grows rapidly with little increase in stress. At this point the microstructure begins to break, at least in certain foods, increasing the strain with little stress, until reach the end point of breakage (A).

The size of the specimen is 2.5mm in diameter and  1.6mm  length,  the material is Ti-6Al-4V alloy, and the flow stress is the compression test results carried out by the split Hoopkinson Bar (SHPB) equipment. The true stress-strain curve is calculated by the SHPB test software, thanks for your help~

Canet has defined the flow stress and his answer is not appropriate for metals, at least the metal of interest.  Given the alloy and its very high strength, failure initiation can be expected at the maximum flow stress because the alloy is running out of work-hardening and shear banding will be initiated.  During the decline of the stress, cavities can be expected to grow in the shear bands, ultimately joining up for complete separation.  So Eisenacher's account of failure is essentially correct.

I think the tension or compression test used in engineering, hardly determines the stresses and strains on microscopic scales. I refer to the problem of scale because the onset of plastic deformation is associated with phenomena such as dislocations, slip planes, among others. For this reason, a curve obtained from a compression test is impossible to determine the value that the questioner wants to know.

Thanks for your valuable answer, if the compression specimen has the crack along shear band, the flow stresses appear as there stages: elastic, stable plastic and softening (failure) stages, can we determine the failure initiation strain as point A, which means the formation of adiabatic shear band, and the start of failure?

To Guang Chen,my answer is yes.  To Juan Velez,my answer is that I interpret the questioner as requiring an engineering answer, not a scientific one in terms of the microscopic mechanisms you mention.  If he wishes to investigate those kinds of details, he would have to broaden his approach into the micro regime.