You may try increase dimensions (height and cross-section) of  testing sample to desirable  sizes by stacking of a number of samples which you can make from rolled sheet copper. 

No problem. Even when you make the compression test, the final shape of the sample will be with a round section. In steel always happen this situation.

I understand you want to produce a deformation gradient across the length of your specimen, one issue related to the square cross section would be that you will loose the radial symmetry around the sample's axis so you will probably have different gradient profiles whether you cut your sample parallel to one of the sides or parallel to the square diagonal, for example, or whether your metallografic section contains the symmetry axis (center of the sample) or not. If you are just willing to get different  values of deformation across the length I would advise you to make sure you have the symmetry axis in the metallographic plane (pay attention you always remove material when you cut and grind your sample) and look only to the center of the sample (disregarding what happens around). If you want to do a more sophisticated analysis, I suggest you combine your measurement with some Finite Element Analysis, otherwise you will have trouble interpreting the deformation gradient. There are quite sophisticated codes (proprietary) designed for forging modelling you could use, but you could also try some cheaper (and not so user friendly) code, like ANSYS.

As long as the geometry and the test condition fulfills plane strain definition, your test data can be valid and the plane strain formulation using slab-method, upper bound method, velocity field etc could be used to convert the measured load-displacement to the effective stress-strain data. Minimizing friction makes the post processing easier and more valid.

you may find the following useful for a simple FEM modeling of the test:

[1] S. Kobayashi, S.-I. Oh, T. Altan, Metal forming and the finite element method, Oxford University press, New York, 1989.

Of course, friction is important factor. See, please, the paper attached concerning that, although there are some techniques to reducing friction forces which really disturb stress - strained state of specimen (liquid and solid  lubricants, concave surface of  specimen and so). But may be take the advantages of contact indentation (macro- or micro-) to analyse variation of mechanical properties of material along rolling?    

I think here friction is a big issue as described by some other researchers on this forum. I think you should read the paper sent by Serge to overcome the problem and obtain the valid tests results.

Thank you for your valuable comments. The issue of friction would still exist even if I use samples for testing with a circular cross-section, wouldn't it? Would the issue of friction more prominent if the samples have a square cross-section?

You'll need to lubricate well in order to reduce the extent of barreling of the deformed sample, and with a square cross section you are going to get severe strain incompatibilities/changes at the corners. Copper being ductile might be quite forgiving whereas other metals might crack on severe deformation of such a sample. That can also depend upon the level (volume fraction) of non-metallic inclusions present in the metal, their type, shape and physical distribution.

You have not given us much information, but it appears you are also going to change the strain path significantly from ~ PS during the rolling deformation to compression at 90 deg. to ND in these tests. (At least that is how it sounds from your description) This will be important at the micromechanical level. In general such factors are not taken into account with a straight continuum mechanics approach, (in J2 plasticity, ABAQUS FEM). Can I suggest you look at a few papers where the effect of such strain path changes has been considered before you embark upon this work. Good luck with the tests.

Jointly with point of view of John Butler, in developing  detail (microlevel) analysis of plasticity and failure processes, the book "Micromechanics and Nanosimulation of Metals and Composites: Advanced Methods ... by Siegfried Schmauder and Leon Mishnaevsky as well as papers of Prof. V. Panin with cooworkers on physical mesomechanics (f.e. Panin V.E. etc Foundations of physical mesomechanics of structurally...) may be recommended for your attention. 

 In course of this discussion the idea came to me as an afterthought about possibility of tension test instead of compression test. No friction, suitable size for sample fabrication... Why not? 

Thank you so much everyone! I will read some from the suggested literature and continue the test to separate out friction and the issue of sample shape.

You may check the foolowing work:

Metallurgical Transactions A

March 1992, Volume 23, Issue 3, pp 935-951

Compression testing techniques to determine the stress/strain behavior of metals subject to finite deformation

M. L. Lovato, M. G. Stout

Did you complete the testing?  Were there any issues?  I am doing a similar test for Stainless steel.

The general issue with compression test is heterogeneity of strain and strain rate in its sample. The more barrelling, the more heterogeneity  and therefore more error in the post processing. The existing solutions for compression test rely on many assumptions.

These assumptions are more valid for the case of axisymmetric sample. So, using square sample could bring serious errors in post processing of the results unless there exist a reasonable solution to post process the results of a square sample (i.e. to covert load displacement to stress strain curve).

As someone who has extensively worked on conversion of the test data to flow curves, I found Avitzur's velocity field for the axisymmetric test more acceptable than other ones. There are un-published works on how to use Avitzur's proposed velocity filed for strain and strain rate and flow curves calculations out of the raw test data.