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Revisiting the Fundamentals and Capabilities of the Stack Compression Test

Article in Experimental Mechanics · November 2011

DOI: 10.1007/s11340-011-9480-5

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 REVISITING THE FUNDAMENTALS AND CAPABILITIES
OF THE STACK COMPRESSION TEST

Luis M. Alves(1), Nielsen C. V. (2) and Paulo A. F. Martins(1,*)

(1)
IDMEC, Instituto Superior Técnico, Univ. Tecn. Lisbon,
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
(2)
Technical University of Denmark, Department of Mechanical Engineering
DTU - Building 425, DK-2800, Kgs. Lyngby, Denmark

(*) Corresponding author. Fax: +351-21-8419058 E-mail: pmartins@ist.utl.pt

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ABSTRACT

Knowledge of the flow curve in metal forming is crucial to analyse formability, to

describe strain-hardening and to set-up the non-linear constitutive equations of metal

plasticity. Commonly available mechanical testing of materials supplied in the form of

sheets and plates, under low loading rates, is limited to small values of strain. As a

result of this, there is a generalized practice, and important source of modelling errors,

of extrapolating the remaining part of the flow curves that are usually determined by

means of tensile and bulge tests.

The aim of this paper is to provide a new level of understanding for the stack

compression test and to evaluate its capability for constructing the flow curves of metal

sheets under high strains across the useful range of material testing conditions.

The presentation draws from the fundamentals of the stack compression test to the

assessment of its overall performance by comparing the flow curves obtained from its

utilisation with those determined by means of compressive testing carried out on solid

cylinder specimens of the same material. Results show that mechanical testing of

materials by means of the stack compression test is capable of meeting the increasing

demand of accurate and reliable flow curves for sheet metals.

Keywords: Mechanical testing of materials, Stack compression test, Experimentation,

Numerical modelling

2
1. INTRODUCTION

From a metal forming point of view, the most important data for modelling material

behaviour is the flow curve because it characterizes strain-hardening and determines

the force and work requirements of a process. In case of cold forming the flow curve

should be available to strain levels above 1 for bulk metal forming, and up to 1, for

sheet metal forming processes.

The compression test performed on solid cylinder specimens [1] is one of the most

widespread mechanical testing methods for determining the flow curve in the field of

metal forming. The capability of evaluating material response to much larger strains

than in tensile tests, due to the absence of necking, in conjunction with the aptitude to

simulate more nearly the operative conditions of real forming processes, such as

forging, rolling and extrusion, which are carried out under high compressive loads, are

seen as the main reasons for its extensive utilization.

The compression test is performed by axially pressing a solid cylinder specimen

between two flat polished, well-lubricated, parallel platens and the flow curve is

determined by combining the experimental values of force and displacement that are

measured by means of transducers. The construction of the flow curve is greatly

influenced by the quality of these measurements and by the overall frictional conditions

at the contact interface between the specimen and the platens. Inaccuracies in the

measurements of force influence the estimate of stresses whereas in the values of

displacement influence both the stresses and the strains. Friction prevents the top and

bottom surfaces from expanding freely and adds extra force that creates difficulties in

the construction of a flow curve with data that are truly and exclusively indicative of the

stress-strain response of the material.

In fact, even when the barrelling contour of the specimens is measured for correcting

friction, there will always be some error in the flow curve. This applies also to the

Rastegaev’s modification of the compression test [2], which diminishes the influence of

friction and guarantees that specimens remain cylindrical up to a high strain, but leads

3
to errors in measuring the height of the specimens due to the fact that walls

surrounding the end faces are bent away and the end faces do not remain plane [3].

The solid cylinder specimens that are utilised in the compression test are limited within

the aspect ratio range 1  h0 d 0  3 of the height h0 to the diameter d 0 of the cross-

sectional area [4, 5]. The upper limit on the aspect ratio prevents failure by buckling or

bending while the lower limit is commonly justified by technical difficulties to operate

extensometers directly on the specimens [6]. This inhibits the utilization of the

compression test for constructing the flow curve of materials available in form of sheets

and plates.

As a result of this, the construction of the flow curve for metal sheets and plates to a

level of strain above 0.2-0.3 (provided by standard tensile testing), is commonly

performed by means of special purpose testing procedures based on hydraulic bulge or

Erichsen’s experimental apparatuses. Although these special purpose tests allow

determination of the flow curves to a level of strain up to 0.6, the overall correlation with

data from tensile testing is considerably influenced by anisotropy whenever normal

anisotropy is considerably different from 1.0.

The stack compression test proposed by Pawelski [7] in 1967 is an alternative

experimental procedure for evaluating the flow curve of raw materials supplied in the

form of sheets and plates. The test makes use of circular discs that are cut out of the

blanks and stacked to form a cylindrical specimen with an aspect ratio in the range of

solid cylinders currently employed in the conventional compression test. On the

contrary to other mechanical testing methods, the stack compression test is not

standardized and has so far received little attention in the open research and technical

literature. The most comprehensive work in the field has been performed by Merklein

and Kuppert [8], who discussed the utilisation of the stack compression test for

evaluating the flow curve of anisotropic materials and by Hochholdinger et al. [9] who

made use of the test for constructing the thermo-mechanical flow curve of a low carbon

steel.

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In what concerns modelling, finite element analysis of the stack compression test has

only been performed, as far as authors are aware, by considering the multi-layered

stacked specimen as a solid cylinder without taking into account the deformation

mechanics of the individual layer components (e.g. [9]).

Under these circumstances the aim of the present paper is threefold; (i) to understand

the reason why there is a lower limit of the aspect ratio in solid cylinder specimens that

inhibits its utilisation directly on disc samples cut from metal sheets or plates, (ii) to

analyse the deformation mechanics of a multi-layer stacked specimen and (iii) to

assess the overall performance of the stack compression test against the conventional

compression test performed on solid cylinder specimens.

The overall methodology was based on independently determined mechanical

properties of the material, experimentation under laboratory conditions and numerical

modelling using an in-house finite element computer program. The presentation is

enriched with numerical and experimental results and is expected to increase the

knowledge on the stack compression test for the benefit of those who perform

mechanical testing of materials in daily practice.

2. EXPERIMENTATION

This section begins with the characterization of the flow curve by means of

compression tests carried out on solid cylinder specimens and follows by presenting

the procedures and the experimental workplan that were utilized for analysing the

fundamentals of the stack compression test and for assessing its overall performance.

2.1 Compression test and flow curve

The raw material utilized in the investigation consisted of an Aluminium alloy AA-2011

that was annealed by heating at 420ºC for one hour and subsequently cooled in air.

The flow curve of AA-2011-O (the symbol ‘O’ denotes O-temper from annealing) was

determined by means of compression tests that were performed by pressing solid

5
cylinder specimens with 25 mm in diameter and height ( h0 d 0  1 ) between two flat

polished, well-lubricated, parallel platens at room temperature. The tests were carried

out on a universal testing machine with a cross-head speed equal to 100 mm/min (1.7

mm/s) and lubrication was performed with Zinc stearate.

The relationship between the true stress and the true strain is shown in Fig. 1. The true

strain was computed from the instantaneous deformation of the specimens using

displacements from crosshead movement of the testing machine whereas the true

stress was computed from the applied load and the instantaneous contact area

between specimens and compression platens.

2.2 Stack compression test

The stack compression tests utilized in the investigation made use of multi-layer

cylinder specimens that were assembled by pilling up circular discs cut from the same

material stock that was used for manufacturing the solid cylinder specimens of the

conventional compression test. The preparation of the discs was critical for ensuring

that all the layers were concentric and had identical cross-sectional area in order to

ensure homogenous deformation under frictionless conditions along the contact

interface with the platens. Fig. 2 shows a solid cylinder and two multi-layer stacked

specimens that were assembled by pilling up two and four discs.

The contact interfaces between the stacked cylinder specimens and the parallel

platens were lubricated in the same manner as in the conventional compression test in

order to ensure that the specimens keep their cylindrical shape during axial

compressive loading. Before applying the lubricant the surfaces of the specimens and

of the compression platens were degreased in order to ensure clean surface

conditions. The experimental workplan is listed in Table I.

The experiments were done in a random order and the influence of friction at the

contact interface between two adjacent disc layers on the overall performance of the

stack compression test will be addressed latter in the presentation.

6
3. FINITE ELEMENT MODELLING

Because the experiments were performed at room temperature under a quasi-static

constant displacement rate (100 mm/min) of the upper-table of the universal testing

machine, no inertial effects on forming mechanisms are likely to occur and, therefore,

no dynamic effects in deformation mechanics were needed to be taken into account.

These operation conditions allowed numerical modelling of the compression and stack

compression tests to be performed with the finite element flow formulation and enabled

the authors to utilize the in-house computer program I-form that has been extensively

validated against experimental measurements of metal forming processes since the

end of the 80’s [10].

The finite element flow formulation implemented in I-form is built upon the following

variational statement,

 |ur | 
  
    dV  21 K  v2 dV  Ti u i dS    0  f du r  dS

(1)
V V ST Sf

where, K is a large positive constant enforcing the incompressibility constraint and V

is the control volume limited by the surfaces SU and ST , where velocity and traction

are prescribed, respectively. In cases where friction exists at the contact interface Sf

between test specimens and parallel platens, it is assumed to be a traction boundary

condition and the additional power consumption term is modelled through the utilization

of the law of constant friction f  mk , where m is the friction factor and k is the shear

flow stress according to the von Mises yield criterion. Stress is related to strain-rate

using the Levy-Mises constitutive equations.

The numerical evaluation of the volume integrals included in equation (1) was

performed by means of a standard discretization procedure that, on account of the

rotational symmetry of the specimens, consisted of the discretization of the initial cross

section by means of four-node axisymmetric quadrilateral elements (Fig. 3). Typical

7
element size that was used in the finite element model is approximately 0.192 mm x

0.195 mm.

The detail on Fig. 3a spotlights the initial gap between the individual discs stacked

upon each other and the enclosed picture in Fig. 3b shows a comparison between the

deformation of a solid cylinder and that of an individual disc of the two-layer stacked

specimen. The deformation of the solid cylinder serves as a reference for the same

amount of displacement of the upper compression platen.

Because the geometry of adjacent individual discs is never truly identical and because

misalignment is always responsible for minor lack of symmetry, the upper and lower

discs of the two-layer stacked specimen in Fig. 3 were modelled with 10 µm difference

in the outside radius. This helps modelling real experimental conditions and, although

dissimilarities are not visible in the initial mesh, they will give rise to significant

differences when the numerical simulation is performed with and without friction along

the contact interface between adjacent discs, as can be seen later in the presentation.

The contact between two individual discs stacked upon each other is modelled by

means of a non-linear procedure based on a penalty approach. The approach is built

upon the normal gap velocity g nk for a nodal point k , contacting an element side ab of

the adjacent disc, (Fig. 4),

g nk  v nk  v na  1 v nb (2)

where, subscript n indicates normal direction and  and 1   are the fractions of the

element side l ab defining the velocity projection of nodal point k on the element side

ab . The penalty contact approach adds the following extra term to equation (1),

 
N
1 k k 2
c   gn (3)
2 k 1

8
where Nk is the total number of contacting points and  is a large positive constant

enforcing the normal gap velocity g nk  0 in order to avoid penetration.

The penalty contact method has the advantage of being purely geometrically based

and therefore no additional degrees of freedom have to be considered as in case of

alternative approaches based on Lagrange multipliers.

The numerical simulation of the process was accomplished through a succession of

displacement increments each of one modelling approximately 0.1% of the initial height

of the test specimens. No remeshing operations were performed and the overall CPU

time for a typical analysis containing 1100 elements was below 1 min on a standard

laptop computer.

4. RESULTS AND DISCUSSION

This section of the article is structured in three main parts. The first part is focused on

the current limit for low aspect ratio solid cylinder specimens, the second part is

centred in the deformation mechanics of stacked cylinder specimens and the third part

performs validation of the flow curves constructed by means of the stack compression

test against those determined by means of the conventional compression test carried

out on solid cylinder specimens.

4.1 Lower limit in the aspect ratio of solid cylinder specimens

Fig. 5 shows the finite element distribution of normal pressure in the compression of

solid cylinder specimens with aspect ratios h0 / d 0 equal to 0.125 and 1 under similar

conditions of friction and axial compressive strain  z . Because friction causes the

normal pressure to increase exponentially towards the center of the specimens in

inverse proportion to the aspect ratio h0 / d 0 (in agreement with the ‘friction hill’ analogy

resulting from slab analysis), it follows that even the smallest amount of friction (say,

m  0.1 ) or the smallest increase in the amount of friction (say, 10% increase) will

9
always give rise to a significant build-up of pressure in case of test specimens with very

small aspect ratios. This may cause lubricant breakdown, followed by an increase in

friction and deviation from homogeneous material flow conditions.

The comparison between the experimental and finite element evolution of the load-

displacement curve for cases 1 to 4 of Table I that is depicted in Fig. 6a corroborates

the aforementioned conclusion and discloses that finite elements can only match the

experimental results of case 1 ( h0 / d 0  0.125) if a friction factor m  0.1 is assumed to

exist at the contact interface (Fig. 6b).

Under these circumstances, the lower limit in the aspect ratio of solid cylinder

specimens is, first and foremost, due to its high sensitivity to friction (refer to Fig. 5

when friction experiences a 10% increase). Technical difficulties to operate

extensometers directly on the specimens cannot be claimed to set the lower limit in the

aspect ratio because displacements can always be acquired directly from the

crosshead movement of the testing machine if frame compliance is accounted for when

calculating the force-displacement curve.

After understanding the reason why solid cylinder specimens with low height to

diameter ratios cut from metal sheets should not be utilized in compression tests it is

important to understand how individual discs deform and interact with each other when

they are stacked into a specimen. With this in mind, the next section will address the

deformation mechanics of stacked cylinder specimens.

4.2 Deformation mechanics of a stacked cylinder specimen

The study on the deformation mechanics of stacked cylinder specimens under

compression loading was built upon two individual discs that were stacked on top of

each other to create a conceptual two-layered finite element model (Fig. 3). It is worth

noting that because material flow and interaction between neighbouring discs will be

similar in two-layer or multi-layer stacked specimens, conclusions drawn from the

10
analysis of the conceptual two-layered model can effectively be extrapolated to multi-

layered specimens with higher aspect ratios.

Fig. 7 shows the initial and computed geometry of the conceptual two-layered finite

element model for 55% reduction of the initial height under frictionless conditions at the

contact interface with the compression platens.

Fig. 7b proves that a stacked specimen is capable of ensuring homogeneous

deformation if sticking conditions prevail along the contact interface between the

adjacent discs pilled upon each other. Conversely, Fig. 7c shows that inhomogeneous

material flow leading to significant differences in the final shape of the discs is likely to

occur whenever sliding is allowed along the adjacent opposed surfaces of the discs

(modelled with 10 µm differences in the outside radius as previously explained in

section 3).

This result justifies the reason why some researchers often glue the discs in an attempt

to guarantee homogeneous deformation of the multi-layer stacked specimens as long

as the elasticity of the gluing film allows it to deform without cracking. In fact, partial

damage and breakdown of the gluing film or any other testing conditions that are

capable of inducing inhomogeneous material flow will inevitably lead to a deformed

profile different from that of a barrelled solid cylinder due to sliding between the

individual disc layers.

The distribution of effective stress in Fig. 7c shows evidence of the inhomogeneous

material flow by presenting a gradient towards the outer boundary of the conceptual

two-layered finite element model. However, the evolution of the compression load with

displacement almost shows no differences against that obtained from the compression

of a solid cylinder or a stacked specimen with sticking conditions along the contact

interface between the discs (Fig. 8).

Taking into account that major differences in the load-displacement curve (Fig. 8) are

below 0.8% for the conceptual model with sliding conditions along the contact interface

between the discs and making use of the values of displacement that are registered

11
from the cross head movement of the testing machine, it follows that the resulting flow

curve is virtually identical when obtained by means of the conventional or the stack

compression test under homogeneous flow conditions. In other words, the

inhomogeneity of material flow due to sliding frictional conditions along the contact

interface between two adjacent stacked discs influences the boundary profile of the

deformed test specimen but does not influence the load-displacement curve and,

therefore, the resulting flow curve.

The experimental assessment of the stack compression test against the compression

test carried out on solid cylinder specimens is performed in the following section.

4.3 Experimental assessment of the stack compression test

Fig. 9 presents the experimental assessment of the stack compression test against the

conventional compression test performed on solid cylinder specimens.

The experimental evolution of the load-displacement curve of stacked cylinder test

specimens with an aspect ratio h0 / d 0  1 , built upon two, four and eight disc layers

(cases 5, 6 and 7 of Table I) is shown in Fig. 9a. As seen, the trend is identical to that

of the conventional compression test performed on solid cylinders (case 4 of Table I).

Major deviation in the compressive load at the end of the test is below 2 %.

The enclosed picture in Fig. 9a shows the initial and deformed specimens in case of a

solid and an eight disc layered cylinder. Again, differences between the two deformed

test specimens are negligible.

As a result of this the flow curve of the Aluminium alloy AA-2011-O obtained by means

of the compression or the stack compression tests is nearly identical (Fig. 9b). This

result confirms the technical viability of the stack compression test for constructing the

flow curve of raw materials supplied in the form of sheets or plates.

12
5. CONCLUSIONS

The stack compression test can be successfully utilized for constructing the flow curve

of raw materials supplied in the form of sheets or plates.

The accuracy and reliability of the results provided by the stack compression tests is

similar to that of the conventional compression test and the current limit for low aspect

ratio stacked cylinder specimens should be identical to that of solid cylinder specimens.

Very small aspect ratios increase sensitivity to friction along the contact interface with

compression platens and may give rise to a significant build-up of pressure that causes

lubricant breakdown and inhomogeneous material flow.

The experimental preparation of stacked cylinder specimens should ensure sticking

friction conditions along the contact interface between adjacent discs in order to

guarantee homogeneous flow conditions. Sliding between adjacent discs induce

inhomogeneous flow that will trigger gradients of strain and stress towards the

boundary of the test specimens. However, the evolution of the load-displacement curve

will not be significantly changed whenever sticking conditions are replaced by sliding

conditions along the contact surface between adjacent stacked discs.

REFERENCES

[1] ASTM Standard E9-09 (1995) Standard test methods of compression testing of

metallic materials at room temperature, ASTM International, West Conshohocken,

USA.

[2] Banabic D., Bunge H.-J., Pöhlandt K., Tekkaya A. E. (2000) Formability of metallic

materials, Springer-Verlag, Berlin, Germany.

[3] Rastegaev M. V. (1940) New method of homogeneous upsetting of specimens for

the determination of flow stress and the coefficient of internal friction, Zavod

Laboratory, 354.

[4] Gunasekera J., Chitty E., Kiridena V. (1989) Analytical and physical modelling of the

buckling behaviour of high aspect ratio billets, Annals of CIRP, 38:249-252.

13
[5] Czichos H., Saito T., Smith L. (2006) Handbook of materials measurement

methods, Springer-Verlag, Berlin, Germany.

[6] House J. W. (2000) Testing machines and strain sensors, in Mechanical testing and

evaluation, ASM International, Materials Park, USA, pp. 225-227.

[7] Pawelski, O. (1967) Über das stauchen von holzylindern und seine eignung zur

bestimmung der formänderungsfestigkeit dünner bleche. Archiv für Eisenhüttenwesen,

38:437-442.

[8] Merklein, M., Kuppert, A. (2009) A method for the layer compression test

considering the anisotropic material behaviour, International Journal of Material

Forming, 12:483-486.

[9] Hochholdinger B., Grass H., Lipp A., Hora P. (2009) Determination of flow curves by

stack compression tests and inverse analysis for the simulation of hot forming, 7th

European LS-DYNA Conference, Salzburg, Austria.

[10] Alves M. L., Rodrigues J. M. C., Martins P. A. F. (2003) Simulation of three-

dimensional bulk forming processes by the finite element flow formulation, Modelling

and Simulation in Materials Science and Engineering – Institute of Physics, 11:803-

821.

14
Fig. 1 - True stress–strain curve of the Aluminium alloy AA-2011-O obtained by means
of the compression test performed on solid cylinder specimens.
Fig. 2 - Initial shape of test specimens that were utilized in conventional and stack
compression tests (cases 4, 5 and 6 of Table I).
 a) b)

Fig. 3 – Finite element model of a two-layer stacked cylinder specimen under

compressive loading. Frictionless boundary condition along the contact interface with
the platens is applied.
a) Initial mesh of the conceptual model with height to diameter ratio of
h0 d 0  0.125 .

b) Computed deformed mesh after 55% reduction in the initial height.

Fig. 4 – Modeling the contact between nodal point k and element side ab of two
individual discs stacked upon each other.
Fig. 5 – Finite element predicted distribution of normal pressure for cases 1 and 4 of
Table I under an axial compressive strain  z  1.0 .
 a)

b)

Fig. 6 – Experimental and finite element predicted evolution of the load vs.
displacement.
a) Cases 1 to 4 of Table I under frictionless conditions. Note that numerical and
experimental results coincide for case 4.
b) Case 1 of Table I under frictionless and frictional conditions.
Fig. 7 – Uniaxial compression of a conceptual two-layered finite element model under
frictionless conditions along the contact interface with the platens.
a) Initial geometry ( h0 / d 0  0.125)
b) Computed deformed geometry after 55% reduction in the initial height under
sticking conditions along the contact interface between the discs.
c) Computed deformed geometry after 55% reduction in the initial height under
sliding conditions along the contact interface between the discs.
Note: Results in b) and c) show the distribution of effective stress in MPa.
Fig. 8 – Computed evolution of the load–displacement curve for the compression of a
solid and a conceptual two-layer stacked cylinder specimens with sticking and sliding
conditions along the contact interface between the discs.
Fig. 9 – Assessment of the stack compression test.
a) Experimental evolution of the load–displacement curve carried out on solid and
stacked cylinder specimens (cases 4 to 7 of Table I).
b) True stress–strain curve of the Aluminium alloy AA-2011-O obtained by means of
the compression (case 4) and stack compression (cases 5 to 7 of Table I) tests.

1
 Case Height Diameter Aspect Ratio Solid / Multi-Layer
h0 (mm) d 0 (mm) h0 / d 0

1 3.125 25 0.125 solid

2 6.25 25 0.25 solid

3 12.5 25 0.5 solid

4 25 25 1 solid

5 25 25 1 2 layers (12.5 mm each)

6 25 25 1 4 layers (6.25 mm each)

7 25 25 1 8 layers (3.125 mm each)

Table I – The experimental workplan.

Note: The lubrication conditions at the upper and lower contact interfaces with the
compression platens were kept equal for all the cases.

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