This document presents the results of a finite element analysis of a tensile loaded shear sample used to characterize the large strain behavior of sheet metals. The analysis validated that the gauge section experiences a state of simple shear. Additional simulations examined the effects of mesh sensitivity, fillets in the gauge section corners to reduce stress concentrations, and a smaller gauge section aspect ratio. The tensile loaded shear sample was concluded to produce a simple shear state in the gauge section.

1. Final Report
PW02 – Design a Planar Simple Shear Test for Characterizing
Large Strain Behaviour of Sheet Metals
Dr. P.D. Wu
Will Peirce
0748205
Marshal Fulford
0554098
April 6, 2011

2. Table of Contents
Abstract........................................................................................................................... 1
Introduction ..................................................................................................................... 1
Tensile Loaded Shear Sample........................................................................................ 2
Planar Simple Shear Test ............................................................................................... 3
Theory – Simple Shear State .......................................................................................... 4
Simulation – Input Parameters........................................................................................ 5
Simulation – Model.......................................................................................................... 6
Results ............................................................................................................................ 7
Verification of the Solution............................................................................................... 9
Mesh Sensitivity........................................................................................................... 9
Simple Shear Loading ............................................................................................... 10
Filleted Sample ............................................................................................................. 11
Smaller Aspect Ratio..................................................................................................... 13
Conclusion .................................................................................................................... 14
References.................................................................................................................... 16

3. 1
Abstract
The tensile loaded shear sample introduced by Kang et al. was subjected to finite
element analysis to determine if the gauge section is in a state of simple shear.
Following that, a mesh sensitivity study was undertaken as well as a model of only the
rectangular gauge section in order to further validate the results. Next, a sample was
introduced which contains fillets in the corners of the gauge section which effectively
reduces the stress concentrations present and thus protects against premature failure of
the specimen. Lastly, it was observed that that a smaller gauge section aspect ratio
produces a more heterogeneous shear distribution in the gauge section. From all of the
above tests, it was concluded that the shear sample introduced by Kang et al. does in
fact produce a state of simple shear in the gauge section.
Introduction
The large strain behaviour of sheet metals is of great interest in the metal forming
industry in order to optimize forming operations. For example, in the aluminum can
industry, reducing the amount of material even slightly leads to less overall cost and
waste. By optimizing forming operations, down gauging may be possible which will
effectively reduce the material needed and consequently reduce the cost and waste.
Hence, there exists a need to characterize the hardening of sheet metals under large
deformations by the use of mechanical testing. However, conventional mechanical tests
prove to be unsuitable. First of all, uniaxial tensile tests cannot achieve high strains
because of early onset of plastic instability [1]. Furthermore, implementation of torsion
tests proves to be difficult when dealing with sheet metal. Therefore, other mechanical

4. 2
testing options must be explored. The test that will be explored in depth in this report is
the planar simple shear test. More specifically, emphasis will be directed towards a
tensile loaded shear sample which can be used in existing tensile test frames.
Tensile Loaded Shear Sample
Figure 1: Specimen geometry used for the model in the numerical simulation [2]
As mentioned previously, we will focus our efforts on the tensile loaded shear
sample. More specifically, we will look at the sample introduced by Kang et al. (refer to
Figure 1). This tensile loaded shear sample contains a rectangular gauge section where
the simple shear state is produced when the specimen is elongated using a tensile
testing device. The main difference between this specimen and other tensile loaded
shear samples is that this specimen utilizes a groove to make the gauge section thinner
than the rest of the specimen which effectively ensures that the deformation primarily
occurs within the gauge section.

5. 3
There are many advantages of using a tensile loaded shear sample as opposed
to other methods of simple shear testing. As mentioned previously, the tensile loaded
shear sample can be used on an existing tensile test frame which offers great
convenience. Secondly, the rectangular gauge section allows for straightforward
calculation of the shear stress and shear strain. Lastly, this test offers high repeatability
since the test relies mostly on the geometry and not the clamping of the specimen as in
other tests.
Of all the advantages there are still some drawbacks to the tensile loaded shear
sample. One of them being that electrical discharge machining (EDM) was used to
machine the grooves in the specimen that was looked at. This produces several
inherent problems. The first problem is that not everyone may have access to EDM
machines. Furthermore, the use of EDM also produces recurring costs for the test which
is not desired. Lastly, EDM uses thermal energy when removing the material which will
most likely affect the material properties in the adjacent areas, thus providing erroneous
results.
Planar Simple Shear Test
If instituted properly, the planar simple shear test can provide empirical data for
evaluation of yield criteria as well as data for determining the anisotropic work
hardening of the material in shear [3]. However, difficulty arises in trying to replicate a
simple shear situation. Due to the nature of the forces acting on the body during simple
shear, a moment is induced on the body. Consequently, the moment is counteracted by
tensile and compressive forces in the corners near the end of the shear zone. These
are known as the end effects.

6. 4
Theory – Simple Shear State
Figure 2: Schematic drawing of the simple shear loading condition showing the
parameters used to calculate shear stress and shear strain
The above figure shows a rectangular body in simple shear. In the simple shear
state, the bottom face is fixed and the top face is given an applied force such that it
deforms the body by a distance of u. From those parameters, the shear stress and
shear strain can be defined. The shear stress is calculated as the applied force divided
by the shear area, which in this case is the length of the shear zone (L) multiplied by its
thickness (t). The shear strain on the other hand is calculated as the displacement (u)
divided by the width of the shear zone (w). These relationships are given as follows:
𝜏 𝑥𝑦 =
𝐹
𝐿𝑡
𝛾𝑥𝑦 =
𝑢
𝑤
However, the shear stress and shear strain by themselves are not very useful. In
order to compare the data from other loading conditions, the shear stress and shear
strain must be converted to effective stress and effective strain. In simple shear loading
it is assumed that the only force acting on the body is the xy shear force. Given that, the
Von Mises equations simplify to the following relationships:
2 𝜎�2
= � 𝜎 𝑦 − 𝜎 𝑧�
2
+ ( 𝜎 𝑧 − 𝜎 𝑥)2
+ � 𝜎 𝑥 − 𝜎 𝑦�
2
+ 6� 𝜏 𝑦𝑧
2
+ 𝜏 𝑧𝑥
2
+ 𝜏 𝑥𝑦
2
�  𝜎� = √3 𝜏 𝑥𝑦
𝜀̅2
= �
2
3
� �𝜀 𝑥
2
+ 𝜀 𝑦
2
+ 𝜀 𝑧
2
� + �
1
3
� �𝛾𝑦𝑧
2
+ 𝛾𝑧𝑥
2
+ 𝛾𝑥𝑦
2
�  𝜀̅ =
𝛾 𝑥𝑦
√3

7. 5
Simulation – Input Parameters
The material used in the simulation was aluminum alloy AA5754 direct chill cast
(DC) sheet metal. The material was assumed to be isotropic although in reality the
material is likely to be anisotropic in nature due to the high degree of forming operations
involved in sheet metal processing. However, the details of the anisotropy need not to
be considered because they would be accounted for in the anisotropic yield criteria,
thus producing the same stress/strain curve as its isotropic counterpart. That being said,
the elasticity was characterized by the Young’s modulus and Poisson ratio with values
of 70 GPa and 0.33 respectively. Additionally, a modified Voce equation was used to
describe the material in the plastic deformation state. The modified Voce equation is:
𝜎 = 𝜎𝑆 �1 − ��1 −
𝜎 𝑦
𝜎𝑠
�
1−𝛼
− (1 − 𝛼)
ℎ 𝑜
𝜎𝑠
𝜀�
1
1−𝛼
�
This equation is fitted to the experimental data to yield values of: σy=94 MPa,
σs=316 MPa, ho=4000 MPa and α=1.13, where σy is the yield strength, σs is the
saturation stress, and ho and α are both parameters to describe the hardening. Also it is
noted that ε in this equation refers to the effective plastic strain. The figure on the
following page shows the curve obtained and the corresponding experimental data.

8. 6
Figure 3: Experimental data obtained by Kang et al. showing the fitting provided by the
modified Voce Equation [2]
It can be seen in the figure that the experimental data is characterized very well
by the modified Voce equation. Furthermore, this data illustrates that the simple shear
test can achieve much larger strains than the uniaxial tensile test.
Simulation – Model
As mentioned previously, the geometry that was modelled was the tensile loaded
shear sample with a groove that was introduced by Kang et al. This geometry was
modelled in 3D using ANSYS Workbench. Additionally, to simplify the model and reduce
the computational time, a symmetry constraint was applied in the thickness direction.
The mesh was generated by the automatic method given in ANSYS with a defined edge
length of 2mm and some refinement in the gauge section. The generated mesh
consisted of predominantly tetrahedral elements (SOLID 187) which are great for
resolving complex geometry. Lastly, the loading conditions implemented in the

9. 7
simulation were such that they replicated the loading of a tensile machine. One of the
ends was fixed and the other end was given a finite displacement in the length direction
while the other directions were held fixed. (Please see Figure 4 for a schematic of the
loading).
Results
Figure 4: Schematic diagram of the loading conditions and parameters used in the
simulation
After we completed the simulation, we needed a way to obtain a stress-strain
curve from the data. For the shear stress we were able to determine the applied force
and divide that by the shear area which is the length of the gauge section divided by the
thickness. From there it was converted to effective stress by multiplying it by √3 as
previously mentioned.

10. 8
For the shear strain, the relative displacement of two points in the middle of the
shear zone was used and divided by the distance between them, which in this case was
the width of the shear zone. Subsequently, the shear strain was converted to effective
strain by dividing it by√3.
Figure 5: Results obtained from the simulation of the original sample
From the results it is evident that the simulation results closely match the input
stress-strain curve that was generated using the modified Voce equation. This
demonstrates that the gauge section is very close to a state of simple shear.
0
50
100
150
200
250
300
350
0 0.2 0.4 0.6 0.8 1
EffectiveStress(MPa)
Effective Strain
Effective Stress vs. Effective Strain
Input
Original
Sample

11. 9
Verification of the Solution
Mesh Sensitivity
In order to verify the merit of the solution, the mesh sensitivity was evaluated
using two different strategies. First the solution was tested for sensitivity to element type
by changing the elements to hexahedral elements (SOLID 186). In changing the
elements, the size was kept consistent with the original in order to ensure only the effect
of element type is observed.
Secondly, the solution was assessed for sensitivity to changes in the element
size by using H-adaptive refinement. H-adaptive refinement starts with a coarse mesh
to solve the simulation. Following that, ANSYS evaluates the solution and implements
local refinement of the mesh in areas where high gradients exist (refer to Figure 6). This
process is continued over several iterations. The problem with H-adaptive refinement is
that it requires a long computational time. Therefore, in a simple geometry such as the
shear sample, it is more practical to apply manual refinement of the mesh.
Figure 6: Close up of the gauge section showing the refinement at the ends of the
gauge section produced by the H-adaptive refinement

12. 10
Figure 7: Results from the mesh sensitivity study
From the plot, the two different mesh strategies show no significant variation from
the original stress-strain curve. Therefore this demonstrates that the solution shows
minimal mesh sensitivity in the range that was considered.
Simple Shear Loading
Next the gauge section was modelled as a single entity with simple shear
loading. The simple shear loading was created much in the same way as in Figure 2 of
the theory section with the bottom face of the rectangular body fixed and the top face
given a finite displacement. Our results previously showed that the shear sample is in
simple shear. If the results from this simulation are consistent with the original results, it
will further confirm the gauge section of the shear sample is in a state of simple shear. If
this is indeed the case, then the stress-strain curves should be relatively similar.
0
50
100
150
200
250
300
350
0 0.2 0.4 0.6 0.8 1
EffectiveStress(MPa)
Effective Strain
Effective Stress vs. Effective Strain
Input
Original Sample
(Tetrahedral
Elements)
Hexahedral
Elements
H-Adaptive
Refinement

13. 11
Figure 8: Results from the rectangular shear zone only model compared to the original
The results obtained are similar to that of the original sample so this further
confirms the state of simple shear. Another noteworthy observation is that the solution
containing only the rectangular shear zone achieves larger strains even though the
displacement was the same for both simulations. This is because the original sample
essentially loses displacement because of elastic deformation that occurs outside of the
gauge section.
Filleted Sample
Figure 9: Sample which contains fillets in the corners of the gauge section
0
50
100
150
200
250
300
350
0 0.5 1 1.5
EffectiveStress(MPa)
Effective Strain
Effective Stress vs. Effective Strain
Input
Original
Sample
Shear Zone
Only

14. 12
In the original simulation, high stress concentrations were observed in the
corners which could result in premature failure and thus provide useless results.
Therefore fillets were added in the corners of the gauge section in an attempt to
alleviate these stress concentrations (refer to Figure 9). To observe the effect of the
fillets, the contour plots of effective plastic strain for each simulation were observed and
compared.
Original Sample
Maximum Strain = 3.0421
Filleted Sample
Maximum Strain = 1.8105
Figure 10: Comparison of the effective plastic strain present in the original sample and
the filleted sample
Upon observation of the contour plots it was found that the filleted sample had a
lower maximum effective plastic strain than its non-filleted counterpart. This
demonstrates that a reduction in stress concentrations did in fact occur as a result of the
added fillets. It is also noted that both samples experienced their maximum value of
effective plastic strain in the corners of the gauge section where one would expect the
stress concentrations to occur. Moreover, both samples showed a similar average
effective plastic strain within the gauge section which shows that the fillets do not affect
the distribution of strain within the gauge section.

15. 13
Smaller Aspect Ratio
Figure 11: Schematic drawing of the gauge section showing the end effects that are
imposed on the body
Next, the effect of changing the aspect ratio of the gauge section was examined.
As shown in the figure, the shear loading induces tensile and compressive forces that
resist the moment. These are known as the end effects. It is well documented that the
end effects are strongly correlated with the aspect ratio, which is the gauge section’s
length divided by its width [4]. So in order to observe this correlation a simulation was
conducted with a sample that had a smaller aspect ratio of 2.5 as opposed to the
original of 7.9375.

16. 14
Figure 12: Results of the smaller aspect ratio compared to the original sample
In the plot it can be seen that the results from the sample with the smaller aspect
ratio differed noticeably from the original results. This is because the end effects in the
sample with the smaller aspect ratio contribute more heterogeneity to the gauge
section.
Conclusion
Finally, through all the work that was done, it has been determined that the shear
sample studied produces near simple shear conditions. Furthermore, it has been shown
that the relative displacement of two points within the shear zone and the applied force
can be used to obtain the effective stress and effective stain and thus produce a stress
strain curve up to large deformations.
0
50
100
150
200
250
300
350
0 0.2 0.4 0.6 0.8 1
EffectiveStress(MPa)
Effective Strain
Effective Stress vs. Effective Strain
Input
Original
Sample
Smaller
Aspect Ratio

17. 15
In reality however, obtaining the relative displacement of two points in the gauge
section would be very tedious. A better solution would be to use the cross-head
displacement of the tensile machine. However, because of the elastic deformation that
occurs outside the gauge section, the cross-head displacement will differ from the
relative displacement of the two points. Thus, further considerations could be to
determine a correction factor such that the cross-head displacement can be used to
determine the stress-strain curve. This would effectively streamline the process of the
shear test.

18. 16
References
1. S. Bouvier, H. Haddadi, P. Levée, C. Teodosiu. “Simple shear tests: Experimental
techniques and characterization of the plastic anisotropy of rolled sheets at large
strains.” Journal of Materials Processing Technology, vol. 172, pp. 96–103, 2006.
2. Jidong Kang, David S. Wilkinson, P. D. Wu, Mike Bruhis, Mukesh Jain, J. David
Embury, Raja K. Mishra. “Constitutive Behavior of AA5754 Sheet Materials at Large
Strains.” Journal of Engineering Materials and Technology, vol. 130, July 2008.
3. Gilmour, K. R., Leacock, A. G., and Ashbridge, M. T. J., “The Determination of the In-
Plane Shear Characteristics of Aluminum Alloys.” Journal of Testing and Evaluation,
JTEVA, Vol. 29, No. 2, pp. 131–137, March 2001.
4. Christian G’Sell, Serge Boni, Suresh Shrivastava. “Application of the plane simple
shear test for determination of the plastic behaviour of solid polymers at large strains.”
Journal of Materials Science, vol. 18, pp. 903-918, August 1982.