Experimental investigations of uniaxial and biaxial cold stretching within PC‐films and bars using optical measurements

Polycarbonate (PC) is an amorphous polymer that is an extremely robust material with a high tenacity, and thus suitable for a lightweight construction with glass‐like transparency. Due to these advantageous properties, PC is often used in industry for example in medical devices, automotive headlamps, sporting equipment, electronics, and a variety of other products. PC is often subjected to uniaxial and biaxial loading conditions. Therefore, reliable material models have to take into account the various resulting experimental effects. For those reasons, we investigate PC specimens under uniaxial and biaxial loading by using different stretch rates and loading scenarios. In addition to that, we propose methods for optical measurement of local stretches to obtain the approximated local true stress. In future work, the displacement fields and the resulting reaction forces will be used for parameter identification of constitutive equations.

In Section 5 we investigate the influence of lateral loading on axial loading by using three different load ratios.Lastly, Section 6 summarizes the main results and gives an outlook for future investigation.

EXPERIMENTAL SETUP
In this work, we investigate PC as film and bar specimens.The films are produced from the material Makrofol DE 1-1 000000 with a thickness of 0.375 mm.The bar specimens are produced from Makrolon 3108 as 3 mm plates.Both materials are characterized by high viscosity and good optical properties.
For uniaxial tests, we use rectangular-shaped films as well as bone-shaped bars as illustrated in Figure 1A and Figure 1D, respectively.Here, the upper clamping jaw is movable and the lower jaw is fixed.For the biaxial tensile test, we use the cross-shaped specimen depicted in Figure 1E.We introduce a hole in the middle of the specimen to localize the strain in this area and to have a dependency between axial and lateral loading.
Similar to the approach presented in ref.
[1], we investigate the induced anisotropy within film specimens through sequential biaxial loading of films.First, the film specimens are uniaxially loaded until the polymer chains are aligned in stretching direction.Second, we cut out three specimens from the middle of the deformed specimens in three different material directions as illustrated in Figure 1B.These three specimens are reloaded until total failure, see Figure 1C.
Uniaxial tensile tests are performed in a servo-hydraulic MTS machine with a 2.5 kN maximum load from MTS in combination with MTS 793/ MPE software, Figure 1F.Biaxial tests are performed using MTS biaxial machine with a 25 kN F I G U R E 1 Specimens geometries and testing equipment.
maximum load combined with the software MTS 793/MPE, Figure 1G, and equipped with four servo-hydraulic arranged in a cruciform manner that enables nonproportional loading along the two axes.
In both testing machines, the force-displacement data are transferred to the digital image correlation system (DIC) GOM Aramis provided by GOM GmbH.This DIC tool is used to measure the 3D deformation at the surface of the specimens during loading.The resolution is frequency-dependent.The measurement system consists of a sensor with two cameras, testing controllers, a lighting system, computers, and Aramis software.

Kinematics
First, we introduce a large deformation framework for the experimental investigations.The deformation gradient tensor is defined as where  and  are the position vectors of an arbitrary material point in the reference and the current configuration, respectively.For the case of pure tension, the deformation gradient in Equation (1) has the following matrix form The Jacobian  is defined as a determinant of the deformation gradient in (1).For the case of pure tension, we define the volume ratio as the determinant of the deformation gradient in matrix form in Equation ( 2) as Now we distinguish between two types of measures, the local ones denoted by • and the global ones denoted by •.
We also define the global stretch rate: In addition, we introduce the engineering strain for further evaluation of the experiment   =   − 1,  = , , . (5)

Approximate local stress
Under the assumption of a homogeneous force distribution within the specimen, we assume for the local and global first Piola-Kirchhoff stress tensor  = P.Moreover, this stress tensor has one component for the uniaxial tensile case, where f is the reaction force and  0  is the cross section area in the initial configuration.The Cauchy stress tensor  is the main subject of the investigations.In terms of the first Piola-Kirchhoff stress tensor it is obtained as Using an upper index  for the quantities obtained from the optical 3D measurement, the last two parts of Equation ( 6) render the local stress at a specific point of the specimen as

Bar specimens
Through measurement of the local stretches with the 3D-optical measurement system, we obtain the volume ratio   and the local stress   .Within bar specimens, we use one GOM system with two cameras.Here, the local strains   and   are measured at the point  from the top of the specimen and the strain   at the point  from the side of the specimen as depicted in We exploit the results of Equation (8) in Equation ( 7) to measure the local stress at point .The obtained   and   for four different stretch rates are depicted in Figure 3A and Figure 3B, respectively.Due to the viscoplastic behavior of the material, we observe a rate dependency, where the obtained stresses increase with increasing stretch rate.In addition to that the obtained local stress   is higher than the global stress   . (B) F I G U R E 3 Results of stresses measurement within bars.

(A) (B)
F I G U R E 4 Measurement methods within bar and film specimen.

Film
In contrast to the tensile bars, we can not evaluate the locale stretch   within film specimens at the point  due to its thin thickness.Therefore, we developed two methods to overcome this problem.In both methods, we measure the local thickness at a specific point of the specimens.
In Method 1 depicted in Figure 4A, we use two GOM systems, where each two cameras are projected on the top surface and the bottom surface of the specimen.We create in GOM-system two fictive lines, in which there are two coordinate systems ( 1 ,  1 ,  1 ) and ( 2 ,  2 ,  2 ), respectively.Each coordinate system calculates the relative displacements  1 and  2 of the points  1 and  ′ 1 .Finally, the current thickness  Opt is obtained as For the measurement method with one GOM-system (Method 2), we use in total two cameras as illustrated in Figure 4B, where we project them on the upper surface and on the bottom surface of the specimen.Here, we measure only the relative displacement  1 .The current thickness  Opt in the deformed configuration, under the assumption that both displacement of upper surface  1 and bottom surface  2 are equal, renders With help of Equation (10) or Equation ( 9) and the initial thickness  0 , we calculate the local stretch  Opt  ( 1 ) in point  1 as We measure the local strains   and   at the point  1 and we obtain the local stretches   and   in Equation ( 5) as Finally, the local stress   at point  1 is obtained with Equation ( 7).
The measurement results of both methods are depicted in Figure 5A.We observe that the resulting local stretch   is almost similar for both methods.We observe that with Method 2, the measured   is greater than one due to difficulties of clamping the thin film specimens, which leads to marginal errors at the beginning of the measurement, but this does not affect the evaluated stresses.Method 2 is more advantageous than Method 1 since it requires less processing time, storage memory, and resources.We validate Method 2 in Figure 5B on bar specimens.We observe that the results obtained for   with Method 2 and Equation (5) are identical.
We evaluate the local stress   within film specimens in the initial loading and reloading phase by using the specimen geometries depicted in Figure 1A and Figure 1C, respectively.The resulting local stress   obtained with both methods and the global stress   for initial loading are depicted in Figure 5C.Similar to the bar specimens, we observe that the local stress   value is higher than the global stress   in the plastic domain.In the elastic domain, both stresses are equal.In addition to that, the local stress values for both methods are almost similar.Therefore, we use Method 2 for further investigations.
In the reloading phase, we plot the resulting global stress   and local stress   in Figure 5D for the material direction  = 0 • ,  = 45 • and  = 90 • .First, we discuss the result of global stress   .In the elastic domain, an isotropic material  behavior is observed.In the plastic domain, an anisotropy is observed and stress-stretch profiles are material direction dependent.Here, the lower the material direction, the higher the global stress   .However, the local stress   increases with increasing loading direction , since the polymer chains are reoriented in loading direction for  = 45 • and  = 90 • , which leads to higher local stretches and as a result to a higher stress   .

BIAXIAL TESTING
For the biaxial testing of PC, we use the specimen geometry in Figure 1E.Similar to the approach presented in ref.
[2], we investigate the influence of lateral direction on the axial direction, we use three different loading cases: Load case 1 with   ∶   = 1 ∶ 0, Load case 2 with   ∶   = 1 ∶ 0.5 and Load case 3 with   ∶   = 1 ∶ 1 as depicted in Figure 6A, Figure 6D, and Figure 6G, respectively.The specimen is clamped in all directions.First, we discuss the results of Load case 1. Due to the lateral strain, we observe reaction force in −direction as illustrated in Figure 6B, and the −shaped necking zone is more oriented in −direction, Figure 6C.
For Load case 2, we increase the load ratio as pictured in Figure 6D, where   = 2  .In Figure 6E we do not observe any change in force level of   in comparison to Load case 1, however, we notice a change in its slope.In addition to that, damage occurs at  = 1.5 s while increasing loading in −direction, Figure 6F.Lastly, Figure 6G depicts the loading in Load case 3. Here, both applied displacement are equal   =   .In Figure 6H we do not observe any change in force level of   in comparison to Load case 1 and 2, but a change in its slope.Furthermore we observe in Figure 6I an early damage at  = 0.6 s.

CONCLUSION
In this paper, we presented different optical measurement methods of the local stretches within uniaxial film and bar specimens.With the help of the framework for finite deformation presented in Section 2, we obtained the Cauchy stress at a specific point of the specimen.In addition to that, we investigated the induced anisotropy within PC-films.We also studied the influence of the lateral direction on the axial direction within biaxial specimens.In future work, we will investigate the influence of varying the stretch in initial loading on stress profiles in the reloading phase within PC-films and the induced anisotropy within biaxial specimens.

Figure 2 .
The local stretches and the corresponding ratio are obtained as

(
D) Local stress σ x in reloading F I G U R E 5 Results of strains and stresses measurement within films.
Results of biaxial testing.
Comparison of both methods