Conversion of engineering stresses to Cauchy stresses in tensile and compression tests of thermoplastic polymers

1 Introduction

Semi-crystalline thermoplastic polymers show a particular behaviour in uniaxial tensile testing. After reaching the initial yield point local necking occurs followed by a cold-drawing plateau, which is associated with the propagation of the neck at the expense of the undrawn regions (Figure 1). The neck propagation is caused by orientation of the originally coiled polymer chains along the stress direction. After necking is completed along the entire specimen strain-hardening occurs until rupture of the specimen at high strains. In uniaxial compression testing of thermoplastic polymers an irregular buckling of the specimen can occur (see Figure 2B). In both tension and compression, non-isochoric behaviour of the polymers during plastic deformation becomes apparent.

Figure 1

Stress-strain behaviour of thermoplastic polymers during tensile testing (according to [1], [2]).

Figure 2

Photographs of specimens made of a PLLA based polymer blend in different present configurations: A) foil specimen during tensile testing, B) cylindrical specimen during compression testing.

The aim is to identify an effective stress along the entire test specimen as input for finite element analyses with polymers. For the determination of Cauchy stresses in material tests the technique of digital image correlation (DIC) is often employed [3], [4]. As an alternative a method for the conversion of engineering stresses to Cauchy stresses for tensile and compression tests is presented, that considers the actual deformations of the test specimens.

2 Methods

To determine Cauchy stresses from engineering stresses in uniaxial tensile and compression tests it is necessary to identify the actual geometrical dimensions of the specimen throughout the test. Therefore pictures of the specimens were taken continuously every second. For the foil specimens in tensile testing one camera is used; for cylindrical specimens in compression testing two cameras were arranged in perpendicular directions. In Figure 2 exemplary pictures of specimens during tensile and compression testing of a Poly(L-lactid) (PLLA) based blend material are shown.

The tests were also performed with other available thermoplastic polymers [tensile test: Polyhydroxybutyrate (PHB) based blend and low-density polyethylene (PE-LD), compression test: Polypropylene (PP)] showing a similar behaviour.

Due to local necking in tensile tests and irregular deformation under compression load the specimen geometry in the photograph of each present configuration is divided into several sections to accomplish the conversion of engineering into Cauchy stresses.

The foil specimen (tensile test) is divided into prismatic (i) and non-prismatic (j) sections according to the actual deformation. In the picture the length of each section l⁽ⁱ⁾ or l^(j) is measured. For prismatic sections one value for the width of the section w⁽ⁱ⁾ is determined; for the non-prismatic sections the width at the beginning w₀^(j) and the end of the section w₁^(j) is measured. As an illustration of the measuring process a schematic present configuration of a tensile specimen with the dimensions to be measured is shown in Figure 3.

Figure 3

Schematic present configuration of a tensile test specimen with the dimensions to be measured.

With the initial width w₀, initial thickness t₀, the actual thickness t and the corresponding measured engineering stress σ₀, an effective tensile Cauchy stress σ_ct for this present configuration can be determined using eq. 1.

The specimen geometry in the compression test is also divided into several arbitrarily long sections appropriate to the deformation. The diameter at the beginning (D_1,0^(k) / D_2,0^(k)) and the end (D_1,1^(k) / D_2,1^(k)) of each section in both views as well as the length of the sections l^(k) is measured (see Figure 4).

Figure 4

Schematic present configuration of a compression test specimen (two arbitrary views) with the dimensions to be measured.

First a mean radius of the measured diameters in both views is determined for the beginning and end of each section using eq. 2.

An effective Cauchy stress for the present configurations σ_cc can be calculated from the mean radii at the beginning R_m0^(k) and the end R_m1^(k) of each section, the corresponding section lengths l^(k), the current specimen length l and the initial radius R₀ as well as the corresponding measured engineering stress σ₀ (eq. 3).

Figure 5 shows exemplary photographs of present configurations of a PLLA based blend specimen with selected prismatic and non-prismatic sections during tensile testing and with selected sections in two views during compression testing.

Figure 5

Selected sections in photographs of exemplary present configurations of PLLA based blend specimens during A) tensile and B) compression testing.

The described measurement and determination of Cauchy stresses is executed for several time points of the tensile and compression test, respectively. For the input of the stress strain data into the material model of the finite element software the engineering strain values (ε₀) have to be converted to “true” strain (ε). Besides the input is restricted to plastic strain, so the elastic portion has to be subtracted. Eq. 4 is used for the determination of true plastic strain [5].

For the validation of the identified stress-strain curves finite element analyses of the tensile and compression tests are performed using the constitutive model SAMP-1 implemented in the finite element software LS-DYNA [5]. The load-displacement curves as approximate solution of the boundary value problem of the material tests should be compared to the load-displacement curves obtained in the experiment. In addition to the determined stress-strain curves a curve for the plastic Poisson’s ratio is entered in the finite element constitutive model.

3 Results

Figure 6 shows the resulting curve for Cauchy stress compared to the engineering stress plotted over the true plastic strain for a PLLA blend specimen for tension and compression, respectively. The curve for the Cauchy stress reveals a significant increase of the slope in the plastic deformation region for the tensile test and in contrary a significant reduction of the yield stresses for the plastic deformation during compression testing. The same tendency was obtained with other polymers (see Figure 7).

Figure 6

Engineering and Cauchy stress plotted against true plastic strain from tensile (t) and compression (c) tests for the PLLA based blend material.

Figure 7

Engineering and Cauchy stress plotted against true plastic strain: from A) tensile test with PHB based blend, B) tensile test with PE-LD and C) compression test with PP.

The deformed geometry of the specimens under tension and compression from finite element analyses compared to the experiment are shown in Figure 8. The deformation of the specimen in the numerical analyses shows good agreement with the geometry occurring in the tensile and compression tests, respectively.

Figure 8

Exemplary deformed specimen of the PLLA based blend under tension and compression from finite element analysis and the experiment.

Figures 9–11 display the resulting load-displacement curves from the finite element analyses compared to the associated experimental results for tension and compression tests. Good agreements between experiment and finite element analysis can be obtained for the tested polymers.

Figure 9

Load-displacement curve from finite element analysis compared to the experiment for the tensile test with the PLLA based polymer blend.

Figure 10

Load-displacement curves from finite element analysis compared to the experiment for tensile tests with A) the PHB based polymer blend and B) PE-LD.

Figure 11

Load-displacement curves from finite element analysis compared to the experiment for compression tests with A) the PLLA based polymer blend and B) PP.

4 Conclusion

A method for the determination of Cauchy stresses in tensile and compression tests could be established, which considers the non-isochoric deformation in plasticity that is typical for thermoplastic polymers. With this method “true” stress-strain curves as input for finite element material models can be identified for arbitrary materials. The simulation of the tensile and compression tests with the determined stress-strain curves by finite element analysis showed good agreement with the experiment concerning the visible deformation of the specimen and the resulting load-displacement curves for the tested polymers.