Multiaxial stress state assessed by 3D X-Ray tomography on semi-crystalline polymers


This work aims at linking the microstructural evolution of semi-crystalline polymers to the macroscopic material behaviour under multiaxial stress state. Tensile tests on notched round bars, interrupted after different stages of deformation before failure, were supposed to have undergone various states of stress in the vicinity of the net cross-section. They were examined using synchrotron radiation tomography. A sample obtained from a test stopped at the end of stress softening stage showed elongated axi-symmetric columns of voids separated by thin ligaments of material. This special morphology allowed investigating the distributions of voids in terms of both volume fraction and orientation. This distribution was compared with the theoretical multiaxial stress/strain field. The combination of the tomographic images analyses and the continuum mechanics approach results in a determination of the principal mechanical parameter driving voids evolution.


In the last years, more and more researchers used the tomography technique to better understand the deformation mechanisms of various materials. Most of studies were dealing with light metals such as aluminium alloys. For a semi-crystalline polymeric material, Laiarinandrasana et al. [1] reported a specific morphology of voids evolution thanks to 3D X-ray tomography. In an initially necked region, void aspect shows elongated shapes separated by thin walls. Taking advantage of the excellent resolution of the images obtained during this campaign, an attempt was made to observe the voids microstructure within an initially notched round bars used to enhance void growth. Tomographic observations were carried out on several polymers. The present paper focuses on the investigations of PolyAmide 6. Before describing the experimental setup (PA6 material, sample preparation, tomography technique), a theoretical background on the mechanical stress/strain state within a notched round bar is given. Then, the radial distribution of voids is discussed in order to highlight the role of the stress triaxiality ratio on void growth. Further analyses of the tomographic images allowed determining void volume fraction gradient according to the longitudinal direction (axis z). Critical analyses of this gradient were performed with the help of the theoretical results about the stress/strain state within the notch region. The last part of the paper describes the void orientation distribution. It is demonstrated that the trajectories of the largest principal stress coincide with the void orientation.

Theoretical background on circumferentially notched specimens

Figure 1a describes the circumferentially notched specimen. Conventional cylindrical coordinates (r, 0, z) are used. Rotational symmetry is considered around longitudinal axis z. R and a are respectively the radius of curvature of the notch root and the outside radius of the minimal cross section. Authors like Bridgman [2-3], Kachanov [4], Davidenkov and Spiridonova [5] evaluated the distribution of true stress/strain in a necked tensile specimens made of metallic materials. Beremin [6] followed the approach by using initially notched specimens with machined radius curvature. The general assumptions required were: i) perfectly plastic material ii) isochoric transformation with a homogeneous axial strain in the minimal cross section. It was then demonstrated that the stress and strain tensors in the vicinity of the minimal cross section could be expressed as follows [6-7]:


where, n = 0.5 ln[1+a/(2R)] and ln is the naeperian logarithm, (r, z) are the current coordinates of the considered point in this plane. Note that in equations (1) and (2) the distribution of stress in the minimal cross section is assumed to be parabolic.

Sketches of a circumferentially notched specimen with the characteristic parameters

Figure 1: Sketches of a circumferentially notched specimen with the characteristic parameters

A lot of papers focused on the minimal cross section (z = 0) because it was the critical zone where damage and fracture occurred. The following equations tractable "by hand" were used in many studies. The radial, hoop and axial stresses are expressed as:


It has to be noticed that any stress component in equation (3) consists of a structural term (function of n) and a constitutive term (the equivalent stress). The multiaxiality of the stress state in the minimal cross section is measured by the stress triaxiality ratio (x0), defined as the mean stress divided by the equivalent von Mises stress.


In addition, strains are assumed to be homogeneous within the cross section. They can be approximated by:


Similarly, the stress/strain gradients with respect to z axis (r = 0) can also be expressed as:


Equations (6-7) were checked by Beremin [6] by comparing these solutions with finite element results.

Experimental preparation of the PA6 notched specimen

Figure 2: Experimental preparation of the PA6 notched specimen

Furthermore, the concept of the trajectories of largest principal stress/strain, discussed in [2-4] has never been exploited since. No experimental verification could be done. Additionally, a true stress cannot be measured and the strain field can be estimated at the surface of the specimen. These trajectories of principal stress were assumed to be circular in a necked sample. Figure 1b illustrates this feature and highlights that at any point the line of iso-principal stress should be perpendicular to the corresponding trajectory of principal stress. Actually, the eigenvector of the principal stress is oriented with an angle a that depends on the coordinates (r,z) of the point of interest. Let p be the curvature radius of the trajectory of the largest principal stress at point M(r, z). Two different expressions of p are given by the authors. According to Kachanov, Davidenkov and Spiridonova [4-5], p = aR/r, whereastmp11-80_thumbfor Bridgman [2-3]. One can notice that p tends to infinity when r = 0, that is on the axis z whereas p = R for r = a on the notch root. In fact, by plotting both expressions, one can realize that the difference is very small. In figure 1b, it can be moticed that : tmp11-82_thumb

Equation (8) clearly shows that the orientation of the eigenvector corresponding to the largest principal stress depends on the considered point. Up to now, there was no clear verification of this theory. The present paper aims at showing some features, observed on tomographic images of a polymer that can match the trends implied by the previous equations.


The material under study is a PolyAmide 6 polymer that was selected due to the quality of the images obtained by Synchrotron Radiation Tomography (SRT) carried out at the European Synchrotron Radiation Facilities (ESRF). The physico-chemical properties, as well as the tomography technique description were detailed in Laiarinandrasana et al. [1]. Following this last reference, a series of interrupted tests were carried on circumferentially notched specimens. In this paper, focus is set on a specimen with initial notch root radius R = 3.5mm and a minimum section radius a = 2mm. This specimen was tested using a traction machine with a load cell and the crosshead displacement measurements. The test was stopped at the end of the stress softening (fig.2) to focus on the voids morphology and distribution assumed to be "frozen" at that time. After unloading and specimen removal, deformed samples were, first photographed (fig.2) in order to locate the volume of interest (VOI within the box in fig.2), then, scanned at ESRF in Grenoble (France).

Locations of volume scan is depicted in fig.1a, symbolized by small rectangles and circles. SRT was carried out using the ID19 tomograph. The local tomographic setup [8] was used to avoid a cutting of the sample. A tomographic scan corresponded to 1500 radiographs, recorded over a 180° rotation of the sample. A radiograph was constituted of 1024 x 512 pixels with an isotropic pixel size of 0.7|im. Hence, in the following tomographic images, the width corresponding to the diameter of the maximum reconstructed 3D volume was of 716 |m whereas the height was of 358|m.

Results and discussions

Radial distribution of voids

Radial distribution and morphology of voids within the neck

Figure 3: Radial distribution and morphology of voids within the neck

Figure 3 describes the morphology and distribution of voids within the minimum cross sections through longitudinal cuts. Voids are observed in black. They exhibit elongated aspect separated by thin walls [1]. This particular morphology allows, at least qualitatively, observations of the variation of the height, the radial expansion and the relative orientation of voids. Indeed, spherical voids would not give such all of these information.

Figure 3 a observed in a VOI located in the centre of the minimal cross section indicates large void expansion in height (distance between two walls and the full height of elongated voids as described in [1]) but also in radial expansion. Note that some patterns indicate coalescence in both directions (radial = flat ellipse and in column = elongated ellipse). Figure 3b shows image of a VOI located close to the sample boundary. No void could be observed in the vicinity of the notch root. Moreover, it can be observed that there is less void with the mean diameter/"height" of voids smaller than at the centre (figure 3a). It can be concluded that in any cross section within the notched region, the void volume fraction (porosity) is maximum in the centre and gradually decreases towards the surface (notch root).

 Normalized stress and stress triaxiality ratio versus normalized radial abscissa for z = 0. R = 3.5mm, a = 2mm.

Figure 4: Normalized stress and stress triaxiality ratio versus normalized radial abscissa for z = 0. R = 3.5mm, a = 2mm. 

This result, reported for many materials in the literature argues that void growth is driven by the stress triaxiality ratio. Indeed, by recalling in figure 4 the plots of the normalized axial/radial stresses (equation 3) as well as the stress triaxiality ratio (equation 4) respect to the normalized radial abscissa, the distribution of the porosity is consistent with the trend of these plots. Conversely, the strain is homogeneous over the whole cross section. Moreover, from equation (2) it can be demonstrated that for any z, 3e/Sr is constant. The contour map of any strain component is "flat" (no gradient). Therefore, the strain cannot be considered as a relevant parameter to be associated with void growth.

Axial distribution of voids

Considering the axial distribution of voids near the axis, from the minimal cross section to the basis of the notch shoulder, figure 5 depicts the tomographic images to highlight the void volume fraction gradient. The same analyses as in the previous subsection are carried out here about the height, width and the amount of voids. All of these characteristics decrease from the centre (minimal cross section) to the notch shoulder.

Distribution of voids along z axis

Figure 5: Distribution of voids along z axis

Following Beremin [6], figure 6 shows plots of normalized stress/strain with respect to z/a according to equations (6-7). The first conclusion is that figure 5 constitutes an experimental verification of the theory. This important result -which can be considered as a novelty from experimental viewpoint- is essentially due to the tomographic images. It can be expected to further assess, for a given material, how far from the basis of the notch/neck shoulder the effect of the notch is present. Nevertheless, it is to be mentioned that unlike the radial distribution of voids (figure 4), the axial distribution (figure 6) cannot indicate which of the stress or the strain is the leading parameter for void growth.

Normalized stress/strain function of z/a for r = 0. R = 3.5mm, a = 2mm.

Figure 6: Normalized stress/strain function of z/a for r = 0. R = 3.5mm, a = 2mm.

Orientation distribution of voids

For the sake of clarity, a zoom of figure 3b is discussed in this section (figure 7). The knowledge of the void morphology [1] enables to draw arrows indicating the orientation of these voids. By superimposing on figure 7 the geometrical construction in figure 1b, three points were considered with respective radii pi, p2, p3. Local voids orientations were symbolized by the corresponding angles p1, p2, P3 respectively measured between the arrows and the vertical z axis. Recall that in figure 1b, the orientation of the largest principal stress is symbolized by the angle a described in equation (8). In figure 7, it turns out that the evolution of angle a is in excellent agreement with the aforementioned p. Indeed, voids angle is null close to the z axis and gradually increases to coincide with the local notch root curvature near the surface. As a matter of fact, these observations were encountered when voids were located outside the minimum cross section (z f 0).

To the authors’ knowledge, such investigations dealing with the mechanical parameters state combined with tomographic observations constitute a novel experimental approach. At this stage, the main important conclusions applied to the PA6 under study consist of:

- voids orientation parallel to the largest principal stress. Therefore this seems to indicate the component of the stress involved in void stretching;

- no relevance of the largest principal strain with voids orientation (flat contour map);

- quantification of voids characteristics allowing the local stress measurement provided a relevant stress scaling methodology (e.g. finite element analysis).


PolyAmide 6 semi-crystalline polymer deformation mechanisms were studied thanks to Synchrotron Radiation Tomography (SRT) carried out at the European Synchrotron Radiation Facilities (ESRF). An initially notched round bar was tested under tension, up to the end of the stress softening. The sample was then released from the traction machine to be observed by SRT. Specific morphology of voids allowed identification of the voids distribution according to the axial/radial direction. Furthermore, voids orientation was observed to be dependent on their location within the notched region. By comparing these features with the theoretical stress/strain fields, it can be concluded that the largest principal stress is the key mechanical parameter that control voids growth. Data collected from image analysis of SRT would be of great importance to be utilised as input in the mechanical analyses (FEA). In particular, material model parameters governing damage evolution should be adjusted to match the experimentally measured void volume fraction distribution.

Distribution of voids orientation along r in a longitudinal cut.

Figure 7: Distribution of voids orientation along r in a longitudinal cut.

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