Time dependent voiding mechanisms in polyamide 6 submitted to high stress triaxiality: experimental characterisation and finite element modelling

Abstract

Double notched round bars made of semi-crystalline polymer polyamide 6 (PA6) were submitted to monotonic tensile and creep tests. The two notches had a root radius of 0.45 mm, which imposes a multiaxial stress state and a state of high triaxiality in the net (minimal) section of the specimens. Tests were carried out until the failure occurred from one of the notches. The other one, unbroken but deformed under steady strain rate or steady load, was inspected using the Synchrotron Radiation Computed Tomography (SRCT) technique. These 3D through thickness inspections allowed the study of microstructural evolution at the peak stress for the monotonic tensile test and at the beginning of the tertiary creep for the creep tests. Cavitation features were assessed with a micrometre resolution within the notched region. Spatial distributions of void volume fraction ($\mathit{Vf}$) and void morphology were studied. Voiding mechanisms were similar under steady strain rates and steady loads. The maximum values of $\mathit{Vf}$ were located between the axis of revolution of the specimens and the notch surface and voids were considered as flat cylinders with a circular basis perpendicular to the loading direction. A model, based on porous plasticity, was used to simulate the mechanical response of this PA6 material under high stress triaxiality. Both macroscopic behaviour (loading curves) and voiding micro-mechanisms (radial distributions of void volume fraction) were accurately predicted using finite element simulations.

Appendix A

• A finite strain approach using updated Lagrangian formulation is used to model large-strain deformation undergone by polymer materials. The polar decomposition of the deformation gradient tensor ${\boldsymbol{F}}$ is written as ${\boldsymbol{F}} = {\boldsymbol{R}} {\boldsymbol{U}}$. ${\boldsymbol{R}}$ and ${\boldsymbol{U}}$ describe a pure rotation and a pure stretch tensor, respectively. The rate of deformation ${\boldsymbol{L}}$ is defined as follows:

  $$ \boldsymbol{L} = \dot{\boldsymbol{F}} \boldsymbol{F}^{-1}. $$

  (1)

  The stretch rate $\boldsymbol{D}$ and the rotation rate ${\boldsymbol{\varOmega}}$ are expressed using this definition of the rate of deformation:

  $$ {\boldsymbol{D}} = \frac{1}{2} \bigl( {\boldsymbol{L}} + {\boldsymbol{L}}^{\mathrm{T}} \bigr),\qquad \boldsymbol{\varOmega} = \frac{1}{ 2} \bigl( {\boldsymbol{L}} - {\boldsymbol{L}}^{\mathrm{T}} \bigr). $$

  (2)

  The stretch rate tensor is transported into a local rotated referential ($\dot{{\boldsymbol{e}}} = {\boldsymbol{R}}^{\mathrm{T}} {\boldsymbol{D}} {\boldsymbol{R}}$) and the large strain is obtained by integrating this stretch rate tensor $\dot{{\boldsymbol{e}}}$. The material behaviour is based on Green–Naghdi transformation of the stress–strain problem into an “equivalent material referential”.

• The total strain is decomposed into elastic (${\boldsymbol{\varepsilon}}^{\mathrm{el}}$) and visco-plastic (${\boldsymbol{\varepsilon}}^{\mathrm{in}}$) components:

  $$ {\boldsymbol{\varepsilon}} = {\boldsymbol{\varepsilon}}^{\mathrm{el}} + {\boldsymbol{\varepsilon}}^{\mathrm{in}}. $$

  (3)

  The elastic part is obtained by the Hooke’s law ${\boldsymbol{\sigma}} = {\boldsymbol{\varLambda}} {\boldsymbol{\varepsilon}}^{\mathrm{el}}$, where ${\boldsymbol{\sigma}}$ is the Cauchy stress tensor and ${\boldsymbol{\varLambda}}$ is the fourth-rank tensor of elastic moduli, constituted by the Young’s elastic modulus $E$ and the Poisson ratio $\nu$. The elastic behaviour is not split into crystalline and amorphous parts and is simplified by assuming unified elastic material coefficients ($E$ and $\nu$).

• The amorphous and crystalline phases are treated as two different phases with different behaviours by taking the crystallinity index into account. However, each phase is subjected to the same stress, namely the macroscopic stress. The amorphous and crystalline phases are distinguished thanks to the indices “a” and “c”, respectively:

  $$\begin{gathered} {\boldsymbol{\varepsilon}}^{\mathrm{in}} = ( 1- \chi ) { \boldsymbol{\varepsilon}}^{\mathrm{a}} + \chi {\boldsymbol{ \varepsilon}}^{\mathrm{c}}, \end{gathered}$$

  (4)
  $$\begin{gathered} {\boldsymbol{\sigma}} = {\boldsymbol{\sigma}}^{\mathrm{a}} = {\boldsymbol{\sigma}}^{\mathrm{c}}. \end{gathered}$$

  (5)

• The evolution of the yield function during the deformation is modelled through isotropic hardening. The hardening parameters are denoted: $R_{0\mathrm{a}}$, $R_{0\mathrm{c}}$, $Q_{\mathrm{a}}$, $Q_{\mathrm{b}}$, $b_{\mathrm{a}}$, and $b_{\mathrm{c}}$. Then

  $$\begin{aligned} \begin{gathered} f^{\mathrm{a}} =J \bigl( {\boldsymbol{ \sigma}}^{\mathrm{a}} \bigr) - b_{\mathrm{a}} Q_{\mathrm{a}} r^{\mathrm{a}} - R_{0\mathrm{a}}, \\ f^{\mathrm{c}} =J \bigl( {\boldsymbol{\sigma}}^{\mathrm{c}} \bigr) - b_{\mathrm{c}} Q_{\mathrm{c}} r^{\mathrm{c}} - R_{0\mathrm{c}}. \end{gathered} \end{aligned}$$

  (6)

  The plastic behaviour is represented by the initial size of the yield surfaces ($R_{0\mathrm{a}}$ and $R_{0\mathrm{c}}$) and by the isotropic hardening parameters linked to the amorphous and crystalline phases, respectively ($Q_{\mathrm{a}}$, $b_{\mathrm{a}}$ and $Q_{\mathrm{c}}$, $b_{\mathrm{c}}$). $R^{\mathrm{a}}$ and $R^{\mathrm{c}}$ denote the size change in elastic domain related to each yield surface and are defined thanks to the internal variables $r^{\mathrm{a}}$ and $r^{\mathrm{c}}$ as follows:

  $$ R^{\mathrm{a}} = b_{\mathrm{a}} Q_{\mathrm{a}} r^{\mathrm{a}}\quad \mbox{and}\quad R^{\mathrm{c}} = b_{\mathrm{c}} Q_{\mathrm{c}} r^{\mathrm{c}}. $$

  (7)

• The visco-plastic behaviour is taken into account through two Norton potentials. The visco-plastic parameters of the Norton power law functions are denoted $K_{\mathrm{a}}$, $K_{\mathrm{c}}$, $n_{\mathrm{a}}$, $n_{\mathrm{c}}$, and the visco-plastic multipliers $\dot{\lambda}$ are the driving forces for the evolution laws:

  $$\begin{gathered} \begin{gathered} \dot{{\boldsymbol{\varepsilon}}}^{\mathrm{a}} = \dot{\lambda}^{\mathrm{a}} \frac{\partial f^{\mathrm{a}}}{\partial {\boldsymbol{\sigma}}^{\mathrm{a}}} = \dot{\lambda}^{\mathrm{a}} {\boldsymbol{n}}^{\mathrm{a}},\qquad \dot{r}^{\mathrm{a}} = \dot{ \lambda}^{\mathrm{a}} \bigl( 1- b_{\mathrm{a}} r^{\mathrm{a}} \bigr), \\ \dot{{\boldsymbol{\varepsilon}}}^{\mathrm{c}} = \dot{ \lambda}^{\mathrm{c}} \frac{\partial f^{\mathrm{c}}}{\partial {\boldsymbol{\sigma}}^{\mathrm{c}}} = \dot{\lambda}^{\mathrm{c}} { \boldsymbol{n}}^{\mathrm{c}},\qquad \dot{r}^{\mathrm{c}} = \dot{ \lambda}^{\mathrm{c}} \bigl( 1- b_{\mathrm{c}} r^{\mathrm{c}} \bigr), \end{gathered} \end{gathered}$$

  (8)
  $$\begin{gathered} \dot{\lambda}^{\mathrm{a}} = \bigl\langle f^{\mathrm{a}} / K^{\mathrm{a}} \bigr\rangle ^{n_{\mathrm{a}}},\qquad \dot{\lambda}^{\mathrm{c}} = \bigl\langle f^{\mathrm{c}} / K^{\mathrm{c}} \bigr\rangle ^{n_{\mathrm{c}}}. \end{gathered}$$

  (9)

  This multi-mechanisms model is coupled to a modified Gurson–Tvergaard–Needleman (GTN) porous media model (Gurson 1977; Tvergaard 1982; Tvergaard and Needleman 1984), then extended by Besson and Guillemer-Neel (2003). The damage variable chosen is the void volume fraction ($\mathit{Vf}$), and the presence of voids within the material is taken into account through the definition of effective quantities. These effective quantities, for example, $a_{*}$, can be defined for any stress tensor ${\boldsymbol{a}}$ and porosity level:

  $$ {\boldsymbol{a}}: \frac{\partial a_{*}}{\partial {\boldsymbol{a}}} = a_{*}. $$

  (10)

• The effective scalar stresses $\sigma_{*}^{\mathrm{a}}$ and $\sigma_{*}^{\mathrm{c}}$ are calculated implicitly via the conditions:

  $$\begin{gathered} G \bigl( {\boldsymbol{\sigma}}^{\mathrm{a}}, p, \sigma_{*}^{\mathrm{a}} \bigr) = \frac{J( \sigma^{\mathrm{a}} )}{ ( \sigma_{*}^{\mathrm{a}} )^{2}} +2 q_{1}. V_{\mathrm{f}}. \cosh \biggl( q_{2} \frac{\sigma_{kk}^{\mathrm{a}}}{\sigma_{*}^{\mathrm{a}}} \biggr) -1- q_{1}^{2}. V_{\mathrm{f}}^{2} =0, \end{gathered}$$

  (11)
  $$\begin{gathered} G \bigl( {\boldsymbol{\sigma}}^{\mathrm{c}}, p, \sigma_{*}^{\mathrm{c}} \bigr) = \frac{J( \sigma^{\mathrm{c}} )}{ ( \sigma_{*}^{\mathrm{c}} )^{2}} +2 q_{1}. V_{\mathrm{f}}. \cosh \biggl( q_{2} \frac{\sigma_{kk}^{\mathrm{c}}}{\sigma_{*}^{\mathrm{c}}} \biggr) -1- q_{1}^{2}. V_{\mathrm{f}}^{2} =0. \end{gathered}$$

  (12)

• The overall polymer is strengthened whilst the deformation increases due to the alignment of polymer chains. This phenomenon is frequently called rheo-hardening and is modelled here using two coefficients $A_{\mathrm{h}}$ and $B_{\mathrm{h}}$, following the work by Challier et al. (2006). The flow potentials for the two phases are then written in their effective forms as follows:

  $$\begin{gathered} \begin{gathered} f_{*}^{\mathrm{a}} = \sigma_{*}^{\mathrm{a}} - R^{\mathrm{a}} - R_{0\mathrm{a}} - R_{\mathrm{h}}, \\ f_{*}^{\mathrm{c}} = \sigma_{*}^{\mathrm{c}} - R^{\mathrm{c}} - R_{0\mathrm{c}} - R_{\mathrm{h}}, \\ R_{\mathrm{h}} = A_{\mathrm{h}} \exp \bigl( B_{\mathrm{h}} \lambda^{\mathrm{a}} \bigr). \end{gathered} \end{gathered}$$

  (13)

  The evolution laws of inelastic strain are finally written as

  $$\begin{gathered} \begin{gathered} \dot{{\boldsymbol{\varepsilon}}}^{\mathrm{a}} = ( 1- V_{\mathrm{f}} ) \dot{\lambda}^{\mathrm{a}} {\boldsymbol{n}}_{*}^{\mathrm{a}}, \quad {\boldsymbol{n}}_{*}^{\mathrm{a}} = \frac{\partial \sigma_{*}^{\mathrm{a}}}{ \partial {\boldsymbol{\sigma}}^{\mathrm{a}}}, \\ \dot{{\boldsymbol{\varepsilon}}}^{\mathrm{c}} = ( 1- V_{\mathrm{f}} ) \dot{\lambda}^{\mathrm{c}} {\boldsymbol{n}}_{*}^{\mathrm{c}}, \quad {\boldsymbol{n}}_{*}^{\mathrm{c}} = \frac{\partial \sigma_{*}^{\mathrm{c}}}{ \partial {\boldsymbol{\sigma}}^{\mathrm{c}}}. \end{gathered} \end{gathered}$$

  (14)