Experimental characterisation of the dilation angle of polymers

Abstract

Despite the wide use of Drucker-Prager plasticity-based models on polymers, the experimental measurement of the dilation angle, a critical parameter to fully describe the plastic potential, has been rarely reported in existing literature. This paper shows, for the first time, the experimental characterisation of the dilation angle of polymers over a wide range of plastic strain. These measurements were obtained from uniaxial compression experiments conducted on poly(methyl methacrylate) (PMMA) and an untoughened epoxy resin. The calculation of the dilation angle relied on the measurements of the compressive force and the strain components obtained via Digital Image Correlation (DIC). Lower values of dilation angle were obtained for the epoxy resin, suggesting that resistance to volumetric change during plastic deformation could be associated to molecular structure and internal forces. The methodology and results presented in this study can be applied to different types of materials and employed for developing and validating constitutive models that incorporate plastic dilation.

Full text

Polymer Testing 125 (2023) 108137
Available online 1 July 2023
0142-9418/© 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Experimental characterisation of the dilation angle of polymers
Gustavo Quino
a
,
b
,
*
, Joseph Gargiuli
b
, Soraia Pimenta
c
, Ian Hamerton
b
, Paul Robinson
a
,
Richard S. Trask
b
a
Department of Aeronautics, Imperial College London, SW7 2AZ, UK
b
Bristol Composites Institute, University of Bristol, Queen’s Building, University Walk, Bristol, BS8 1TR, UK
c
Department of Mechanical Engineering, Imperial College London, SW7 2AZ, UK
ARTICLE INFO
Keywords:
Polymer
Plasticity
Dilation
Compression
Test method
ABSTRACT
Despite the wide use of Drucker-Prager plasticity-based models on polymers, the experimental measurement of
the dilation angle, a critical parameter to fully describe the plastic potential, has been rarely reported in existing
literature. This paper shows, for the rst time, the experimental characterisation of the dilation angle of polymers
over a wide range of plastic strain. These measurements were obtained from uniaxial compression experiments
conducted on poly(methyl methacrylate) (PMMA) and an untoughened epoxy resin. The calculation of the
dilation angle relied on the measurements of the compressive force and the strain components obtained via
Digital Image Correlation (DIC). Lower values of dilation angle were obtained for the epoxy resin, suggesting that
resistance to volumetric change during plastic deformation could be associated to molecular structure and in-
ternal forces. The methodology and results presented in this study can be applied to different types of materials
and employed for developing and validating constitutive models that incorporate plastic dilation.
1. Introduction
Polymers are widely used in a variety of applications due to their
excellent mechanical properties, lightweight, and relatively low cost. To
design and analyse polymer-based structures or materials (e.g., solid
polymers, foams, polymer-based composites, etc.), it can be important to
capture their behaviour accurately under different loading regimes. One
of the challenges in modelling the behaviour of polymers is their com-
plex and nonlinear response arising from their plastic, viscoelastic, and
time-dependent nature.
Among the different constitutive models capable to reproduce the
plastic behaviour of polymers, Drucker-Prager plasticity-based models
are widely used due to their exibility and ease of implementation
[1–4]. One of the parameters required to fully dene the plastic po-
tential function and the ow rule in these models is the dilation angle
ψ
.
Physically, the dilation angle correlates with the volume change during
plastic deformation, commonly observed in polymers [5,6]. A higher
dilation angle indicates higher dilatation or volume increase during
plastic ow. In some cases, due to the lack of experimental data or for
simplicity, the dilation angle
ψ
is considered equal to the friction angle β
(associated ow rule) [1,7,8]. However, this assumption overlooks the
actual volumetric dilation that may occur during plastic ow. To
accurately describe plastic dilation, a non-associated ow rule may be
necessary, requiring the measurement of the dilation angle parameter.
To the knowledge of the authors, the literature reporting the char-
acterisation of the dilation angle of polymers is limited. Some authors
reported on the plastic Poisson’s ratio, directly related to the dilation
angle as will be shown in Section 2.3. Dean and Crocker measured the
plastic Poisson’s ratio of a toughened epoxy adhesive from tensile ex-
periments, combining strain measurements obtained via contact exten-
someters and strains indirectly measured using analytical corrections for
machine’s compliance [9]. However, owing to practical limitations of
the extensometers, large strains could not be achieved. In addition,
tensile experiments and contact extensometers do not provide valid re-
sults whenever strain localisation such as necking occurs. Morelle et al.
reported on the plastic Poisson’s ratio of the commercial epoxy resin
RTM6, based on a highly cross-linked tetra-epoxide, obtained from axial
and transverse measurements of strains with a linear variable differen-
tial transformer (LVDT) and analysis of images taken during the ex-
periments [10]. No further details on the image analysis were provided,
and the plastic Poisson’s ratio was only reported for a plastic strain of
3%.
In this paper, we propose a methodology to measure the dilation
angle of polymers over a large range of strains using non-contact full
* Corresponding author. Department of Aeronautics, Imperial College London, SW7 2AZ, UK.
E-mail address: g.quino-quispe@imperial.ac.uk (G. Quino).
Contents lists available at ScienceDirect
Polymer Testing
journal homepage: www.elsevier.com/locate/polytest
https://doi.org/10.1016/j.polymertesting.2023.108137
Received 26 May 2023; Received in revised form 28 June 2023; Accepted 30 June 2023

Polymer Testing 125 (2023) 108137
2
eld strain measurements obtained via Digital Image Correlation (DIC).
A thermoset and a thermoplastic polymer were selected to demonstrate
the measurement protocol. The aim is to generate a procedure to mea-
sure the dilation angle of polymers, with the possible application to
other materials. This methodology will improve the accuracy and reli-
ability of constitutive models with non-associated ow rules. The results
of this study will also contribute to a better understanding of the plastic
deformation behaviour of polymers and improve the design and analysis
of polymer-based structures and products. The accurate measurement of
this parameter can be particularly relevant in micromechanical model-
ling of composites which has recently gained popularity in the com-
posites community [1,8,11–13]. The matrix, conned between the bres
or other reinforcements, dilates and develops hydrostatic stresses under
external loads which due to their pressure sensitivity, is relevant to
calculate the different responses, especially when non-linearity and
plastic deformation occur.
The following section provides a detailed description of the mate-
rials, specimen design, experimental methods, and data reduction pro-
cedures used in this study. The results obtained for the two materials are
then presented and discussed in Section 3. A sensitivity analysis was
conducted to evaluate the potential impact of errors in the measure-
ments of the elastic properties. The concluding remarks, implications
and future research directions are summarised in the nal section.
2. Experimental
2.1. Materials and specimens
Two materials, a thermoplastic and a thermoset, were considered in
this study. Poly(methyl methacrylate) (PMMA) was purchased in the
form of a 15 mm thick commercial plate. A proprietary untoughened
epoxy resin (ex Solvay, Tg=200◦C) was cast into cylindrical rods of
12.7 mm diameter, following the curing cycle recommended by the
manufacturer (2 h @ 180◦C).
All compression specimens were machined to cylinders of 6 mm
diameter and 6 mm height as per the geometry shown in Fig. 1. This 1:1
aspect ratio geometry was found to display less barrelling in comparison
to longer aspect ratios [14,15], improving the stress uniaxiality of the
experiments at large deformations. To have statistically representative
results, at least ve specimens of each material were tested. The cylin-
drical surfaces of the coupons were prepared with a ne airbrush
generated black speckle on white paint to conduct DIC.
2.2. Mechanical characterisation
The experimental setup is shown in Fig. 2. Quasi-static tests were
conducted in the servo-hydraulic universal testing machine Instron 8872
(Instron, USA). Compression platens of 30 mm diameter and mirror
surface nish were especially designed to t the machine and to uni-
formly transmit the compressive load onto the specimens. The experi-
ments were conducted in quasi-static regime under displacement
control, at a prescribed constant crosshead speed of 0.006 mm/s, cor-
responding to a nominal strain rate of 0.001/s. Tests were manually
stopped after a strain of at least 0.6 was reached. To minimise the effects
of friction, Teon tape 3 M 5480 (3 M, USA) was placed between the
loading platens and the at ends of the specimen as in Ref. [15]. The use
of Teon tape showed a higher reduction of barrelling than other
lubricants.
The load histories were obtained from the Instron 25 kN (Instron,
USA) load cell mounted in the testing apparatus. During the experi-
ments, images of the speckled specimens were acquired with two USB3
cameras Flir S BFS–U3–123S6M −C (Teledyne Flir, USA) at a rate of 1
image per second with 12.3 MP resolution. Articial illumination was
provided by two LED lights conveniently set to improve contrast and
depth of eld in the images. The average speckle size was approximately
4.8 pixels. To assess the quality of the speckle pattern, the mean in-
tensity gradient was measured as in Refs. [16,17] and found to be 21.2
which corresponds to a good quality of speckle pattern. The series of
images were postprocessed using stereo digital image correlation (subset
size 31 pixels, step size 10 pixels) with the software DaVis 10 (LaVision,
Germany).
2.3. Data reduction
Strains and stresses reported and discussed in this manuscript are
true (i.e., logarithmic) strains and true stresses. The strains were ob-
tained via DIC analysis. A representative virtual strain gauge (VSG)
located in the centre of the specimen, of approximately 70 ×70 pixels
size (0.37 mm ×0.37 mm), was used to extract the strain components.
The true stress
σ
z
t was calculated by dividing the (negative) load F by the
true cross-sectional area:
σ
z
t=F
Aoe2
ε
r
t
(1)
Where Ao is the initial cross-sectional area, and
ε
r
t is the radial true
strain.
The elastic modulus E was measured from the initial linear part of the
axial strain
ε
z
t vs. stress
σ
z
t curves, via a linear regression within the axial
strain range 0 <
ε
z
t<0.015, within the linear response. The Poisson’s
ratio
ν
was obtained from a linear regression of the radial and axial
Fig. 1. Dimensions of the compression specimens (mm).
Fig. 2. Experimental setup: i) Testing machine, ii) Loading platens, iii) Spec-
imen, iv) USB3 Cameras, and v) LED lights.
G. Quino et al.

Polymer Testing 125 (2023) 108137
3
strains
ε
r
t and
ε
z
t, also within the axial strain range 0 <
ε
z
t<0.015.
To calculate the dilation angle, the entire histories of strains and
axial stresses are required. The analysis starts with the decomposition of
the total true strain
ε
t into elastic
ε
e and plastic
ε
p components using the
additive decomposition:
ε
t=
ε
e+
ε
p(2)
Dening a cylindrical coordinate system with axis coincident with
the specimen’s axis, the axial and radial true strain components,
ε
z
t and
ε
r
t
respectively, can also be decomposed into their elastic and plastic parts
as shown in Eqs. (3) and (4).
ε
z
t=
ε
z
e+
ε
z
p(3)
ε
r
t=
ε
r
e+
ε
r
p(4)
The true axial strain
ε
z
t and true hoop strain
ε
θ
t can directly be ob-
tained from the DIC analysis of the cylindrical surface. The radial
component
ε
r
t, under the assumption of axisymmetric deformation, is
always equal to the hoop strain
ε
θ
t. An analytical proof and an experi-
mental validation of this statement can be found in Appendix A.
For an isotropic material under uniaxial true compressive stress
σ
z
t,
using Eqs. (3) and (4), the plastic axial and radial components of the
strain,
ε
z
p and
ε
r
p respectively, can be calculated as:
ε
z
p=
ε
z
t−
σ
z
t
E(5)
ε
r
p=
ε
r
t+
νσ
z
t
E(6)
where E and
ν
are the elastic modulus and Poisson’s ratio of the polymer,
respectively.
The plastic Poisson’s ratio
ν
p, under uniaxial load, is dened as the
negative ratio between the transverse and the axial plastic strain:
ν
p= −
ε
r
p
ε
z
p
(6a)
Finally, the dilation angle
ψ
can be evaluated as a function of the plastic
Poisson’s ratio
ν
p using Eq. (7), as in Refs. [9,10,18]:
ψ
=tan−1−31−2
ν
p
21+
ν
p(7)
The equations above show that the dilation angle
ψ
, under uniaxial
loading, only depends on the plastic Poisson’s ratio. This parameter can
be obtained from the history of load
σ
z
t, the history of strain components
ε
z
t,
ε
r
t; and the elastic material constants E, and
ν
, as shown in Eqs. (4)–
(6).
3. Results and discussion
3.1. Compressive behaviour
Fig. 3 shows the DIC full eld measurements of axial strain
ε
z
t for one
representative specimen of each material, at different levels of
compressive strain. The fact that the lateral edges of the specimens
remained nearly parallel during the test conrms that use of Teon tape
prevented signicant barrelling effects in the experiments. Fig. 3 also
shows the uniformity of the strain eld, suggesting that the whole vol-
ume of the specimen mainly stayed under uniaxial loading conditions
during the test, even up to large deformations. The uniformity observed
in the strain maps also validates the assumption of axisymmetric
deformation, required for the data reduction procedure described in
Section 2.3.
Fig. 4a and b shows the true stress vs. true strain curves for PMMA
Fig. 3. Full eld measurements of axial strain
ε
z
t obtained from DIC.
G. Quino et al.

Polymer Testing 125 (2023) 108137
4
and epoxy resin respectively. Both strains and stresses are compressive
and therefore represented by negative values; for PMMA and epoxy
resin, the initial part of the plot is linear. With increasing axial strain, the
material yields with a progressive reduction of the tangent modulus,
followed by a local maximum stress. After this local maximum stress, at
strain
ε
s
o, the softening regime starts, with a continuous decrease of the
compressive stress until strain
ε
s
f, possibly driven by the restructuring of
molecular chains [19]. The next regime in the strain vs. stress curve is
the strain hardening, where compressive stress increases with strain,
mainly driven by the resistance to chain alignment [20–22]. This type of
behaviour has been previously reported for various types of polymers in
the literature [19,23]. The untoughened epoxy resin exhibits similar
overall behaviour, but yields, softens, and hardens at higher stresses
(Fig. 4b). None of the specimens fractured in the experiments.
The average measurements and standard deviations of elastic
moduli, Poisson’s ratio, and softening strains are summarised in Table 1.
The average measured PMMA’s elastic modulus of 3.20 GPa is consistent
with previously reported values [21]. The measured Poisson’s ration of
PMMA was 0.45. The elastic modulus obtained for the epoxy resin of
3.24 GPa, and the Poisson’s ratio of the epoxy resin of
ν
=0.43 fall
within the ranges previously reported for other epoxy resin systems
[24].
3.2. Dilation angle
From Eqs. (1)–(5) and the elastic parameters from Table 1, the plastic
components of the strain were calculated. Representative sets of radial
and axial components of plastic strain vs. total axial strain obtained from
PMMA and epoxy resin samples are shown in Fig. 5a and b respectively.
Initially, both plastic axial and radial components are zero during the
linear regime, where there is no plasticity. After an initial non-linear
region, both components follow an almost linear trend with respect to
the total axial strain, approximately starting from the beginning of the
softening regime; a similar trend was also observed in the epoxy resin.
Since the dilation angle is only dened when there is plasticity, we
present it as a function of the plastic strain. The dilation angles vs. plastic
axial strain
ε
z
p, for PMMA and epoxy resin, are shown in Fig. 6a and b
respectively. The dilation angle of PMMA displays a decreasing trend at
small plastic strains. The values of the dilation angle of PMMA stabilise
after a plastic strain of approximately 0.15, after which is becomes linear
with respect to the axial strain. Fig. 6b shows that the dilation angle of
the epoxy resin with respect to the plastic axial strain. It also exhibits a
decreasing trend at low plastic strains. In this case, stabilisation occurs
after a plastic strain of approximately 0.2, after which the dilation angle
becomes linear with the plastic strain. Since the dilation angle does not
show a stable value at small plastic strains or even within part of the
softening region, a representative value of this parameter should better
be measured after full softening has taken place.
Even after the dilation angle becomes stable, there seems to be a
consistent increase with axial strain. To the authors’ knowledge, most
Drucker-Prager plasticity models currently implemented in commercial
FE software cannot account for a variable dilation angle or even negative
as was observed in the epoxy resin. The slope within of the dilation angle
within the stable region has a magnitude of 6.7◦per unit of strain in the
case of the PMMA, and 9.4◦per unit of strain for epoxy resin, measured
from the slopes of the dilation angle within the stable region. For the
purposes of comparison, we select a representative value of the dilation
angle at plastic strain of 0.3.
The average dilation angle of the PMMA at a plastic strain of 0.3 was
7.84 ±1.31◦. This value is smaller than the friction angle β=20◦found
by Rueda-Ruiz et al. via inverse modelling [25]. This mismatch agrees
with previously reported ndings showing that associated ow rules
(
ψ
=β) overestimate the plastic dilatancy of polymers [26]. Therefore, a
non-associative ow rule should be used to model the plastic behaviour
of PMMA.
The average dilation angle of the untoughened epoxy resin at a
plastic strain of 0.3 was −1.50 ±0.52◦. In Refs. [9,10], respectively, the
plastic Poisson’s ratios for an epoxy adhesive (measured in tension) and
RTM6 epoxy resin (measured in compression) were reported. The cor-
responding dilation angles were 28.51 ◦and 0 ◦respectively. In another
study, Sorini et al. [27] calibrated their constitutive model accounting
for tension/compression asymmetry in RTM6. They found different
values of dilation angle under tension and compression, 14.28 ◦and
−0.001 ◦, respectively. Comparison of our measurements with those
described above show that the dilation angle of thermoset epoxy resins is
close to zero under compression. These low values of the compression
dilation angles indicate that the plastic volumetric ow is close to
isochoric.
The variation in plastic dilatancy between PMMA and epoxy resin
may be related to differences in molecular structure and internal forces.
From the structural perspective, the aliphatic nature of PMMA is ex-
pected to enable greater chain rotations compared to the epoxy resin,
where the highly cross-linked network causes limited segmental motion
[28,29]. The internal forces in these polymers are also different. The
PMMA would predominantly have dipole-dipole interactions, while in
the epoxy resin there are hydrogen bonds, stronger than the
dipole-dipole forces [30]. Molecular dynamics simulations conducted by
Peng et al. have suggested that polymers with longer chains, and
Fig. 4. Axial true stress
σ
z
t vs. true strain
ε
z
t curves for a) PMMA and b)
untoughened epoxy resin.
Table 1
Elastic material constants and softening strains evaluated from the experiments.
Material E [GPa]
ν
[−]
ε
s
o [−]
ε
s
f [−]
PMMA 3.20 ±0.11 0.45 ±0.03 0.084 ±0.003 0.282 ±0.001
Epoxy resin 3.24 ±0.07 0.43 ±0.01 0.100 ±0.002 0.275 ±0.004
G. Quino et al.

Polymer Testing 125 (2023) 108137
5
consequently higher internal forces, exhibit smaller volume dilation
[31]. The highly cross-linked structure and the higher internal forces in
the epoxy resin could therefore cause its limited segmental motion and
higher resistance to volume change, leading to the observed reduced
plastic dilation.
3.3. Sensitivity study
The evaluation of the dilation angle
ψ
depends on the elastic prop-
erties such as elastic modulus E and Poisson’s ratio
ν
(Equations (1)–(7))
and could then be sensitive to errors in their characterisation. A para-
metric analysis was conducted to assess the sensitivity of the presented
method to possible errors in the measured values of elastic parameters. A
representative PMMA test was selected for this exercise. The analysis
considered a parametric sweep of different values of Poisson’s ratio, in
steps of 0.04, and three values of elastic modulus in steps of 0.3 GPa with
central values equal to the averaged experimental measurements.
Fig. 7 shows how the dilation angle is affected by these varying
elastic parameters. Overall, lower values of elastic modulus and Pois-
son’s ratio lead to higher dilation angles. The curves are mostly affected
at smaller plastic strains. For the parametric sweep, the dilation angle
shows a standard deviation of ±4.41◦at a small plastic strain of 0.02.
However, at a plastic strain of 0.3, the standard deviation reduces to ±
0.36 ◦. This smaller uncertainty also supports the earlier observation
that a more representative value of the dilation angle can be obtained
after the strain softening regime.
4. Conclusions
In this paper, uniaxial compression tests were conducted to experi-
Fig. 5. Typical plastic strain components vs. axial strain extracted from a representative test on a) PMMA, and b) Epoxy resin.
Fig. 6. Dilation angle
ψ
vs. plastic axial strain
ε
z
p for a) PMMA and b)
untoughened epoxy resin.
Fig. 7. Sensitivity analysis of the dilation angle with respect to the variability
of elastic properties E and
ν
.
G. Quino et al.

Polymer Testing 125 (2023) 108137
6
mentally measure the dilation angle
ψ
, a relevant parameter in Drucker-
Prager plasticity models with non-associated ow, and directly linked to
plastic dilatancy. One thermoset and one thermoplastic polymer were
selected to demonstrate the proposed methodology. Digital image cor-
relation was employed to obtain a full map of the strain components. A
parametric study was also conducted to assess the inuence of possible
measurement errors in the method herein presented. The main conclu-
sions are:
•The presented technique allows for the calculation of the dilation
angle of polymers from uniaxial compression up to large strains.
•The dilation angle of PMMA is smaller than its friction angle.
Therefore, it follows a non-associated ow rule.
•The untoughened epoxy resin displays a plastic ow nearly isochoric
under compressive loads.
•The untoughened epoxy resin has a smaller angle of dilation than the
PMMA, possibly due to its highly cross-linked polymeric network.
•While not perfectly constant as the axial strain increases, the dilation
angle shows less variability after full material softening, where
possible errors in the measured elastic modulus or Poisson’s ratio
have minimal inuence.
The methodology herein presented will help to build more reliable
and accurate constitutive models for polymers. Our future research ac-
tivities include the incorporation of these experimental measurements in
the development of high-delity micromechanical models for bre
composite materials. Some relevant questions that remain open include
the tension-compression asymmetry, effects of temperature, strain rate
upon plastic dilation.
CRediT author statement
Gustavo Quino: Conceptualization, Methodology, Validation, Formal
analysis, Investigation, Writing- Original draft preparation, Writing -
Review & Editing, Visualization. Joseph Gargiuli: Investigation, Writing
- Review & Editing. Soraia Pimenta: Conceptualization, Methodology,
Writing - Review & Editing, Supervision. Ian Hamerton: Resources,
Writing - Review & Editing, Writing - Review & Editing, Supervision,
Project administration. Paul Robinson: Conceptualization, Methodol-
ogy, Writing - Review & Editing, Supervision. Richard S. Trask:
Conceptualization, Methodology, Resources, Writing - Review & Edit-
ing, Supervision, Project administration, Funding acquisition.
Declaration of competing interest
The authors declare that they have no known competing nancial
interests or personal relationships that could have appeared to inuence
the work reported in this paper.
Data availability
Data will be made available on request.
Acknowledgements
Authors acknowledge the support of by UK Engineering and Physical
Sciences Research Council (EPSRC) programme Grant EP/T011653/1,
Next Generation Fibre-Reinforced Composites: a Full Scale Redesign for
Compression, a collaboration between Imperial College London and the
University of Bristol. We thank Solvay (Wrexham) for the provision of
the untoughened epoxy resin and assistance with sample manufacturing
and Dr Mark Harriman and Dr Jon Meegan for useful discussions. For the
purpose of open access, the author has applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version
arising.
Appendix A. Radial and hoop strain components
If the circular cross-sections remain circular during the compression tests, the radial component and the hoop component of the strain tensor
measured on the surface of the cylindrical specimens must be the same.
Given an initial radius of the undeformed cross section r1 that grows to r2, the true hoop strain
ε
θ
t is:
ε
θ
t=ln 2
π
r2
2
π
r1(A.1)
After simplication of the 2
π
term in both numerator and denominator, the hoop component is equal to the radial component as we initially
intended to proof:
ε
θ
t=lnr2
r1=
ε
r
t(A.2)
The validity of this assumption was cross-checked against experimental measurements obtained with DIC (Fig. A.1), using the same processing
settings described in Section 2.2. One virtual extensometer with start and end at opposite edges of the specimen was created to measure the radial
strain
ε
r
t . In addition, a virtual strain gauge of 70 ×70 pixels size, approximately 2.3 times the subset and 7 times the step size, was created in the
middle of the extensometer to measure the hoop component of strain
ε
θ
t. Fig. A.1 shows that these two sources provide with the same values during all
the experiment.
G. Quino et al.

Polymer Testing 125 (2023) 108137
7
Fig. A.1. Histories of radial strain obtained from the virtual extensometer, and hoop strain measured from a virtual strain gauge at the centre of the speci-
men’s surface.
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