Available via license: CC BY-NC-ND 4.0
Content may be subject to copyright.
Rate- and temperature-dependent strain hardening in
glassy polymers: Micromechanisms and constitutive
modeling based on molecular dynamics simulations
Wuyang Zhao
Institute of Applied Mechanics, Friedrich-Alexander-Universität Erlangen-Nürnberg,
Egerlandstraße 5, 91058 Erlangen, Germany
Abstract
We perform molecular dynamics simulations under uniaxial tension to investi-
gate the micromechanisms underlying strain hardening in glassy polymers. By
decomposing the stress into virial components associated with pair, bond, and
angle interactions, we identify the primary contributors to strain hardening as the
stretching of polymer bonds. Interestingly, rather than the average bond stretch,
we find that the key contributions to stress response come from a subset of bonds
at the upper tail of the stretch distribution. Our results demonstrate that the stress
in the hardening region can be correlated with the average stretch of the most
extended bonds in each polymer chain, independent of temperatures and strain
rates. These bonds, which we denote as load-bearing bonds, allow us to define a
local load-bearing deformation gradient in continuum mechanics that captures their
contribution to the hardening stress tensor. Building on this insight, we incorporate
the load-bearing mechanism into a constitutive framework with orientation-induced
back stress, developing a model that accurately reproduces the stress response of
the molecular systems over a wide range of temperatures and strain rates in their
glassy state.
Keywords: Glassy polymers, Strain hardening, Constitutive modeling, Molecular
dynamics simulations, Back stress
Email address: wuyang.zhao@fau.de (Wuyang Zhao)
Preprint submitted to Elsevier November 13, 2024
arXiv:2411.07811v1 [cond-mat.soft] 12 Nov 2024
1. Introduction
Glassy polymers are essentially amorphous polymers at temperatures below
their glass transition temperature
Tg
, exhibiting solid-like mechanical behavior
due to restricted atomic mobility from neighboring interactions. The polymer
molecules, typically with chain-like structures, can undergo strain hardening at
large deformations, effectively increasing the toughness of the materials. Accu-
rately modeling the mechanical response in the hardening region is essential for
studying the failure mechanisms of glassy polymers. Unlike rubbers, which are
amorphous polymers above
Tg
, the strain hardening behavior of glassy polymers is
sensitive to strain rate and temperature as evident in various experiments [
1
,
2
,
3
]
and molecular dynamics (MD) simulations [
4
,
5
]. However, the mechanisms un-
derlying the rate- and temperature-dependence of strain hardening remain not fully
understood.
For decades, the hardening mechanism in glassy polymers has often been
interpreted through entropic elasticity theory developed for rubbers [
6
,
7
], which at-
tributes stress increases to the reduction of entropy during the elongation of polymer
chains. While this theory applies well to rubbers, various experiments [
8
,
9
] and
MD simulations [
4
] reveal inconsistencies when applied to glassy polymers. This
discrepancy probably arises from a core assumption in entropic elasticity theory
that polymer segments are free to adjust their conformations to accommodate defor-
mation. However, in glassy polymers, atomic mobility is restricted, preventing such
free adjustment. Consequently, the relaxation time associated with the evolution
of polymer segments becomes non-negligible, leading to rate-dependent behavior
in the hardening region of glassy polymers. This limited mobility also introduces
temperature sensitivity, as relaxation times are typically temperature-dependent.
Thus, understanding strain hardening in glassy polymers involves addressing two
key questions: (i) which structures primarily contribute to the stress response, and
(ii) the relaxation of which structures most significantly affects stress behavior?
To address the first question, various assumptions have been tested via constitu-
tive modeling, given that experimental observation of the microscopic structure
during deformation is challenging. Although the precise origin of hardening stress
remains unclear, most constitutive models adopt the Lee-Kröner decomposition
F=FeFp
[
10
,
11
] of the deformation gradient
F
, where
Fe
and
Fp
represent the
elastic and viscoplastic components of
F
, respectively. In this framework,
Fe
is
assumed to contribute to the stress, which drives the evolution of
Fp
, characterized
by the viscosity
η
. To capture strain hardening, Arruda et al. [
12
] introduced a
back stress term to account for chain orientation effects. Anand et al. [
13
,
14
]
2
generalized this formulation by incorporating a unimodular orientation tensor
A
as an internal variable of the free energy density for orientation and proposed that
the evolution of
A
is subjected to a dynamic recovery term driven by the back
stress. Additional approaches have also been proposed, including models with
deformation-dependent viscosity [
1
,
3
], a viscous model that uses strain rate as
an external variable to capture steady-state mechanical behavior [
15
], and models
that represent orientation through shear transformation zones (STZs) [
16
,
17
,
18
].
While these models successfully replicate the stress response of glassy polymers in
the hardening region under various conditions, they remain largely phenomenolog-
ical, each grounded in different assumptions about the origin of hardening stress.
A deeper understanding of the underlying micromechanisms is needed to further
elucidate the nature of strain hardening in glassy polymers.
In addressing the second question, it is critical to determine whether the pri-
mary relaxation mechanism influencing strain hardening occurs at the scale of
polymer chains, such as disentanglement, or at the scale of local segments, like
bond rotations. MD simulations have been used to study the mechanisms of dis-
entanglement by tracking the evolution of the entanglement lengths
Ne
during
deformation [
4
,
19
,
17
]. Hoy and Robbins [
4
] observed that hardening increases
with decreasing
Ne
only when
Ne
is relatively small; beyond a critical value
Ncr
e
,
hardening saturates with
Ne
. This suggests that disentanglement is relevant primar-
ily at smaller
Ne
. This finding may help to reconcile the conflicting results reported
in [
19
] and [
17
], where
Ne
remains constant in [
19
] but evolves in [
17
] during
deformation. Furthermore, the critical value
Ncr
e
is likely temperature-dependent,
as increased temperatures accelerate polymer chain relaxation. This implies that
disentanglement only occurs at sufficiently high temperatures for glassy polymers,
while at low temperatures, the dominant relaxation mode is likely limited to local
segmental rotations. These include backbone chain rotations (
α
-relaxation) or side
group rotations (
β
-relaxation) for certain glassy polymers [
20
,
21
,
22
]. Experi-
mental studies on the time-temperature superposition of glassy polymers at large
strains [
23
,
24
,
25
] support this assumption. These studies demonstrate that the
stress-strain curves of glassy polymers at specific temperatures and strain rates can
coincide in the hardening region, with derived shift factors matching those from
small strains. This suggests that similar relaxation mechanisms are active across
both small and large strain regimes in these materials. Since disentanglement is
not the primary relaxation mechanism at small strains before yielding, it is likely
not the dominant mechanism at large strains in the hardening region either, based
on these experimental results. Using identical shift factors for both small and large
strains, Xiao and Tian [
26
] employed the back stress model [
14
] to successfully
3
predict pre-deformation effects in the strain hardening region of glassy polymers.
This paper aims to investigate the microscopic mechanisms underlying strain
hardening in glassy polymers using MD simulations, with a focus on capturing its
rate- and temperature-dependent behavior. To achieve this, we first decompose the
virial stress components into contributions from pair, bond, and angle interactions,
identifying bond and angle stresses as the primary contributors to strain hardening.
Subsequently, we examine the microscopic quantities that can be correlated with
these bond and angle stresses within the hardening region. Our key finding is
that these stress terms are correlated with the averaged values of the largest bond
stretch in each polymer chain, independent of temperature and strain rate. We
denote these bonds as load-bearing bonds and represent this microscopic quantity
through a deformation gradient, capturing the macroscopic local load-bearing
deformation, denoted as
Fl
. Building on this result, we propose a decomposition
of the total deformation gradient,
F=FlFr
, where
Fr
represents the resisting
component of the deformation gradient. Finally, we develop a constitutive model
within the framework established by [
14
], incorporating orientation-induced back
stress. Compared to the model proposed by [
14
] and various other hardening
models [
26
,
1
,
3
,
16
,
17
,
18
,
15
], the improvement presented in this paper lies in
providing a clear physical basis for the stress response in the hardening region, as
well as a well-defined interpretation of the deformation gradient decomposition
F=FlFr
, where the contribution of
Fl
to the stress can be directly extracted
from MD simulations.
The remaining part of this paper is organized as follows. In Section 2, we
introduce the MD systems used in this study. Section 3 presents the analysis of the
microscopic mechanisms underlying strain hardening, followed by a discussion on
the decomposition
F=FlFr
in Section 4 based on the findings from this analysis.
Section 5 derives the constitutive model, including parameter identification and the
validation. Finally, Section 6 concludes this study.
2. Methods and models
2.1. Systems and samples
We use the coarse-grained (CG) potential of atactic polystyrene (PS) developed
by Qian et al. [
27
] for MD simulations, which maps each chemical monomer to a
CG bead at its center of mass. The CG potential consists of pair, bond, and angle
interactions. Similar to most bottom-up CG models [
28
,
29
], this model can only
represent the structural properties of its atomic counterpart in a limited temperature
range, which is
400 −500 K
for the CG PS [
27
]. Below this temperature, the
4
structural properties is not transferable and its dynamics is accelerated due to the
reduction of degree of freedom [
30
], resulting in a glass transition temperature
of
Tg≈170 K
, much lower than the the values in atomistic simulations [
31
] and
experiments [
32
]. Therefore, this model is considered a generic model of glassy
thermoplastics below its
Tg
. The temperature transferablility could be improved
by the recently proposed energy renormalization approach [
33
,
34
]. However,
as we focus on general properties of glassy polymers, an arbitrary traditional
CG bead-spring model is sufficient for the purpose of this study. Compared to
other widely-used generic bead-spring models such as the the Kremer-Grest model
[
35
] and the morse model [
36
], the primary reason for using such a CG model
is that various previous work [
37
,
38
,
39
,
15
,
40
] has provided substantial useful
information for choosing its parameters.
The process of preparing the MD systems has been described elsewhere in detail
[
39
,
15
]. In brief, each MD system is generated using the self-avoiding random
walk algorithm [
41
] and first equilibrated at a high temperature of
590 K
and then
cooled down to the target temperature under NPT conditions. Different from our
previous work [
39
,
15
], we choose a much faster cooling rate of
˙
Tc=−50 K ns−1
and do not perform a further equilibrium simulation at the cooled target temperature.
This operation significantly reduces the effects of strain softening, which increases
with the state of equilibrium in glassy systems [
42
,
43
], making it possible to focus
on the effects of strain hardening. We consider cubic MD systems with periodic
boundary conditions (PBC), comprising 500 polymer chains with 200 CG beads in
each chain each. Temperatures from 10 K to 150 K are considered.
All the MD simulations are performed using LAMMPS [
44
] with a time step
size of
5 fs
. The Nosé-Hoover thermostat and barostat [
45
,
46
,
47
] with coupling
times of 0.5 ps and 5 ps are used, respectively.
2.2. Uniaxial tensile simulations
We conduct uniaxial tensile deformation in
x
-direction with constant true strain
rate of
˙ϵ= 1%/ns −100%/ns
up to a maximum stretch of
λx= 5
, where
λx
is
defined as
λx=lx/Lx
with
lx
and
Lx
being the current and initial edge length of
the MD system in the
x
-direction, respectively. The true strain rate is defined as
˙ϵ=˙
λx/λx
. The purpose of using true strain rate instead of engineering strain rate
is to keep it consistent with the expression of deformation velocity
l=˙
F F −1
used
in continuum mechanics. For convenience, we also describe the stretch using the
engineering normal strain as
εxx(t) = λx(t)−1 = exp( ˙ϵt)−1.(1)
5
The surfaces in the lateral directions are subjected to NPT conditions with
p= 1
atm. It is notable that voids could form at large strains, typically above the engi-
neering strain of
300%
, resulting in a drop of stress with increasing deformation,
which is neglected in this paper.
2.3. Virial stress contribution
We use the virial formulation [48] to evaluate the stress tensor, defined as
σ=−1
V
NA
X
I=1
mIvI⊗vI−1
V
NA
X
I=1
fI⊗rI(2)
with the volume of simulation box
V
and the total number of CG beads
NA
. The
first term refers to the kinematic contribution
σke
, where
mI
and
vI
denote the
mass and velocity vector of bead
I
. The second term represents the virial stress
tensor
σvirial
, where
rI
and
fI
are position vector and the resultant force vector on
atom
I
. It has been derived within statistical mechanics that the virial expression
(2)
is equivalent to the Cauchy stress tensor with a sufficient large
NA
[
49
,
50
], which
connects the macroscopic and microscopic mechanical responses.
To decompose the stress response between inter- and intra-chain interactions
in glassy polymers, we consider the components of force
fI
due to pair, bond,
and angle interactions as
fI=fpair
I+fbond
I+fangle
I
. The virial stress is then
decomposed as
σvirial =1
V
NA
X
I=1 −fpair
I⊗rI
| {z }
=:σpair
+1
V
NA
X
I=1 −fbond
I⊗rI
| {z }
=:σbond
+1
V
NA
X
I=1 −fangle
I⊗rI
| {z }
=:σangle
,(3)
giving the definition of pair stress
σpair
, bond stress
σbond
, and angle stress
σangle
.
2.4. Microscopic properties
We characterize the microscopic structure of local segments and global polymer
chain of the MD system. For local segments, the bond length
lb
, bond angle
θa
between two adjacent bond vectors, and bond orientation
PI
are considered.
Following [5], the bond orientation PIis defined as
PI=1
23 cos2θb
I−1.(4)
6
where
θb
I
is the angle between bond vector
I
and the loading direction, which is the
x
direction in this paper. These quantities are typically used in a thermodynamic
average sense. For example, the averaged bond orientation is denoted as
⟨P⟩=
PNB
I=1 PI/NB
with
NB
the number of bonds considered.
⟨P⟩
ranges from
−0.5
to
1
, which gives the value of
−0.5
if all bonds are orthogonal to the loading direction,
while equals
1
if they are all parallel to the loading direction. The value
⟨P⟩= 0
means random distribution of all the bond orientations.
At the scale of polymer chains, we consider their elongation during deformation
following the definition given by Hoy and Robbins [51], described as
εchain
xx (t) = q⟨R2
ee,x(t)⟩ − q⟨R2
ee,x(0)⟩
q⟨R2
ee,x(0)⟩
,(5)
where
Ree,x(t)
denotes the
x
-component of the end-to-end vector of a polymer
chain at time twhile ⟨R2
ee,x(t)⟩represents the thermodynamically-averaged value
of its squares.
3. Analysis of molecular dynamics results
3.1. Dominating relaxation mechanisms in strain hardening
We start with identifying the primary relaxation mechanisms of strain hardening
in glassy polymers, which result in their rate- and temperature-dependent behavior
as shown in Figure 1. Basically, the solid curves in Figure 1 show rate-dependent
hardening behavior at a constant temperature of
120 K
, indicating the existence of
relaxation mechanisms in the hardening region. The dashed curves in Figure 1 are
presented to repeat the time-temperature correlation (TTC) behavior as discussed in
our previous work [
39
] with a minimum set of simulations to compare the relaxation
mechanisms at small and large strains, exhibiting different TTC behavior between
small and large deformation at different temperatures.
At lower temperatures, the simulations under conditions of
[T= 120 K,˙ϵ=
100%/ns]
and
[T= 80 K,˙ϵ= 10%/ns]
result in approximately the same stress-
strain curves at both small and large deformation, suggesting that the relaxation
mechanisms in the hardening region should be the same as at small strains, i.e.,
before yielding. Assuming that the relaxation at the scale of polymer chains is not
involved at small strains before yielding due to frozen mobility, it is reasonable to
speculate that the rate-dependent strain hardening behavior at lower temperatures
7
0 100 200 300 400
0
50
100
150
200
εxx
(
%
)
σxx
(MPa)
120K,1%/ns
120K,10%/ns
120K,100%/ns
80K,10%/ns
150K,10%/ns
150K,20%/ns
Figure 1: Stress–strain curves of the MD systems in uniaxial tension simulations at
different temperatures and true strain rates.
is also primarily determined by the relaxation of local structures, i.e., the structure
composed of atoms close to each other.
In contrast, at higher temperatures close to
Tg
, the MD system exhibits different
TTC behavior at small and large strains. Specifically, the stress-strain curve at
[T= 120 K,˙ϵ= 1%/ns]
coincide with the curve at
[T= 150 K,˙ϵ= 20%/ns]
at
strains below
80%
but approximately coincide with the curve at
[T= 150 K,˙ϵ=
10%/ns]
at strains above
300%
. This different TTC behavior at small and large
strains means that the dominating relaxation mechanisms in the hardening region
at higher temperatures is not only the relaxation of local structures. Most likely, the
relaxation of polymer chains such as disentanglement start contributing to the rate-
and temperature-dependence of strain hardening at this temperature. As discussed
in Section 1, this conjecture can well interpret the contradictory results of MD
simulations for different glassy polymer models reported in [
19
,
17
], where the
entanglement length
Ne
remains approximately constant in [
19
] but evolves with
increasing strain in the hardening region in [17].
To observe the effects of disentanglement at higher temperatures, we fol-
low [
51
] to compare the stretch of polymer chains under different conditions during
deformation as depicted in Figure 2, where the averaged stretch of polymer chains
is scaled in the form of strain as defined in Equation
(5)
. The chain stretch
εchain
xx
can follow the macroscopic strain
εxx
with increasing deformation, indicating
that the effects of relaxation at the scale of polymer chains play a minor role in
strain hardening. Instead, polymer chains are stretched approximately affine to
the macroscopic deformation. It is the relaxation of local structures such as chain
8
segments that dominates the rate- and temperature-dependent strain hardening. At
different temperatures and strain rates, the chain stretch
εchain
xx
coincide at small
strains and exhibit slight deviations at large strains, where the chain stretch at
higher temperatures and lower strain rates becomes smaller than that at lower
temperatures and higher strain rates, indicating that disentanglement happens under
these conditions, but only resulting in a small contribution to the relaxation at the
scale of polymer chains. This result is consistent with the observation in [
51
] and
the results shown in Figure 1. In summary, the stress increase in strain hardening
is caused by chain stretch, primarily mediated by relaxation of local structures at
temperatures below
Tg
, and slightly influenced by the relaxation at larger length
scales such as disentanglement at the temperatures close to Tg.
0 100 200 300 400
0
100
200
300
400
εxx
(
%
)
εchain
xx
(
%
)
120K,1%/ns
120K,10%/ns
120K,100%/ns
80K,10%/ns
150K,10%/ns
150K,20%/ns
360 380 400
320
340
360
Figure 2: The averaged stretch of polymer chains as defined in Equation
(5)
as a
function of applied engineering strain at different temperatures and true strain rates.
3.2. Primary contributions to strain hardening
We further attempt to identify which local structure is the main resource to
strain hardening in glassy polymers. To this end, we consider the virial components
of stress responses during deformation as defined in Equation
(3)
. We focus on the
simulations at lower temperatures to avoid the influence of disentanglement. Figure
3(a) depicts the
x
components of bond, angle, and pair stresses during deformation
at the conditions of
[T= 120 K,˙ϵ= 100%/ns]
and
[T= 80 K,˙ϵ= 10%/ns]
,
which is adopted from the simulation sets resulting in coinciding stress-strain
curves in Figure 1. It is evident in Figure 3(a) that the virial components also
coincide in these simulations, suggesting that the dominating relaxation mechanism
also has very similar influence on each virial component. In Figure 3(a), the pair
9
stress initially increases to a yield point and then decreases continuously, suggesting
that the pair interaction is not the primary contribution to strain hardening. In
contrast, the strain hardening should be attributed to the bond and angle stresses as
they increase in the hardening region.
(a)
0 100 200 300 400
0
50
100
150
εxx (%)
∆σxx (MPa)
σbond,80K,10%/ns
σangle,80K,10%/ns
σpair,80K,10%/ns
σbond,120K,100%/ns
σangle,120K,100%/ns
σpair,120K,100%/ns
(b)
0 100 200 300 400
0
20
40
60
εxx (%)
∆σdev
xx (MPa)
σbond,80K,10%/ns
σangle,80K,10%/ns
σpair,80K,10%/ns
σbond,120K,100%/ns
σangle,120K,100%/ns
σpair,120K,100%/ns
Figure 3: The virial stress contributions as defined in Equation
(3)
as a function of
engineering strain in the form of (a) the normal component in the loading direction
and (b) the deviatoric part. The
y
axis
∆σ
denoted the in crease of stress compared to
its initial value as ∆σ(t) = σ(t)−σ(0) at time t.
To explain the drop of the
x
component of the pair stress in Figure 3(a), we
further consider the deviatoric part of the stress, which is defined as
σdev
xx =
[2σxx −σyy −σzz ]/3
in uniaxial deformation, as depicted in Figure 3(b). Here,
the deviatoric pair stress remains approximately constant after yielding, indicating
that pair interactions also has no negative contributions to strain hardening. As
explained in our previous work [
15
], the decrease of pair stress in the loading
direction could be attributed to bond orientation, which squeezes the space in
the loading direction but releases more space along the lateral directions for pair
interactions. Similar to their
x
components, the deviatoric part of bond and angle
stresses also increases in the hardening region, justifying that the strain hardening
is caused by bond and angle interactions in the deviatiric point of view.
To further explore the exact source contributing to the stress response in strain
hardening, we compare the evolution of different quantities representing the micro-
scopic structures in Figure 4. Figure 4(a) presents the evolution of the averaged
bond length
⟨lb⟩
during deformation at a strain rate of
˙ϵ= 100%/ns
at different
temperatures. A nonmonotonic dependence of
⟨lb⟩
on the temperature suggests that
⟨lb⟩
is not an appropriate variable to describe the microscopic structure associated
with strain hardening. Instead, we use the averaged value of the maximum bond
length in each polymer chain
⟨lb
max⟩
to characterize the microscopic structure. As
10
shown in Figure 4(b), the variable
⟨lb
max⟩
monotonically increases with decreasing
temperatures in the hardening region. Furthermore, the transition similar to yield-
ing can be observed in the curves in Figure 4(b). These features indicate that
⟨lb
max⟩
could be a good candidate to characterize the microscopic structure associated
with strain hardening. Moreover, it is reasonable to imagine that the traction of
an elongated chain should be balanced by its longest bond if all bonds are con-
nected in series. We denote this bond as load-bearing bond. We further denote
the local deformation associated with the load-bearing bonds as local load-bearing
deformation.
(a)
0 100 200 300 400
4.8
5
5.2
εxx
(
%
)
⟨lb⟩
(Å)
40K
60K
80K
100K
120K
(b)
0 100 200 300 400
5.8
5.9
6
6.1
εxx
(
%
)
⟨lb
max⟩
(Å)
40K
60K
80K
100K
120K
(c)
0 100 200 300 400
0
0.1
0.2
εxx (%)
⟨P⟩
40K
60K
80K
100K
120K
(d)
0 100 200 300 400
75
80
85
90
95
εxx (%)
⟨θb⟩(◦)
40K
60K
80K
100K
120K
Figure 4: The averaged microscopic quantities of the MD system along deformation
with a strain rate of
100%/ns
at different temperatures: (a) averaged bond length, (b)
averaged value of the largest bond length in each polymer chain, (c) averaged bond
orientation as defined in Equation (4), and (d) averaged bond angle.
Figure 4(c-d) presents the evolution of averaged chain orientation
⟨P⟩
and
bond angle
⟨θb⟩
during deformation, respectively, which increases with decreasing
temperature in the hardening region. These two variables can represent the relax-
ation of the chain segments, where a slower increase during deformation means a
11
stronger relaxation, caused by higher temperatures. This relaxation accommodates
the chain stretch to mediate the stress increase.
3.3. The quantitative effects load-bearing bond
Following the finding that the stress in the hardening region is primarily con-
tributed by the longest bond in each chain, i.e., the load-bearing bonds, we explore
their quantitative effects on strain hardening in this section.
In Figure 5(a-b), the evolution of bond stress and angle stress are plotted against
the change of the averaged lengths of load-bearing bonds, which are represented
in the form of relative stretch
⟨lb
max⟩/⟨lb
max(0)⟩
. Here,
⟨lb
max(0)⟩⟩
denotes the value
of
⟨lb
max⟩
before deformation. It is evident that both the bond and angle stresses
approximately coincide at larger stretch of load-bearing bonds, corresponding to
the hardening region. This result suggest that the bond and angle stresses are
dominated by the stretch of load-bearing bonds, independent of the relaxation of
chain segments, for example, characterized by the evolution of ⟨P⟩and ⟨θb⟩.
(a)
1 1.01 1.02 1.03 1.04 1.05 1.06
0
50
100
150
200
⟨lb
max⟩/⟨lb
max(0)⟩
∆σbond
xx
(MPa)
40K,100%/ns 40K,10%/ns
40K,1%/ns 60K,100%/ns
60K,10%/ns 60K,1%/ns
80K,100%/ns 80K,10%/ns
80K,1%/ns 100K,100%/ns
100K,10%/ns 100K,1%/ns
120K,100%/ns 120K,10%/ns
120K,1%/ns
(b)
1 1.01 1.02 1.03 1.04 1.05 1.06
0
20
40
60
⟨lb
max⟩/⟨lb
max(0)⟩
∆σangle
xx (MPa)
Figure 5: The response of (a) bond stress and (b) angle stress as a function of
the stretch of the averaged value of the longest bonds in each chain at different
temperatures and strain rates.
We relate the stretch of load-bearing bonds to continuum mechanics by repre-
senting them using a local deformation gradient, denoted as
Fl
. In uniaxial defor-
mation,
Fl
comprises only the diagonal components as
Fl
ii =⟨lb
max,i⟩/⟨lb
max,i(0)⟩
with
i=x, y, z
, where
lb
max,i
denotes the projection of
lb
max
in direction
i
. Then we
have the left Cauchy-Green tensor
bl=FlFlT
with
FlT =FlT
. Its isochoric
component of the deviatoric part is given by
bl∗,dev =Jl−2
3bl−1
3itr Jl−2
3bl(6)
12
with the Jacobian determinant
Jl= det Fl
and the second-order unit tensor in
the deformed configuration
i
. Figure 6 depicts the deviatoric part of the bonded
Kirchhoff stresses with respect to the evolution of the
x
components of
bl∗,dev
,
showing that the bonded deviatoric Kirchhoff stresses are almost proportional to
bl∗,dev
xx
in the hardening region, independent of temperatures and strain rates. This
relation can be formulated as
τbonded,dev =µbdev bl∗,(7)
which is equivalent to the formulation of Neo-Hookean (NH) model utilizing
τ=Jσ
with
J= det F
. Here, the sum of the bond and angle stress is denoted as
bonded stress as they exhibit similar dependence on the local load-bearing stretch
as shown in Figure 5. Through fitting the curves in the hardening region in Figure
6, the value of the bond shear modulus is obtained as
µb= 131 MPa
is obtained.
Based on the relation of the bonded stress and load-bearing stretch in the hardening
region, we assume that µbis the intrinsic hardening modulus.
0 0.511.5 2
0
50
100
150
bl∗,dev
xx
∆τdev,bond
xx + ∆τdev,angle
xx (MPa)
40K,100%/ns
40K,10%/ns
40K,1%/ns
60K,100%/ns
60K,10%/ns
60K,1%/ns
80K,100%/ns
80K,10%/ns
80K,1%/ns
100K,100%/ns
100K,10%/ns
100K,1%/ns
120K,100%/ns
120K,10%/ns
120K,1%/ns
Figure 6: Deviatoric part of the bonded stress (sum of the bond and angle stress) as
a function of the deviatoric part of the left Cauchy-Green deformation tensor for the
load-bearing stretch in the loading direction as defined in Equation (6).
In summary, the results in this section suggests that the strain hardening is
caused by the stretch of chain segments, primarily mediated by the relaxation
of local structures at temperatures far below
Tg
and also slightly influenced by
disentanglement at temperatures close to
Tg
. The stress increase during strain
hardening is primarily contributed by bonded interactions, i.e., the bond stress and
angle stress, while the pair stress only has negligible influence in the hardening
13
region. The bonded stress is mainly contributed by the stretch of local load-bearing
bond, which can be related to continuum mechanics by defining a local load-
bearing deformation gradient
Fl
. Such a definition results in a relation between
the bonded stress and
Fl
, independent of temperatures and strain rates, leading to
the NH constitutive model characterized by the bond shear modulus µb.
4. Decomposition of the deformation gradient
Similar to the decomposition of the virial stress, we assume that the total stress
in continuum mechanics comprises the contribution from pair, bonded and an
elastic term as represented by the rheological model in Figure 7:
σ=σpair +σbonded +σe,(8)
where
σbonded
is the stress due to bond and angle interactions. The elastic term
σe
is defined for completeness of the constitutive model, which might be mainly
attributed to volumetric deformation. We assume these stress terms are independent
and our focus in this section is on the analysis of the bonded branch.
σe
σpair ηp
σbonded Mback
ηb
σ σ
Figure 7: Rheological representation of the constitutive model.
4.1. Decomposition of deformation gradient
Based on the results in Section 3.3 that the bonded stress is merely contributed
by the local load-bearing deformation gradient
Fl
, it is reasonable to assume a
decomposition of the total deformation into two parts as
F=FlFr
, where
Fr
is
denoted as the resistance part of the deformation gradient. This decomposition
corresponds to the illustration of the bonded branch in Figure 7, where
Fl
con-
tributes to
σbonded
while the evolution of
Fr
is driven by
σbonded
. As the part
of chain segments corresponding to
Fr
also has stress caused by bonded inter-
actions, which resists itself to be stretched, the resultant driving force should be
the difference between
σbonded
and the resistant stress. In convention, we denote
14
this resistance stress as back stress. The driving stress leads to the evolution of
Fr
, balanced by the friction due to sliding of polymer segments. The friction
is considered as rate-dependent and characterized by the viscosity
ηb
. While in
mathematical representation, our decomposition
F=FlFr
is the same as the
Lee-Kröner decomposition
F=FeFp
[
10
,
11
], the physical meaning becomes
clear in the decomposition F=FlFrfor glassy polymers.
The velocity gradient of the resistant part is defined as
lr=˙
FrFr−1,
and the corresponding rate of deformation tensor is given by
Dr= [lr+lrT]/2
.
Assuming isochoric resistance deformation
Fr
analogous to most study of inelastic
flow in glassy polymers, i.e.,
det Fr= 1
, the relation between
Dr
and the driving
effective Mandel stress Meff,dev =Mb onded,dev −Mback,dev is given by
Dr=Meff,dev
2ηb,(9)
where
Mbonded,dev =FlT τbonded,dev Fl−T
is the pull-back of the bonded Kirch-
hoff stress
τbonded,dev =Jσbonded,dev
to the intermediate configuration. The
superscript “dev” denotes the deviatoric part of the stress tensor.
4.2. Deformation of the resistance stretch
Based on the results obtained in Section 3, the isochoric part of the load-
bearing deformation gradient
Fl∗
can be recovered using the relation
(7)
with
given bond shear modulus
µb
and bonded stress in MD simulations. Then, it
is straightforward to calculate the components of
Fr
and its rate
˙
Fr
using the
decomposition
F=FlFr
. Figure 8(a) presents the evolution of the stretch of the
resistance part in the loading direction
λr
x
at various temperatures and strain rates,
where
λr
x
is the
x
component in
Fr
. An evident rate- and temperature-dependence
of
λr
x
can be observed. The resistance stretch in simulations at higher temperature
and lower strain rate shows more tendency to follow the total deformation of the
systems. Figure 8(b) illustrates the
x
component of
Dr
scaled by the loading
rate during deformation at the same temperature and rate conditions as in Figure
8(a). It is striking that all the scaled curves approximately coincide during the
whole deformation process, suggesting that the flow rate of the resistance stretch
is independent of temperature. Instead, it can follow the stretch rate of the whole
system
˙ϵ
in a scaled point of view. A yield point can be evidently observed at the
highest point of the curves, where
Dr
xx ≈˙ϵ
, indicating that at the yield point only
15
the resistance part is deformed while the load-bearing stretch
λl
x
is fixed. After the
yield point, the decrease of
Dr
xx
corresponds to the increase of the load-bearing
stretch ˙
λl
x, leading to the increase of bonded stress in the hardening region.
(a)
0 100 200 300 400
1
2
3
εxx (%)
λr
x
40K,100%/ns
40K,10%/ns
40K,1%/ns
80K,100%/ns
80K,10%/ns
80K,1%/ns
120K,100%/ns
120K,10%/ns
120K,1%/ns
(b)
0 100 200 300 400
0
0.5
1
εxx (%)
Dr
xx/˙ϵ
Figure 8: Estimated (a) stretch and (b) rate of deformation of the resistance part of
the deformation gradient
Fr
in the loading direction. In (b), the rate of deformation is
scaled by the prescribed true strain rate.
4.3. Yielding at small deformation
To analyze the stress arising from the flow of the resistance deformation, we
consider the yield behavior of the bonded stress at small strains. Figure 9 shows
the stress-strain curves of the bonded contributions in MD simulations at strains
below
30%
, where the yield points are marked by solid points. The yield stress is
estimated as the highest point on the curves at strains below
20%
. As the bonded
stress flattens after yielding, the errors of the yield stresses resulting from this
estimation method are acceptable.
The estimated yield stresses are used to identify the parameters in the viscosity
model ηbin Equation (9). Here, we consider the Eyring model
ηb=ηb
0(T)Qmeq
Tsinh Qmeq
T−1
≈ηb
0(T)2Qmeq
Texp Qmeq
T−1
(10)
with the specific activation volume
Q
and equivalent stress
meq =pMeff,dev :Meff ,dev /2
.
The variable
ηb
0(T)
is a temperature-dependent viscosity at zero stress, denoted
as static viscosity. The approximation sign is Equation
(10)
is valid if
meq
is
16
0 10 20 30
0
5
10
15
εxx (%)
∆τdev,bond
xx + ∆τdev,angle
xx (MPa)
40K,1%/ns
40K,10%/ns
40K,100%/ns
80K,1%/ns
80K,10%/ns
80K,100%/ns
120K,1%/ns
120K,10%/ns
120K,100%/ns
Figure 9: The response of bonded stress of the MD system during deformation at
small strains marked by the yield points.
sufficiently large. In uniaxial deformation alone
x
-axis,
meq =√3|Meff,dev
xx |/2 =
√3|τeff,dev
xx |/2.
Assuming that before yielding there is no back stress, we have
Meff,dev =
Mbonded,dev
at small strains. Utilizing the approximation at the yield point that
Dr
xx,y≈˙ϵ
as demonstrated in Figure 8(b) and combining Equations
(9)
and
(10)
,
we have
Mbonded,dev
xx,y=ηb
0(T)2√3Q|Mbonded,dev
xx,y|
Texp Qmeq
y
T−1
˙ϵ, (11)
where the subscript “y” denotes the stress at the yield point. Given
Mbonded,dev
xx,y=
τbonded,dev
xx,y>0
in uniaxial tension as evident in the MD simulation results, Equation
(11) can be transformed into
meq
y
T=1
Qln 2√3ηb
0(T)Q˙ϵref
T!+1
Qln ˙ϵ
˙ϵref ,(12)
where the variable
˙ϵref
is an arbitrary constant reference strain rate. The dependence
of
meq
y/T
on
ln(˙ϵ/ ˙ϵref )
at different temperatures and strain rates are plotted in
Figure 10(a), where
˙ϵref = 1%/ns
is used. The specific activation volume
Q
can
be fitted, giving an averaged value of Q= 80.54 K MPa−1.
We further identify the temperature dependent static viscosity
ηb
0(T)
by intro-
ducing a shift factor
a(T)
that
ηb
0(T) = a(T)ηb
g0
with the static viscosity
ηb
g0
at
Tg
.
Inserting this relation into Equation
(12)
, the shift factor
a(T)
can be expressed as
ln(a(T)) = Qmeq
y
T−ln Tg
T−ln 2√3ηb
g0Q˙ϵref
Tg!−ln ˙ϵ
˙ϵref ,(13)
17
(a)
0 1 2 3 4
0.1
0.2
0.3
ln(˙ϵ/˙ϵref)
meq,bonded
y/T
(MPa/K)
40K
60K
80K
100K
120K
(b)
0.008 0.012 0.016 0.02 0.024
5
10
15
1/T
(1/K)
Q meq,bonded
y(˙ϵref)/T
(MPa/K)
sim
t
Figure 10: (a) Bonded yield stress as a function of scaled strain rates at different
temperatures with
˙ϵref = 1%/ns
and their linear fitting (dashed). (b) The quantity of
Q meq,bonded
y/T
at the strain rate of
˙ϵref
as a function of
1/T
from simulations (solid
curve) and the linear fit (dashed curve).
where the last two terms on the right hand side are independent of temperature.
Using Arrhenius function that
a(T) = exp (A/T −A/Tg)
, we can identify the
constant
A
by fitting the curves of
Qτeq
y/T −ln (Tg/T )
with respect to
1/T
with
a linear function as shown in Figure 10(b). We obtain
A= 1008 K
and the relation
Qmeq
y
T−ln Tg
T=A
T+B(14)
with
B=−6.749
. Comparing Equations
(13)
and
(14)
, the static viscosity at
Tg
can be expressed as
ηb
g0 =Tg
2√3Q˙ϵref
exp B+A
Tg,(15)
which results in ηg0 = 26.86 MPa ns = 2.686 ×10−8MPa s.
It is notable that the values of
Q
,
A
, and
ηb
g0
are calculated based on the follow-
ing assumptions at the yield point: i) the back stress has negligible contributions to
the flow of the resistance deformation; ii) the flow rate of the resistance deformation
is the same as the prescribed strain rate; iii) The Eyring model and Arrhenius func-
tion are applicable for the MD systems at temperatures below
Tg
at small strains
around the yield point. While the results derived from the last two assumptions
seem to be reasonable as evident from the MD simulation results shown in Figures
8(b) and 10(a-b), the first assumption could result in oversimplification and errors
of the estimated values. The values of these quantities will be further checked in
constitutive models discussed in Section 5.
18
4.4. Back stress
The remaining component in the bonded branch in Figure 7 is the expression
of the back stress. We write the xcomponent in Equation (9) as
Meff,dev
xx = 2ηbDr
xx =ηb
g0a(T)2Qmeq
Tsinh Qmeq
T−1
Dr
xx.(16)
In uniaxial tension, we can solve this equation utilizing the relation
meq =
√3Meff,dev
xx /2, which gives
Meff,dev
xx =2
√3
T
Qsinh−1 √3ηb
g0 a(T)QDr
xx
T!.(17)
We plot the results of the estimated
Meff,dev
xx
as a function of strain at different
temperatures and strain rates in Figure 11(a) and found that the effective Mandel
stress flattens in the hardening region. While this is based on the assumption that
the viscosity
ηb
is independent of the shape of the system, these curves cannot
provide quantitative results for the analysis of the microscopic mechanisms of strain
hardening bur only serve to provide some insights for the subsequent constitutive
modeling.
(a)
0 100 200 300 400
0
5
10
15
εxx
(
%
)
Meff
xx
(MPa)
(b)
0 2 4 6 8
0
50
100
150
br∗,dev
xx
Mback
xx
(MPa)
40K,100%/ns
40K,10%/ns
40K,1%/ns
80K,100%/ns
80K,10%/ns
80K,1%/ns
120K,100%/ns
120K,10%/ns
120K,1%/ns
Figure 11: Estimated (a) effective Mandel stress and (b) back stress in the deviatoric
form.
Subtracting the estimated effective stress from the total bonded Mandel stress,
we obtain the estimated back stress
Mback
xx
. We plot them as a function of the
deviatoric part of the left Cauchy green tensor for the resistance deformation
br∗,dev
xx
in Figure 11(b). The definition of
br∗,dev
xx
is similar to their counterpart
bl∗,dev
xx
given
19
in Equation
(6)
. The rate- and temperature-dependence of the curves in Figure
11(b) indicates that the back stress
Mback
is not a unary function of
Fr
. Instead,
it is appropriate to introduce an internal variable to construct the function of the
back stress, such as the widely used orientation tensor
A
[
14
]. Furthermore, the
shape of the curves suggests that the relation between
Mback
and
A
should not be
the NH model as the slopes increases with deformation. A Langevin-based model
could be considered as used in [26].
5. Constitutive modeling
In this section, we synthesize the results obtained in previous sections in a con-
stitutive model as schematically illustrated in Figure 7 to validate the mechanism
of local load-bearing deformation that primarily induces strain hardening in glassy
polymers. In the pair branch, the conventional Lee-Kröner decomposition of the
deformation gradient
F=FeFp
[
10
,
11
] is used while in the bonded branch,
the decomposition
F=FlFr
discussed in Section 3 is assumed. We start with
deriving the mathematical expression in Section 5.1 and 5.2, followed by parameter
identification and validation in Section 5.3 and 5.4, respectively.
5.1. Thermodynamics
The requirement of the constitutive model for thermodynamic consistency is
firstly derived in this subsection. Different from the conventional Coleman-Noll
procedure [
52
,
53
], which considers a single continuum body
Ω
subjected external
source of force, heat flux, and entropy flux, we follow the method proposed
by Bouchbinder [
54
] to derive the expression of the first and second laws of
thermodynamics by taking into account the effects of the reservoir, which is
thermally connected with Ω. Then, the first and second laws read
ZΩ
˙udV+˙
UR=ZΩ
S:1
2˙
CdV(18)
ZΩ
˙sdV+˙
SR≥0,(19)
where
u
and
s
denote the energy density and entropy density in
Ω
while
UR
and
SR
are the total energy and entropy in the reservoir, respectively. The variables
S
and
C=FTF
denote the Piola-Kirchhoff stress tensor and the right Cauchy-Green
tensor, respectively. As the reservoir is in equilibrium state, its temperature can be
20
defined as
TR=∂UR/∂SR
, resulting in the relation
˙
SR=˙
UR/TR
. Inserting this
relation and the first law (18) into the second law (19), it can be transformed as
−1−T
TR˙
UR+ZΩS:1
2˙
C−˙
Ψ−˙
T sdV≥0,(20)
where the free energy density
Ψ
is operationally defined as
Ψ = u−T s
to avoid
the explicit dependence of the energy on entropy u=u(s).
The variable
−˙
UR
in Equation
(20)
equals the heat flux from the reservoir to
the system Ω. This process controls the temperature Tin Ωthrough adjusting TR,
which is independent of what happens in
Ω
. Therefore, the inequality
(20)
must be
satisfied for arbitrary ˙
UR, requiring
−1−T
TR˙
UR≥0,(21)
which can be fulfilled by specifying the direction of the heat flux
Q:=−˙
UR=
−K[T−TR]
with an effective scalar coefficient
K > 0
. The second term in
Equation
(20)
also must be satisfied for all possible
˙
C
and
˙
T
, independent of the
heat heat flux into Ω, and also for arbitrary Ω, leading to the inequality
S:1
2˙
C−˙
Ψ−s˙
T≥0,(22)
essentially identical to the conventional Clausius-Duhem inequality.
Based on the MD simulation results discussed in Section 3 and 4, the free
energy density
Ψ
is assumed to be a function of temperature
T
, the right Cauchy-
Green tensor
C
, the visco-plastic part of the deformation gradient
Fp
for the pair
contributions, the stretch resisting part of the deformation gradient
Fr
for the
contribution from bonded interactions, and an internal variable
A
associated with
the back stress, which has been mostly assumed to represent the orientation of
polymer chains and is symmetric and unimodular [
14
]. The rate of the free energy
is then expressed as
˙
Ψ = ∂Ψ
∂T ˙
T+∂Ψ
∂C:˙
C+∂Ψ
∂Fp:˙
Fp+∂Ψ
∂Fr:˙
Fr+∂Ψ
∂A:˙
A
=∂Ψ
∂T ˙
T+ 2 ∂Ψ
∂C:1
2˙
C−2Ce∂Ψ
∂Ce:Dp−2Cl∂Ψ
∂Cl:Dr+∂Ψ
∂A:˙
A,(23)
21
where
Dp
is the symmetric part of the velocity gradient
lp=˙
FpFp−1
. Substituting
Equation (23) into (22), the Clausius-Duhem inequality becomes
S−2∂Ψ
∂C:1
2˙
C−s+∂Ψ
∂T ˙
T+ 2Ce∂Ψ
∂Ce:Dp
+ 2Cl∂Ψ
∂Cl:Dr−∂Ψ
∂A:˙
A≥0,(24)
which must be satisfied for all possible
˙
C
and
˙
T
, resulting in the constitutive model
S= 2 ∂Ψ
∂C,(25)
s=−∂Ψ
∂T .(26)
These expressions simplifies the inequality (24) to
2Ce∂Ψ
∂Ce:Dp+ 2Cl∂Ψ
∂Cl:Dr−∂Ψ
∂A:˙
A≥0.(27)
As assumed that the pair and bonded contributions are independent, the inequality
(27)
must be satisfied for all possible
Dp
, requiring that the first term in Equation
(27) to be non-negative. This can be realized by defining the flow law as
Dp=1
2ηp2Ce∂Ψ
∂Ce
|{z }
=:Mpair
=Mpair
2ηp(28)
with the viscosity ηp>0, where Mpair denotes the pair Mandel stress tensor.
For the same reason, the last two terms in Equation
(27)
must also be non-
negative as they are variables associated with the bonded contributions. Adopting
the following evolution equation for A
˙
A=ADr+DrA−f(A)A(29)
as widely used [
14
,
55
,
26
], where the function
f(A)
represents the recovery
relaxation of
A
due to the back stress originated from
A
. The variable
A
would
be associated with the left Cauchy-Green tensor
br=FrFrT
if the recovery term
f(A)
vanishes [
14
]. Using the equations of motion
(29)
, the last two terms in
inequality (27) can be simplified as
2Cl∂Ψ
∂Cl−2A∂Ψ
∂A:Dr+A∂Ψ
∂A:f(A)≥0.(30)
22
The simplified inequality
(30)
can be satisfied by requiring a stronger form that
each term is non-negative. Similar to Ref. [26], we adopt the simplest form that
Dr=1
2ηb2Cl∂Ψ
∂Cl−2A∂Ψ
∂A
| {z }
=:Meff
=Meff
2ηb,(31)
f(A) = 1
2ηori 2A∂Ψ
∂A
| {z }
=:Mback
=Mback
2ηori (32)
with the viscosity
ηb>0
and
ηori >0
, as well as the Mandel back stress
Mback
and the Mandel effective stress
Meff
. The equilibrium relations
(25)
,
(26)
and
the evolution equations
(28)
,
(31)
,
(29)
, as well as the expression for the recovery
function
(32)
constitute the formulation of the constitutive model satisfying the
thermodynamic laws.
5.2. Individual models
We assume the free energy density
Ψ
can be decomposed into a thermal part
Ψth(T)
, an equilibrium part
Ψe(C)
, an inequilibrium part accounting for the
pair contribution
Ψpair(Ce)
, an inequilibrium part for the bonded contribution
Ψbond(Cl), and an orientation part Ψori (A)as
Ψ = Ψth(T)+Ψe(C)+Ψpair(Ce)+Ψbond (Cl)+Ψori(A).(33)
The expression for
Ψth
is not of interest in the present paper as we focus on
the simulations under isothermal conditions. The equilibrium term is assumed to
be only dependent on the volume change as there are no permanent cross-links in
the present MD models for thermoplastics. The pair term
Ψpair
is assumed to be
subjected to the NH model for simplicity. The bonded term is also considered as
an NH model as justified in Section 3.3, specifically, as evident in Figure 6. The
expression of
Ψori
is assumed to be a Langevin-related model as suggested by the
results in Figure 11(b), where we take the widely used Eight-Chain model [
12
].
23
Therefore, the individual models can be summarized as
Ψe=κe
2[J−1]2,(34)
Ψpair =µp
2[tr (Ce∗)−3] + κp
2[Je−1]2,(35)
Ψbonded =µb
2tr Cl∗−3+κb
2Jb−12,(36)
Ψori =µbackλLλ∗β+λLln β
sinh β−Ψori
0(37)
with
λ∗=ptr(A∗)/3
,
β=L−1λ∗/λL
and a constant
Ψori
0
. The parameter
λL
represents the locking stretch while the function
L−1
is the inverse of the Langevin
function
L(•) = coth(•)−(•)−1
. The coefficient
µback
can be considered as shear
modulus accounting for the chain orientation. As
Je=Jb=J
, the effective total
bulk modulus can be defined as κ=κe+κp+κb.
Utilizing the constitutive relation
(25)
, the Cauchy stress tensor can be calcu-
lated as
σ=1
JF SF T=µp
Jdev (be∗)
| {z }
=:σpair,dev
+µb
Jdev bl∗
| {z }
=:σbonded,dev
+κ[J−1]
| {z }
=:σvol
.(38)
The back stress reads
Mback =µback L−1λ∗/λL
3λ∗/λLdev (A∗),(39)
which recovers to the NH formulation at small ratio of
λ∗/λL
as
L−1(x)≈3x
for
sufficiently small positive
x
. The Mandel stresses associated with the bonded and
pair terms are given by
Mbonded = 2Cl∂Ψ
∂Cl=µbdev Cl∗,(40)
Mpair = 2Ce∂Ψ
∂Ce=µpdev (Ce∗).(41)
The flow rates in the associated branches are defined in Equations
(28)
and
(31)
with Meff =Mbonded −Mback, while the flow rate of Ais expressed as
˙
A=ADr+DrA−Mback
2ηori A.(42)
24
As discussed in Section 4, the Eyring model is adopted for the viscosity associ-
ated with bonded term as in Equation
(10)
. The same model is also adopted for the
pair term. They are expressed as
ηb=ηb
g0ab(T)Qbmeff
eq
T"sinh Qbmeff
eq
T!#−1
,(43)
ηp=ηp
g0ap(T)Qpmpair
eq
Tsinh Qpmpair
eq
T−1
(44)
with
Qb
,
ηb
g0
, and
meff
eq
being the specific activation volume, the stress-independent
viscosity at
Tg
, and the norm of
Meff
, respectively, associated with the bonded term.
Accordingly, the quantities
Qp
,
ηb
g0
, and
mpair
eq
have the same meaning associated
with the pair term. The Arrhenius function is taken for both terms as
ab(T) = exp Ab
T−Ab
Tg,(45)
ap(T) = exp Ap
T−Ap
Tg(46)
with constant parameters
Ab>0
and
Ap>0
. The viscosity associated with chain
orientation
ηori
is assumed to be a function of
ηb
and the shape of the system,
taking the form
ηori =c ηbλ∗
max
λ∗
min m
(47)
with
λ∗
max = max {λ∗
1, λ∗
2, λ∗
3}
and
λ∗
min = min {λ∗
1, λ∗
2, λ∗
3}
, where
λ∗2
1
,
λ∗2
2
,
λ∗2
3
denote the eigenvalues of
C∗
. The coefficient
c
is a constant parameter denoting
the ratio between
ηori
and
ηb
. The exponent
m
accounts for the effect of shape
on the viscosity for chain orientation, which increases with the stretch of polymer
chains.
5.3. Parameter identification
The parameters of the constitutive model are listed in Table 1 together with their
identified values, categorized into the groups of pair, bonded, and back stresses,
as well as the bulk modulus for the total stress. The bulk modulus
κ
is roughly
estimated according to the response of the pressure to the volume change
J
, which
is not a main focuses of the present paper. The parameters in the pair group are
25
identified according to the stress-strain curves of the pair contributions of the virial
stress as presented in Section 3.2, where the shear modulus
µp
is estimated to be the
slope of the stress-strain curves at small strains while the remaining parameters are
identified using the same process as discussed in Section 4.3. In the bonded group,
the shear modulus
µb
is assumed to be identical to be the bond shear modulus
fitted in Section 3.3 while the activation temperature
Ab
and the specific activation
volume
Qb
are adopted from the fitted values presented in Section 4.3. However,
this process is based on the assumption that the back stress vanishes before yielding,
which could result in deviations between the results from the constitutive model
and the MD simulations. We attempt to reduce this deviation by adjusting as few
parameters as possible. Here, only the static viscosity
ηb
g0
is set to be adjustable.
In the back group, we assume the shear modulus
µback
is identical to
µb
as they
originates from the same chain segments while the remaining three parameters
have to be identified by fitting the stress-strain curves.
Table 1: Parameters of the constitutive model and their identified values.
Parameter value unit physical meaning stress groups
κ300 [MPa] total bulk modulus total
µp148 [MPa] shear modulus
pair
ηp
g0 0.003 [MPa s] static viscosity at Tg
Ap3393 [K] activation temperature
Qp118 [K MPa−1]spcific activation volume
µb131 [MPa] shear modulus
bonded
ηb
g0 2.686E-8 [MPa s] static viscosity at Tg
Ab1008 [K] activation temperature
Qb80.54 [K MPa−1]spcific activation volume
µback 131 [MPa] shear modulus
backλL1.1 [−]locking stretch
c0.049 [−]viscosity ratio
m2.39 [−]shape exponent
Specifically, the parameters
ηb
g0
,
λL
,
c
, and
m
are identified by minimizing the
deviations of the deviatoric part of the bonded stresses between the constitutive
model and the MD simulation results. We consider the conditions at temperatures
of
60 K
and
100 K
, each for strain rates of
1%/ns
and
1%/ns
. As shown in Figure
12, the constitutive model can well reproduce the MD results at
60 K
and at the
strain rate of
100%/ns
for
100 K
. For the simulation at
100 K
and
1%/ns
, the
26
deviations are small at intermediate strains but start increasing from the strain of
300%
, which might be caused by the effects of disentanglement as discussed in
Section 3.2. As the main focus of the paper is the mechanisms of local load-bearing
stretch on strain hardening, the effects of disentanglement are not further discussed
here and will be systematically studied in future research.
0 100 200 300 400
0
50
100
150
εxx
(
%
)
τdev,bonded
xx
(MPa)
60 K, 1%/ns
60 K, 100%/ns
100 K, 1%/ns
100 K, 100%/ns
Figure 12: Parameter identification: Bonded Kirchhoff stress during deformation of
the constitutive model (solid curves) and MD simulations (dashed curves).
5.4. Validation
We compare the stress responses of the constitutive model with identified
parameters summarized in Table 1 and the MD simulation results in uniaxial tension
at temperatures of
T= 20 K −120 K
for every
20 K
below
Tg
with true strain rates
of
˙ϵ= 1%/ns −100%/ns
. It is noticeable that the temperatures considered in the
validation are beyond the limits of temperatures used for parameter identification
to challenge the constitutive model.
Firstly, the bonded stresses are compared as illustrated in Figure 13, demon-
strating an excellent agreement between the constitutive model and the MD results,
except for the deviations at the temperatures of 100 K and 120 K with strain rates
of
1%/ns
as shown in Figure 13(a). This can be attributed to the effects of disen-
tanglement that are not considered in this paper as discussed in Section 5.3, which
will be pursued in future work.
Furthermore, we also present the results of the pair stress and the total stress
for the completeness of evaluating the constitutive model. As shown in Figure
14, both the pair stress and the total stress of the constitutive model closely match
the MD results. However, some deviations are noticeable for the pair stress. The
27
(a) ˙ϵ= 1%/ns
0 100 200 300 400
0
50
100
150
εxx
(
%
)
τdev,bonded
xx
(MPa)
20 K
40 K
60 K
80 K
100 K
120 K
(b) ˙ϵ= 10%/ns
0 100 200 300 400
0
50
100
150
εxx
(
%
)
τdev,bonded
xx
(MPa)
20 K
40 K
60 K
80 K
100 K
120 K
(c) ˙ϵ= 50%/ns
0 100 200 300 400
0
50
100
150
εxx
(
%
)
τdev,bonded
xx
(MPa)
20 K
40 K
60 K
80 K
100 K
120 K
(d) ˙ϵ= 100%/ns
0 100 200 300 400
0
50
100
150
εxx
(
%
)
τdev,bonded
xx
(MPa)
20 K
40 K
60 K
80 K
100 K
120 K
Figure 13: Comparison between the bonded Kirchhoff stress during deformation of
the constitutive model (solid curves) and MD simulations (dashed curves) at different
strain rates and temperatures.
deviations in the yield region could be attributed to the oversimplification of the
Eyring model, which is merely a thermally-activated model and does not include
structural relaxation of glassy materials. This deviation could be reduced by
employing a more complex plasticity model, e.g., the phenomenological softening
model [
56
], the shear transformation zone (STZ) model [
57
,
58
], or the effective
temperature model [
59
,
60
,
61
]. Another deviation can be observed at large strains
for simulations at high strain rates and low temperatures, where the MD results
exhibit a bit strain hardening behavior. This could be interpreted by the saturation
of the STZ for glassy materials, which can induce strain hardening for metallic
glasses [
57
]. However, the deviations in pair stresses is not further studied as the
primary purpose of the constitutive model is to validate the mechanism of local load-
bearing deformation-induced strain hardening in glassy polymers. Furthermore, the
deviations resulted from pair stress at large strains are acceptable in the total stress
as illustrated in Figures 14(b,d,f) as the bonded stress contributes significantly
28
more than the pair stress in the hardening region.
(a) ˙ϵ= 1%/ns
0 100 200 300 400
0
10
20
30
εxx
(
%
)
τdev,pair
xx
(MPa)
20 K
40 K
60 K
80 K
100 K
120 K
(b) ˙ϵ= 1%/ns
0 100 200 300 400
0
100
200
εxx
(
%
)
σxx
(MPa)
20 K
40 K
60 K
80 K
100 K
120 K
(c) ˙ϵ= 10%/ns
0 100 200 300 400
0
10
20
30
εxx
(
%
)
τdev,pair
xx
(MPa)
20 K
40 K
60 K
80 K
100 K
120 K
(d) ˙ϵ= 10%/ns
0 100 200 300 400
0
100
200
εxx
(
%
)
σxx
(MPa)
20 K
40 K
60 K
80 K
100 K
120 K
(e) ˙ϵ= 100%/ns
0 100 200 300 400
0
10
20
30
40
εxx
(
%
)
τdev,pair
xx
(MPa)
20 K
40 K
60 K
80 K
100 K
120 K
(f) ˙ϵ= 100%/ns
0 100 200 300 400
0
100
200
εxx
(
%
)
σxx
(MPa)
20 K
40 K
60 K
80 K
100 K
120 K
Figure 14: Comparison between the pair Kirchhoff stress (a,c,e) and the total Cauchy
stress (b,d,f) during deformation of the constitutive model (solid curves) and MD
simulations (dashed curves) at different strain rates and temperatures.
29
6. Discussion and outlook
We have conducted MD simulations to investigate the physical origin of strain
hardening and its dependence on strain rate and temperature in glassy polymers.
The main findings and contributions are summarized as follows:
•
Primary mechanisms of strain hardening: The principal contributors to strain
hardening are identified as bond and angle stresses, with negligible influence
from the pair term.
•
Load-bearing bond stretches: Bond and angle stress components are gov-
erned primarily by the average stretch of the maximum bond in each polymer
chain, denoted as
⟨lbmax⟩/⟨lbmax(0)⟩
(referred to as load-bearing bonds),
and are largely independent of strain rate and temperature in the hardening re-
gion. This suggests that elongation forces stem primarily from load-bearing
bonds, while the remaining chain structure resists the stretch.
•
Decomposition of deformation gradient: We propose decomposing the de-
formation gradient into a local load-bearing part,
Fl
, and a stretch-resistance
part,
Fr
, such that
F=FlFr
. Here,
Fl
primarily contributes to stress
response in the hardening region, while
Fr
resists chain segment elongation.
•
Local load-bearing approximation: The local load-bearing component
Fl
can be approximated by
⟨lbmax⟩/⟨lbmax(0)⟩
in uniaxial tension simulations.
Results indicate that the bonded stress (i.e., the combined bond and angle
stress) and the left Cauchy-Green tensor for
Fl
align well with the NH model
in the hardening region, allowing us to identify the shear modulus
µb
from
MD simulations.
•
Constitutive modeling: Based on these mechanisms, we developed a consti-
tutive model comprising elastic, pair, and bonded branches, which demon-
strates good agreement with simulation results across a wide range of tem-
peratures and strain rates.
While the constitutive model presented in this paper effectively captures the
temperature- and rate-dependent stress response in the hardening region at temper-
atures well below
Tg
, deviations become apparent as the temperature approaches
Tg
. This discrepancy is likely due to disentanglement effects, as indicated by the
inability of polymer chain stretches to fully follow the macroscopic deformation of
the entire system as discussed in Section 3.1. This deviation could be reduced by in-
troducing a local deformation gradient
Fd
to account for the inelastic deformation
induced by disentanglement such that
F=FlFrFd
, where the rate of
Fd
is driven
30
by the bonded stress. The model incorporating the effects of disentanglement will
be systematically studied in future work. Although the current constitutive model
has only been validated against MD simulations, several models with a similar
structure that incorporate orientation-induced back stress have been successfully
validated against experimental data for a variety of glassy polymers [
14
,
55
,
26
]. A
quantitative comparison of material parameters between real-world polymers and
MD simulations will require more accurate CG MD or atomistic models.
Acknowledgment
This work is funded by the German Research Foundation (DFG) projects
396414850 and 549959345. The author thanks Sylvain Patinet from PMMH, Paris,
for insightful discussions. The author also gratefully acknowledges the support by
the FAU Emerging Talents Initiative project and the HPC resources provided by
the Erlangen National High Performance Computing Center (NHR@FAU) of the
FAU under the NHR project b136dc. The hardware is funded by the DFG project
440719683.
References
[1]
M. Wendlandt, T. A. Tervoort, U. W. Suter, Non-linear, rate-dependent strain-
hardening behavior of polymer glasses, Polymer 46 (25) (2005) 11786–11797.
doi:10.1016/j.polymer.2005.08.079.
[2]
D. J. A. Senden, J. A. W. van Dommelen, L. E. Govaert, Strain hardening and
its relation to bauschinger effects in oriented polymers, Journal of Polymer
Science Part B: Polymer Physics 48 (13) (2010) 1483–1494.
doi:10.
1002/polb.22056.
[3]
D. J. A. Senden, S. Krop, J. A. W. van Dommelen, L. E. Govaert, Rate-
and temperature-dependent strain hardening of polycarbonate, Journal of
Polymer Science Part B: Polymer Physics 50 (24) (2012) 1680–1693.
doi:
10.1002/polb.23165.
[4]
R. S. Hoy, M. O. Robbins, Strain hardening of polymer glasses: Effect of
entanglement density, temperature, and rate, Journal of Polymer Science Part
B: Polymer Physics 44 (24) (2006) 3487–3500.
doi:10.1002/polb.
21012.
31
[5]
D. Hossain, M. Tschopp, D. Ward, J. Bouvard, P. Wang, M. Horstemeyer,
Molecular dynamics simulations of deformation mechanisms of amorphous
polyethylene, Polymer 51 (25) (2010) 6071–6083.
doi:10.1016/j.
polymer.2010.10.009.
[6]
L. R. G. Treloar, The physics of rubber elasticity, Oxford : Clarendon Press ;
New York : Oxford University Press, 2005.
[7]
R. N. Haward, The physics of glassy polymers, Springer Science & Business
Media, 2012.
[8]
H. van Melick, L. Govaert, H. Meijer, On the origin of strain hardening
in glassy polymers, Polymer 44 (8) (2003) 2493–2502.
doi:10.1016/
S0032-3861(03)00112-5.
[9]
C. Tian, R. Xiao, J. Guo, An Experimental Study on Strain Hardening of
Amorphous Thermosets: Effect of Temperature, Strain Rate, and Network
Density, Journal of Applied Mechanics 85 (10), 101012 (07 2018).
doi:
10.1115/1.4040692.
[10]
E. Kröner, Allgemeine kontinuumstheorie der versetzungen und eigenspan-
nungen, Archive for Rational Mechanics and Analysis 4 (273) (1959).
doi:10.1007/BF00281393.
[11]
E. H. Lee, Elastic-Plastic Deformation at Finite Strains, Journal of Applied
Mechanics 36 (1) (1969) 1–6. doi:10.1115/1.3564580.
[12]
E. M. Arruda, M. C. Boyce, H. Quintus-Bosz, Effects of initial anisotropy on
the finite strain deformation behavior of glassy polymers, International Jour-
nal of Plasticity 9 (7) (1993) 783–811.
doi:10.1016/0749-6419(93)
90052-R.
[13]
L. Anand, M. E. Gurtin, A theory of amorphous solids undergoing large
deformations, with application to polymeric glasses, International Jour-
nal of Solids and Structures 40 (6) (2003) 1465–1487.
doi:10.1016/
S0020-7683(02)00651-0.
[14]
L. Anand, N. M. Ames, V. Srivastava, S. A. Chester, A thermo-mechanically
coupled theory for large deformations of amorphous polymers. part i: For-
mulation, International Journal of Plasticity 25 (8) (2009) 1474–1494.
doi:10.1016/j.ijplas.2008.11.004.
32
[15]
W. Zhao, P. Steinmann, S. Pfaller, Modeling steady state rate- and
temperature-dependent strain hardening behavior of glassy polymers, Me-
chanics of Materials (2024) 105044
doi:10.1016/j.mechmat.2024.
105044.
[16]
G. Z. Voyiadjis, A. Samadi-Dooki, Constitutive modeling of large inelastic
deformation of amorphous polymers: Free volume and shear transformation
zone dynamics, Journal of Applied Physics 119 (22) (2016) 225104.
doi:
10.1063/1.4953355.
[17]
P. Zhu, J. Lin, R. Xiao, H. Zhou, Unravelling physical origin of the
bauschinger effect in glassy polymers, Journal of the Mechanics and Physics
of Solids 168 (2022) 105046. doi:10.1016/j.jmps.2022.105046.
[18]
J. Lin, P. Zhu, C. Tian, H. Zhou, R. Xiao, Physically-based interpretation of
abnormal stress relaxation response in glassy polymers, Extreme Mechanics
Letters 52 (2022) 101667. doi:10.1016/j.eml.2022.101667.
[19]
Jatin, V. Sudarkodi, S. Basu, Investigations into the origins of plastic flow
and strain hardening in amorphous glassy polymers, International Journal
of Plasticity 56 (2014) 139–155.
doi:10.1016/j.ijplas.2013.11.
007.
[20]
C. Bauwens-Crowet, J. C. Bauwens, G. Homès, Tensile yield-stress behavior
of glassy polymers, Journal of Polymer Science Part A-2: Polymer Physics
7 (4) (1969) 735–742. doi:10.1002/pol.1969.160070411.
[21]
C. Bauwens-Crowet, The compression yield behaviour of polymethyl
methacrylate over a wide range of temperatures and strain-rates, Journal of
Materials Science 8 (7) (1973) 968–979. doi:10.1007/BF00756628.
[22]
C. Siviour, S. Walley, W. Proud, J. Field, The high strain rate compressive
behaviour of polycarbonate and polyvinylidene difluoride, Polymer 46 (26)
(2005) 12546–12555. doi:10.1016/j.polymer.2005.10.109.
[23]
J. Diani, P. Gilormini, J. Arrieta, Direct experimental evidence of time-
temperature superposition at finite strain for an amorphous polymer network,
Polymer 58 (2015) 107–112.
doi:10.1016/j.polymer.2014.12.
045.
33
[24]
C. Federico, J. Bouvard, C. Combeaud, N. Billon, Large strain/time dependent
mechanical behaviour of pmmas of different chain architectures. application
of time-temperature superposition principle, Polymer 139 (2018) 177–187.
doi:10.1016/j.polymer.2018.02.021.
[25]
C. Bernard, N. Bahlouli, D. George, Y. Rémond, S. Ahzi, Identification of
the dynamic behavior of epoxy material at large strain over a wide range of
temperatures, Mechanics of Materials 143 (2020) 103323.
doi:10.1016/
j.mechmat.2020.103323.
[26]
R. Xiao, C. Tian, A constitutive model for strain hardening behavior of
predeformed amorphous polymers: Incorporating dissipative dynamics of
molecular orientation, Journal of the Mechanics and Physics of Solids 125
(2019) 472–487. doi:10.1016/j.jmps.2019.01.008.
[27]
H.-J. Qian, P. Carbone, X. Chen, H. A. Karimi-Varzaneh, C. C. Liew,
F. Müller-Plathe, Temperature-transferable coarse-grained potentials for ethyl-
benzene, polystyrene, and their mixtures, Macromolecules 41 (24) (2008)
9919–9929. doi:10.1021/ma801910r.
[28]
D. Reith, M. Pütz, F. Müller-Plathe, Deriving effective mesoscale potentials
from atomistic simulations, Journal of Computational Chemistry 24 (13)
(2003) 1624–1636. doi:10.1002/jcc.10307.
[29]
P. K. Depa, J. K. Maranas, Speed up of dynamic observables in coarse-grained
molecular-dynamics simulations of unentangled polymers, The Journal of
Chemical Physics 123 (9) (2005) 094901. doi:10.1063/1.1997150.
[30]
M. Rahimi, I. Iriarte-Carretero, A. Ghanbari, M. C. Böhm, F. Müller-
Plathe, Mechanical behavior and interphase structure in a silica–polystyrene
nanocomposite under uniaxial deformation, Nanotechnology 23 (30) (2012)
305702. doi:10.1088/0957-4484/23/30/305702.
[31]
A. V. Lyulin, M. Michels, Molecular dynamics simulation of bulk atactic
polystyrene in the vicinity of tg, Macromolecules 35 (4) (2002) 1463–1472.
doi:10.1021/ma011318u.
[32]
S. K. Kaliappan, B. Cappella, Temperature dependent elastic–plastic be-
haviour of polystyrene studied using afm force–distance curves, Polymer
46 (25) (2005) 11416–11423.
doi:10.1016/j.polymer.2005.09.
066.
34
[33]
W. Xia, J. Song, C. Jeong, D. D. Hsu, F. R. J. Phelan, J. F. Douglas,
S. Keten, Energy-renormalization for achieving temperature transferable
coarse-graining of polymer dynamics, Macromolecules 50 (21) (2017) 8787–
8796. doi:10.1021/acs.macromol.7b01717.
[34]
W. Xia, N. K. Hansoge, W.-S. Xu, F. R. Phelan, S. Keten, J. F. Dou-
glas, Energy renormalization for coarse-graining polymers having differ-
ent segmental structures, Science Advances 5 (4) (2019) eaav4683.
doi:
10.1126/sciadv.aav4683.
[35]
K. Kremer, G. S. Grest, Dynamics of entangled linear polymer melts: A
molecular-dynamics simulation, The Journal of Chemical Physics 92 (8)
(1990) 5057–5086. doi:10.1063/1.458541.
[36]
P. M. Morse, Diatomic molecules according to the wave mechanics. ii. vibra-
tional levels, Phys. Rev. 34 (1929) 57–64.
doi:10.1103/PhysRev.34.
57.
[37]
M. Ries, G. Possart, P. Steinmann, S. Pfaller, Extensive CGMD simulations of
atactic ps providing pseudo experimental data to calibrate nonlinear inelastic
continuum mechanical constitutive laws, Polymers 11 (11) (2019).
doi:
10.3390/polym11111824.
[38]
W. Zhao, M. Ries, P. Steinmann, S. Pfaller, A viscoelastic-viscoplastic consti-
tutive model for glassy polymers informed by molecular dynamics simula-
tions, International Journal of Solids and Structures 226-227 (2021) 111071.
doi:10.1016/j.ijsolstr.2021.111071.
[39]
W. Zhao, R. Xiao, P. Steinmann, S. Pfaller, Time–temperature correlations
of amorphous thermoplastics at large strains based on molecular dynamics
simulations, Mechanics of Materials 190 (2024) 104926.
doi:10.1016/
j.mechmat.2024.104926.
[40]
W. Zhao, Y. Jain, F. Müller-Plathe, P. Steinmann, S. Pfaller, Investigating
fracture mechanisms in glassy polymers using coupled particle-continuum
simulations, Journal of the Mechanics and Physics of Solids 193 (2024)
105884. doi:10.1016/j.jmps.2024.105884.
[41]
A. Ghanbari, M. C. Böhm, F. Müller-Plathe, A simple reverse mapping
procedure for coarse-grained polymer models with rigid side groups, Macro-
molecules 44 (13) (2011) 5520–5526. doi:10.1021/ma2005958.
35
[42]
W. Zhao, Multiscale modeling of the fracture behavior of glassy poly-
mers across the atomistic and continuum scale, Ph.D. thesis, Friedrich-
Alexander-Universität Erlangen-Nürnberg (2023).
doi:10.25593/
open-fau-401.
[43]
H. van Melick, L. Govaert, B. Raas, W. Nauta, H. Meijer, Kinetics of ageing
and re-embrittlement of mechanically rejuvenated polystyrene, Polymer 44 (4)
(2003) 1171–1179. doi:10.1016/S0032-3861(02)00863-7.
[44]
A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown,
P. S. Crozier, P. J. in ’t Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen,
R. Shan, M. J. Stevens, J. Tranchida, C. Trott, S. J. Plimpton, Lammps - a
flexible simulation tool for particle-based materials modeling at the atomic,
meso, and continuum scales, Computer Physics Communications 271 (2022)
108171. doi:10.1016/j.cpc.2021.108171.
[45]
M. Parrinello, A. Rahman, Polymorphic transitions in single crystals: A
new molecular dynamics method, Journal of Applied Physics 52 (12) (1981)
7182–7190. doi:10.1063/1.328693.
[46]
G. J. Martyna, D. J. Tobias, M. L. Klein, Constant pressure molecular dynam-
ics algorithms, The Journal of Chemical Physics 101 (5) (1994) 4177–4189.
doi:10.1063/1.467468.
[47]
W. Shinoda, M. Shiga, M. Mikami, Rapid estimation of elastic constants by
molecular dynamics simulation under constant stress, Physical Review B 69
(2004) 134103. doi:10.1103/PhysRevB.69.134103.
[48]
A. P. Thompson, S. J. Plimpton, W. Mattson, General formulation of pressure
and stress tensor for arbitrary many-body interaction potentials under peri-
odic boundary conditions, The Journal of Chemical Physics 131 (15) (2009)
154107. doi:10.1063/1.3245303.
[49]
J. A. Zimmerman, E. B. WebbIII, J. J. Hoyt, R. E. Jones, P. A. Klein, D. J.
Bammann, Calculation of stress in atomistic simulation, Modelling and
Simulation in Materials Science and Engineering 12 (4) (2004) S319.
doi:
10.1088/0965-0393/12/4/S03.
[50]
A. K. Subramaniyan, C. Sun, Continuum interpretation of virial stress in
molecular simulations, International Journal of Solids and Structures 45 (14)
(2008) 4340–4346. doi:10.1016/j.ijsolstr.2008.03.016.
36
[51] R. S. Hoy, M. O. Robbins, Strain hardening in polymer glasses: Limitations
of network models, Phys. Rev. Lett. 99 (2007) 117801.
doi:10.1103/
PhysRevLett.99.117801.
[52] B. D. Coleman, W. Noll, The thermodynamics of elastic materials with heat
conduction and viscosity, Archive for Rational Mechanics and Analysis 13 (1)
(1963) 167–178. doi:10.1007/BF01262690.
[53]
B. D. Coleman, M. E. Gurtin, Thermodynamics with Internal State Variables,
The Journal of Chemical Physics 47 (2) (1967) 597–613.
doi:10.1063/
1.1711937.
[54]
E. Bouchbinder, J. S. Langer, Nonequilibrium thermodynamics of driven
amorphous materials. i. internal degrees of freedom and volume deforma-
tion, Phys. Rev. E 80 (2009) 031131.
doi:10.1103/PhysRevE.80.
031131.
[55]
N. M. Ames, V. Srivastava, S. A. Chester, L. Anand, A thermo-mechanically
coupled theory for large deformations of amorphous polymers. part ii: Ap-
plications, International Journal of Plasticity 25 (8) (2009) 1495–1539.
doi:10.1016/j.ijplas.2008.11.005.
[56]
M. C. Boyce, D. M. Parks, A. S. Argon, Large inelastic deformation of glassy
polymers. part i: rate dependent constitutive model, Mechanics of Materials
7 (1) (1988) 15–33. doi:10.1016/0167-6636(88)90003-8.
[57]
J. Langer, Shear-transformation-zone theory of deformation in metallic
glasses, Scripta Materialia 54 (3) (2006) 375–379, viewpoint set no:
37. On mechanical behavior of metallic glasses.
doi:10.1016/j.
scriptamat.2005.10.005.
[58]
J. Lin, J. Qian, Y. Xie, J. Wang, R. Xiao, A mean-field shear transformation
zone theory for amorphous polymers, International Journal of Plasticity 163
(2023) 103556. doi:10.1016/j.ijplas.2023.103556.
[59]
K. Kamrin, E. Bouchbinder, Two-temperature continuum thermomechanics
of deforming amorphous solids, Journal of the Mechanics and Physics of
Solids 73 (2014) 269–288. doi:10.1016/j.jmps.2014.09.009.
37
[60]
R. Xiao, T. D. Nguyen, An effective temperature theory for the nonequilib-
rium behavior of amorphous polymers, Journal of the Mechanics and Physics
of Solids 82 (2015) 62–81. doi:10.1016/j.jmps.2015.05.021.
[61]
R. Xiao, C. Tian, Y. Xu, P. Steinmann, Thermomechanical coupling in glassy
polymers: An effective temperature theory, International Journal of Plasticity
156 (2022) 103361. doi:10.1016/j.ijplas.2022.103361.
38
