1. Introduction
Because of its practical importance, understanding and modeling the nonlinear behavior of elastomers has been a major research in chemistry, materials, and continuum mechanics for a century. A major step toward this goal has been the introduction of the entropic statistical theory of polymers, which explained the nature of the nonlinear behavior and the shape of the stress–strain curve [
1,
2,
3,
4].
However, for more than 75 years, the failure of the statistical theory to explain some aspects of the observed behavior, such as the experimentally observed slope in the Mooney plots [
5,
6,
7], has been disappointing [
8], and hundreds of physics-based and phenomenological models have been proposed to overcome the limitations; in particular, the second invariant has been incorporated [
6,
7]. However, despite some improvements, problems remained and have been manifested by conflicting claims and unsolved issues [
9]. Some of them have been: (1) the need for more than one test to characterize an isotropic incompressible material when only one modulus (one test) is needed to define the linear material [
10]; (2) the need for the introduction of a second invariant or chain transverse (or tube) constraints [
11]; (3) the failure of the full network model, i.e., sphere integration of the chain behavior to obtain the continuum one [
12,
13]; (4) the need for the modification of the chain stretch(es) (longitudinal and transverse) by averaging in the sphere [
11,
14]; (5) and a conceptual contradiction of affine deformations with the statistical theory [
15,
16]. The lack of sufficient understanding and the difficulty in selecting the appropriate model resulted in tens of papers comparing the predictive power of different models when parameters are characterized by a multitude of approaches. Some well-known comparative studies are [
10,
12,
14,
17,
18,
19,
20,
21,
22,
23].
Based on the consistency of the 3D extension of the statistical theory [
9], and on some insights obtained from machine learning [
24], a new micro–macro connection for the chain stretch has been proposed, where the stretch tensor replaces the Cauchy–Green deformation tensor from the original affine theory [
4]. This results in an orientationally non-affine chain stretch, but which is consistent with neglecting the entropy changes from the network reorientation, as usually assumed, where only chain entropies are considered. It has been demonstrated that the new micro–macro relation solves and explains many standing issues like the slope in the Mooney plots [
9]. The resulting model is also characterized from a (any) single stress–strain curve and results in accurate 3D predictions [
9,
24]. However, as also therein mentioned, it is expected that the network entropy changes when chains approach locking, so a more orientationally affine behavior is expected. A simple three-parameter full network model has been proposed recently under these considerations [
25].
The purpose of this paper is to provide further insights into the model. There are four main contributions. (1) A closed form, simple, analytical expression for the model for moderately large stretches (within the Gaussian zone) is given. This closed form is important in developing many analytical studies and derivations. It is often considered an asset for many models, as it is for the Neo-Hookean model. (2) A detailed comparison of the present model with the Neo-Hookean model is performed, demonstrating the relevance of the orientationally non-affine deformations assumption in reproducing the experimental observations for general multiaxial loadings with parameters obtained from a single test. (3) It is demonstrated how the parameters of the model may be easily obtained from the Mooney space, or alternatively from the Mooney–Rivling constants, revealing also the importance of the lowest range of large stretches. (4) The model is verified against a large variety of experimental results for different elastomers. These data include true biaxial tests with different stretch ratios, different treatments, and different stretch levels. In all the predictions using the model, the three material parameters have been ontained only from a tensile test.
2. The Orientationally Non-Affine Chain Stretch
As above explained, the initial success of the statistical theory and the Neo-Hookean model by Wall [
4]—who first noted that it entailed a Hookean behavior in shear—in explaining the shape of the stress–strain uniaxial relation was not followed by a satisfactory extension to 3D. Many researchers, starting from Mooney [
5] and followed by Rivlin and co-workers [
6,
7,
10,
26,
27,
28,
29,
30], highlighted the failure of a theory based only on the first invariant
of the Cauchy–Green deformation tensor, so they phenomenologically proposed the incorporation of an additional term. The Neo-Hookean model results in a constant in the Mooney
plot
where
is the uniaxial stretch and
is the nominal stress. However,
is not constant in experiments but has a slope in the order of
—depending on the polymer; polyurethane elastomers may have slopes of the order of
or even 10 times
[
31]. Mooney’s solution has been to incorporate such slope by adding a
term
using the Rivlin’s Cauchy–Green tensor invariants
,
, the strain energy is
However, Rivlin [
6,
7] also noted that the parameters
could not be considered as constants but were functions of the invariants themselves, i.e.,
,
.
represents the invariants of the right Cauchy–Green deformation tensor
,
is the deformation gradient, and
is the right stretch tensor. Noteworthy, the Neo-Hookean model is just the affine full integration in the sphere
of the chain function
where
is the chain stretch obtained from the right Cauchy–Green deformation tensor
and the chain direction in the reference configuration
, and
is the classical Neo-Hookean shear modulus. A corrected “Neo-Hookean” model using
, is
where
is a shear-like modulus—see below for the correspondence with the classical
. This model provides much more accurate results and, importantly, the correct 3D tendencies, including the observed slope in Mooney plots [
9]. Model (
5) is also physically consistent with the neglected entropy terms regarding the reorientation of the chains. Building upon this model, a new three-parameter model that incorporates two experimentally observed effects has been proposed: (1) a constant term to account for internal energy effects at low deformation levels [
8,
32] and (2) a chain-locking behavior which incorporates an increasingly orientationally affine deformation (assuming that chains near locking deform under more affine conditions). In the remaining part of the paper, important insights into the model, the Mooney representation of the model, and its predictive power for different elastomers are given.
3. Non-Affine Model with Three Parameters
Using Langevin distributions [
8], where
is the inverse Langevin function, the derivative of the chain energy
with respect to the chain stretch
can be written as:
where
is the effective chain stretch—see below—and
is the chain locking. The variable
is the conversion factor from the observed continuum uniaxial referential locking stretch
to the chain locking, computed as—see motivation in [
25]
where
stands for the first invariant of
; see
Figure 1. At very large stretches, the following values are obtained.
These values are consistent with the approximate relations between locking stretches for those types of experiments. An estimation of the reference locking stretch for the chain is
, where
is the macroscopic locking stretch obtained during a tensile test, and
is the value of the
function at that stretch
for the uniaxial test. Equation (
7) has two addends. The second addend corresponds to the classical statistical (Langevin) theory [
8,
18,
33] with the exception of the presence of the loading mode factor
accounting for chain constraints. The first addend in the chain tension
—term
in Equation (
7)—corresponds to an internal energy contribution. This contribution can be considered approximately constant, and it is relevant only for relatively small stretches (e.g., about 50–100%), see [
8,
32] p. 32, but dominates the tension near the infinitesimal range—below 10% of stretches. It is noteworthy that this constant term at small strains has also been obtained in the data-driven determination of the chain function from experiments; see Ref. [
24]. Furthermore,
can take negative values and that still
. Hence, even when this term is neglected at large deformations, it cannot be neglected when determining the constants if the shear modulus is obtained in that regime.
Several works have used the affine and non-affine behavior of polymer chains to characterize the transition between the microscopic constitutive model and the continuous solid [
11,
25,
34,
35], but in most of them, the consideration of non-affinity does not refer to the non-affinity in the orientation of the chains, but rather refers to the amount of the effective stretch in a given chain direction with respect to the continuum one. For example, the continuum deformation tensor to compute the chain stretch is typically the Cauchy–Green (quadratic) deformation tensor. The non-affine stretch
is computed herein from the continuum stretch tensor
and the chain direction
(which is treated as a spatial direction, not a specific chain direction) as
in this expression,
are the principal continuum stretches and
are, respectively, the azimuthal and polar spherical angles of the chain with respect to the principal directions. The microstretch
is the one consistent with biaxial experimental data at moderately large stretches; see [
24]. However, near locking, it is to be expected that chains reorient statistically toward the stretched directions because locking behavior seems experimentally more consistent with the affine assumption. Unfortunately, there is still no experimentally verified theory which incorporates the network reorientation in the entropy, so the increasing relevance of that term with very large deformations results in the effective average reorientation of chains. Then, this effect is incorporated phenomenologically by considering an average effective reoriented chain with stretch—
corresponds to the orientationally non-affine case, verified experimentally up to moderate stretches, and
corresponds to the limit affine case expected at chain locking.
The parameters
,
and
are the material fitting parameters with a clear physical interpretation. For deformations sufficiently small, the locking effect can be neglected (typically 30% of the locking stretch), which is known as the Gaussian distribution case, e.g., [
35,
36].
The model considers a full network of chains isotropically oriented, so a chain oriented in a given direction represents all chains oriented in that direction. Then, the derivative of the continuum stored energy
is computed from the chain rule as
where the last line is the numerical integration of
points of quadrature, with
being the weights of integration (such that
). In our case, we use the quadrature points proposed by Bazant and Oh [
37] with
, which is the same one used by Miehe et al. in their non-affine model [
11]. The non-affine stretch is
where
A main problem of Langevin statistical models is the evaluation of the inverse of the Langevin function. There is no analytical expression for that inverse function. Furthermore, it is difficult to accurately evaluate the inverse Langevin function because of the asymptotic behavior near locking. Thus, some studies are dedicated to this issue [
38,
39,
40,
41,
42,
43,
44]. However, in the present case, it is relevant to separate the Gaussian linear zone from the nonlinear locking one. The Petrosyan [
45] approximation to the inverse Langevin function (with a maximum error of
) conveniently splits the linear (Gaussian) and nonlinear parts
the nonlinear contribution is
then, the approximation symbol is used because of the consideration of
in the linear part and
in the nonlinear one and
where the second line is the non-Gaussian contribution [
46,
47], and with
the derivative
is
It is important to remark here that in contrast to the formulation in [
25], the Gaussian case is integrated exactly, and only the non-Gaussian contribution needs to be integrated numerically. This is relevant because Mooney plots are only relevant in the Gaussian zone. If
denotes the pressure-like Lagrange multiplier of the incompressible case,
is the identity tensor, and
denotes the Green–Lagrange strain tensor, while the incompressible case gives the following second Piola–Kirchhoff and Piola stress tensors, respectively,
and
:
with
where
represents the eigenvectors of
. i.e., the Piola stress is
where
is the rotation from the right polar decomposition of
.
In the typical quasi-incompressible case, the stored energy can be written as
with
being the determinant of the deformation gradient tensor
and
being the isochoric stretches. Taking into account that
it is obtained
where
is given in Equation (
11) by replacing
by
, and
depends on the choice for the penalty function. For the Gaussian case
and
where
are the eigenvectors of the left Cauchy–Green deformation tensor.
A relevant case is that of homogeneous deformation. For any given state, we can assume there are deformations in the principal axis. In most tests, one of the directions—label it as the third one—remains unloaded, so the stress state is biaxial, and the stretch in that axis is given by the incompressibility condition; namely
. It is in the interest of simplifying analytical derivations in homogeneous tests to consider the incompressible case. In this case,
and
For the Gaussian range of deformations, the explicit expression
is obtained, whereas the non-Gaussian case gives the additional term
Predictions for the typical experiments are obtained using these formulae, employing
Uniaxial test: (uniaxial stretch), and ;
Equibiaxial test: (equibiaxial stretch), and ;
Pure shear: (strip test stretch), and .
However, Equation (
28) is valid for any test in which one axis—labeled as the third one—is unloaded. In incompressible cases, since the pressure comes from equilibrium, one axis may be taken as the zero reference.
In the case of uniaxial tests, it is typical to plot the experimental data, and hence the model fit, in the
axes. The effective uniaxial modulus can be obtained by setting
, where
is the infinitesimal strain. In this case, the relevant Gaussian case gives
to compare, the classical Neo-Hookean model gives
The comparison of both models for infinitesimal strains
give the relation between the moduli of both models
this relation guarantees the same initial slope in the predictions by both models in a tensile test. Additionally, for a given stretch
, the slope for the tensile test is
so for very large strains—recall that we are considering the Gaussian case
which is to be compared to the Neo-Hookean value
—cf. Equation (
31) for
Remarkably,
affects the initial slope—Equation (
33)—but not the behavior at large stretches—Equation (
35). Note that for
, Equation (
33) gives
and Equation (
36) gives
(again the
correction). In summary, from the initial slope and the intermediate slope (large moderate stretches, so the locking effect is not important), the two parameters of the model, namely
and
, can be determined, the former from Equation (
35) and the latter with the computed
and Equation (
33).
4. Mooney Space Representation
Mooney’s plot is just another way of plotting the same tensile test experimental databut weighting visually the initial part of the experiment by using the representation in Equation (
1). The Neo-Hookean model contradicts experimental evidence, where a slope in the order of
is observed [
8,
28]. This problem motivated Mooney’s phenomenological proposal of using a
constant over
(or equivalently the
invariant) which corrected the statistical theory to accommodate the experimental slope in that plot. The relevance of the
invariant has been explained in many papers [
8,
48,
49,
50]. The slope in Mooney plots has also been the center of attention in fitting constitutive models [
29,
51,
52,
53]. In the herein proposed model, the Mooney slope is obtained naturally from the statistical theory. The model’s slope at
can be computed by considering the power series in
(
for a tensile test), where
, so
. To this end, the Mooney plot function is
whose expansion series in
is
For
, the previous expression of
with
given in Equation (
36) is recovered. Then, the pursued slope is
whose expansion is
now, at
, the slope is
then, Mooney plots may be used to identify the parameters of the model in a more simple way from the
-value at
—call it
, the Neo-Hookean constant, and the slope—call it
, the Mooney constant. The solution is
If
, as in the Neo-Hookean model, a nonvanishing initial slope
is still obtained, which is of the order of
. The slope changes in general with deformation, but an almost constant slope is obtained for
, and there is a vanishing initial one for
. Of course, using Equations (
37) and (
39), the combination of function and slope values at any stretch, or two values at different stretches in the Gaussian zone, may be used to determine
and
by solving the linear system of equations.