1. Introduction
Polymer-bonded composites materials (PBMs) have drawn increasing attention in recent years due to their advantage in complex shaping, bi-injection molding, and ease of machining. By compounding different filler materials with polymer, new functional materials can be made, providing a viable technology for on-demand material design and engineering [
1]. Applications of PBMs include, but are not limited to, polymer-bonded magnetics [
2,
3], energetics [
4,
5], aluminum [
6], fiber [
7,
8], and many others [
9,
10,
11]. To utilize those functional materials in critical applications, understanding their mechanical properties and deformation behaviors under different loading and temperature conditions is of great importance in meeting the basic design criteria.
The mechanical properties such as compliance, modulus, toughness, and strength of polymer materials depend largely on strain (rate) and temperature [
12,
13]. The stress–strain behavior of PBMs is usually that of viscoelastic plastic, which can be approximated as an elastic (linear) regime followed by a plastic (nonlinear) regime [
14] in phenomenological view. The deformation is homogeneous in the elastic regime and can be related to the applying stress by Hooke’s law [
15]. Once the stress of the material reaches a certain stress level, the material will produce irreversible strain and plastic deformation [
16]. The maximum stress below which the resulting strain remains proportional to the applying stress is the elastic limit of the polymer materials. However, the filler materials can drastically alter the elastic limit and change the stress–strain behavior in both elastic and plastic regimes [
17,
18,
19]. In addition, the plastic regime of the PBM can also be vastly different to that of the regular polymer materials due to the use of different filler materials.
The need to incorporate temperature effects into the deformation behavior of PBMs has been identified in several experimental studies on a variety of materials, for example, EDC37 in the temperature range of −65
C to 60
C by Williamson et al. [
20], Rowanex between −60
C and 60
C by Walley et al. [
21], HTPB at low temperatures by Chen et al. [
22], short carbon-fiber-reinforced polyether-ether-ketone (SCFR/PEEK) composites between 20
C and 235
C by Zheng et al. [
23], epoxy mortar at temperature between 80
C to 210
C by Vecchio et al. [
24], nanoparticle/epoxy nanocomposites between 23
C to 53
C by Unger et al. [
25], and many others [
26,
27]. Several models have been reported to describe the mechanical responses of PBM under different temperatures, including the elastic–plastic fracture-mechanism-based model using the glass-transition temperature (
) as a piecewise knot point with a total number of 39 parameters [
28], temperature-dependent mesoscale interface-based model [
29], and the temperature-dependent visco-hyperelastic model [
30]. Models based on the concept of temperature correction factor and its variants have been seen, for example, in [
31,
32]. Experimental evidence indicates that the mechanical behaviors of PBMs under tension and compression are quite different, and the stress–strain behavior under tension and compression may be described separately [
20,
21,
28]. The aforementioned models in [
29,
30,
31] were established for a single mode, and their applicability to the other mode is unknown. Other models such as the one reported in [
28] and its variants requires a dozen parameters, demanding much more testing data for parameter calibration, which may not be realistic in all cases.
The goal of this study is to develop a general temperature-dependent stress–strain constitutive model for PBMs suitable for both tension and compression loading modes with a minimal set of parameters. Compression and tension testing in the temperature range from −40 C to 75 C is performed to obtain stress–strain response data. Based on the observation of the stress–strain curves at individual temperatures, a nonlinear model based on the Ramberg–Osgood (RO) relationship is proposed to describe the stress–strain under a given temperature. By correlating the model parameter with temperature, a general temperature-dependent constitutive model can be formulated. The reminder of the study is organized as follows. First, the experimental work is presented in detail and the data in compression and tension are acquired. Next, the constitutive model is developed based on the testing data. Following that, the model validation using both an independent set of testing data and third-party data is performed. The effectiveness of the model is demonstrated, and the performance of the model is compared with a reference model having a comparable number of parameters. The developed constitutive model is further implemented as a user material in a finite element package for structural applications. Finally, conclusions are drawn based on the current results.
5. A User Material Implementation of the Model
To use the proposed temperature-dependent stress–strain model for structural applications, a user material subroutine for the proposed model is developed for finite element analysis. The dimensional of the structural model is the same as the actual specimens. The structural models for the tension and compression specimens are created and meshed using quadratic hexahedron elements with an average size of 2 mm.
For the tension specimen, a constant stress of 2 MPa is applied to the top face with the bottom face fixed in all degrees of freedom, representing a tension test condition. For the compression specimen, a constant stress of −20 MPa is applied on the top face, and the same boundary conditions are used for the bottom face.
Figure 18 demonstrates the resulting normal stresses after applying the load at −40
C, which are shown for the tension and compression specimens in
Figure 18a,b, respectively. Under tension, it can be found that the stress on the top face is about 2 MPa, which is the same as the applied load, and the maximum stress occurs in the middle section of the structural part. Under compression, the maximum stress appears at the fixed position. From the top face, the stress gradually decreases along the UX direction. The stress on the top face is the same as the applied load of 20 MPa. For both cases, the resulting stresses are consistent with the actually applied loads.
Figure 19 presents the FEA results using the specimen structural models in
Figure 18 with the developed user material routine for temperatures of −40
C, 20
C, and 75
C and compares them with the theoretical results of the proposed temperature-dependent stress–strain model. Both tension and compression results are compared. The maximum relative error (RE) is 0.01239, occurring at −40
C for the tension case, and the maximum RE is 0.01537 at 75
C for the compression case, indicating the effectiveness of the developed user material routine for structural applications.
6. Conclusions
A general temperature-dependent stress–strain constitutive model for polymer-bonded composite materials is proposed, developed, validated, and compared. Experimental testing data were acquired in the temperature range of −40–75 C for both tension and compression. The model under a given temperature is proposed based the linear accumulation of an elastic term and a plastic term, resembling the Ramberg–Osgood relationship. The model parameters are obtained by fitting the testing data at each of the temperatures, and the temperature dependence of the model parameters can be established using the response surface method. The proposed model is validated using independent sets of testing data and third-party data, and is further compared with an existing model with a comparable number of parameters. A user material routine of the proposed model is implemented and verified for structural applications. Based on the current results, the following conclusions are drawn.
The proposed method can unify the temperature effect by modeling the temperature dependence of the constitutive model parameters, thus providing an alternative to the existing temperature correction factor-based methods. Furthermore, such a treatment allows a minimal number of fitting parameters compared to the existing models.
The proposed method provides a general approach to model the stress–strain behaviors of PBMs. The basic assumption that the stress–strain response is an accumulation of an elastic term and a power-law plastic term is observed in many other materials such as various metals. The effectiveness and generality of the model are validated and demonstrated by two independent preponderate data sets in the temperature range −40–75 C.
It is worth mentioning that the developed model is a semiempirical data-driven model, and it does not deal with the microscopic mechanism. The dynamic mechanical analysis on the material can be carried out to enhance the understanding of the mechanism, and SEM imaging may be used to reveal the interface interaction between the filler particles and the binder materials. The constitutive model established in this study does not incorporate the strain rate effect. The strain-rate-induced hardening phenomenon has been reported in several studies, e.g., [
41,
42]. The incorporation of the strain rate effect to the proposed model will be studied in future work. The proposed model considers the PBM as a homogeneous material in macroscopic perspective. The local variations of the material properties due to the microscopic effects of filler materials and polymer binder interface [
43,
44,
45] may be explored in the future. In addition, the effectiveness of the model is verified using the presented experimental data acquired in this study and a third-party published data. Due to the limit on currently accessible data, the verification of the developed model using other PBMs will be investigated in the future. The current modeling adopts a deterministic approach, and a probabilistic approach can also be employed for reliability analysis [
46,
47,
48].