 Research
 Open Access
Modelling of thermomechanical behaviour of fibrous polymeric composite materials subject to relaxation transition in the matrix
 Valeriy Pavlovich Matveenko^{1},
 Nikolay Alexandrovich Trufanov^{2}Email author,
 Oleg Yurievich Smetannikov^{2},
 Igor Nikolaevich Shardakov^{1} and
 Igor Nikolaevich Vasserman^{1}
https://doi.org/10.1186/s4075901600158
© The Author(s). 2016
 Received: 7 October 2016
 Accepted: 9 November 2016
 Published: 25 November 2016
Abstract
Background
Fiber–reinforced polymer composite materials are widely used in different branches of industry due to their distinctive features such as high specific strength and stiffness and due to as considerable opportunity to formulate materials with controllable variation of properties in response to the action of external factors (smartmaterials). A distinguishing feature of products made of composite materials is that the processes of product and material fabrication are inseparable. Therefore the estimation of composite properties based on the composite architecture and properties of the reinforcing fibers and matrix is a very actual task.
Methods
The model of polymer behavior at glass transition recently developed by the authors was generalized to the case of fiberreinforced polymer matrix composites using two approaches: one is base on the concept of free specific energy, the other – on the growth of matrix stiffness. For homogeneous materials these two approaches are of equal worth, whereas for composite materials they give different results under deformation in the transverse direction. The stiffness growth approach is more accurate, but is very expensive computationally and, is highly sensitive to the experimental data errors.
Results
Using the finite element method and averaging technique the thermoelastic constants of composites containing different types of fibers in the glassy and highelastic states were calculated based on the fiber and matrix properties. Softening of the matrix has an insignificant effect on the longitudinal modulus of a composite but leads to a considerable decrease of the transverse and shear moduli. The coefficient of thermal expansion in the transverse direction is much higher than the coefficient of thermal expansion in the longitudinal direction, especially when the composite is in the highelastic state.
Conclusion
The model of polymer behavior at glass transition recently developed by the authors can be generalized to the case of fiberreinforced polymer matrix composites. The thermoelastic constants of composites containing different types of fibers can be calculated from the fiber and matrix properties using the finite element method and averaging technique.
Keywords
 Fiber–reinforced composites
 Glass transition
 Glass and highelastic state of polymer
Background
Fiber–reinforced polymer composite materials are widely used in different branches of industry due to their distinctive features Among technical characteristics of considerable commercial importance are high specific strength and stiffness, low specific density, high fracture toughness and fatigue strength, as well as considerable opportunity to formulate materials with controllable variation of properties in response to the action of external factors (smartmaterials).
The materials under consideration consist of fibrous filling material, which when combined with a polymer binding material, forms a fiber–reinforced polymer matrix composite. As a rule, reinforcing fibers have high longitudinal stiffness and tensile strength, while a composite matrix has high toughness, which provides superior mechanical characteristics of composites. The structure of these materials is responsible to the anisotropy of their physicomechanical properties.
A distinguishing feature of products made of composite materials is that the processes of product and material fabrication are inseparable. It can be said with assurance that each product is unique in showing properties that may be different in different its parts. It means that physicomechanical properties of the product cannot be estimated by a standard approach from the test data of specimens made of the same material. Therefore at present, the problem of current concern is the estimation of composite properties based on the composite architecture and properties of the reinforcing fibers and matrix.
The process of composite formation proceeds in a certain temperature range. The temperature regimes arising in this interval are accompanied by relaxation transitions to glassy and rubberlike states in the polymer matrix. Organic plastic materials also demonstrate a strong dependence of fiber mechanical properties on temperature. All these thermomechanical processes give rise to residual/technological stresses.
Predictive modeling of the composite behavior based on the properties of its components is the objective of many studies (Christensen 1979; Hill 1964a, b; 1965a, b; 1966; Malmeister et al. 1980; Pobedrya 1964; Sokolkin Yu and Tashkinov 1984; Vildeman et al. 1997). Thus, works (Christensen 1979; Hill 1964a, b; 1965a, b; 1966; Malmeister et al. 1980; Vildeman et al. 1997) are devoted to modeling the thermomechanical behavior of fiberreinforced composites. The focus of authors’ attention in these studies are such problems as estimation of elastic moduli of composite materials based on the properties of their components (Christensen 1979; Malmeister et al. 1980; Pobedrya 1964; Sokolkin Yu and Tashkinov 1984, Dinzart and Lipiński 2010, Dupaix and Boyce 2006, Ge et al. 2012), modeling of nonlinear and transient processes in a composite material (Vildeman et al. 1997, Bugakov 1989), damage accumulation and fracture of composites (Sokolkin Yu and Tashkinov 1984; Vildeman et al. 1997, Bugakov 1989, Christensen 1979). However, the above studies are concerned mainly with the processes of deformation and fracture in finished composites.
On the other hand, there are a lot of papers (Askadskiy 1973; Bartenev and Zelenev 1976; Begishev et al. 1997; Brader et al. 2009; Buckley and Jones 1995; Bugakov 1989; Dinzart and Lipiński 2010; Dupaix and Boyce 2006; Henann and Anand 2008; Liu et al. 2006; Lustig et al. 1996; Nguen et al. 2008; Qi et al. 2008; Scalet et al. 2015; Shardakov and Golotina 2009; Sollich 1998; Tervoort et al. 1996; Xiao et al. 2013) (including the works by the present authors (Matveenko et al. 2012; 2015), which investigate the thermomechanical behavior of homogeneous polymers (not composites) at glass transition/softening temperatures. The effect of shape memory, which is a frequent consequence of thermorelaxation transitions, is investigated in (Dinzart and Lipiński 2010; Liu et al. 2006; Lustig et al. 1996; Nguen et al. 2008; Qi et al. 2008; Scalet et al. 2015; Shardakov and Golotina 2009; Xiao et al. 2013).
Simulation of thermorelaxation transitions in composite materials is a less explored area, which at present is being studied intensively. Most of recently published works on this subject are concerned with composite smart materials exhibiting the shape memory effect (Ge et al. 2012; 2014; Mulina and Sawant 2009; Tan et al. 2014, Srivastava et al. 2010a, b, Tervoort et al. 1996), in which the relaxation and phase transitions can occur both in the matrix (Ge et al. 2014; Mulina and Sawant 2009; Tan et al. 2014) and in the reinforcing fibers (Ge et al. 2012).
In this paper, we generalize a thermomechanical model of composite material undergoing glass transition developed early by the authors in (Matveenko et al. 2012; 2015) to the case of fiberreinforced composite materials with polymeric matrix, which due to variation of temperature can undergo transition from the highelastic to the glassy state and back. The relations constructed in these papers are based on the specific free energy of the composite, which is used as a scalar measure of the state of the material undergoing glass transition. We introduce a technique for determining the equation parameters in terms of the effective properties of the composite. We also develop a more general model, in which glass transition is considered as the process of enhancement of the matrix stiffness. We compare the two models to show their advantages and drawbacks. A numerical technique has been proposed to determine the effective material constants and functions for unidirectional fiberreinforced composites with epoxy matrix undergoing glass transition due to temperature variation. The investigation has been performed taking no account of viscous elasticity.
Methods
Polymer matrix
The polymer matrix consists of very long, linear, branched or crosslinked macromolecules, which comprise a large number of repeatable chains. The structure of the macromolecule is specified by configuration and conformation. Configuration is the mutual arrangement of atoms in the molecule, which cannot be changed without breaking the existing chemical bonds. Conformation is the relative spatial arrangement of atoms in the molecule, which can vary under the action of thermal motion by rotation around single carboncarbon bonds. Depending on the structure the polymers can be classified as thermoplastics and reactoplastics. Thermoplastics have a linear or branched structure and can transform from a plastic to a solid state and change back. Reactoplastics possess a spatial crosslinked structure, in which the neighboring chains are linked by chemical covalent bonds, the strength of which is as high as the strength of the bonds in the polymer chains. Polymers of this group are unable to pass into a reversible plastic state. They can only decompose chemically when heated to certain temperatures. The molecules of polymers can form supermolecular structures, one being knows as a fluctuation network. Physically, this implies that some segments of molecules in microvolumes form the shortrange order structures with higher density and more intensive intermolecular interaction (fluctuation network nodes). Subject to thermal or mechanical loads these structures may decay in some places and emerge in other places.
The above mentioned features of the molecular structure specifies the behavior of solid polymers, which depending on temperature and mechanical loads can exist in two states – glassy and highelastic. At low temperatures the polymer behaves as glass. Under the action of applied stresses the conformational motion of polymer chains does not occur, which means that the shape of macromolecules does not change. Deformation in this state is small and is related to changes in the interatomic distances and bond angles and also to a change in the average intermolecular space.
Beginning with some temperature the deformability of a polymer increases causing its transition into highelastic state. In this state the strain can be as high as a few hundred percent and is associated with the conformational movements of macromolecules, the failure of fluctuation network nodes in some places and their formation in other less strained places. It should be noted that a transition from the glassy to the highelastic state is not an instantaneous process. It occurs within a certain temperature range, which suggests that there exists some transitional region T _{ g2} ≤ T ≤ T _{ g1}.
At temperature lower than the glass transition temperature and high level of stresses the polymer undergoes forced highelasticdeformation, which involves unfreezing of segmental mobility of macromolecules.
 1)
The characteristic times of external actions are much shorter than the relaxation times of a polymer in the glassy state.
 2)
Characteristic times of external actions are far beyond the limits of the relaxation spectrum of a polymer in the highelastic state.
In this case, outside the glass transition temperature range the polymer behaves as an elastic material. The behavior of the polymer matrix in the glass transition temperature range at decreasing temperature is specified by increase in the mechanical stiffness of the polymer due to a decrease in the segmental mobility of molecules and a growth of the energy of intermolecular interaction. At the phenomenological level this looks like a superposition of additional elastic bonds onto the original polymer network in the highelastic state. Note that each of these bonds at the time of its formation is assumed to be nondeformed.
where γ _{ L } is the parameter determining the width of the glass transition temperature range.
Reinforcing fibers
Constitutive relations for fiber reinforced composites
The model developed by the authors in (Matveenko et al. 2012) can be generalized to the description of fiber reinforced composites undergoing glass transition. As in the case of isotropic material, two approaches can be applied. The former relies on the concept of free energy and the latter – on the concept stiffness enhancement. In the isotropic case, these two approaches yield identical results. In the anisotropic case, the results are found to be different.
\( {C}_{ijkl}^{(1)},\kern0.5em {\alpha}_{kl}^{(1)} \) are the components of the effective stiffness tensors and coefficients of linear temperature expansion of the composite at T ≥ T _{ g1}; \( {C}_{ijkl}^{(2)},\kern0.5em {\alpha}_{kl}^{(2)} \) are the same quantities for bonds formed in the process of vitrification of the matrix. The procedure of evaluating \( {C}_{ijkl}^{(2)},\kern0.5em {\alpha}_{kl}^{(2)} \) in terms of the composite properties in the highelastic \( {C}_{ijkl}^{(1)},\kern0.5em {\alpha}_{kl}^{(1)} \) and glassy \( {C}_{ijkl}^g,\kern0.5em {\alpha}_{kl}^g \) states is given below.
For approximate calculations it is allowable to take \( {\alpha}_{11}^{(2)}={\alpha}_{11}^g,\kern0.5em {\alpha}_{22}^{(2)}={\alpha}_{33}^{(2)}=0,5\left({\alpha}_{22}^{(1)}+{\alpha}_{22}^g\right) \).
The above technique of constructing the physical relations for composite materials with a vitrifying matrix (12) is based on the concept of free energy. The value of the free energy is invariant to the choice of coordinate system and does not take into account the anisotropy of the contribution of the vitrification degree to polymer stiffness. The isotropic polymers are lost to such effect, whereas composite materials characterized by a considerable directional variation of properties are rather sensitive to the conversion degree. The second stiffnessbased approach, which like the free energy approach, relies on the same physical and phenomenological considerations, allows us to eliminate this drawback.
where ΔC _{ N } = ∂Ĉ/∂NΔN; ΔC _{ T } = dĈ/dtΔt − ΔC _{ N }. The value of the second term is specified by an increase in the stiffness of vitrifying composite due to the formation of new intermolecular bonds in the binding agent. Their unstrained (natural) state coincides with the current (actual) state of the material at the time of their formation (stiffness enhancement without tension (Wang et al. 2001)). Therefore, the increment of the stress tensor can be expressed as
\( \varDelta \sigma \left({t}_1\right)=\varDelta {\widehat{C}}_T\left(N\left({t}_1\right),T\left({t}_1\right)\right):\overline{\widehat{\varepsilon}}\left({t}_1\right)+\widehat{C}\left(N\left({t}_1\right),T\left({t}_1\right)\right):\varDelta \overline{\widehat{\varepsilon}}\left({t}_1\right) \),
The stress–strain relations for polymer composites undergoing glass transition, which are derived based on the stiffnessgrowth method (23), (25), are more general compared to the “energy” relations (12) and therefore more accurate in describing the glass transition processes. However, they are awkward to handle in practice because of the necessity for accurate evaluation of the stiffness tensor derivative with respect to the conversion degree and calculation of tensor relations \( \widehat{C}(N),\kern0.5em {\widehat{\varepsilon}}^o(N) \). In numerical computation of thermomechanical characteristics of composite materials this leads to unreasonably high time costs and memory consumption and what is more undesirable to a loss of accuracy in the case of performing calculations for boundary value technological problems due to inevitably rough estimation of the derivative while doing time discretization. Therefore in practice it is more appropriate to use simpler expression such as (12).
Let us define the class of composite materials, for which the “energy” relations (12) give an accurate description, i.e., are coincide with (23, 25). To this end, we represent both types of equations in the componentwise form:
Thus, to ensure that in the description of behavior of composite material undergoing glass transition the energytype relations coincide with the stiffnesstype relations, the dependence of the components of the elastic tensor on the degree of material conversion should obey the law (29)
Expression (31) describes the dependence of temperature deformation on time providing going from the “stiffness growth” relations to “the energy” relations and back.
Expressions (29) (31) allow us to single out the class of composites, for which both approaches give the same results. The most demonstrative example is the composite containing no filling material. In other materials such similarities might be the result of the random combination of matrix and fiber properties. Therefore it is essential to estimate the errors that occur in the calculations due to the application of simplified relations (12). To this end, we carry out a comparative analysis of the “energy” and “stiffness” models with the aim of averaging the thermomchanical properties of the composite material based on the Voigt and Reuss methods.
where ε(T _{ k }), is the strain in the cooled specimen after unloading: ε(T _{ k }) = − σ ^{ eg }(T _{ k })/(E ^{(1)} + E ^{(2)}); ε ^{ s }(T _{ k }) = − σ ^{ sg }(T _{ k })/E(T _{ k }). The values of the effective thermomechanical characteristics in (32)–(35) depend on the method of averaging the properties of the composite.
Here \( {E}_M^{(1)} \) is the equilibrium modulus of the binding agent; \( {E}_M^{(2)}={E}_M^g{E}_M^{(1)};\kern0.5em {E}_a,{\alpha}_{a,}\kern0.5em {E}_{M,}{\alpha}_M \) are the moduli and liner coefficients of thermal expansion of reinforcing and binding materials, respectively; \( {E}_M={E}_M^{(1)}+{E}_M^{(2)}N;\kern0.5em \overline{\psi}=1\psi \).
Solving Eqs. (37) and (38) taking into account the deformation conditions and temperature variation, we obtain the effective stresses that agree with (36). Thus, for the specified computational scheme both the ‘energy” and “stiffness” relations yield an exact solution.
It is seen that the maximum error is observed in the glass transition interval. Its value increases with the growth of ψ and reaches its maximum value at ψ = 0,96. In the temperature range corresponding to the operating conditions of the end product (lower than T _{ g2}), the calculation error does not exceed 10%. For polymer materials reinforced with isotropic fibers (glass, boronplastics, etc.) with \( {E}_M^{(2)}<<1 \) the maximum error in the estimates of transverse stresses is as high as 3% (Fig. 4b, curve 1). This suggests that the “energy” relations can be readily used for description of the behavior of composite materials in the examined range of material properties reducing thereby the computational costs related with the evaluation of unknown thermomechanical constants.
Results and discussion
Predicting of the effective material functions and constants for fiberreinforced composite based on the results of numerical experiments
One of the challenging problems in the mechanics of composite materials is the estimation of their mechanical properties based on the properties of their components. In our case it is necessary to determine two sets of parameters, i.e., the parameters corresponding to the highelastic state of the binder  \( {C}_{ijkl}^{(1)},\;{\alpha}_{ii}^{(1)} \), and the parameters corresponding to the glassy state  \( {C}_{ijkl}^g,\;{\alpha}_{ii}^g \).
Consider the unlimited, continuously reinforce medium whose structure is formed by straightened and equally oriented cylindrical fibers having equal circular cross sections. The space between the fibers is filled with the binding medium – the matrix whose characteristics differ from the characteristics of the fiber.
The emphasis is on the prediction of the effective properties of the idealized homogeneous medium based on the properties of the phases and their geometrical characteristics, using for this purpose the averaging procedure.
The volume fraction of fibers in the composite is defined by the radius of inclusion: . Hence \( R=\sqrt{\psi \frac{2 ab}{\pi }} \). Since we consider a hexagonal cell, we have \( a=\frac{b}{\sqrt{3}} \)
The general formulation of the problem of the linear elasticity theory is reduced to the following system of equations:

Here σ*_{ ij }(x), ε*_{ ij }(x), are the components of the strain and stress tensors at the structural level. The elastic moduli of the matrix G ^{ m }, B ^{ m } are taken to be G _{1}, B _{1} in the highelastic state and G _{ g }, B _{ g }  in the glassy state.
The boundary conditions depend on what components of the elasticity and temperature deformation tensors we need to determine.
Consider a composite material consisting of the isotropic matrix and transverselyisotropic fibers. The composite is assumed to be transversely isotropic such that the isotropy axis coincides with the fiber direction (zaxis (1)) and the x(3)y(2)plane is referred to as the isotropy plane.
These characteristic can be found by solving four boundary value problems of the elasticity theory: uniaxial tension along the fiber length, transverse deformation in the plane normal to the fiber direction, shear deformation in the direction of the fiber axis and free temperature deformation.
The value of \( {U}_1^0 \) is specified, and U _{2} and U _{3} are the unknown parameters found by solving the problem. The composite deformation in the longitudinal direction is constant and equal to \( {\varepsilon}_{11}=\varepsilon ={U}_1^0/c \). From the solution of the problem we determine the average longitudinal stresses σ _{11} and average transverse strains ε _{22} = U _{2}/b and ε _{33} = U _{3}/a.
The value of \( {U}_2^0 \) is specified, and U _{1} and U _{3} are the unknown parameters found by solving the problem. The composite deformation in the longitudinal direction is constant and equal to \( {\varepsilon}_{22}={U}_2^0/b \). From the solution of the problem we determine the average transverse normal stresses σ _{22} and average strains ε _{11} = U _{1}/c and ε _{33} = U _{3}/a.
Elastic characteristics of the composite components
E_{1}  E_{2}  ν_{12}  ν_{23}  G_{12}  

Glass fiber  9,32E + 10    0,24     
Boron fiber  3,70E + 11    0,15     
Carbon fiber  2,26E + 11  1,29E + 10  0,31  0,2  6,00E + 10 
Organic fiber (T > T_{g})  6,10E + 10  1,60E + 09  0,27  0,27  1,30E + 09 
Organic fiber(T < T_{g})  1,21E + 11  3,35E + 09  0,27  0,27  2,60E + 09 
Coefficient of linear thermal expansion of fibers
α_{1},K^{−1}  α_{2},K^{−1}  

Glass fiber  5E6  5E6 
Carbon fiber  −5E7  2.7E5 
Organic fiber  −6,3E6  8,5E5 
From the comparison of the figures it is evident that the character of the dependence of all parameters on the coefficient of the volume fraction of fiber remains practically unchanged. Softening of the binding agent in the highelastic state has negligible effect on the longitudinal modulus of the composite (Fig. 7), but leads to a considerable decrease of the transverse and shear moduli, as well as to weakening of the influence of fiber properties on these quantities (Figs. 8 and 9).
To sum up, we can say that the proposed two approaches can be used to construct the constitutive relations in the form of (12) for composites undergoing glass transition. The effective characteristics of the examined composites in the steady states (highelastic and glassy) can be conveniently predicted based on the known properties of the matrix and fiber.
Conclusion

The model of polymer behavior at glass transition recently developed by the authors was generalized to the case of fiberreinforced polymer matrix composites using two approaches: one is base on the concept of free specific energy, the other – on the growth of marix stiffness. The analysis has shown that for homogeneous materials these two approaches are of equal worth, whereas for composite materials they give different results under deformation in the transverse direction.

The stiffness growth approach is more accurate, because it takes into account the anisotropy of the influence of the degree of conversion on the composite stiffness. However its application involves large computational costs. Moreover, it is highly sensitive to the experimental data errors.

For composites under longitudinal deformation (the load is applied along the fiber axis) the two approaches agree fairly well. In the case of transverse deformation (the load is applied transverse to the fiber axis) the maximal difference is observed in the glass transition temperature range. In the range of temperatures corresponding to operating conditions of the end product this difference is inessential.

Using the finite element method and averaging technique a numerical algorithm for calculating the thermoelastic constants of composites in the glassy and highelastic states has been developed. The thermomechanical properties of the fiber and matrix and volume fraction of fibers are used as the input data.

The longitudinal, transverse and shear moduli of composites containing different types of fibers have been plotted against the volume fraction of the fiber in the case when the matrix is in the glassy or highelastic state. It has been shown that softening of the matrix has an insignificant effect on the longitudinal modulus of a composite but leads to a considerable decrease of the transverse and shear moduli.

The coefficients of thermal expansion of composites containing different types of fibers have been plotted against the volume fraction of the fiber in the case when the matrix is in the glassy or highelastic state. It has been shown that the coefficient of thermal expansion in the transverse direction is much higher than the coefficient of thermal expansion in the longitudinal direction, especially when the composite is in the highelastic state.
Declarations
Acknowledgements
This article was prepared under a grant of Russian Foundation for Basic Research #160100474.
Authors’ contributions
VPM  Methods, Abstract. NAT  Methods, Conclusion. OYS  Methods, Results, Conclusion. INS  Methods, Results. INV  Translate, Editing. All authors read and approved the final manuscript.
Competing interests
00000: Materials Science, general
14026: Computational Science and Engineering
18000: Ceramics, Glass, Composites, Natural Materials
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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