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Three-dimensional modeling and experimental validation of thermomechanical response of FRP composites exposed to one-sided heat flux
Three-dimensional modeling and experimental validation of thermomechanical response of FRP composites exposed to one-sided heat flux Shengbo Shi, Linjie Li, Guodong Fang, Jun Liang, Fajun Yi, Guochang Lin PII: DOI: Reference:
S0264-1275(16)30383-5 doi: 10.1016/j.matdes.2016.03.098 JMADE 1576
To appear in: Received date: Revised date: Accepted date:
20 December 2015 15 March 2016 18 March 2016
Please cite this article as: Shengbo Shi, Linjie Li, Guodong Fang, Jun Liang, Fajun Yi, Guochang Lin, Three-dimensional modeling and experimental validation of thermomechanical response of FRP composites exposed to one-sided heat flux, (2016), doi: 10.1016/j.matdes.2016.03.098
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ACCEPTED MANUSCRIPT Three-dimensional modeling and experimental validation of thermomechanical response of FRP composites exposed to one-sided heat flux
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Shengbo Shi1,3, Linjie Li2,3, Guodong Fang3, Jun Liang3, Fajun Yi3, Guochang Lin3
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1. National Key Laboratory of Aerospace Flight Dynamics, Northwestern Polytechnical University,
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Xi’an 710072, P.R. China
2. China Helicopter Research and Development Institute, Jingdezhen 333000, P.R. China 3. Science and Technology on Advanced Composites in Special Environments Key Laboratory, Harbin
*Corresponding
author:
Shengbo
Shi.
Tel./fax:
+86
29
88492783.
E-mail
address:
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[email protected] (S. Shi).
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Institute of Technology, Harbin 150001, P.R. China
Abstract: The heat transfer, gas diffusion process and thermomechanical deformation are generally
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coupled and associated with the chemical decomposition for fiber reinforced polymer composites at
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elevated temperatures. The three-dimensional (3-D) governing differential equations for the coupled problem
of
porous
elastomers
were
developed.
The
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temperature-diffusion-deformation
thermomechanical behavior of a silica/phenolic composite material was predicted using the
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mathematical model. The spatially dependent temperature and pore pressure, displacement, and stress contours of silica/phenolic composites exposed to one-sided radiant heat flux were investigated. Based on the digital image correlation technique, a non-contact high temperature deformation measurement test was conducted. The temperature profiles were measured by the thermocouples embedded in different depths of the specimen, while the full-field displacements and strains were provided by correlating the two digital images of the specimen surface in the un-deformed and deformed states, respectively. The accuracy of the proposed model was assessed by comparing the predicted temperatures and displacements with experimental values for the same boundary and initial conditions.
Keywords: Polymer-matrix composites; Thermomechanical behavior; High temperature deformation; 1
ACCEPTED MANUSCRIPT Gas diffusion; Digital image correlation. 1 Introduction
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Fiber reinforced polymer (FRP) composites present attractive characteristics for space vehicles and
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nozzles of solid rocket motor where light-weight materials with superior thermal protection performances and high mechanical properties are needed [1-3]. The continually expanding use of FRP composites in large structural applications requires a better understanding of the interdependent thermal
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and mechanical responses of the material when it is subjected to elevated and high temperatures [4]. For
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FRP composites, irreversible changes in thermochemical and thermomechanical properties will occur under high temperatures, which adversely affect the structural integrity and reliability of these
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composites [5]. In order to design structures with FRP components, it is necessary to accurately model
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the variation of the thermomechanical responses over a broad temperature range.
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Significant research has been conducted to understand the thermomechanical behavior of decomposing materials, and mathematical models and numerical methods have been developed [6-15].
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McManus and Springer [8,9] presented a finite difference method to describe the thermal and mechanical responses of one-dimensional (1-D) thin plate when rapidly heated to high temperatures. Their work included modeling of material ablation and mechanical damage to simulate the delamination of fiber plies. A similar model was developed by Wu and Katsube [10-12], in which the coupling between the thermochemical decomposition and thermomechanical deformation was made clear throughout the formulation. Combining the poroelasticity theory and thermochemical decomposition analysis, Sullivan and Salamon [13-15] proposed a finite element method to analyze the material response of a glass-phenolic composite. For two-dimensional (2-D) plane problems, free thermal expansion tests and restrained thermal growth tests were numerically simulated. Bai and Keller et al. [16,17], Matsuura et al.
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ACCEPTED MANUSCRIPT [18], McGurn and DesJardin et al. [19], Li et al. [20], Luo et al. [21], Martias et al. [22] further extended the prediction method of thermomechanical responses of FRP composites and their predicted results
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were verified against the experimental data. Recently, Gibson, Mouritz, Feih, et al. [23-28] discussed the
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high temperature and fire behavior of polymer matrix composites under load in fire. In their work, the temperature profiles in polymer laminates exposed to fire were calculated using thermal models. Also, mechanical models were developed to analyse the fire structural response and failure of laminates and
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sandwich composites under compression or tension loading.
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Moreover, high temperature deformation measurement of the material has been performed using the optical technique of digital image correlation (DIC) [29-33]. Lyons et al. [29], Pan et al. [30,31]
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established a simple, easy-to-implement yet effective high-temperature DIC method to measure the
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full-field in-plane surface displacements and strains of the material at elevated temperatures. The ability
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of this DIC method was evaluated by a series of experiments. Cholewa and Summers et al. [32] developed an integrated infrared thermography and digital image correlation (TDIC) technique to
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simultaneously measure spatial and temporal distributions of temperatures and displacements. The TDIC system was demonstrated using a series of one-sided heat exposure experiments performed on compressively loaded sandwich composites. Thermally induced chemical reactions occur when FRP composites are exposed to high temperatures. The resin system degrades to form gaseous products and carbonaceous char. These decomposition gases are trapped because of the low porosity and permeability of the material and this may cause high internal pressures [34]. As the pyrolysis reactions proceed, the gases begin to flow through the material attenuating the conduction of heat to the reaction zone [35]. Additionally, high thermal stress exists in the charred material due to the difference of thermochemical expansion of the
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ACCEPTED MANUSCRIPT composite matrix and the second phase, which results in thermal deformation of the material. In other words, the heat-mass transfer processes and thermomechanical deformation are generally coupled and
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associated with the thermochemical decomposition of polymer matrix. The effects of these processes
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must be considered in the formulation of the mathematical model to predict the thermomechanical behavior of FRP composite materials at high temperatures. Moreover, for complex structural component application with FRP composites, a three-dimensional (3-D) model is required.
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In the above-cited work, the investigations focused on the 1-D and 2-D thermomechanical response.
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Also, in our previous work [36], prediction of the 2-D thermomechanical behavior of FRP composites was only given. However, 3-D thermomechanical model and high temperature deformation
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measurement test which used to validate the model have been rarely reported, especially for the complex
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structural component application. In contrast, in this paper, the mathematical formulation for 3-D
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thermomechanical model was established and further implemented using Comsol Multiphysics software, which is applicable for complex structural component. This method was validated using high
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temperature deformation measurement tests performed on silica/phenolic composite specimens. 2 Governing equations of the coupled temperature-diffusion-deformation problem 2.1 Problem statement Since the heat-mass transfer processes and thermomechanical deformation are coupled and associated with the chemical decomposition, the thermomechanical behavior of FRP composites under high temperatures can be treated as a coupled temperature-diffusion-deformation (TDD) problem of the porous elastomers. Fig. 1 presents the geometrical model, loading and boundary conditions of the solving problem. It assumes that one side of a cuboid composite specimen in the through-thickness direction is heated with a radiant heat flux. Another side of the specimen (x-z plane shown in Fig. 1) is sprayed with
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ACCEPTED MANUSCRIPT the black and white paints where the thermal deformation is measured. This surface is in air and a light source is applied on it to ensure good ray condition for taking images. Other surfaces of the specimen are
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assumed adiabatic.
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The model also assumes that: (1) the composite material is transversely isotropic, and x-y plane is the isotropy plane; (2) the isotropy plane of the porous material is coincident with the isotropy plane of the solid material [13]; (3) the gas flow is governed by the Darcy’s law. With these assumptions, the
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governing differential equations for the coupled TDD problem which involve mass conservation
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equation of gas phase, motion equation of solid phase, and energy conservation equation can be established.
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2.2 Mathematical formulas for the 3-D thermomechanical model
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2.2.1 Mass conservation equation of gas phase
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Thermal decomposition of polymer matrix occurs when the material reaches moderately high temperatures (about 350 oC). This pyrolysis reaction can be modeled by a Arrhenius equation in which
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the degree of conversion for the pyrolysis reaction, denoted by c , is defined as the mass loss that is to occur before the completion of the pyrolysis reaction over the total mass loss for this reaction. The value of c varies from 0 at the beginning of the pyrolysis reaction to 1 at the completion of the pyrolysis reaction. The Arrhenius kinetic reaction equation for the pyrolysis reaction can be given by
dc E A0 c n exp a dt RT
(1)
where A0 , Ea and n are the reaction rate constant, activation energy, and order of reaction for the pyrolysis reaction, respectively. And they can be determined by thermogravimetic analysis (TGA). R is the universal gas constant. The time rate of gas generation for the thermal decomposition is calculated by
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ACCEPTED MANUSCRIPT mpg t
re sin virg
dc dt
(2)
where re sin is the volume fraction of the resin matrix. Using Darcy’s law, the volume average gas
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velocity vector, vg , can be expressed in terms of internal gas pressure, p , permeability of solid
vg
T
p p p i T j L k T (T ) x T (T ) y L (T ) z
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materials, , and gas viscosity, , as follows,
(3)
where the subscripts T and L indicate the material properties in the isotropy plane (x-y plane) and in
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the perpendicular direction (z direction), respectively.
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According to the principles of mass conservation for gas phase, the differential equation governing the diffusion of pyrolysis gases through the porous solid material can be derived. The time rate of gas
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storage in the solid materials is equal to the sum of the time rate of gas generation and the rate of change
t
mpg t
g vg
(4)
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mg
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of the gas mass flux through the volume element.
where mpg is the mass of the generated gas during thermal decomposition. The variation of gas mass per
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unit bulk volume can be defined in space and time as a function of three primary dependent variables, involving the porous solid elastic strain e , the pore pressure p , and the temperature T which describe the decomposition process [13,14]. The left side of equation (4) can be expanded as follows, mg t
mg e mg p mg T e t p t T t
(5)
with, mg e eyy e ezz g11 xx g 33 e t t t t
(6)
mg p g p p t M t
(7)
mg T T g g g T t t
(8)
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ACCEPTED MANUSCRIPT where ij is a constitutive coefficient which relates the pore pressure to the total stress; g and g are the volume fraction of decomposition gases trapped in pores for the control volume and the
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coefficient of thermal expansion of the gas, respectively; M is the constitutive coefficient of the
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transversely isotropic material and its general expression was given by Biot [37]. 2.2.2 Equation of motion of solid phase
Based on the classical form of the equilibrium equation, the equations of motion of the porous
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elastomers in the absence of body forces and under quasi-static conditions is expressed as follows, (9)
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ij , j 0
where ij is the total stress tensor. Similar to Sullivan and Wu’s work [10,13,15], the effective stress
(10)
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ij Cijkl ekl ij p
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be written as follows,
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law for overall strain developed by Carroll [38] is also used in this paper. The total stress tensor, ij , can
It is noted that the poroelastic parameter, ij , in Eq. (10) is used to evaluate the effect of the pyrolysis
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gas pressures in the pores on the total stress of the porous material. The value of ij varies from 0 to 1. For silica fiber reinforced polymer composites, the values of 11 and 33 are taken as 0.9 in the present study. The elastic strains are related to the total and thermal strains and its expressions for the present solution case are,
exx xx T T (T ) eyy yy T T (T ) ezz zz T L (T ) exz xz eyz yz exy xy
(11)
where T and L are the coefficients of thermal expansion of the solid material in the plane of isotropy and in the perpendicular direction, respectively, which are functions of temperature. Substituting Eqs. (11) and (10) into Eq. (9) and employing the strain-displacement relations, the equations of motion for the solid phase can be rewritten in x, y and z direction, respectively. Five
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ACCEPTED MANUSCRIPT independent elastic constants ( C11 , C12 , C13 , C33 , and C44 ) for transversely isotropic materials are
1 C11 C12 ux, yy u y , xy C44 ux, zz uz , xz 2 C11 T (T ) C12 T (T ) C13 L (T ) T, x 11 p, x 0
(12)
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C11ux , xx C12 u y , yx C13u z , zx
T
also considered. In x direction, Eq. (9) reduces to,
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Similarly, the equations of motion in y and z directions respectively become, 1 C11 C12 ux, yx u y , xx C44 u y , zz uz , yz 2 C11 T (T ) C12 T (T ) C13 L (T ) T, y 11 p, y 0
C11u y , yy C12u x , xy C13u z , zy
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C13ux , xz C13u y , yz C33uz , zz C44 u x, zx u z , xx C44 u y , zy u z , yy
2C13 T (T ) C33 L (T )T, z 33 p, z 0
(13)
(14)
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In Eqs. (12)-(14), a contracted notation is used to denote the elastic constants Cijkl . Furthermore, it is
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revealed that the thermal deformations u x , u y and u z , the temperature T , the pore pressure p are
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coupled.
2.2.3 Energy conservation equation
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A control volume is chosen to analyze the heat and mass transfer processes of FRP composites at
by,
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high temperatures. Based on Laplace’s equation, the energy conservation in the control volume is given
T T T T T (T ) i T (T ) j L (T ) k t x y z dc g Cpg (T )vg T hp s dt
s Cps (T )
(15)
where the term on the left side gives the rate of change of internal energy in the control volume, the three terms on the right side are the diffusion term, the convection term and the chemical decomposition term, respectively. In Eq. (15), Cps and Cpg are the heat capacity for the solid material and for the gas, respectively, which are functions of temperature; T and L are the thermal conductivity coefficient for the material in the plane of isotropy and in the perpendicular direction, respectively; hp is the heat of reaction for the decomposition reaction, which has a negative value.
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ACCEPTED MANUSCRIPT 2.2.4 Initial and boundary conditions The initial pore pressure p0 is taken as 1 atm and the initial temperature T0 is taken as 20 oC. The
z
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T z, t
qw z 0
ux x, y, z z H 0 u y x, y, z
zH
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L (T )
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following heat flux and mechanical boundary conditions are applied in the present model.
0 uz x, y, z z H 0
(16)
(17)
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where qw is the radiant heat flux perpendicular to the heated surface, H is the thickness of the geometrical model.
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3 Non-contact high temperature deformation measurement test
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3.1 Experimental procedure
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Digital image correlation (DIC) has proven to be a powerful non-contact optical technique for accurate measurement of surface deformations, either at a point or over a field containing strain gradients
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[30]. In the DIC method, a video camera and personal computer acquire digitized images of a random speckle pattern on the surface of a specimen before and after deformation. Displacements are calculated
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directly by correlating the two digital images [29]. In order to accurately measure both the full-field displacements and temperatures of structural materials exposed to radiant heat flux, a non-contact high-temperature thermal deformation measuring system which involves DIC testing system, thermocouples temperature measurement system, radiant heating simulation device, and heat flux measurement system was carried out in the present study, as shown in Fig. 2(a). The DIC testing system (VIC-3D, American Correlated Solutions Inc., US) consists of two CCD camera (1624×1224 pixel), white light source and computer system. Before the test, a random speckle pattern on the specimen surface was artificially made (Fig. 2(b)), and a calibration process of the CCD camera was also conducted to obtain the internal parameters. The thermocouples
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ACCEPTED MANUSCRIPT embedded at different depth of the specimen were used to monitor the in-depth temperature. After the thermal environment of the radiant heating simulation device had been steady, the radiant heat flux was
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measured by the heat flux sensor firstly, as shown in Fig. 2(c). Then, the specimen surface was heated at
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a heat flux of 155 kW/m2 (Fig. 2(d)). During the test, the images carried the deformation information of the specimen were acquired at each 100 ms. To correlate the deformed image to the undeformed reference image, each image was divided into small subsets.
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3.2 Materials
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A silica/phenolic composite, kindly provided by Aerospace Research Institute of Materials and Processing Technology (Beijing, China), was selected and characterized in this study. This composite
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was 24 wt% silica fiber (Shaanxi HuaTek Fiberglass Material Group, China), 20.4 wt% phenolic fiber
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(Tongshan chemical plant, China), and 55.6 wt% phenolic resin (Tongshan chemical plant, China).
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Properties of the phenolic fiber were similar to the phenolic resin. The silica/phenolic composite specimen for deformation measurement was 14×14×33.7 mm. The basic material properties,
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temperature-dependent thermal and transport properties of silica/phenolic composites have been summarized in Table 1. The kinetic parameters of decomposition for this composite were determined by TGA. The thermal and transport properties of the virgin and char components were collected from a number of sources. Based on these data, material properties of the charred composite can be calculated as a weighted function of degree of conversion, c . The temperature-dependent stiffness properties were measured by high temperature compression tests reported in our earlier work [39]. 4 Results and discussion Based on Comsol Multiphysics software, the mathematical model developed in Section 2 was used to analyze the thermomechanical responses of silica/phenolic composites subjected to one-sided heat
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ACCEPTED MANUSCRIPT flux of 155 kW/m2. The coupled model was verified against available experimental data measured by the high temperature deformation test. Fig. 3 shows the temperature distribution of silica/phenolic
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composites along the heating direction. As expected, the temperature decreases with increasing distance
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from the heated surface. There is a large temperature gradient in the internal material during heating, which can induce the high thermal stress. The experimental temperatures for initial depths of 0.96, 3.98, 7.73, and 12.26 mm are also presented in Fig. 3 as data points. It can be found that the calculated
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temperature curves at heating time of 100 s and 200 s are in good agreement with the experimental values,
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respectively.
Comparisons between the calculated and experimental displacement u z versus the distance from
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the heated surface for silica/phenolic composites at heating time of 100 s and 200 s are shown in Fig. 4.
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The curves show that the displacement u z increases with decreasing distance from the heated surface. It
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can be seen from Fig. 4 (a) and (b) that the displacement u z also increases with increasing heating time. The absolute maximum displacement is about 0.17 mm at 100 s, and this value increases with increasing
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heating time to about 0.23 mm at 200 s. As can be seen from Fig. 4 (b), there is good agreement between the calculated and experimental values at 200 s. The calculated displacements roughly coincide with the experimental results at 100 s except the displacement at the spatial location less than 8 mm away from the heated surface. The maximum displacement difference for all the data points shown in Fig. 4(a) is 0.05 mm. The discrepancies are thought to be due to the fact that the sufficiently high temperatures of these locations may significantly affect the stability or accuracy of the experimental results. It is important to point out that, because of the failure of the random speckle pattern on specimen surface for depths of 0-4.54 mm at 200 s due to high temperatures, the measured displacement data of these locations are unavailable. Fig. 5 presents the displacement contours of silica/phenolic composite specimen during
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ACCEPTED MANUSCRIPT heating obtained by the DIC technique. From the figure, it is possible to visually observe the transition in displacement from the heated surface to the back surface.
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Fig. 6 shows the simulated full-field and section total strain ( zz ) contours of silica/phenolic
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composite specimen during heating. The figure demonstrates non-monotonicity of total strain zz of the material along heating direction. The total strain has a local peak during heating. This peak is not at the heated surface, but at the interface of charring layer where the pyrolysis reaction reaches completion and
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pyrolysis zone where the intensive thermal decomposition occurs. Furthermore, the peak strain moves
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towards the back surface of the material with increasing heating time. The maximum strain is 0.017 at 100 s.
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Fig. 7 shows the pore pressure-depth curves of silica/phenolic composite specimen along the
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heating direction at heating times from 50 to 200 s. As can be seen, the pore pressure has a local peak
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during heating and exhibits a parabola tendency in the through-thickness direction of the material. This peak is shifted to the back surface of the material with increasing heating time. Comparison of Figs. 3 and
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7 shows that the peak pore pressure occurs just after the onset of the pyrolysis reaction. This, of course, results because the porosity and permeability of the material is still low at this early stage of the pyrolysis reaction.
Fig. 8 illustrates the full-field total stress ( zz ) contour of silica/phenolic composite specimen at heating times from 50 to 200 s. Similar to the total strain zz , the total stress zz exhibits non-monotonicity and the peak stress does not appear at the heated surface. With increasing heating time, this peak is shifted to the back surface of the material. Fig. 9 shows the variation curves of the total stress
zz versus distance from the heated surface at different heating time. It is revealed that the peak total stress decreases with increasing heating time. This is due to the fact that the degradation in stiffness
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ACCEPTED MANUSCRIPT properties of the thermal protection material occurs during heating. Additionally, an apparently maximum value appears at depths of 2.7, 7.2 and 9.4 mm for heating times of 100, 150 and 200 s,
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respectively, and this may be attributed to the effect of the pore pressure on the material response.
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Fig. 10 presents the photograph of silica/phenolic composite specimen after heating. From the figure, it is possible to roughly distinguish different regions in a post-test specimen, involving charring layer where the pyrolysis reaction reached completion, pyrolysis zone where most of the material is
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undergoing decomposition, and virgin material. Fig. 11 shows the morphology of virgin material and
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heated surface for silica/phenolic composites. It is obvious that the microstructure of the charred composite material at the heated surface does not contain the polymer resin phase due to the completion
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of the pyrolysis reaction. Thus, the silica fibers and carbonaceous char left on the heated surface are not
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bonded together and easy to be stripped off by the high speed stream. Moreover, a number of pores and
5 Conclusions
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cracks are observed in the porous skeleton.
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The 3-D thermomechanical model was developed and further implemented using Comsol Multiphysics, which was applicable for complex structural component. The model considers the combined effects of thermochemical decomposition, gas diffusion and thermomechanical deformation on the thermal performance of the material. The model can predict the temperature, pore pressure and deformation behavior of FRP composites exposed to one-sided radiant heat flux. The DIC system can measure the temperature profiles, thermally induced surface displacements and strains. There is good agreement between the predicted and experimental values for spatially dependent temperature of silica/phenolic composites. The experimental displacements roughly coincide with the calculated results except the displacement at the spatial location near the heated surface. The total strain has a local peak
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ACCEPTED MANUSCRIPT during heating and this peak appears at the interface of charring layer and pyrolysis zone. The stress exhibits non-monotonicity along the heating direction. The peak stress is shifted to the back surface and
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decreases with increasing heating time due to degradation in stiffness properties of FRP composites at
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elevated temperatures. Acknowledgement
The present work is supported by the National Natural Science Foundation of China under Grant No.
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11402202, the Fundamental Research Funds for the Central Universities of China under Grant No.
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3102014JCQ01001, and the Supporting Technology Funds for Aerospace under Grant No. 2014-HT-XGD.
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References
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[1] P. Luangtriratana, B.K. Kandola, P. Myler, Ceramic particulate thermal barrier surface coatings for glass fibre-reinforced epoxy composites, Mater. Des. 68, 2015, 232-244.
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[2] B.K. Kandola, M.H. Akonda, A.R. Horrocks, Fibre-reinforced glass/silicate composites: effect of fibrous reinforcement on intumescence behaviour of silicate matrices as a fire barrier application, Mater. Des. 86, 2015, 80-88.
AC
[3] D. Teixeira, M. Giovanela, L.B. Gonella, J.S. Crespo, Influence of injection molding on the flexural strength and surface quality of long glass fiber-reinforced polyamide 6.6 composites, Mater. Des. 85, 2015, 695-706.
[4] Y. Bai, T. Keller, T. Vallée. Modeling of stiffness of FRP composites under elevated and high temperatures. Compos. Sci. Technol. 68, 2008, 3099-3106. [5] K. Wei, R. He, X, Cheng, et al., A lightweight, high compression strength ultra high temperature ceramic corrugated panel with potential for thermal protection system applications, Mater. Des. 66, 2015, 552-556. [6] E. Kandare, B.K. Kandola, P. Myler, G. Edwards, Thermo-mechanical responses of fiber-reinforced epoxy composites exposed to high temperature environments, Part I: Experimental data acquisition, J. Compos. Mater. 44, 2010, 3093-3114.
14
ACCEPTED MANUSCRIPT [7] M. Derradji, N. Ramdani, T. Zhang, et al., Mechanical and thermal properties of phthalonitrile resin reinforced with silicon carbide particles, Mater. Des. 71, 2015, 48-55. [8] L.N. McManus, G.S. Springer, High temperature thermomechanical behavior of carbon-phenolic
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and carbon-carbon composites, I. Analysis, J. Compos. Mater. 26, 1992, 206-229. [9] L.N. McManus, G.S. Springer, High temperature thermomechanical behavior of carbon-phenolic
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and carbon-carbon composites, II. Results, J. Compos. Mater. 26, 1992, 230-251. [10] Y. Wu, N. Katsube, A constitutive model for thermomechanical response of decomposing composites under high heating rates, Mech. Mater. 22, 1996, 189-201.
NU
[11] Y. Wu, N. Katsube, A thermomechanical model for chemically decomposing composites, I. Theory, Int. J. Eng. Sci. 35, 1997, 113-128.
MA
[12] Y. Wu, N. Katsube, A thermomechanical model for chemically decomposing composites, II. Application, Int. J. Eng. Sci. 35, 1997, 129-139.
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[13] R.M. Sullivan, N.J. Salamon, A finite element method for the thermochemical decomposition of
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polymeric materials, I. Theory, Int. J. Eng. Sci. 30, 1992, 431-441. [14] R.M. Sullivan, N.J. Salamon, A finite element method for the thermochemical decomposition of
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polymeric materials, II. Carbon phenolic composites, Int. J. Eng. Sci. 30, 1992, 939-951. [15] R.M. Sullivan, A coupled solution method for predicting the thermostructural response of decomposing, expanding polymeric composites, J. Compos. Mater. 27, 1993, 408-434.
AC
[16] Y. Bai, T. Valle´e, T. Keller, Modeling of thermal responses for FRP composites under elevated and high temperatures, Compos. Sci. Technol. 68, 2008, 47-56. [17] Y. Bai, T. Keller, Modeling of mechanical response of FRP composites in fire, Compos. Part A: Appl. S. 40, 2009, 731-738. [18] Y. Matsuura, K. Hirai, T. Kamita, et al., A challenge of modeling thermomechanical response of silica-phenolic composites under high heating rates, In: Proceedings of 49th AIAA Aerospace Sciences Meeting, Orland, Florida, 2011. [19] M.T. McGurn, P.E. DesJardin, A.B. Dodd, Numerical simulation of expansion and charring of carbon-epoxy laminates in fire environments, Int. J. Heat Mass Tran. 55, 2012, 272-281. [20] H. Li, E. Kandare, S. Li, et al., Integrated thermal, micro- and macro-mechanical modelling of post-fire flexural behaviour of flame-retarded glass/epoxy composites, Comp. Mater. Sci. 59, 2012, 22-32.
15
ACCEPTED MANUSCRIPT [21] C. Luo, J. Lua, P.E. DesJardin, Thermo-mechanical damage modeling of polymer matrix sandwich composites in fire, Compos. Part A: Appl. S. 43, 2012, 814-821. [22] C. Martias, Y. Joliff, C. Favotto, Effects of the addition of glass fibers, mica and vermiculite on the
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mechanical properties of a gypsum-based composite at room temperature and during a fire test, Compos. Part B: Eng. 62, 2014, 37-53.
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[23] A.G. Gibson, T.N.A. Browne, S. Feih, A.P. Mouritz, Modeling composite high temperature behavior and fire response under load, J. Compos. Mater. 46, 2012, 2005-2022. [24] A.G. Gibson, M.E. Otheguy Torresa, T.N.A. Brownea, et al., High temperature and fire behaviour
NU
of continuous glass fibre/polypropylene laminates, Compos. Part A: Appl. S. 41, 2010, 1219-1231. [25] A.P. Mouritz, S. Feih, E. Kandare, et al., Review of fire structural modelling of polymer composites,
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Compos. Part A: Appl. S. 40, 2009, 1800-1814.
[26] A.P. Mouritz, S. Feih, E. Kandare, A.G. Gibson, Thermal-mechanical modelling of laminates with
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fire protection coating, Compos. Part B: Eng. 48, 2013, 68-78.
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[27] S. Feih, Z. Mathys, A.G. Gibson, A.P. Mouritz, Modelling the tension and compression strengths of polymer laminates in fire, Compos. Sci. Technol. 67, 2007, 551-564.
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[28] S. Feih, A.P. Mouritz, S.W. Case, Determining the mechanism controlling glass fibre strength loss during thermal recycling of waste composites, Compos. Part A: Appl. S. 76, 2015, 255-261. [29] J.S. Lyons, J. Liu, M.A. Sutton, High-temperature deformation measurements using digital-image
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correlation, Exp. Mech. 36, 1996, 64-70. [30] B. Pan, K. Qian, H. Xie, A. Asundi, Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review, Meas. Sci. Technol. 20, 2009, 062001, 1-17. [31] B. Pan, D. Wu, J. Gao, High-temperature strain measurement using active imaging digital image correlation and infrared radiation heating, J. Strain Anal. Eng. Des. 49, 2013, 224-232. [32] N. Cholewa, P.T. Summers, S. Feih, A.P. Mouritz, B.Y. Lattimer, S.W. Case, A technique for coupled thermomechanical response measurement using infrared thermography and digital image correlation (TDIC), Exp. Mech. 56, 2016, 145-164. [33] X. Chen, L. Yang, N. Xu, et al., Cluster approach based multi-camera digital image correlation: Methodology and its application in large area high temperature measurement, Opt. Laser Technol. 57, 2014, 318-326. [34] J.B. Henderson, T.E. Wiecek, A mathematical model to predict the thermal response of
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ACCEPTED MANUSCRIPT decomposing, expanding polymer composites, J. Compos. Mater. 21, 1987, 373-393. [35] S. Shi, J. Liang, F. Yi, G. Fang, Modeling of one-dimensional thermal response of silica-phenolic composites with volume ablation, J. Compos. Mater. 47, 2013, 373-393.
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[36] S. Shi, J. Liang, G. Lin, G. Fang, High temperature thermomechanical behavior of silica-phenolic composite exposed to heat flux environments. Compos. Sci. Technol. 87, 2013, 204-209.
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[37] M.A. Biot, Mechanics of deformation and acoustic propagation in porous media, J. Appl. Phys. 33, 1962, 1482-1498.
[38] M.M. Carroll, An effective stress law for anisotropic elastic deformations, J. Geophys. Res. 84,
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1979, 7510-7512.
[39] S. Shi, L. Gu, J. Liang, G. Fang, et al., A mesomechanical model for predicting the degradation in
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stiffness of FRP composites subjected to combined thermal and mechanical loading, Mater. Des. 89,
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2016, 1079-1085.
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Figure and Table Captions
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Fig. 1. Illustration of the geometrical model, loading and boundary conditions for the coupled TDD
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problem.
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Fig. 2. High-temperature deformation measurement platform. (a) Experiment facility; (b) Random speckle pattern on the specimen surface; (c) Heat flux measurement system; (d) Fixture and insulation of the specimen.
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Fig. 3. Comparison of calculated and experimental spatially dependent temperature profiles for silica/phenolic composites.
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Fig. 4. Comparison of calculated and experimental spatially dependent displacement ( u z ) profiles of silica/phenolic composites at heating time of (a) 100 s and (b) 200 s.
50 s; (b) 100 s; (c) 150 s; (d) 200 s.
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Fig. 5. The experimental displacement contours of silica/phenolic composite specimen during heating. (a)
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Fig. 6. The simulated total strain ( zz ) contours of silica/phenolic composites during heating. (a)
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Full-field strain contour at 100 s; (b) Full-field strain contour at 200 s; (c) Section strain contour at 100 s; (d) Section strain contour at 200 s. Fig. 7. Pore pressure-depth profiles of silica/phenolic composites along the heating direction at different
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heating time.
Fig. 8. Full-field total stress ( zz ) contour of silica/phenolic composites at different heating time. (a) 50 s; (b) 100 s; (c) 150 s; (d) 200 s. (Unit: Pa) Fig. 9. The spatially dependent total stress ( zz ) curves of silica/phenolic composites along heating direction at different heating time. Fig. 10. The schematic of physical delamination for silica/phenolic composite specimen after heating. Fig. 11. Morphology of virgin material and heated surface for silica/phenolic composites. (a) Virgin material; (b) Heated surface.
Table 1. Material properties of silica/phenolic composites.
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Fig. 1. Illustration of the geometrical model, loading and boundary conditions for the coupled TDD
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problem.
Fig. 2. High-temperature deformation measurement platform. (a) Experiment facility; (b) Random speckle pattern on the specimen surface; (c) Heat flux measurement system; (d) Fixture and insulation of the specimen.
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Fig. 3. Comparison of calculated and experimental spatially dependent temperature profiles for
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silica/phenolic composites.
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(a)
(b)
Fig. 4. Comparison of calculated and experimental spatially dependent displacement ( u z ) profiles of silica/phenolic composites at heating time of (a) 100 s and (b) 200 s.
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Fig. 5. The experimental displacement contours of silica/phenolic composite specimen during heating. (a)
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50 s; (b) 100 s; (c) 150 s; (d) 200 s.
Fig. 6. The simulated total strain ( zz ) contours of silica/phenolic composites during heating. (a) Full-field strain contour at 100 s; (b) Section strain contour at 100 s; (c) Full-field strain contour at 200 s; (d) Section strain contour at 200 s.
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Fig. 7. Pore pressure-depth profiles of silica/phenolic composites along the heating direction at different
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heating time.
Fig. 8. Full-field total stress ( zz ) contour of silica/phenolic composites at different heating time. (a) 50 s; (b) 100 s; (c) 150 s; (d) 200 s. (Unit: Pa)
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Fig. 9. The spatially dependent total stress ( zz ) curves of silica/phenolic composites at different heating
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Fig. 10. The schematic of physical delamination for silica/phenolic composite specimen after heating.
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Fig. 11. Morphology of virgin material and heated surface for silica/phenolic composites. (a) Virgin
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material; (b) Heated surface.
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Table 1. Material properties of silica/phenolic composites.
Value
Density of solid material, s (kg/m ) 3
Source
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Properties
Activation energy, Ea ( J/mol)
7.88E4 3
Reaction rate constant, A0 (kg/m s)
Manufacturer TGA TGA
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1.94E5
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1270
1.0
TGA
-4.187E5
TGA
6.18×10-18
[14,15]
4.85×10-15
[14,15]
1.48×10-5+2.50×10-8T
[34]
4.51×10-5
[34]
Coefficient of thermal expansion of char material, char (1/K)
-1.28×10-4
[34]
Specific heat of solid material, Cps (J/kgK)
1090+1.09T
[34]
2390+1.05T
[34]
0.8+2.76×10-4T
[34]
0.96+8.42×10-4T-4.07×10-6T2
[34]
Order of reaction, n (-)
Permeability of virgin material, virg (m2) Permeability of char material, char (m2)
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Gas viscosity, (kg/ms)
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Heat of decomposition, Q (J/kg)
Coefficient of thermal expansion of virgin
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material, virg (1/K)
Specific heat of decomposition gas, Cpg (J/kgK)
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Thermal conductivity of virgin material,
virg (W/mK)
Thermal conductivity of char material,
char (W/mK)
Elastic modulus, EL (GPa)
9.36
20 oC
2.92
450 oC
6.02
100 oC
2.76
500 oC
3.32
135 oC
1.04
550 oC
2.75
200 oC
1.06
600 oC
2.77
350 oC
1.03
650 oC
2.85
400 oC
0.94
800 oC
Tests
Poisson's ratio, T (-)
0.2
Tests
Poisson's ratio, TL (-)
0.3
Tests
Poisson's ratio, LT (-)
0.25
Tests
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Graphical abstract
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Highlights
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Thermomechanical behavior of polymer matrix composites can be treated as a coupled
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temperature-diffusion-deformation problem.
A 3-D model was developed and further implemented using Comsol Multiphysics.
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Digital image correlation technique can accurately measure full-field thermally induced surface displacements and strains.
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Through-thickness variation in thermomechanical response is evident.
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S0264-1275(16)30383-5 doi: 10.1016/j.matdes.2016.03.098 JMADE 1576
To appear in: Received date: Revised date: Accepted date:
20 December 2015 15 March 2016 18 March 2016
Please cite this article as: Shengbo Shi, Linjie Li, Guodong Fang, Jun Liang, Fajun Yi, Guochang Lin, Three-dimensional modeling and experimental validation of thermomechanical response of FRP composites exposed to one-sided heat flux, (2016), doi: 10.1016/j.matdes.2016.03.098
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Three-dimensional modeling and experimental validation of thermomechanical response of FRP composites exposed to one-sided heat flux
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Shengbo Shi1,3, Linjie Li2,3, Guodong Fang3, Jun Liang3, Fajun Yi3, Guochang Lin3
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1. National Key Laboratory of Aerospace Flight Dynamics, Northwestern Polytechnical University,
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Xi’an 710072, P.R. China
2. China Helicopter Research and Development Institute, Jingdezhen 333000, P.R. China 3. Science and Technology on Advanced Composites in Special Environments Key Laboratory, Harbin
*Corresponding
author:
Shengbo
Shi.
Tel./fax:
+86
29
88492783.
address:
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[email protected] (S. Shi).
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Institute of Technology, Harbin 150001, P.R. China
Abstract: The heat transfer, gas diffusion process and thermomechanical deformation are generally
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coupled and associated with the chemical decomposition for fiber reinforced polymer composites at
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elevated temperatures. The three-dimensional (3-D) governing differential equations for the coupled problem
of
porous
elastomers
were
developed.
The
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temperature-diffusion-deformation
thermomechanical behavior of a silica/phenolic composite material was predicted using the
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mathematical model. The spatially dependent temperature and pore pressure, displacement, and stress contours of silica/phenolic composites exposed to one-sided radiant heat flux were investigated. Based on the digital image correlation technique, a non-contact high temperature deformation measurement test was conducted. The temperature profiles were measured by the thermocouples embedded in different depths of the specimen, while the full-field displacements and strains were provided by correlating the two digital images of the specimen surface in the un-deformed and deformed states, respectively. The accuracy of the proposed model was assessed by comparing the predicted temperatures and displacements with experimental values for the same boundary and initial conditions.
Keywords: Polymer-matrix composites; Thermomechanical behavior; High temperature deformation; 1
ACCEPTED MANUSCRIPT Gas diffusion; Digital image correlation. 1 Introduction
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Fiber reinforced polymer (FRP) composites present attractive characteristics for space vehicles and
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nozzles of solid rocket motor where light-weight materials with superior thermal protection performances and high mechanical properties are needed [1-3]. The continually expanding use of FRP composites in large structural applications requires a better understanding of the interdependent thermal
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and mechanical responses of the material when it is subjected to elevated and high temperatures [4]. For
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FRP composites, irreversible changes in thermochemical and thermomechanical properties will occur under high temperatures, which adversely affect the structural integrity and reliability of these
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composites [5]. In order to design structures with FRP components, it is necessary to accurately model
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the variation of the thermomechanical responses over a broad temperature range.
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Significant research has been conducted to understand the thermomechanical behavior of decomposing materials, and mathematical models and numerical methods have been developed [6-15].
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McManus and Springer [8,9] presented a finite difference method to describe the thermal and mechanical responses of one-dimensional (1-D) thin plate when rapidly heated to high temperatures. Their work included modeling of material ablation and mechanical damage to simulate the delamination of fiber plies. A similar model was developed by Wu and Katsube [10-12], in which the coupling between the thermochemical decomposition and thermomechanical deformation was made clear throughout the formulation. Combining the poroelasticity theory and thermochemical decomposition analysis, Sullivan and Salamon [13-15] proposed a finite element method to analyze the material response of a glass-phenolic composite. For two-dimensional (2-D) plane problems, free thermal expansion tests and restrained thermal growth tests were numerically simulated. Bai and Keller et al. [16,17], Matsuura et al.
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ACCEPTED MANUSCRIPT [18], McGurn and DesJardin et al. [19], Li et al. [20], Luo et al. [21], Martias et al. [22] further extended the prediction method of thermomechanical responses of FRP composites and their predicted results
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were verified against the experimental data. Recently, Gibson, Mouritz, Feih, et al. [23-28] discussed the
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high temperature and fire behavior of polymer matrix composites under load in fire. In their work, the temperature profiles in polymer laminates exposed to fire were calculated using thermal models. Also, mechanical models were developed to analyse the fire structural response and failure of laminates and
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sandwich composites under compression or tension loading.
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Moreover, high temperature deformation measurement of the material has been performed using the optical technique of digital image correlation (DIC) [29-33]. Lyons et al. [29], Pan et al. [30,31]
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established a simple, easy-to-implement yet effective high-temperature DIC method to measure the
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full-field in-plane surface displacements and strains of the material at elevated temperatures. The ability
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of this DIC method was evaluated by a series of experiments. Cholewa and Summers et al. [32] developed an integrated infrared thermography and digital image correlation (TDIC) technique to
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simultaneously measure spatial and temporal distributions of temperatures and displacements. The TDIC system was demonstrated using a series of one-sided heat exposure experiments performed on compressively loaded sandwich composites. Thermally induced chemical reactions occur when FRP composites are exposed to high temperatures. The resin system degrades to form gaseous products and carbonaceous char. These decomposition gases are trapped because of the low porosity and permeability of the material and this may cause high internal pressures [34]. As the pyrolysis reactions proceed, the gases begin to flow through the material attenuating the conduction of heat to the reaction zone [35]. Additionally, high thermal stress exists in the charred material due to the difference of thermochemical expansion of the
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associated with the thermochemical decomposition of polymer matrix. The effects of these processes
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must be considered in the formulation of the mathematical model to predict the thermomechanical behavior of FRP composite materials at high temperatures. Moreover, for complex structural component application with FRP composites, a three-dimensional (3-D) model is required.
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In the above-cited work, the investigations focused on the 1-D and 2-D thermomechanical response.
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Also, in our previous work [36], prediction of the 2-D thermomechanical behavior of FRP composites was only given. However, 3-D thermomechanical model and high temperature deformation
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measurement test which used to validate the model have been rarely reported, especially for the complex
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structural component application. In contrast, in this paper, the mathematical formulation for 3-D
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thermomechanical model was established and further implemented using Comsol Multiphysics software, which is applicable for complex structural component. This method was validated using high
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temperature deformation measurement tests performed on silica/phenolic composite specimens. 2 Governing equations of the coupled temperature-diffusion-deformation problem 2.1 Problem statement Since the heat-mass transfer processes and thermomechanical deformation are coupled and associated with the chemical decomposition, the thermomechanical behavior of FRP composites under high temperatures can be treated as a coupled temperature-diffusion-deformation (TDD) problem of the porous elastomers. Fig. 1 presents the geometrical model, loading and boundary conditions of the solving problem. It assumes that one side of a cuboid composite specimen in the through-thickness direction is heated with a radiant heat flux. Another side of the specimen (x-z plane shown in Fig. 1) is sprayed with
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assumed adiabatic.
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The model also assumes that: (1) the composite material is transversely isotropic, and x-y plane is the isotropy plane; (2) the isotropy plane of the porous material is coincident with the isotropy plane of the solid material [13]; (3) the gas flow is governed by the Darcy’s law. With these assumptions, the
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governing differential equations for the coupled TDD problem which involve mass conservation
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equation of gas phase, motion equation of solid phase, and energy conservation equation can be established.
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2.2 Mathematical formulas for the 3-D thermomechanical model
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2.2.1 Mass conservation equation of gas phase
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Thermal decomposition of polymer matrix occurs when the material reaches moderately high temperatures (about 350 oC). This pyrolysis reaction can be modeled by a Arrhenius equation in which
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the degree of conversion for the pyrolysis reaction, denoted by c , is defined as the mass loss that is to occur before the completion of the pyrolysis reaction over the total mass loss for this reaction. The value of c varies from 0 at the beginning of the pyrolysis reaction to 1 at the completion of the pyrolysis reaction. The Arrhenius kinetic reaction equation for the pyrolysis reaction can be given by
dc E A0 c n exp a dt RT
(1)
where A0 , Ea and n are the reaction rate constant, activation energy, and order of reaction for the pyrolysis reaction, respectively. And they can be determined by thermogravimetic analysis (TGA). R is the universal gas constant. The time rate of gas generation for the thermal decomposition is calculated by
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re sin virg
dc dt
(2)
where re sin is the volume fraction of the resin matrix. Using Darcy’s law, the volume average gas
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velocity vector, vg , can be expressed in terms of internal gas pressure, p , permeability of solid
vg
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p p p i T j L k T (T ) x T (T ) y L (T ) z
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materials, , and gas viscosity, , as follows,
(3)
where the subscripts T and L indicate the material properties in the isotropy plane (x-y plane) and in
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the perpendicular direction (z direction), respectively.
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According to the principles of mass conservation for gas phase, the differential equation governing the diffusion of pyrolysis gases through the porous solid material can be derived. The time rate of gas
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storage in the solid materials is equal to the sum of the time rate of gas generation and the rate of change
t
mpg t
g vg
(4)
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mg
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of the gas mass flux through the volume element.
where mpg is the mass of the generated gas during thermal decomposition. The variation of gas mass per
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unit bulk volume can be defined in space and time as a function of three primary dependent variables, involving the porous solid elastic strain e , the pore pressure p , and the temperature T which describe the decomposition process [13,14]. The left side of equation (4) can be expanded as follows, mg t
mg e mg p mg T e t p t T t
(5)
with, mg e eyy e ezz g11 xx g 33 e t t t t
(6)
mg p g p p t M t
(7)
mg T T g g g T t t
(8)
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coefficient of thermal expansion of the gas, respectively; M is the constitutive coefficient of the
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transversely isotropic material and its general expression was given by Biot [37]. 2.2.2 Equation of motion of solid phase
Based on the classical form of the equilibrium equation, the equations of motion of the porous
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elastomers in the absence of body forces and under quasi-static conditions is expressed as follows, (9)
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ij , j 0
where ij is the total stress tensor. Similar to Sullivan and Wu’s work [10,13,15], the effective stress
(10)
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ij Cijkl ekl ij p
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be written as follows,
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law for overall strain developed by Carroll [38] is also used in this paper. The total stress tensor, ij , can
It is noted that the poroelastic parameter, ij , in Eq. (10) is used to evaluate the effect of the pyrolysis
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gas pressures in the pores on the total stress of the porous material. The value of ij varies from 0 to 1. For silica fiber reinforced polymer composites, the values of 11 and 33 are taken as 0.9 in the present study. The elastic strains are related to the total and thermal strains and its expressions for the present solution case are,
exx xx T T (T ) eyy yy T T (T ) ezz zz T L (T ) exz xz eyz yz exy xy
(11)
where T and L are the coefficients of thermal expansion of the solid material in the plane of isotropy and in the perpendicular direction, respectively, which are functions of temperature. Substituting Eqs. (11) and (10) into Eq. (9) and employing the strain-displacement relations, the equations of motion for the solid phase can be rewritten in x, y and z direction, respectively. Five
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1 C11 C12 ux, yy u y , xy C44 ux, zz uz , xz 2 C11 T (T ) C12 T (T ) C13 L (T ) T, x 11 p, x 0
(12)
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C11ux , xx C12 u y , yx C13u z , zx
T
also considered. In x direction, Eq. (9) reduces to,
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Similarly, the equations of motion in y and z directions respectively become, 1 C11 C12 ux, yx u y , xx C44 u y , zz uz , yz 2 C11 T (T ) C12 T (T ) C13 L (T ) T, y 11 p, y 0
C11u y , yy C12u x , xy C13u z , zy
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C13ux , xz C13u y , yz C33uz , zz C44 u x, zx u z , xx C44 u y , zy u z , yy
2C13 T (T ) C33 L (T )T, z 33 p, z 0
(13)
(14)
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In Eqs. (12)-(14), a contracted notation is used to denote the elastic constants Cijkl . Furthermore, it is
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revealed that the thermal deformations u x , u y and u z , the temperature T , the pore pressure p are
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coupled.
2.2.3 Energy conservation equation
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A control volume is chosen to analyze the heat and mass transfer processes of FRP composites at
by,
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high temperatures. Based on Laplace’s equation, the energy conservation in the control volume is given
T T T T T (T ) i T (T ) j L (T ) k t x y z dc g Cpg (T )vg T hp s dt
s Cps (T )
(15)
where the term on the left side gives the rate of change of internal energy in the control volume, the three terms on the right side are the diffusion term, the convection term and the chemical decomposition term, respectively. In Eq. (15), Cps and Cpg are the heat capacity for the solid material and for the gas, respectively, which are functions of temperature; T and L are the thermal conductivity coefficient for the material in the plane of isotropy and in the perpendicular direction, respectively; hp is the heat of reaction for the decomposition reaction, which has a negative value.
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z
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T z, t
qw z 0
ux x, y, z z H 0 u y x, y, z
zH
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L (T )
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following heat flux and mechanical boundary conditions are applied in the present model.
0 uz x, y, z z H 0
(16)
(17)
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where qw is the radiant heat flux perpendicular to the heated surface, H is the thickness of the geometrical model.
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3 Non-contact high temperature deformation measurement test
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3.1 Experimental procedure
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Digital image correlation (DIC) has proven to be a powerful non-contact optical technique for accurate measurement of surface deformations, either at a point or over a field containing strain gradients
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[30]. In the DIC method, a video camera and personal computer acquire digitized images of a random speckle pattern on the surface of a specimen before and after deformation. Displacements are calculated
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directly by correlating the two digital images [29]. In order to accurately measure both the full-field displacements and temperatures of structural materials exposed to radiant heat flux, a non-contact high-temperature thermal deformation measuring system which involves DIC testing system, thermocouples temperature measurement system, radiant heating simulation device, and heat flux measurement system was carried out in the present study, as shown in Fig. 2(a). The DIC testing system (VIC-3D, American Correlated Solutions Inc., US) consists of two CCD camera (1624×1224 pixel), white light source and computer system. Before the test, a random speckle pattern on the specimen surface was artificially made (Fig. 2(b)), and a calibration process of the CCD camera was also conducted to obtain the internal parameters. The thermocouples
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measured by the heat flux sensor firstly, as shown in Fig. 2(c). Then, the specimen surface was heated at
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a heat flux of 155 kW/m2 (Fig. 2(d)). During the test, the images carried the deformation information of the specimen were acquired at each 100 ms. To correlate the deformed image to the undeformed reference image, each image was divided into small subsets.
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3.2 Materials
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A silica/phenolic composite, kindly provided by Aerospace Research Institute of Materials and Processing Technology (Beijing, China), was selected and characterized in this study. This composite
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was 24 wt% silica fiber (Shaanxi HuaTek Fiberglass Material Group, China), 20.4 wt% phenolic fiber
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(Tongshan chemical plant, China), and 55.6 wt% phenolic resin (Tongshan chemical plant, China).
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Properties of the phenolic fiber were similar to the phenolic resin. The silica/phenolic composite specimen for deformation measurement was 14×14×33.7 mm. The basic material properties,
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temperature-dependent thermal and transport properties of silica/phenolic composites have been summarized in Table 1. The kinetic parameters of decomposition for this composite were determined by TGA. The thermal and transport properties of the virgin and char components were collected from a number of sources. Based on these data, material properties of the charred composite can be calculated as a weighted function of degree of conversion, c . The temperature-dependent stiffness properties were measured by high temperature compression tests reported in our earlier work [39]. 4 Results and discussion Based on Comsol Multiphysics software, the mathematical model developed in Section 2 was used to analyze the thermomechanical responses of silica/phenolic composites subjected to one-sided heat
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composites along the heating direction. As expected, the temperature decreases with increasing distance
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from the heated surface. There is a large temperature gradient in the internal material during heating, which can induce the high thermal stress. The experimental temperatures for initial depths of 0.96, 3.98, 7.73, and 12.26 mm are also presented in Fig. 3 as data points. It can be found that the calculated
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temperature curves at heating time of 100 s and 200 s are in good agreement with the experimental values,
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respectively.
Comparisons between the calculated and experimental displacement u z versus the distance from
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the heated surface for silica/phenolic composites at heating time of 100 s and 200 s are shown in Fig. 4.
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The curves show that the displacement u z increases with decreasing distance from the heated surface. It
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can be seen from Fig. 4 (a) and (b) that the displacement u z also increases with increasing heating time. The absolute maximum displacement is about 0.17 mm at 100 s, and this value increases with increasing
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heating time to about 0.23 mm at 200 s. As can be seen from Fig. 4 (b), there is good agreement between the calculated and experimental values at 200 s. The calculated displacements roughly coincide with the experimental results at 100 s except the displacement at the spatial location less than 8 mm away from the heated surface. The maximum displacement difference for all the data points shown in Fig. 4(a) is 0.05 mm. The discrepancies are thought to be due to the fact that the sufficiently high temperatures of these locations may significantly affect the stability or accuracy of the experimental results. It is important to point out that, because of the failure of the random speckle pattern on specimen surface for depths of 0-4.54 mm at 200 s due to high temperatures, the measured displacement data of these locations are unavailable. Fig. 5 presents the displacement contours of silica/phenolic composite specimen during
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Fig. 6 shows the simulated full-field and section total strain ( zz ) contours of silica/phenolic
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composite specimen during heating. The figure demonstrates non-monotonicity of total strain zz of the material along heating direction. The total strain has a local peak during heating. This peak is not at the heated surface, but at the interface of charring layer where the pyrolysis reaction reaches completion and
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pyrolysis zone where the intensive thermal decomposition occurs. Furthermore, the peak strain moves
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towards the back surface of the material with increasing heating time. The maximum strain is 0.017 at 100 s.
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Fig. 7 shows the pore pressure-depth curves of silica/phenolic composite specimen along the
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heating direction at heating times from 50 to 200 s. As can be seen, the pore pressure has a local peak
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during heating and exhibits a parabola tendency in the through-thickness direction of the material. This peak is shifted to the back surface of the material with increasing heating time. Comparison of Figs. 3 and
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Fig. 8 illustrates the full-field total stress ( zz ) contour of silica/phenolic composite specimen at heating times from 50 to 200 s. Similar to the total strain zz , the total stress zz exhibits non-monotonicity and the peak stress does not appear at the heated surface. With increasing heating time, this peak is shifted to the back surface of the material. Fig. 9 shows the variation curves of the total stress
zz versus distance from the heated surface at different heating time. It is revealed that the peak total stress decreases with increasing heating time. This is due to the fact that the degradation in stiffness
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respectively, and this may be attributed to the effect of the pore pressure on the material response.
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Fig. 10 presents the photograph of silica/phenolic composite specimen after heating. From the figure, it is possible to roughly distinguish different regions in a post-test specimen, involving charring layer where the pyrolysis reaction reached completion, pyrolysis zone where most of the material is
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undergoing decomposition, and virgin material. Fig. 11 shows the morphology of virgin material and
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heated surface for silica/phenolic composites. It is obvious that the microstructure of the charred composite material at the heated surface does not contain the polymer resin phase due to the completion
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of the pyrolysis reaction. Thus, the silica fibers and carbonaceous char left on the heated surface are not
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bonded together and easy to be stripped off by the high speed stream. Moreover, a number of pores and
5 Conclusions
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cracks are observed in the porous skeleton.
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The 3-D thermomechanical model was developed and further implemented using Comsol Multiphysics, which was applicable for complex structural component. The model considers the combined effects of thermochemical decomposition, gas diffusion and thermomechanical deformation on the thermal performance of the material. The model can predict the temperature, pore pressure and deformation behavior of FRP composites exposed to one-sided radiant heat flux. The DIC system can measure the temperature profiles, thermally induced surface displacements and strains. There is good agreement between the predicted and experimental values for spatially dependent temperature of silica/phenolic composites. The experimental displacements roughly coincide with the calculated results except the displacement at the spatial location near the heated surface. The total strain has a local peak
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decreases with increasing heating time due to degradation in stiffness properties of FRP composites at
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elevated temperatures. Acknowledgement
The present work is supported by the National Natural Science Foundation of China under Grant No.
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11402202, the Fundamental Research Funds for the Central Universities of China under Grant No.
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3102014JCQ01001, and the Supporting Technology Funds for Aerospace under Grant No. 2014-HT-XGD.
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References
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[1] P. Luangtriratana, B.K. Kandola, P. Myler, Ceramic particulate thermal barrier surface coatings for glass fibre-reinforced epoxy composites, Mater. Des. 68, 2015, 232-244.
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[2] B.K. Kandola, M.H. Akonda, A.R. Horrocks, Fibre-reinforced glass/silicate composites: effect of fibrous reinforcement on intumescence behaviour of silicate matrices as a fire barrier application, Mater. Des. 86, 2015, 80-88.
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[3] D. Teixeira, M. Giovanela, L.B. Gonella, J.S. Crespo, Influence of injection molding on the flexural strength and surface quality of long glass fiber-reinforced polyamide 6.6 composites, Mater. Des. 85, 2015, 695-706.
[4] Y. Bai, T. Keller, T. Vallée. Modeling of stiffness of FRP composites under elevated and high temperatures. Compos. Sci. Technol. 68, 2008, 3099-3106. [5] K. Wei, R. He, X, Cheng, et al., A lightweight, high compression strength ultra high temperature ceramic corrugated panel with potential for thermal protection system applications, Mater. Des. 66, 2015, 552-556. [6] E. Kandare, B.K. Kandola, P. Myler, G. Edwards, Thermo-mechanical responses of fiber-reinforced epoxy composites exposed to high temperature environments, Part I: Experimental data acquisition, J. Compos. Mater. 44, 2010, 3093-3114.
14
ACCEPTED MANUSCRIPT [7] M. Derradji, N. Ramdani, T. Zhang, et al., Mechanical and thermal properties of phthalonitrile resin reinforced with silicon carbide particles, Mater. Des. 71, 2015, 48-55. [8] L.N. McManus, G.S. Springer, High temperature thermomechanical behavior of carbon-phenolic
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and carbon-carbon composites, I. Analysis, J. Compos. Mater. 26, 1992, 206-229. [9] L.N. McManus, G.S. Springer, High temperature thermomechanical behavior of carbon-phenolic
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and carbon-carbon composites, II. Results, J. Compos. Mater. 26, 1992, 230-251. [10] Y. Wu, N. Katsube, A constitutive model for thermomechanical response of decomposing composites under high heating rates, Mech. Mater. 22, 1996, 189-201.
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[11] Y. Wu, N. Katsube, A thermomechanical model for chemically decomposing composites, I. Theory, Int. J. Eng. Sci. 35, 1997, 113-128.
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[12] Y. Wu, N. Katsube, A thermomechanical model for chemically decomposing composites, II. Application, Int. J. Eng. Sci. 35, 1997, 129-139.
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[13] R.M. Sullivan, N.J. Salamon, A finite element method for the thermochemical decomposition of
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polymeric materials, I. Theory, Int. J. Eng. Sci. 30, 1992, 431-441. [14] R.M. Sullivan, N.J. Salamon, A finite element method for the thermochemical decomposition of
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polymeric materials, II. Carbon phenolic composites, Int. J. Eng. Sci. 30, 1992, 939-951. [15] R.M. Sullivan, A coupled solution method for predicting the thermostructural response of decomposing, expanding polymeric composites, J. Compos. Mater. 27, 1993, 408-434.
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[16] Y. Bai, T. Valle´e, T. Keller, Modeling of thermal responses for FRP composites under elevated and high temperatures, Compos. Sci. Technol. 68, 2008, 47-56. [17] Y. Bai, T. Keller, Modeling of mechanical response of FRP composites in fire, Compos. Part A: Appl. S. 40, 2009, 731-738. [18] Y. Matsuura, K. Hirai, T. Kamita, et al., A challenge of modeling thermomechanical response of silica-phenolic composites under high heating rates, In: Proceedings of 49th AIAA Aerospace Sciences Meeting, Orland, Florida, 2011. [19] M.T. McGurn, P.E. DesJardin, A.B. Dodd, Numerical simulation of expansion and charring of carbon-epoxy laminates in fire environments, Int. J. Heat Mass Tran. 55, 2012, 272-281. [20] H. Li, E. Kandare, S. Li, et al., Integrated thermal, micro- and macro-mechanical modelling of post-fire flexural behaviour of flame-retarded glass/epoxy composites, Comp. Mater. Sci. 59, 2012, 22-32.
15
ACCEPTED MANUSCRIPT [21] C. Luo, J. Lua, P.E. DesJardin, Thermo-mechanical damage modeling of polymer matrix sandwich composites in fire, Compos. Part A: Appl. S. 43, 2012, 814-821. [22] C. Martias, Y. Joliff, C. Favotto, Effects of the addition of glass fibers, mica and vermiculite on the
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mechanical properties of a gypsum-based composite at room temperature and during a fire test, Compos. Part B: Eng. 62, 2014, 37-53.
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[23] A.G. Gibson, T.N.A. Browne, S. Feih, A.P. Mouritz, Modeling composite high temperature behavior and fire response under load, J. Compos. Mater. 46, 2012, 2005-2022. [24] A.G. Gibson, M.E. Otheguy Torresa, T.N.A. Brownea, et al., High temperature and fire behaviour
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of continuous glass fibre/polypropylene laminates, Compos. Part A: Appl. S. 41, 2010, 1219-1231. [25] A.P. Mouritz, S. Feih, E. Kandare, et al., Review of fire structural modelling of polymer composites,
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Compos. Part A: Appl. S. 40, 2009, 1800-1814.
[26] A.P. Mouritz, S. Feih, E. Kandare, A.G. Gibson, Thermal-mechanical modelling of laminates with
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fire protection coating, Compos. Part B: Eng. 48, 2013, 68-78.
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[27] S. Feih, Z. Mathys, A.G. Gibson, A.P. Mouritz, Modelling the tension and compression strengths of polymer laminates in fire, Compos. Sci. Technol. 67, 2007, 551-564.
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[28] S. Feih, A.P. Mouritz, S.W. Case, Determining the mechanism controlling glass fibre strength loss during thermal recycling of waste composites, Compos. Part A: Appl. S. 76, 2015, 255-261. [29] J.S. Lyons, J. Liu, M.A. Sutton, High-temperature deformation measurements using digital-image
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correlation, Exp. Mech. 36, 1996, 64-70. [30] B. Pan, K. Qian, H. Xie, A. Asundi, Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review, Meas. Sci. Technol. 20, 2009, 062001, 1-17. [31] B. Pan, D. Wu, J. Gao, High-temperature strain measurement using active imaging digital image correlation and infrared radiation heating, J. Strain Anal. Eng. Des. 49, 2013, 224-232. [32] N. Cholewa, P.T. Summers, S. Feih, A.P. Mouritz, B.Y. Lattimer, S.W. Case, A technique for coupled thermomechanical response measurement using infrared thermography and digital image correlation (TDIC), Exp. Mech. 56, 2016, 145-164. [33] X. Chen, L. Yang, N. Xu, et al., Cluster approach based multi-camera digital image correlation: Methodology and its application in large area high temperature measurement, Opt. Laser Technol. 57, 2014, 318-326. [34] J.B. Henderson, T.E. Wiecek, A mathematical model to predict the thermal response of
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ACCEPTED MANUSCRIPT decomposing, expanding polymer composites, J. Compos. Mater. 21, 1987, 373-393. [35] S. Shi, J. Liang, F. Yi, G. Fang, Modeling of one-dimensional thermal response of silica-phenolic composites with volume ablation, J. Compos. Mater. 47, 2013, 373-393.
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[36] S. Shi, J. Liang, G. Lin, G. Fang, High temperature thermomechanical behavior of silica-phenolic composite exposed to heat flux environments. Compos. Sci. Technol. 87, 2013, 204-209.
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[37] M.A. Biot, Mechanics of deformation and acoustic propagation in porous media, J. Appl. Phys. 33, 1962, 1482-1498.
[38] M.M. Carroll, An effective stress law for anisotropic elastic deformations, J. Geophys. Res. 84,
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1979, 7510-7512.
[39] S. Shi, L. Gu, J. Liang, G. Fang, et al., A mesomechanical model for predicting the degradation in
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stiffness of FRP composites subjected to combined thermal and mechanical loading, Mater. Des. 89,
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2016, 1079-1085.
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Figure and Table Captions
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Fig. 1. Illustration of the geometrical model, loading and boundary conditions for the coupled TDD
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problem.
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Fig. 2. High-temperature deformation measurement platform. (a) Experiment facility; (b) Random speckle pattern on the specimen surface; (c) Heat flux measurement system; (d) Fixture and insulation of the specimen.
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Fig. 3. Comparison of calculated and experimental spatially dependent temperature profiles for silica/phenolic composites.
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Fig. 4. Comparison of calculated and experimental spatially dependent displacement ( u z ) profiles of silica/phenolic composites at heating time of (a) 100 s and (b) 200 s.
50 s; (b) 100 s; (c) 150 s; (d) 200 s.
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Fig. 5. The experimental displacement contours of silica/phenolic composite specimen during heating. (a)
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Fig. 6. The simulated total strain ( zz ) contours of silica/phenolic composites during heating. (a)
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Full-field strain contour at 100 s; (b) Full-field strain contour at 200 s; (c) Section strain contour at 100 s; (d) Section strain contour at 200 s. Fig. 7. Pore pressure-depth profiles of silica/phenolic composites along the heating direction at different
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heating time.
Fig. 8. Full-field total stress ( zz ) contour of silica/phenolic composites at different heating time. (a) 50 s; (b) 100 s; (c) 150 s; (d) 200 s. (Unit: Pa) Fig. 9. The spatially dependent total stress ( zz ) curves of silica/phenolic composites along heating direction at different heating time. Fig. 10. The schematic of physical delamination for silica/phenolic composite specimen after heating. Fig. 11. Morphology of virgin material and heated surface for silica/phenolic composites. (a) Virgin material; (b) Heated surface.
Table 1. Material properties of silica/phenolic composites.
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Fig. 1. Illustration of the geometrical model, loading and boundary conditions for the coupled TDD
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Fig. 2. High-temperature deformation measurement platform. (a) Experiment facility; (b) Random speckle pattern on the specimen surface; (c) Heat flux measurement system; (d) Fixture and insulation of the specimen.
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Fig. 3. Comparison of calculated and experimental spatially dependent temperature profiles for
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(a)
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Fig. 4. Comparison of calculated and experimental spatially dependent displacement ( u z ) profiles of silica/phenolic composites at heating time of (a) 100 s and (b) 200 s.
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Fig. 5. The experimental displacement contours of silica/phenolic composite specimen during heating. (a)
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50 s; (b) 100 s; (c) 150 s; (d) 200 s.
Fig. 6. The simulated total strain ( zz ) contours of silica/phenolic composites during heating. (a) Full-field strain contour at 100 s; (b) Section strain contour at 100 s; (c) Full-field strain contour at 200 s; (d) Section strain contour at 200 s.
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Fig. 7. Pore pressure-depth profiles of silica/phenolic composites along the heating direction at different
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Fig. 8. Full-field total stress ( zz ) contour of silica/phenolic composites at different heating time. (a) 50 s; (b) 100 s; (c) 150 s; (d) 200 s. (Unit: Pa)
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Fig. 9. The spatially dependent total stress ( zz ) curves of silica/phenolic composites at different heating
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Fig. 10. The schematic of physical delamination for silica/phenolic composite specimen after heating.
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Fig. 11. Morphology of virgin material and heated surface for silica/phenolic composites. (a) Virgin
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Table 1. Material properties of silica/phenolic composites.
Value
Density of solid material, s (kg/m ) 3
Source
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Properties
Activation energy, Ea ( J/mol)
7.88E4 3
Reaction rate constant, A0 (kg/m s)
Manufacturer TGA TGA
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1.94E5
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1270
1.0
TGA
-4.187E5
TGA
6.18×10-18
[14,15]
4.85×10-15
[14,15]
1.48×10-5+2.50×10-8T
[34]
4.51×10-5
[34]
Coefficient of thermal expansion of char material, char (1/K)
-1.28×10-4
[34]
Specific heat of solid material, Cps (J/kgK)
1090+1.09T
[34]
2390+1.05T
[34]
0.8+2.76×10-4T
[34]
0.96+8.42×10-4T-4.07×10-6T2
[34]
Order of reaction, n (-)
Permeability of virgin material, virg (m2) Permeability of char material, char (m2)
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Gas viscosity, (kg/ms)
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Heat of decomposition, Q (J/kg)
Coefficient of thermal expansion of virgin
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Specific heat of decomposition gas, Cpg (J/kgK)
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Thermal conductivity of virgin material,
virg (W/mK)
Thermal conductivity of char material,
char (W/mK)
Elastic modulus, EL (GPa)
9.36
20 oC
2.92
450 oC
6.02
100 oC
2.76
500 oC
3.32
135 oC
1.04
550 oC
2.75
200 oC
1.06
600 oC
2.77
350 oC
1.03
650 oC
2.85
400 oC
0.94
800 oC
Tests
Poisson's ratio, T (-)
0.2
Tests
Poisson's ratio, TL (-)
0.3
Tests
Poisson's ratio, LT (-)
0.25
Tests
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Graphical abstract
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Highlights
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Thermomechanical behavior of polymer matrix composites can be treated as a coupled
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temperature-diffusion-deformation problem.
A 3-D model was developed and further implemented using Comsol Multiphysics.
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Digital image correlation technique can accurately measure full-field thermally induced surface displacements and strains.
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Through-thickness variation in thermomechanical response is evident.
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