Additional data for “Understanding and predicting defect formation in ...

Additional data for “Understanding and predicting defect formation in automated fibre placement pre-preg laminates”, published in Composites Part A: Applied Science and Manufacturing Jonathan P.-H. Belnoue, Tassos Mesogitis, Oliver J. Nixon-Pearson, James Kratz, Dmitry S. Ivanov, Ivana K. Partridge, Kevin D. Potter and Stephen R. Hallett

A.1 Cure kinetics model The cure reaction rate, 𝑥̇ of 8552 epoxy resin can be computed as follows [1]: 1 1 −1 𝑥̇ = ( + ) 𝑥̇ 𝑘 𝑥̇ 𝑑

(1)

Here 𝑥̇ 𝑘 is the chemically controlled part of the reaction, defined as follows [1]: 𝑥̇ 𝑘 = (

−1 1 1 + ) + 𝑥̇ 𝑒 𝑥̇ 𝑐1 𝑥̇ 𝑖2 + 𝑥̇ 𝑐2

(2)

where for each reaction 𝑥̇ 𝑐1 , 𝑥̇ 𝑖2 , 𝑥̇ 𝑐2 and 𝑥̇ 𝑒 𝑚𝑖 𝐸𝑎𝑖 1 𝑥̇ 𝑖 = 𝐾0𝑖 𝑒 − 𝑅𝑇 (1 − 𝑥)𝑙𝑖 ( − 𝑥) (𝑥 𝑛2𝑖 + 𝑏𝑖 )𝑛𝑖 𝑟

(3)

Here 𝑥 is the instantaneous degree of cure. 𝑥̇ 𝑑 expresses the diffusion controlled part of the reaction and is defined as follows [1]: 𝑥̇ 𝑑 = 𝑘𝑑0 𝑒

−

𝐵 𝑓

(4)

where 𝑓 = 𝑎𝑓 (𝑇 − 𝑇𝑔 ) + 𝑏𝑓

(5)

Here 𝑎𝑓 and 𝑏𝑓 are calculated as follows: 𝑎𝑓 = 𝑎1 𝑎𝑓 = 𝑆𝑎 𝑇𝑔 + 𝐶𝑎 𝑎𝑓 = 𝑎2

if if if

𝑇𝑔 < 𝑇𝑔𝑎1 𝑇𝑔𝑎1 < 𝑇𝑔 < 𝑇𝑔𝑎2 𝑇𝑔 > 𝑇𝑔𝑎2

(6)

where 𝑎2 − 𝑎1 𝑇𝑔𝑎2 − 𝑇𝑔𝑎1

(7)

𝐶𝑎 = 𝑎2 − 𝑆𝑎 𝑇𝑔𝑎2

(8)

𝑆𝑎 =

and

Similarly, 𝑏𝑓 = 𝑏1 𝑏𝑓 = 𝑆𝑏 𝑇𝑔 + 𝐶𝑏 𝑏𝑓 = 𝑏2

if if if

𝑇𝑔 < 𝑇𝑔𝑏1 𝑇𝑔𝑏1 < 𝑇𝑔 < 𝑇𝑔𝑏2 𝑇𝑔 > 𝑇𝑔𝑏2

(9)

where 𝑏2 − 𝑏1 𝑇𝑔𝑏2 − 𝑇𝑔𝑏1

(10)

𝐶𝑏 = 𝑏2 − 𝑆𝑏 𝑇𝑔𝑏2

(11)

𝑆𝑏 =

and

Here 𝑇𝑔 represents the glass transition temperature and is described by the DeBenedetto equation as follows: 𝑇𝑔 = 𝑇𝑔0 +

𝜆𝑥(𝑇𝑔∞ − 𝑇𝑔0 ) 1 − (𝜆 − 𝑥)𝑥

(12)

The total heat of reaction, 𝐻𝑇 of the resin system of this study is 600 J/g [1]. In addition an initial degree of cure, 𝑥0 of 0.05 was assumed [1]. Table A.1 summarises all the parameters used in the Eqns. (1)-(12).

Table A.1. Cure kinetics parameters.

Parameter 𝑘0 𝐸𝑎 𝑙 𝑟 𝑚 𝑛2 𝑏 𝑛 𝑘𝑑0 𝐵 𝑎1 𝑎2 𝑇𝑔𝑎1 𝑇𝑔𝑎2 𝑏1 𝑏2 𝑇𝑔𝑏1 𝑇𝑔0 𝑇𝑔∞ 𝑇𝑔𝑏2 𝜆

Value 𝑥̇ 𝑐1 153900.5 64.929.5 2.347 1 0 1 0.1594 1.413

𝑥̇ 𝑖2 1000 0 0 1 0 0 1 0

Units 𝑥̇ 𝑐2 1000 0 0 1 0 0 1 0

𝑥̇ 𝑒 3.963E+11 133168.3 1.029 1 0 1 0 5.586

4 0.21 4.8E-04 4.8E-04 0 100 0.021 0.031 120 195 -7 250 0.78

1/s J/mol

1/s 1/⁰C 1/⁰C ⁰C ⁰C

⁰C ⁰C ⁰C ⁰C

A.2 Specific heat capacity The resin specific heat capacity is defined as follows [1]: 𝑐𝑝𝑟 = 𝑐𝑝𝑟𝑏 +

𝑐𝑝𝑔 − 𝑐𝑝𝑟𝑏 1−𝑒

(13)

𝑘[(1−𝑇𝑔 )−∆𝑇𝑐 ]

where 𝑐𝑝𝑖 = (1 − 𝑥)𝑐𝑝𝑖0 + 𝑥𝑐𝑝𝑖∞

for 𝑖 = 𝑟𝑏, 𝑔

(14)

and 𝑐𝑝𝑖𝑗 = 𝑠𝑖𝑗 𝑇 + 𝑐𝑖𝑗

for 𝑖 = 𝑟𝑏, 𝑔 and 𝑗 = 0, ∞

(15)

Here 𝑇 is the temperature. The fibre specific heat capacity, 𝑐𝑝𝑓 is calculated as follows [2]: 𝑐𝑝𝑓 = 750 + 2.05𝑇

(16)

The specific heat capacity of the composite is expressed using the rule of mixtures: 𝑐𝑝 = 𝑤𝑓 𝑐𝑝𝑓 + (1 − 𝑤𝑓 )𝑐𝑝𝑟

(17)

Here 𝑤𝑓 is the fibre weight fraction and is defined as:

𝑤𝑓 =

𝑣𝑓 𝜌𝑓

(18)

𝜌

where 𝑣𝑓 is the fibre volume fraction, 𝜌𝑓 and 𝜌 is the fibre density and composite density, respectively. A 𝑣𝑓 of 0.55 was assumed in this study. The composite density is computed using the rule of mixtures as follows:

𝜌 = 𝑣𝑓 𝜌𝑓 + (1 − 𝑣𝑓 )𝜌𝑟

(19)

where 𝜌𝑟 is the density of the resin. The parameters in Eqns. (13)-(19) are listed in Table A.2.

Table A.2. Specific heat capacity parameters.

Parameter 𝜌𝑟 𝜌𝑓 𝑠𝑔0 𝐶𝑔0 𝑠𝑔∞ 𝑐𝑔∞ 𝑠𝑟0 𝐶𝑟0 𝑠𝑟∞ 𝐶𝑟∞ 𝑘 ∆𝑇𝑐

value 1301 1790 3.775 730 3.4 830 3.27 1088 2 1350 0.278 -1.5

Units kg/m3 kg/m3 J/(kg ⁰C 2) J/(kg ⁰C) J/(kg ⁰C 2) J/(kg ⁰C) J/(kg ⁰C 2) J/(kg ⁰C) J/(kg ⁰C 2) J/(kg ⁰C) 1/⁰C ⁰C

A.3 Thermal conductivity The thermal conductivity of the resin is calculated as follows [1]: 𝑘𝑟 = 𝐴𝑘𝑟 + 𝐵𝑘𝑟 𝑇 + 𝐶𝑘𝑟 𝑥

(20)

The fibre thermal conductivity in the longitudinal direction is defined as [2]: 𝑘𝑙𝑓 = 𝐴𝑘𝑙𝑓 + 𝐵𝑘𝑙𝑓 𝑇

(21)

The thermal conductivity of the fibre in the transverse direction is [2]: 𝑘𝑡𝑓 = 𝐴𝑘𝑡𝑓 + 𝐵𝑘𝑡𝑓 𝑇

(22)

The thermal conductivity of the composite in the longitudinal direction is computed using the rule of mixtures as follows: 𝑘11 = 𝑣𝑓 𝑘𝑙𝑓 + (1 − 𝑣𝑓 )𝑘𝑟

(23)

In the transverse direction the thermal conductivity can be computed as follows [3]: 𝑘𝑡𝑓

𝑘22 = 𝑘33 = 𝑣𝑓 𝑘𝑟 (

𝑘𝑟

− 1) +

1 𝑘𝑟 ( 2

−

𝑘𝑡𝑓 2𝑘𝑟

𝑘𝑡𝑓

) + 𝑘𝑟 (

𝑘𝑟

𝑘𝑡𝑓

− 1) √𝑣𝑓2 − 𝑣𝑓 +

(

𝑘𝑟

2

2𝑘𝑡𝑓

(

𝑘𝑟

(24)

+1) 2

−2)

Table A.3 summarises all the parameters used in Eqns. (20)-(24).

Table A.3. Thermal conductivity parameters.

Parameter 𝐴𝑘𝑟 𝐵𝑘𝑟 𝐶𝑘𝑟 𝐴𝑘𝑙𝑓 𝐵𝑘𝑙𝑓 𝐴𝑘𝑡𝑓 𝐵𝑘𝑡𝑓

value 0.148 3.43E-04 6.07E-02 7.69 1.56E-02 2.4 5.07E-04

Units W/m/⁰C W/m/⁰C2 W/m/⁰C W/m/⁰C W/m/⁰C2 W/m/⁰C W/m/⁰C2

A.4 Consolidation model parameters

The material parameters used in the model for consolidation of the material system IM7-8552 are given in Table A.4. The procedure for parameter extraction from simple consolidation tests is described in [4]. The parameters a and b control the behaviour of the rate dependent term of the apparent viscosity (𝑏𝑠𝑞𝑢𝑒𝑒𝑧𝑒 and 𝑏𝑏𝑙𝑒𝑒𝑑 are the values of b before and after locking respectively). The parameter k controls the size of the inter-fibre channels at the micro-scale.

Table A.4. Consolidation model parameters.

Tempertaure (°C) 30 40 50 60 70 80 90

k 0.949073376 0.919238734 0.884086642 0.850373593 0.823853122 0.806058649 0.795357049

a -0.9302 -0.9124 -0.8822 -0.856 -0.8434 -0.8391 -0.8378

𝑏𝑠𝑞𝑢𝑒𝑒𝑧𝑒 -16.51 -15.01 -14.12 -13.58 -13.27 -13.08 -12.96

𝑏𝑏𝑙𝑒𝑒𝑑 -33.24 -31.74 -30.85 -30.31 -30 -29.81 -29.69

References [1] Johnston A. An integrated model of the development of process-induced deformation in autoclave processing of composite structures: University of British Columbia; 1997. [2] Van Ee D, Poursartip A. HexPly 8552 MATERIAL PROPERTIES DATABASE for use with COMPRO CCA and Raven. . 2009. [3] Farmer J, Covert E. Thermal conductivity of a thermosetting advanced composite during its cure. Journal of Thermophysics and Heat Transfer(USA) 1996;10(3):467-475. [4] Belnoue JPH, Nixon-Pearson OJ, Ivanov D, Hallett SR. A novel hyper-viscoelastic model for consolidation of toughened prepregs under processing conditions. Mechanics of Materials. 2016;97:118-34.