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Hybrid phase field simulation of dynamic crack propagation in functionally gradedTo appear in: Composites Part B Received Date: 14 March 2016 Revised Date: 25 May 2016 Accepted Date: 3 J

Hybrid phase field simulation of dynamic crack propagation in functionally graded

To appear in: Composites Part B

Received Date: 14 March 2016

Revised Date: 25 May 2016

Accepted Date: 3 June 2016

Please cite this article as: Doan DH, Bui TQ, Duc ND, Fushinobu K, Hybrid phase field simulation of

dynamic crack propagation in functionally graded glass-filled epoxy, Composites Part B (2016), doi:

10.1016/j.compositesb.2016.06.016.

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.

Tels.: +81-(0)3-5734-2945 (D H Doan); +81-7021506399 (T Q Bui)

Email: doan.d.aa.eng@gmail.com (D.H.Doan); buiquoctinh@duytan.edu.vn; tinh.buiquoc@gmail.com (T Q Bui)

Fig 1 Non-uniform microscopically inhomogeneous structure of the NiCoCrAlY-YSZ composite

five layered functionally graded material [4]

crack located on the stiffer side; and (b) FG beam with a crack located on the compliant side Our definition

of the stiffer side and compliant one is exactly the same as that in [3]

1400 1600 1800

(a)

0 10 20 30 40 4

6 8 10

y (mm)

Real data Fitting

(b)

1.4 1.6 1.8 2 2.2

Elastic modulus (GPa)

Fig 4 Schematic of two impact loading profiles: (a) A constant displacement velocity (or a Heaviside step

loading) and (b) a linear displacement velocity (or a Heaviside step loading with a finite rise time)

Fig 5 Schematic of final crack paths (black solid lines) of two FGM beams (a), (b) and one homogeneous

beam (c) made by the experiments [3] A vertical line (marked in blue color) is located 10 mm away from

the crack tip, which just helps to establish the scale

Partial length of beam [mm]

Exp Num

(b)

Fig 6 Comparison of the final crack paths (a) and (b) of an FGM beam with a crack located on the stiffer side

obtained by the experiments [3] and the hybrid phase field formulation, taking the constant displacement velocity

Fig 7 shows the same numerical comparison between two approaches but the linear

Fig 7 Comparison of the final crack paths of an FGM beam with a crack located on the stiffer side obtained

by the experiments [3] and the hybrid phase field formulation, taking the linear displacement velocity.

Fig 8 Deformation of FG beams with a crack placed on the stiffer side at two different time steps, taking the

constant displacement velocity (red color represents the phase field, blue color represent the crack path)

Crack propagation time ( µ s)

Experiment This work

(a)

x 10−40

50 100 150 200 250

Crack propagation time (s)

This work This work (Smoothed) Experiment

(b)

Fig 9 FG beam with a crack located on stiffer side: Crack length versus crack propagation time (a) and

crack velocity versus crack propagation time

8

Partial length of beam [mm]

v=3.5 m/s v=5 m/s

Fig 10 Effect of different impact velocities on the crack paths and initial kink angles of stiffer cracked

(a) Calculation result with the constant displacement velocity

(b) Calculation result with the linear displacement velocity

Fig 11 Comparison of final crack paths of an FG beam with a pre-crack located on the compliant side

between the phase field method and experimental data

(a) Constant velocity (b) linear velocity

Fig 12 Comparison of the final crack paths of a homogeneous beam between the numerical phase field

model and experimental data: (a) constant and (b) displacement velocities

50 100 150 200 250 300

Time (s)

Stiffer Compliant Homo

(a)

10 20 30 40 50 60 70

Time (s)

Stiffer Compliant Homo

Fracture energy Elastic energy

(b)

Fig 13 Comparison of evolution of various energies in simulation of dynamic crack propagation for both FG

beams and homogenous one: (a) Kinetic energy and (b) elastic bulk and fracture energies

0 5 10 15 20 25 30 35 40 45 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x (mm)

Real data Fitting