Damage evolution and failure mechanism of three-dimensional braided composite subjected to varied high strain-rate loadings

Abstract

Three-dimensional (3D) braided composite exhibit the excellent integrity of a braided structure, which gives the composites the advantages of high-impact damage tolerance, excellent delamination resistance, and high fatigue resistance. They have been widely used in the fields of aerospace and transport. Composites are always subjected to high strain-rate loading in these application cases. Braided composites show significantly different damage evolution under varied high strain-rate loadings. A bridge which can connect the strain rate variation to the different damage evolution of composites is urgently needed for composite strength design. This research demonstrated how the strain rate variation induced the change of damage evolution and failure mode of 3D braided composite. A high-speed camera system was used to capture damage process information. A micro-scale rate-dependent finite-element model (FEM) was established to study the underlying mechanisms. The results show that the velocity gap between yarns and resin becomes more obvious at higher strain-rate loading. Damage morphology of composites under lower strain rate resulted from the higher transverse stress at the output-bar area. Higher strain-rate loading induces plastic deformation, which occurs earlier in composites. The deformation distributes more haphazardly under higher strain-rate loading. Optimizing suggestions are given to improve the impact resistance of 3D braided composites.

Keywords

3D braided composite strain rate damage evolution finite element analysis

Three-dimensional braided composites are reinforced with three-dimensional (3D) braided preform. The interlaced yarns can effectively prevent cracks extending in composites,^1,2 which impart 3D braided composites with the advantages of excellent fatigue resistance,³ delamination resistance, and high impact tolerance. Thus, they have been widely used in the fields of aerospace^4,5 and transport.⁶ During their service life, composites are prone to suffer impact load due to the fast motion of automotive or aircraft.^7,8 The strength of composites should be specially designed to withstand the potential impact load in practical use. Damage evolution and failure mechanisms of 3D braided composite under impact load is an import basis in its dynamic strength optimization. However, damage initiation, development, and failure mechanisms of 3D braided composite under impact load are not very clear yet. An investigation of this issue is urgently needed for the 3D braided composite’s strength design.

The mechanical performances of composites under high strain-rate loading are quite different from those under quasi-static loading.⁹ In recent years, the dynamic behavior of composites under varied strain-rate load has attracted plenty of interest.¹⁰ Composites go through short-term loading experience under high strain-rate impact load. When the duration of impact load is shorter than the structure’s dynamic response time, the mechanical behavior of composites cannot be analyzed with quasi-static equilibrium, and the propagation process of stress wave must be considered.¹¹ In this case, the strain rate can significantly influence the mechanical performance of the composite.^12,13 With increasing the strain rate, the yield stress of resin increases,¹⁴ accompanied with the plateau stress¹⁵ and the dynamic compressive strength¹⁶ of the composite increases. A positive linear relationship between strength and strain rate¹⁷ or logarithmic strain rate¹⁸ has been found. The modulus of composites also changes under different strain-rate loads.¹⁹ However, tendency to change may be varied along different loading directions.²⁰ The failure strain of composites may increase²¹ or decrease²² significantly with the increase of strain rate, depending on composite structures. Besides, the interfacial strength of composites increases with increasing strain rate because the atoms do not get enough time to relax at higher strain rates.²³ Mechanical performance of composites shows varied sensitivities to strain rate when the fiber contents, reinforcement type, and matrix strength change. There is a strain-rate threshold for strength and energy absorption capacity. As the strain rate increases, the strength and energy absorption capacity of composite increase within this threshold²⁴ and decrease after exceeding this threshold.²⁵

The variation of strain-rate load can induce different damage process and failure modes of composites. In impact tests, composites experience short-term loading processes. The duration of loading is shorter than the composites’ dynamic response time. This means parts of the composite may have been loaded and deformed under impact load. Other parts of the composite maintain their still state owing to their inherent property of inertia.²⁶ Thus, highly uneven deformation distribution occurred in the composite.²⁷ This is quite different from the deformation process under static loading, which shows much uniform deformation distribution.²⁸ When the strain rate increased, the response time of the material reduced, and hence the crack propagates much more rapidly through the specimen in a brittle manner.²⁹ Ultimately composites show more severe brittle failure²² and less ductile failure.³⁰ The deformation primarily concentrates near the fracture crack and the location of possible failure.¹⁸ The increased interfacial bonding strength due to high strain-rate load results in increased fiber breakages and decreased fiber pull-out.^31,32 With the increase of strain rate, longer and deeper cracks are formed along the braiding path³³ and more fibers are crushed into fragments.³⁴

Many studies have reported the significant influence of strain rate on stress–strain response, failure process, and failure modes of composite. For braided composite, the interlaced braided yarns generate an intricate heterogeneous interface in the composite. The stress wave simultaneously reflects, refracts, and transmits in the interface.³⁵ This makes stress distribution and damage evolution of the composite more complex under different strain-rate loadings.³⁶ However, there is a lack of studies on the underlying mechanisms of how the change of strain rate leads to the variation of composite damage evolution. A bridge which can connect the strain rate variations to different damage processes of braided composite is urgently needed for composite strength design. The aim of this research is to reveal damage evolution and failure mechanisms of 3D braided composite under impact load. We focus on the effect of strain rate on the damage evolution of 3D braided composite under different strain-rate impact loading. Dynamic testing with varied strain rates was conducted and a high-speed camera system was used to capture the damage process information. A micro-scale rate-dependent finite element model (FEM) was established to analyze the stress and strain distribution, the proportion of elastic/plastic deformation, and the velocity gap between yarns and resin under different strain rates. The damage process and failure modes of composites under various strain rates are thoroughly discussed. The research conclusion aims to provide the designers with useful guidance for strength optimization of 3D braided composite. This can increase the service life and safe reliability of 3D braided composite parts.

Experiment

Preparation of 3D braided composite

The carbon fiber was supplied by Toray Inc., Japan. The epoxy resin JC-02A was supplied by Changshu Jiafa chemical Inc. (China). Table 1 and Table 2 present the mechanical parameters of carbon fiber and epoxy resin, respectively. Figure 1 presents the manufacturing process of the 3D braided composite specimen. Three-dimensional braided preform was fabricated on a four-step braided machine (RQX-3D/B, supplied by Tianjin Qianxing Machine Factory) and impregnated with resin using a vacuum-assisted resin transfer molding (VARTM) technique. The composite was solidified in the temperature sequence of 90°C-2h, 110°C-1h, and 130°C-4h. After being cooled to room temperature, the composite was demolded and cut into test specimens. The fiber volume fraction in this research is 38%. According to ASTM 6856-03, at least two represented braided unit cells should be contained in the specimen for dynamic test of braided composite. In order to observe the damage evolution more clearly, we fabricated the specimen with four-unit cells in width and thickness direction and two-unit cells in the length direction. The final size of the specimen is 12 × 12 × 12 mm.

Table 1.

Mechanical property of carbon fiber

Type	E₁₁ (GPa)	E₂₂ = E₃₃ (GPa)	G₁₂ = G₁₃ (GPa)	G₂₃ (GPa)	$υ_{12}$ = $υ_{13}$	$υ_{23}$	XT (GPa)	$ρ$ (g/cm³)
T700	230	14	9	5	0.25	0.3	4.9	1.8

Table 2.

Mechanical property of epoxy resin

Property	Magnitude
Compressive modulus (GPa)	3.0
Compressive strength (MPa)	102
Shear modulus (GPa)	0.89
Shear strength (MPa)	113
Yield strength (MPa)	69
Poisson’s ratio (MPa)	0.3
Density (g/cm³)	1.13

Impact test with different strain rates

The high-velocity impact test with varied strain rates was conducted on a split Hopkinson pressure bar (SHPB) as shown in Figure 2(a). Figure 2(b) shows the schematic diagram of the SHPB apparatus. The test apparatus consists of gas container, striker bar, incident bar, transmission bar, and data analysis system. The material used for the striker bar and other bars is maraging steel with a high yield strength of 1830 MPa. The diameter of all bars is 30 mm. The length of the striker bar, incident bar, and transmission bar is 400 mm, 2000 mm, and 2000 mm, respectively.

The striker bar is driven by the high-pressure nitrogen gas and shoot out from the gas chamber when the test starts. The striker bar hits the incident bar and a stress pulse is generated in the interacting surface. The stress pulse propagates along the incident bar and part of it propagates through the interacting surface into the specimen, while another part of stress pulse reflects back to incident bar. The assumption of one-dimensional wave propagation theory which assumes the stress wave only propagates without dispersion is adopted³⁷ to calculate the stress and strain of the specimen. The strain-rate $\dot{ε} (t)$ , strain $ε (t)$ , and stress $σ (t)$ of the specimen are calculated by the following equations:³⁷

\dot{ε} (t) = - \frac{2 C_{0}}{L_{s}} ε_{R} (t)

(1)

ε (t) = - \frac{2 C_{0}}{L_{s}} \int_{0}^{t} ε_{R} d t

(2)

σ (t) = \frac{E_{b} A_{b}}{A_{s}} ε_{T} (t)

(3)

where

C_{0} = \sqrt{E_{b} / ρ_{b}}

is the velocity of generated stress pulse,

L_{s}

represents the specimen length,

ε_{R} (t)

and

ε_{T} (t)

are the reflected and transmitted part of the stress pulse, respectively.

E_{b}

is the elastic modulus of the bars,

A_{b}

and

A_{s}

are the cross-sectional area of the bar and specimen, respectively. Equations (1) to (3) are based on the assumption that the force on both sides of the specimen is equal. The reflected and transmitted parts of the stress pulse were measured by the strain gage (with resistance of 240 Ω and gage factor of 2.02) mounted on the incident and transmission bar, respectively. A super dynamic strain indicator (WA-02, manufactured by University of Science and Technology of China) was used to record the strain signals. Then the signals were input to the computer by the data acquisition card (PC12406, manufactured by Hua Cai Ltd, China) for subsequent analysis. The data-sampling rate was 40 MHz for all experiments in the present work.

The varied strain-rate load is realized by altering the pressure magnitude of the nitrogen gas. Based on practical application conditions of composite in service,^38–40 strain rates of 500⁻¹, 1000⁻¹, and 1500⁻¹ are selected in this study. At least three tests were conducted under each strain rate. A high-speed camera system (IX Camera Ltd, United Kingdom) was triggered simultaneously with SHPB apparatus to capture the damage process of specimen at a frequency of 100,000 fps.

The force at specimen-incident surface $F_{1}$ and specimen-transmission bar surface $F_{2}$ can be expressed as follows:⁴¹

F_{1} = E A_{b} (ε_{I} + ε_{R})

(4)

F_{2} = E A_{b} ε_{T}

(5)

where

E

and

A_{b}

are the elastic modulus and cross-section area of the bar, respectively.

ε_{I}

ε_{R}

, and

ε_{T}

are the strain induced by incident stress wave, reflection stress wave, and transmission stress wave, respectively. Dynamic equilibrium is achieved when

F_{1} = F_{2}

. The shape of stress wave has a significant influence on the time required for the dynamic equilibrium. The rectangular stress wave requires the shortest time for stress equilibrium.⁴² Thus, we choose a rectangular stress wave as the incident pulse as seen in Figure 3. A parameter R as introduced in literature⁴¹ can be used to evaluate the dynamic equilibrium state in the SHPB test. The value of R is 0.036 in our test. This means the force difference between specimen-incident bar and specimen-transmission bar is very small. The dynamic equilibrium is realized and dynamic test results can represent the real strength of 3D braided composite.

Test results

The strength and modulus of 3D braided composite under different strain rates are presented in Figure 4. The strength and modulus of 3D braided composite both increase with the strain rate. This is in accordance with the results found by Sun et al.³⁷ The error bar of the strength and modulus has been provided. The maximum standard deviation of strength and modulus is 4.6% and 4.3%, respectively. This is within the accepted range of experimental error.

Figure 5(a) shows damage evolution of 3D braided composite at varied strain rates. The duration of impact load lasts 170 $μ s$ . The specimen is divided into three areas – input-bar area, output-bar area, and middle area – as demonstrated in Figure 5(b) in order to describe the damage process clearly. The input-bar and output-bar area are the two side-parts of specimen which are close to the input bar and output bar, respectively. The middle area is the part does not interact with the input bar or output bar.

When the specimen is loaded with 500⁻¹, the damage initiated from the output-bar area at 60 $μ s$ . Then the cracks extended and the composite was finally split off at the output-bar area at 170 $μ s$ . At the strain rate of 1500⁻¹, the composite firstly fractured in the input-bar area at 60 $μ s$ , followed by the damage extension to the adjacent area. The composite finally shattered near the input-bar area. At strain rate of 1000⁻¹, the composite was damaged in both the input-bar and output-bar area. This is similar to Zhang et al.’s findings in the damage process of 3D angle-interlock reinforced composite under varied strain-rate loading.⁴³

Finite element model

A rate-dependent micro-scale FEM was established. The geometry model was built up based on the realistic path of braided yarns. A rate-dependent constitutive model of both resin and yarns was used to define their elastic–plastic behavior. Damage initiation and evolution were defined to simulate the failure process. Boundary conditions and load were applied according to the test conditions.

Geometry model

Geometry model of 3D braided composite was set up with the software of CATIA V-5. Figure 6 shows the geometry model of 3D braided composite. The braided preform consists of interior cell, surface cell, and corner cell. A geometry model of each cell was created based on the real space trace of braided yarns. The matrix was formed by Boolean operation. A geometry model of 3D braided composite was finally built up by integrating the preform and matrix together.

Rate-dependent constitutive laws

The resin is assumed as isotropic material which obeys J-2 isotropic hardening plastic theory. The rate-dependent behavior of resin involves rate-dependent property of elastic modulus and plastic flow laws.

The rate-dependent property of elastic modulus is defined as the following empirical formulation⁴³

H (\dot{ε}) = {α_{1} ({\ln \dot{ε}}^{*})}^{β_{1}} + γ_{1}

(6)

where

\dot{ε^{*}} = \frac{\dot{ε}}{\dot{ε_{0}}}

\dot{ε}

is the applied strain rate,

\dot{ε_{0}}

is the reference strain rate.

α_{1}

β_{1}

, and

γ_{1}

are the material constants, which can be obtained by experimental test. A dynamic test of epoxy resin under different strain rates has been studied in our previous work.⁴⁴ The value of

α_{1}

β_{1}

γ_{1}

used in this model is 500, 0.4216, and 3000, respectively. A modified Johnson–Cook (MJC) model⁴⁵ was established to define the rate-dependent plastic flow behavior of resin as shown in equation (7).

σ = (A 2+ B ε^{n}) [1 + (C_{1} + C_{2} \dot{ε}) {\ln \dot{ε}}^{*}]

(7)

where A represents the yield strength of epoxy resin under reference strain rate,

B

is the strain-hardening constant,

ε

is the equivalent plastic strain, n is the strain-hardening coefficient,

C_{1}

and

C_{2}

are the strain rate strengthening coefficients. Figure 7 shows the test result of epoxy resin under different strain rates obtained in our previous study.⁴⁴ We substituted the values of

σ

and

ε

at the strain rate of 0.0025/s into equation (7), the value of

A

B

, and n could be derived. Then, we substituted the values of

σ

and

ε

at other different strain rates, the value of parameter

C_{1}

and

C_{2}

could be derived. The value of these parameters is given in Table 3.

Table 3.

Parameters of modified Johnson–Cook (MJC) model

Parameter	A (MPa)	$B$	n	$C_{1}$	$C_{2}$
Value	101	7.2	0.4444	0.011	3 × 10^-8

The braided yarns which exist in the form of fiber bundle impregnated with resin are regarded as transversely isotropic laminate materials. The bridging model developed by Huang⁴⁶ is used to predict the elastic constants in the present work. The compliance matrix for braiding yarn can be defined as follows

[S] = (V_{f} [S^{f}] + V_{m} [S^{m}] [A]) {(V_{f} [I] + V_{m} [A])}^{- 1}

(8)

where

[S^{f}]

and

[S^{m}]

refer to the compliance matrix of carbon fiber and epoxy matrix, respectively.

V_{f}

and

V_{m}

are the volume fraction of the fiber and matrix in braiding yarns, respectively.

[I]

is a unit matrix.

The bridging matrix $[A]$ is given by following expression:

[A] = [\begin{matrix} a_{11} & a_{12} & a_{13} & 0 & 0 & 0 \\ 0 & a_{22} & 0 & 0 & 0 & 0 \\ 0 & 0 & a_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & a_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & a_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & a_{66} \end{matrix}]

(9)

The elements of the bridging matrix are given as follows:

a_{11} = E^{m} / E_{11}^{f}

(10)

a_{22} = a_{33} = a_{44} = 0.5 (1 + E^{m} / E_{22}^{f})

(11)

a_{55} = a_{66} = 0.5 (1 + G^{m} / G_{12}^{f})

(12)

a_{12} = a_{13} = (S_{12}^{f} - S_{12}^{m}) (a_{11} - a_{22}) / (S_{11}^{f} - S_{11}^{m})

(13)

where

E^{m}

and

G^{m}

refer to elastic modulus and shear modulus of the matrix, respectively.

E_{11}^{f}

and

E_{22}^{f}

are the elastic modulus of fiber in axial and transverse direction, respectively.

G_{12}^{f}

is the shear modulus of fiber.

S_{12}^{f} = - \frac{ν_{12}^{f}}{E_{11}^{f}}

S_{11}^{f} = \frac{1}{E_{11}^{f}}

S_{11}^{m} = \frac{1}{E^{m}}

S_{12}^{m} = - \frac{ν^{m}}{E^{m}}

ν_{12}^{f}

is Poisson’s ratio of fiber in axial direction and

ν^{m}

is Poisson’s ratio of the matrix.

Then the stiffness matrix of yarns can be obtained as follows

[C^{y}] = {[S^{y}]}^{- 1}

(14)

The volume fractions of the fiber in braiding yarns ( $V_{f}$ ) can be defined as

V_{f} = \frac{π D_{y}^{2}}{4 Ω}

(15)

where

Ω

is the equivalent area of yarn cross-section and

D_{y}

refers to the equivalent diameter of the braiding yarn, which can be defined as

D_{y} = \sqrt{\frac{4 λ}{π ρ}}

(16)

where

λ

represents linear density of yarn and

ρ

refers to the fiber density. Elastic constants of braided yarns under different strain rates can be obtained by substituting elastic constants of resin and fiber into equation (8) and are shown in Table 4.

Table 4.

Elastic parameters of braided yarns under different strain rates

Property	$E_{11} (GPa)$	$E_{22} (GPa)$	$E_{33} (GPa)$	$G_{12} (GPa)$	$G_{13} (GPa)$	$G_{23} (GPa)$	$ν_{12}$	$ν_{13}$	$ν_{23}$
500⁻¹	166.93	8.10	8.10	3.45	3.45	2.92	0.28	0.28	0.40
1000⁻¹	175.35	8.50	8.50	3.59	3.59	3.07	0.28	0.28	0.40
1500⁻¹	180.44	9.02	9.02	3.84	3.84	3.31	0.28	0.28	0.40

Hill’s anisotropic plastic model is adopted to define the yielding and plastic flow of the braided yarns.⁴⁷ Six parameters are introduced based on the von-mises yield theory to define the anisotropic yield status in Hill’s model. The plastic potential function can be expressed by equation (17) in the cartesian coordinate system.⁴⁸

2 f \equiv F {(σ_{y y} - σ_{z z})}^{2} + G {(σ_{z z} - σ_{x x})}^{2} + H {(σ_{x x} - σ_{y y})}^{2} + 2 L {σ_{y z}}^{2} + 2 M {σ_{x z}}^{2} + 2 N {σ_{x y}}^{2} = 1

(17)

Parameters $F$ , $G$ , $H$ , $L$ , $M$ , and $N$ are defined by the uniaxial yield stress and can be expressed as follows:⁴⁸

F = \frac{{(σ^{0})}^{2}}{2} (\frac{1}{{\bar{σ}}_{22}^{2}} + \frac{1}{{\bar{σ}}_{33}^{2}} - \frac{1}{{\bar{σ}}_{11}^{2}}) = \frac{1}{2} (\frac{1}{R_{22}^{2}} + \frac{1}{R_{33}^{2}} - \frac{1}{R_{11}^{2}})

(18)

G = \frac{{(σ^{0})}^{2}}{2} (\frac{1}{{\bar{σ}}_{11}^{2}} + \frac{1}{{\bar{σ}}_{33}^{2}} - \frac{1}{{\bar{σ}}_{22}^{2}}) = \frac{1}{2} (\frac{1}{R_{11}^{2}} + \frac{1}{R_{33}^{2}} - \frac{1}{R_{22}^{2}})

(19)

H = \frac{{(σ^{0})}^{2}}{2} (\frac{1}{{\bar{σ}}_{22}^{2}} + \frac{1}{{\bar{σ}}_{11}^{2}} - \frac{1}{{\bar{σ}}_{33}^{2}}) = \frac{1}{2} (\frac{1}{R_{22}^{2}} + \frac{1}{R_{11}^{2}} - \frac{1}{R_{33}^{2}})

(20)

L = \frac{3}{2} {(\frac{τ^{0}}{{\bar{σ}}_{23}})}^{2} = \frac{3}{2 R_{23}^{2}}

(21)

M = \frac{3}{2} {(\frac{τ^{0}}{{\bar{σ}}_{13}})}^{2} = \frac{3}{2 R_{13}^{2}}

(22)

N = \frac{3}{2} {(\frac{τ^{0}}{{\bar{σ}}_{12}})}^{2} = \frac{3}{2 R_{12}^{2}}

(23)

where

{\bar{σ}}_{i j}

is the measured yield stress value when

σ_{i j}

is applied as the only nonzero stress component.

σ^{0}

is the user-defined reference yield stress specified for the material plasticity definition. We defined the longitudinal direction of fiber as the 1 direction. The transverse yield stress is taken as the reference yield stress.⁴⁹ A representative unit cell (RUC) was established to determine the plastic constants as shown in Figure 8(a). Uniaxial loads

σ_{11}, σ_{22}, τ_{13}

, and

τ_{23}

with different strain rate were applied to the RUC as the only nonzero stress to derive the yield stress in each uniaxial direction. The simulation results are shown in Figure 8(b). The value of

{\bar{σ}}_{11}

{\bar{σ}}_{22}

{\bar{σ}}_{33}

{\bar{σ}}_{12}

{\bar{σ}}_{13}

, and

{\bar{σ}}_{23}

under different strain rates are obtained from the stress–strain curves in Figure 8 (b). Then, the value of

F

G

H

L

M

, and

N

can be derived according to equations (18) to (23).

For transversely anisotropic materials, ${\bar{σ}}_{22}$ = ${\bar{σ}}_{33}$ , ${\bar{σ}}_{12}$ = ${\bar{σ}}_{13}$ . $R_{11}$ , $R_{22}$ , $R_{33}$ , $R_{12}$ , $R_{13}$ , and $R_{23}$ are the anisotropic yield stress ratios. $τ^{0} = \frac{σ^{0}}{\sqrt{3}}$ . The yield stress ratios are defined as follows:

R_{11} = \frac{{\bar{σ}}_{11}}{σ^{0}} {, R}_{22} = \frac{{\bar{σ}}_{22}}{σ^{0}} {, R}_{33} = \frac{{\bar{σ}}_{33}}{σ^{0}} {, R}_{12} = \frac{{\bar{σ}}_{12}}{τ^{0}} {, R}_{13} = \frac{{\bar{σ}}_{13}}{τ^{0}} {, R}_{23} = \frac{{\bar{σ}}_{23}}{τ^{0}}

The yield stress ratios of braided yarns under different strain rates are presented in Table 5. The yield stress ratios are inputted into FEM to define the anisotropic plastic property of the braided yarns.

Table 5.

Yield stress ratio of yarns under different strain rates

Stress ratio	R₁₁	R₂₂	R₃₃	R₁₂	R₁₃	R₂₃
500⁻¹	13.3	1.0	1.0	1.06	1.06	0.83
1000⁻¹	6.8	1.0	1.0	1.04	1.04	0.79
1500⁻¹	4.76	1.0	1.0	1.03	1.03	0.76

Von-Mises yield criterion indicates that there are the following relations between stress tensor $σ_{i j}$ , strain increment $d ε_{i j}$ and plastic potential function $f$ :⁴⁸

d ε_{i j} = \frac{\partial f}{\partial σ_{ij}} d λ

(24)

where

d λ

is the proportionality coefficient. By substituting equation (17) into (24) and ignoring the change of material volume in the process of plastic deformation, the strain flow criterion of material after yield can be obtained.

d ε_{x x} = d λ [H (σ_{x x} - σ_{y y}) + G (σ_{x x} - σ_{z z})],

(25)

d ε_{y y} = d λ [F (σ_{y y} - σ_{z z} + H (σ_{y y} - σ_{x x})]

(26)

d ε_{z z} = d λ [G (σ_{z z} - σ_{x x} + H (σ_{z z} - σ_{y y})]

(27)

d ε_{y z} = d λ L σ_{y z}

(28)

d ε_{z x} = d λ M σ_{z x}

(29)

d ε_{x y} = d λ N σ_{x y}

(30)

The dissipated energy due to elastic and plastic deformation can be calculated in FEM. Hence, the influence of strain rate on energy dissipation due to elastic and plastic deformation can be analyzed.

Damage initiation and evolution

For 3D braided composite, the failure modes under impact load include fiber breakage, fiber buckling, resin fracture, and interface debonding.⁵⁰ Failure mode of fiber buckling can be visually observed in the simulation results. Failure modes of fiber breakage, resin fracture, and interface debonding are all considered in the current FEM.

Ductile damage criterion combined with shear damage criterion⁴⁴ is used to define the damage initiation and evolution of resin and yarns. Ductile criterion for damage initiation is met when the following condition is satisfied:⁵¹

ω_{D} = \int \frac{d {\bar{ε}}^{p l}}{{\bar{ε}}^{p l}_{D} (η, {\dot{\bar{ε}}}^{p l})} = 1

(31)

where

ω_{D}

is a state variable that increases monotonically from 0 to 1 with plastic deformation, which is calculated at each increment and outputted in Abaqus as variable “DUCTCRT”.

η = - p / q

is the stress triaxially,

p

is the pressure stress,

q

is the Mises equivalent stress,

{\bar{ε}}^{p l}_{D}

is the equivalent plastic strain at the onset of damage, and

{\dot{\bar{ε}}}^{p l}

is the equivalent plastic strain rate. The equivalent plastic deformation in integrals can be derived with the following equation:

{\bar{u}}^{p l} = L {\bar{ε}}^{p l}

(32)

where L is the characteristic length of the element.

{\bar{ε}}^{p l}

is the relative strain of element.

The shear criterion for damage initiation is met if the following formula⁵¹ is satisfied:

ω_{S} = \int \frac{d {\bar{ε}}^{p l}}{{\bar{ε}}^{p l}_{S} (θ_{S}, {\dot{\bar{ε}}}^{p l})} = 1

(33)

where

ω_{S}

is a state variable that increased monotonically from 0 to 1 with plastic deformation, which is calculated at each increment and outputted in Abaqus as variable “SHRCRT”.

θ_{S} = (q + k_{s} p) / τ_{\max}

is the shear stress ratio,

τ_{\max}

is the maximum shear stress, and

k_{s}

is a material constant. The value of

k_{s}

used in this paper is

k_{s} = 0.3

, based on work by Zhang et al.⁵²

Surface-based cohesive behavior is defined at the yarn/resin interface. It is assumed that the interface is separated in a linear elastic manner. The interface stress tensors $T$ can be decomposed to vertical component $T_{n}$ and two in-plane components $T_{s}$ and $T_{t}$ . $δ_{n}$ , $δ_{s}$ , and $δ_{t}$ are the corresponding interface displacement tensors. The interface stress–strain response can be expressed as

T = \{\begin{matrix} T_{n} \\ T_{s} \\ T_{t} \end{matrix}\} = [\begin{matrix} K_{n n} & K_{n s} & K_{n t} \\ K_{n s} & K_{s s} & K_{s t} \\ K_{n t} & K_{s t} & K_{t t} \end{matrix}] \{\begin{matrix} δ_{n} \\ δ_{s} \\ δ_{t} \end{matrix}\} = K δ

(34)

Damage initiates when the maximum contact stress ratio reaches 1 as shown in equation (35).

max \{\frac{T_{n}}{T_{n}^{0}}, \frac{T_{s}}{T_{s}^{0}}, \frac{T_{t}}{T_{t}^{0}}\} = 1

(35)

where

T_{n}^{0}

T_{s}^{0}

T_{t}^{0}

represent the peak values of the contact stress when the separation is either purely normal to interface or purely in the first or the second shear direction, respectively. Parameters of the cohesive model are obtained by experimental test. The details of cohesive model parameter determination of this material system can be found in work by Zhang et al.⁵³ The value of the parameters is presented in Table 6. The overall damage status of cohesive interface D_I, which monotonically evolves from 0 to 1, is calculated at each increment and outputted in Abaqus as variable “CSDMG”.

Table 6.

Parameters of cohesive model

$K_{n}$ (N/mm³ $)$	$K_{s} = K_{t}$ (N/mm³)	$τ_{n}^{0}$ (MPa)	$τ_{s}^{0} = τ_{t}^{0}$ (MPa)	$G_{n}^{C}$ (N/mm)	$G_{s}^{C} = G_{t}^{C}$ (N/mm)	$β$
4 × 106	10 × 106	120	150	0.25	1.0	1.0

Once the corresponding initiation criterion is reached, the stress tensor ( $σ$ ) in the material is given by the scalar damage equation

σ = (1 - D) \bar{σ}

(36)

where D is the overall damage variable ranging from 0 to 1 and

\bar{σ}

is the effective (or undamaged) stress tensor computed in the current increment.

\bar{σ}

is the stress that would exist in the material in the absence of damage. The evolution of the damage variable D is specified in linear form with the relative plastic displacement

{\bar{u}}^{p l}

as shown in Figure 9.

{\bar{u}}_{f}^{p l}

is the ultimate failure displacement of element. The element loses its load-carrying capacity and is removed from the mesh when D = 1.

Boundary condition and mesh

The FEM was established in commercial universal finite element software Abaqus 6.14. Figure 10 shows the diagram of meshing and impact model. The yarns, input bar, and output bar were partitioned with hexahedron-shaped mesh. The resin was partitioned with a tetrahedron-shaped mesh. The impact load was defined at the end of the input bar. The freedom of the two bars was completely fixed except along the impact direction. No extra restriction was applied on the specimen.

Results and discussion

Validation of FEM

Figure 11 shows the results obtained from simulation and experiments. Figure 11(a) presents stress–strain curves obtained from FEM and experiment results. The FEM can predict the stress history of composite with accepted accuracy. Stress–strain curves obtained from simulation are a little higher than that from experiments, as Wan et al. also found in literature.⁴⁸ This is because several kinds of defects generated during the manufacturing of composite. Figure 11(b) presents the damage morphology of composite obtained from FEM and experimental results. The damage morphologies obtained from FEM under different strain rates match well with the experimental results. This indicates the validity of the proposed FEM.

Damage process and failure mechanism

Three-dimensional braided composites experienced dramatically different failure processes under varied strain rates as presented in the section “Test results”. This is attributed to the changed stress distribution in composite. Figure 12 shows simulation results of damage and stress distribution in composite under different strain rates at 20 μs. Input-bar area undertakes higher load in the beginning of impact stage due to “inertial effect”.²⁷ For composite loaded at 1500⁻¹, the stress of input-bar area is higher than that loaded with 500⁻¹, which consequently induces ductile damage (DUCTRT) and shear damage (SHRCRT) initiating and obvious interface damage (CSDMG) occurred in input-bar area. This leads to the specimen finally splitting off in the input-bar area.

For composite loaded with 500⁻¹, the stress carried by the input-bar area is not high enough to fracture the resin. Composite did not show any damage in the input-bar area at all. The stress wave propagates through the specimen and into the output bar. During the propagation, the stress wave was transversely distracted due to the multi-directionally oriented yarns. Figure 13 presents simulation results of composite loaded with 500⁻¹ at 70 $μ s$ . The length of arrows in Figure 13(a) represents the magnitude of related stress. It can be found that the transverse stress (S₁₁ and S₂₂ for resin, S₃₃ and S₂₂ for yarns) distributed in the output-bar area is higher than that in the input-bar area, which results in higher strain along out-of-plane direction (LE₁₁ and LE₂₂) and more severe damage occurred in output-bar area than input-bar area as shown in Figure 13(b). Consequently, the output-bar area firstly fractured instead of the input-bar area for composite loaded with 500⁻¹.

The stress undertaken by composite loaded at 1000⁻¹ is between the other two specimens (loaded at strain rates of 500⁻¹ and 1500⁻¹). Thus, the damage initiates in both input-bar and output-bar area.

It can be seen from the above analysis that when composite is loaded at lower strain rate, the extra reinforcement in the output-bar area can protect the structure from splitter, while at higher strain-rate loading, composite should be reinforced in the input-bar area to improve its impact resistance.

Local strain distribution

The varied stress leads to the unevenness of strain distribution in composite. Figure 14 presents the local strain of composite. In order to analyze the distribution of local deformation, the specimen was evenly divided into nine small sections as shown in Figure 14(a). Section P₁ is close to the input bar and P₉ is next to the output bar. When the total strain of composite is 1%, the strain of each section is shown in Figure 14(b). As can be seen, the strain of P₁ and P₉, which is larger than the overall average strain of composite, is significantly larger than that of the interior sections. Table 7 shows the largest strain gap among nine sections under different strain rates. Notably, the maximum strain of sections loaded at 1500⁻¹ is 1.61%, which is 133% higher than the minimum strain; While the largest strain gap among sections loaded at 1000⁻¹ and 500⁻¹ is 107% and 70%, respectively. The higher the strain rate, the larger is the strain gap among sections. Uniform stress and strain distribution in composite is found by Zhang et al.⁵⁴ under static load. This indicates the strain rate has significant influence on damage evolution of composite. When the impact duration is shorter than the composite’s response time, strain distribution in composite will change obviously.

Table 7.

The largest strain gap among subareas

Strain rate	Maximum strain of subarea	Minimum strain of subarea	Largest strain gap (%)
1500⁻¹	1.61	0.69	133
1000⁻¹	1.53	0.74	107
500⁻¹	1.48	0.89	70

In order to investigate the local strain distribution further, the strain of each point in top surface is extracted. Then the strain map of the top surface was replotted with a “waterfall curve” as shown in Figure 15(a). The replotted strain map was shown in Figure 15(b). The strain map of the top surface of composite loaded at 1500⁻¹ shows more fluctuation than the other two specimens (loaded at lower strain rates), especially in the section near the input bar and output bar. The strain map of composite loaded at 1000⁻¹ is much flatter; however, the section near the input bar still shows continuous fluctuation. For composite loaded at 500⁻¹, besides input-bar and output-bar areas, the strain map in the central area is much more even than that of the other two strain-rate loading cases. This indicates that at varying strain rates, although the total strain of composites is identical, the strain distribution in composite is considerably varied. As the strain rate increased, the unevenness of strain distribution in composite also increased, which accelerated the damage of composite.

Elastic and plastic deformation

The different stress history under varied strain rates also induces distinct characteristics of elastic and plastic deformation. Figure 16 shows the plastic deformation of resin at the same strain level of composite. There is no plastic deformation in resin at the strain of 2% when loaded under 500⁻¹. The resin went through totally elastic deformation. For composite loaded under 1500⁻¹ and 1000⁻¹, a little plastic deformation can be seen in resin at the strain of 2%. At the strain of 3.2% and 4.5%, the composite loaded under 1500⁻¹ presents more plastic deformation than that loaded with the lower strain rates. The higher the strain rate loaded on composite, the more the plastic deformation occurred.

Deformation ratio $γ$ is defined to analyze the elastic and plastic deformation proportion further. The value of $γ$ can be calculated with the following equation.

γ = \frac{ε_{p}}{ε_{e}}

(37)

where

ε_{e}

and

ε_{p}

represents the elastic and plastic deformation of composite, respectively.

Figure 17 presents the value of $γ$ under different strains. As can be seen, the value of $γ$ remains 0 before the strain of 1%, which demonstrates that composites went through completely elastic deformation during this period. After the strain of 1%, the $γ$ of composite loaded under strain rate of 1500⁻¹ firstly increases as the plastic deformation occurred. The $γ$ of composite loaded under 1000⁻¹ increases at the strain of 1.1%. The $γ$ of composite loaded under 500⁻¹ does not significantly increase until the strain of composite exceeds 3%. When the strain of composite exceeds 1%, at the same strain level, the $γ$ of composite loaded under 1500⁻¹ strain rate is always bigger than that of 1000⁻¹ and 500⁻¹. This indicates that although the total strain of composite is identical, the constitution of elastic and plastic deformation varies a lot under different strain-rate loading. Higher strain rate induces more plastic deformation. Higher plastic deformation is more likely to lead to the damage initiating in composite. Thus, special attention should be paid in composite design when the composite may be subjected to higher strain-rate load.

Velocity gap between the yarns and resin

The density and modulus difference of yarns and resin results in unequal stress wave propagation speed in components of composite. This can induce movement mismatch of the yarns and resin at the interface.³³ Figure 18 shows the average velocity of the yarns and resin in composite. The average velocity of the yarns and resin is derived from the absorbed kinetic energy by each of them. The absorbed kinetic energy can be calculated in the FEM.

Figure 18(a) presents the average velocity of resin and yarns under different strain rates. It can be found that at initial impact stage, the velocity of yarns is almost the same as that of resin. After 15 μs, the yarns loaded under 1000 s⁻¹ and 1500 s⁻¹ travel more rapidly than the resin, while under the strain rate of 500⁻¹, no obvious speed difference between yarns and resin is observed until 30 μs. Figure 18(b) presents the velocity gap between the yarns and resin at initial impact stage (0–35 μs). As can be seen, the velocity gap enlarges with the increase of strain rate. At 30 μs, the velocity gap of composite loaded at 1500 s⁻¹ reaches as high as 48 m/s, which is 1.05 and 2.4 times larger than that of composites loaded under 1000 s⁻¹ and 500 s⁻¹, respectively. The movement mismatch resulted from the velocity gap finally induces the interface debonding. The larger the velocity gap, the more severely the interface may debond.

Figure 19(a) shows the velocity distribution in the yarns and resin under different strain rates when composites absorbed identical kinetic energy (530 mJ). With increasing the strain rate, the velocity of yarns and the velocity gap between yarns and resin both increase. Figure 19(b) shows the distribution map of velocity components in the yarns and resin. V₃ represents the velocity component along the loading direction, while V₁ and V₂ represent two velocity components perpendicular to the loading direction. It can be found that the magnitude of V₃ in the yarns is higher than that in resin; while V₁ and V₂ of the yarns and resin are almost identical. Thus, the velocity gap between the yarns and resin is mainly resulted from the velocity difference along loading direction. It should be noted that the change of fiber volume fraction extends or reduces the velocity-gap area in composite. This can result in different damage initiation. Thus, the variation of fiber volume matters for the damage evolution. Further investigation is necessary when determining damage evolution of composite with other fiber volume fractions.

Conclusion

The damage evolution of 3D braided composite under different strain rate is investigated in this research. The impact test was conducted and a micro-scale rate-dependent FEM was established to reveal the underlying damage mechanisms. Damage process and failure modes of composites under various strain rates are thoroughly discussed. The main conclusions are listed below.

The stress distribution of composite is obviously different under varied strain-rate loading, which significantly affects the damage evolution of composite. The damage evolutions of 3D braided composite show distinct difference under varied strain rates.

The velocity gap between resin and yarns increases with increasing strain rate. The largest velocity gap between resin and yarns reaches 48 m/s, which is more likely to induce interface debonding during the impact process.

Although the total strain is identical, 3D braided composite goes through different elastic/plastic deformation history under varied strain rate. The deformation distribution shows more obvious unevenness with higher strain-rate impact loading, which can accelerate the damage initiation of composite.

Optimizing suggestions are derived to improve the impact resistance of 3D braided composite under different strain rates.

Footnotes

Data availability

The raw data required to reproduce these findings are available to download from []. The processed data required to reproduce these findings are available to download from https://data.mendeley.com/drafts/kfzf478jwd].

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: the Natural Science basic Research Program of Shaanxi Province [grant numbers 2021JQ-659, 2020JQ-819], Research Fund for the Doctoral Program of Xi’an Polytechnic University [grant numbers 107020527], Beilin Science and Technology Planning Project [grant numbers GX2210], National Natural Science Foundation of China [grant number 12102144, 12002248], The Jiaxing Public Welfare Technology Application Research Project [grant number 2021AD10012], The Open Project Program of Key Laboratory of Yarn Materials Forming and Composite Processing Technology of Zhejiang Province [grant number MTC2021-04, MTC2020-22].

ORCID iDs

Xingzhong Gao

Tao Liu

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