Experimental investigation and numerical simulation of unidirectional carbon fiber composite under multi-axial loadings

Accepted Manuscript Experimental investigation and numerical simulation of unidirectional carbon fiber composite under multi-axial loadings Yangyang Qiao, Chiara Bisagni, Yuanli Bai PII:

S1359-8368(16)31082-4

DOI:

10.1016/j.compositesb.2017.05.034

Reference:

JCOMB 5057

To appear in:

Composites Part B

Received Date: 19 June 2016 Revised Date:

19 November 2016

Accepted Date: 11 May 2017

Please cite this article as: Qiao Y, Bisagni C, Bai Y, Experimental investigation and numerical simulation of unidirectional carbon fiber composite under multi-axial loadings, Composites Part B (2017), doi: 10.1016/j.compositesb.2017.05.034. 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

Experimental investigation and numerical simulation

Yangyang Qiao1, Chiara Bisagni2, Yuanli Bai1* 1

Department of Mechanical and Aerospace Engineering,

2

Faculty of Aerospace Engineering,

M AN U

University of Central Florida, Orlando, FL 32816, USA

SC

axial loadings

RI PT

of unidirectional carbon fiber composite under multi-

Delft University of Technology, Kluyverweg 1, 2629HS Delft, The Netherlands

Abstract

TE D

Unidirectional carbon fiber composites were tested under multi-axial loading conditions including tensile/compression/shear loadings along and perpendicular to fiber directions. Compression dominated tests showed brittle fracture mode like local kicking/buckling, while tension

EP

dominated tests showed fracture mode like delamination and fiber breakage. Pure shear tests with displacement control showed material hardening and softening before total failure. Tsai-Wu

AC C

failure criterion with Mises-Hencky interaction term was used as the yield criterion, and an associated flow rule (AFR) was used to describe the plastic flow of this material. The modified Tsai-Wu failure criterion was able to predict the failure of the composites under different loading conditions. A new material post-failure softening rule was proposed, where a new parameter was introduced to represent different loading conditions of the composite. This model was implemented to ABAQUS/Explicit as a user material subroutine (VUMAT). Numerical

*

Corresponding author, email: [email protected] Page 1 of 47

ACCEPTED MANUSCRIPT

simulations using finite element method well duplicated the elasticity and plasticity of this material. Failure features like delamination was simulated using cohesive surface feature. This model showed good capacity in predicting both failure initiations and fracture modes for the

RI PT

unidirectional fiber composites. This model has good application potentials to large-scale

AC C

EP

TE D

M AN U

SC

engineering problems.

Page 2 of 47

ACCEPTED MANUSCRIPT

1. Introduction Polymer matrix composites (PMCs) materials have been used as aircraft primary and secondary structures (Chowdhury, Chiu et al. 2016, Guermazi, Tarjem et al. 2016). It is reported that more

RI PT

than 50% of the primary structure of the Boeing 787 Dreamliner is made of carbon fiber reinforced plastic and other composites (Company 2008). Carbon fiber composites have advantages over metals in many aspects. High strength to weight ratio is one of the most valuable

SC

advantages since it reduces weight and therefore increases fuel efficiency (Tekalur, Shivakumar et al. 2008). In addition, PMCs also have high energy absorbing capability (Buitrago, García-

M AN U

Castillo et al. 2013, Luo, Yan et al. 2016). However, there are still tough challenges in the industrial applications of PMCs. One of the most complicated problems is the failure mechanisms which are quite different from metallic materials.

The failure mechanisms of PMCs are complex because the modes of failure depend upon stress state, specimen geometry, fiber direction, material property and possible manufacturing defects.

TE D

In addition, local failure initiations occur way before final failure. During the stage of damage evolution, the material can still sustain load before catastrophic fracture. Researchers have been dedicated to developing models for predicting the behavior of PMCs especially failure features,

EP

for instance, as summarized in the following paragraphs. The modeling of PMCs behavior can be classified into several catalogs, which include failure

AC C

criterion method, continuum damage mechanics method and plasticity method. The failure criterion method mainly considers the initial or final failure locus of PMCs while the continuum damage mechanics method takes degradation of modulus into account. The plasticity method considers the material non-linearity to be plasticity. Some phenomenological failure criteria describing initial failure of composite laminate have been postulated based on material strength. Some of the most known ones are Maximum stress/strain, Tsai-Hill (Tsai 1968), Hoffman (Hoffman 1967), Franklin-Marin (Franklin 1968), Tsai-Wu (Tsai Page 3 of 47

ACCEPTED MANUSCRIPT

and Wu 1971) and Hashin (Hashin and Rotem 1973) criteria. These criteria can be further classified into two groups, non-interactive failure criteria and interactive failure criteria. If a criterion has no interaction between stress or strain components, it is defined as a non-interactive

RI PT

failure criterion which compares individual stress or strain component with the corresponding material strength, e.g. the maximum stress/strain criterion. On the contrary, most of the phenomenological criteria like Tsai-Hill, Hoffman, Franklin-Marin, Tsai-Wu and Hashin belong

SC

to interactive failure criteria, which are interactive failure criteria. Failure criteria can also be classified based on whether it is associated with failure modes or not. Some of the failure criteria

M AN U

utilize stress or strain polynomial expressions to describe the failure locus, such as Tsai-Hill, Tsai-Wu and Hoffman. These criteria did not distinguish between different failure modes. Other criteria like maximum stress/strain, Hashin, Yamada and Sun (1978), Hart-Smith (Lin and Lo 2001) and Puck (Usui 1997) specify particular failure modes for different loading conditions. Continuum damage mechanics (CDM) models are based on the observation that as the failure

TE D

evolution of fiber reinforced composites, continuous stiffness degradation is shown in materials. CDM uses internal variables to describe the progressive loss of rigidity. Kachanov (1958) firstly developed a continuum damage mechanics framework to study the creep rupture of metals.

EP

Schapery (1990), Murakami and Kamiya (1997), Hayakawa, Murakami et al. (1998), Olsson and Ristinmaa (2003) and Maimí, Camanho et al. (2007) proposed stiffness degradation and damage

AC C

evolution models using a second or fourth order damage tensor. The damage tensors are related to damage mechanisms and dissipation energy which controls the evolution of damage state. Kwon and Liu (1997), Matzenmiller, Lubliner et al. (1995), Schipperen (2001), Maa and Cheng (2002) and Camanho, Maimí et al. (2007) proposed thermodynamic models to describe the progressive failure properties. These thermodynamic models were limited to plane structures. Pinho, Iannucci et al. (2006) proposed a three-dimensional failure criterion for laminated fiber-reinforced composites based on the physical model for each failure mode. Donadon, De Almeida et al. (2009)

Page 4 of 47

ACCEPTED MANUSCRIPT

developed a fully three-dimensional failure model to predict damage in composite structures subjected to multi-axial loading. Bandaru, Chavan et al. (2016) has proposed a CDM model with strain rate effects and validated with the ballistic impact response of S2 glass/epoxy laminate.

RI PT

Wang, Liu et al. (2015) modified the Linde’s criterion considering the shearing effect for anisotropic progressive damage to describe the elastic-brittle behavior of fiber-reinforced in the framework of CDM.

SC

The plasticity method is mainly used for composite materials that exhibit ductile behavior like boron/aluminum, graphite/PEEK and other thermoplastic composites. Recently, a number of

M AN U

plasticity models have been used to describe the non-linear behavior with micromechanics approaches (Shih and Yang 1993, Huang and Black 1996). This method was also used in structural level. A 3-D plastic potential function was proposed to describe the nonlinear behavior in anisotropic fiber composites by Sun and Chen (1989). Vaziri, Olson et al. (1991) proposed an orthotropic plane stress material model that combines the classical flow theory of plasticity with a

TE D

failure criterion. Xie and Adams (1995) developed a three-dimensional plasticity model to describe the plastic response of unidirectional composites. A three-dimensional finite element analysis demonstrated the application of Xie and Adams’s model on compression and short-beam

EP

shear tests. Car, Oller et al. (2000) proposed a generalized anisotropic elasto-plastic constitutive model for large strain analysis of fiber-reinforced composite with the frame of mixing theories.

AC C

Vyas, Pinho et al. (2011) presents an elasto-plastic model framework which incorporated a nonassociative flow rule for unidirectional plies. This method computationally homogenize the directional properties of composites materials. And it helps in deriving a formulation applicable for such materials and amenable for easy implementation in a finite element simulation (Obikawa and Usui 1996). Liu and Zheng (2010) pointed out that the elastic/damage coupling constitutive model may be insufficient in order to accurately describe the damage initiation and evolution of laminated fiberPage 5 of 47

ACCEPTED MANUSCRIPT

reinforced composites. Some damage/plasticity coupled nonlinear models were also introduced to describe the interactive effect of plastic deformations with damage properties. Chow and Yang (1997) outlined a constitutive model for mechanical response in inelastic composite materials due

RI PT

to damage. Lin and Hu (2002) developed an elasticity-plasticity/damage coupled constitutive model together with a mixed failure criterion for single lamina. Barbero and Lonetti (2002) also presented a damage/plasticity model for an individual lamina and then assembled it to describe

SC

the behavior of polymer matrix composite laminates. Cherniaev and Telichev (2015) utilized a von Mises J2 function as the yield surface couple with material damage to model the interlaminar

M AN U

regions and validated by impact tests. Ferry, Perreux et al. (1998) proposed a new model describing the behavior of composite laminates with a plastic model and an anisotropic micromechanical damage model.

Delamination is an important mode of failure inside a ply or between plies. Numerous numbers of criteria have been proposed to predict the initiation and propagation of delamination. These

TE D

criteria use different combinations of transverse tension, shear and sometimes tension along fiber direction, in a linear or quadratic form. Maximum stress criterion sets transverse tensile strength, and two shear strengths as the limit to predict failure initiation. Hashin (1980) used a quadratic

EP

form that incorporated the transverse tensile stress and two shear stresses. Lee (1982) proposed a model similar to maximum stress criterion which combines two shear stresses in a quadratic form.

AC C

Brewer and Lagace (1988) utilized compressive stress to predict the delamination initiation. Tong (1997) included tensile stress along fiber direction in either linear or quadratic forms. Curve fitting was also introduced in delamination prediction by Goyal, Johnson et al. (2004). Delamination propagation is another concern in model prediction. Criteria have been proposed mainly based on the three crack separation modes and their corresponding critical strain energy release rate

. The simplest mode assumes no interaction between three crack separation modes

and that delamination grows when any one of the three strain energy release rate reaches its Page 6 of 47

ACCEPTED MANUSCRIPT

corresponding limit (Orifici, Herszberg et al. 2008). Hahn (1983) proposed a criterion which considers the interaction between mode I and mode II crack. Hahn and Johannesson (1984), Donaldson (1985), Hashemi, Kinloch et al. (1990) and Benzeggagh and Kenane (1996)

RI PT

incorporated parameters which need curve fitting to describe the delamination propagation. World-Wide Failure Exercises (WWFE) (Hinton, Kaddour et al. 2004) have been conducted to evaluate the postulated models. Plenty of models have been applied to predict the failure of fiber-

constitutes failure initiation of a composite structure.

SC

reinforced-polymer composites. It is recognized that there is no universal definition for what

M AN U

This paper presents a three-dimensional elasticity-plasticity coupled with a damage evolution criterion for fiber-reinforced polymer matrix composites. Different from crossbeam, flat plate, circular and thin-walled tube specimens, a plate with grooves on both sides have been tested under biaxial loading conditions to validate the proposed model. Biaxial loading conditions on the unidirectional carbon fiber composites can better reveal the anisotropic behaviors of this material.

TE D

By introducing grooves/notches into the specimen, deformation and damage can be focused at the desired areas while delamination is prone to happen. A cohesive surface method is used to simulate the delamination feature. An interesting phenomenon is found in these carbon fiber

EP

composites. The Young’s moduli in tension and compression are different for the same material. This phenomenon was also noticed and studied by van Dreumel (1982), Van Dreumel and Kamp

AC C

(1977), Furuyama, Higuchi et al. (1993), Melanitis, Tetlow et al. (1994), Đorđević, Sekulić et al. (2007) and Djordjević, Sekulić et al. (2010). This phenomenon is also taken into account in the model. A material post-failure softening model depending on loading conditions is postulated to predict the fracture propagation. This model is implemented to the ABAQUS/Explicit as a material user subroutine (VUMAT) to simulate all the failure modes observed in the tests with good accuracy. The developed model is very suitable for large-scale practical engineering applications. Page 7 of 47

ACCEPTED MANUSCRIPT

2. Experiments 2.1.Experimental setup The specimens investigated in this research are rectangular plates, which are 94.50 mm long, 20

RI PT

mm wide and 4mm thick. A notch of radius 12 mm is introduced on each side and the minimum thickness of the plate is 1.5 mm in the center. Specimens are made of unidirectional IM7 graphite fiber with 8552 epoxy. In order to reduce stress concentration and investigate the failure of the

SC

notched part, four aluminum tabs were attached to both sides of the specimens where clamps will be on. The fiber direction is along the axial direction. See Figure 1.

M AN U

The tests were conducted at the Impact and Crashworthiness Laboratory in the Department of Mechanical Engineering at Massachusetts Institute of Technology, with an Instron load frame with both vertical and horizontal actuators for biaxial testing. See Figure 2. Combined loadings of tension/compression and shear are applied to the specimens. Along the fiber direction, tensile or compressive loadings is applied. Meanwhile, shear loading is applied on the transvers direction.

TE D

Combination of the two axis loadings generated a series of loading conditions and a schematic plot is provided to illustrate the loading path in Figure 1(c). resultant total force

and the horizontal force

represents the angle between the

. It can be calculated through

=

°

atan(

EP

). Totally, seven loading conditions were designed to test this material as listed in Table 1.

/

Most of these tests were conducted under the load control mode except for uniaxial compression

AC C

and shear tests, which had been done by both load control and displacement control methods on two different actuators.

Page 8 of 47

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

EP

Figure 1: A picture of a real specimen (a) front view (b) side view and (c) an illustration of loading condition with angle .Note that specimens are spray painted for digital image recording

AC C

and analysis.

Page 9 of 47

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Figure 2: Instron dual-actuator loading frame

TE D

Table 1: Summary of loading conditions Loading conditions

Uniaxial compression

EP

Compression + Shear Compression + Shear

AC C

Shear

Tension + Shear Tension + Shear

Uniaxial tension

Repeat tests 90°

3

45°

3

70° 0°

70° 80° 90° Total:

3 3 2 3 3

20

Page 10 of 47

ACCEPTED MANUSCRIPT

For cases that angle

needs to be maintained during a process, force control testing method

(mode) is applied to the specimen. In order to capture the deformation and failure modes, an optical measurement system recording images during tests was used to accurately measure

RI PT

displacement and strain fields. Images were captured by two cameras: one in the front view and the other one in the side view. Digital image correlation (DIC) technique was used to post-process the image data. Semi-gloss black and white paint were sprayed in small dots randomly on the

SC

specimen surfaces. Note that many specimen pictures in this paper show sprayed patterns.

2.2.Experimental results

M AN U

2.2.1.Uniaxial compression Three specimens were tested under uniaxial compression loading condition. Two of them were tested with displacement control and one was tested with force control. Tests curves show linear elasticity before failure with no obvious plasticity. The tests show good repeatability in regard to failure modes and material strength. All tests demonstrated brittle failure modes with local

TE D

buckling during tests. See Figure 3. The measured forces at total failure are in a range of 23.76 to 29.40 kN. The recorded force-displacement curve is shown in Figure 4, where test results of other

AC C

EP

loading conditions are also included.

Figure 3: Brittle failure modes observed under compression test (a) side view (b) front view

Page 11 of 47

ACCEPTED MANUSCRIPT

70 60 50

)

Tension + Shear

RI PT

Uniaxial Tension

30 20 10 0

SC

Vertical Force (

40

-10 -20

Uniaxial Compression

-30 -0.5

0

(a)

7

3

2

2.5

)

TE D EP

) Shear Force (

4

0.5 1 1.5 Vertical Displacement (

Tension + Shear

6 5

M AN U

Compression + Shear

Shear

AC C

2 1

Compression + Shear

0

0 (b)

0.5 1 1.5 Horizontal Displacement (

2

2.5 )

Page 12 of 47

ACCEPTED MANUSCRIPT

Figure 4: A collection of force-displacement curves of all seven loading conditions in (a) vertical and (b) horizontal directions

=

70° loading

RI PT

2.2.2.Compression + Shear with = 70° Three specimens were tested under combined compression and shear with

condition. Experiments show good repeatability with force-displacement curves following similar trend and similar failure modes. In vertical direction, material presents linear elasticity of fibers

SC

and a small range of nonlinearity before failure. In horizontal direction, materials show a typical elastic-plastic behavior of polymer matrix. All tests show brittle failure modes during tests. The force-displacement curves of vertical and horizontal directions are also shown in Figure 4.

M AN U

2.2.3.Compression + Shear with = 45° Three specimens were tested under combined compression and shear with

condition. Similar behaviors as the ones with

=

=

45° loading

70° were observed. Experiments show good

repeatability in terms of force-displacement curves and failure modes. In vertical direction,

TE D

materials show linear elasticity till failure. In horizontal direction, materials show a typical elastic-plastic behavior of polymer matrix. All tests show brittle failure modes during tests with a skew failure surface. For the two combined compression and shear loading, one can see that the

EP

shear loading greatly influence the strength in compressive direction. This is due to the instability introduced by shear loading which accelerate the buckling.

AC C

2.2.4.Shear Three specimens were tested under shear loading condition. Two of them were tested with

displacement control and the third one was tested with force control. For the two displacement control tests, material presents typical elastic-plastic behavior (due to matrix deformation) before failure initiation. After failure initiation, a long range of material softening occurs. At this stage, the matrix gradually degraded and the fiber takes the main load. For the one test with force control, material behaves the same before failure initiation. After reaching the material strength limit, failure initiates and material cannot sustain increasing loads in the horizontal shear direction, Page 13 of 47

ACCEPTED MANUSCRIPT

and therefore total failure was observed. This phenomenon revealed that the matrix gradually degraded under displacement control mode and catastrophically failed under force control mode. A comparison of specimens between displacement control and force control shear tests is shown

M AN U

SC

RI PT

in Figure 5.

Figure 5: Comparison of last test step in shear tests between (a) displacement control and (b) force control tests

TE D

2.2.5. Tension + Shear with = 70° Two specimens were tested under combined tension and shear with

= 70° loading condition.

For both vertical and horizontal directions, material shows an elastic-linear plastic behavior. One thing to mention here is that these two tests didn’t run until total failure, because they were

EP

stopped when some delaminations were observed from the side view. See Figure 6. Due to high fiber tensile strength, delamination is the main degradation. It should be noted that the material

AC C

can still sustain more loads under the same loading path if tests continued.

Page 14 of 47

SC

RI PT

ACCEPTED MANUSCRIPT

Figure 6: Delamination failure features observed under tests of combined tension and shear with

M AN U

β=70°. (a) Side view after tests (b) Front view during tests

2.2.6.Tension + Shear with = 80° Three specimens were tested under combined tension and shear with

= 80° loading condition.

Experiments showed good repeatability. For the vertical direction, the material presents a linear elastic range and then the displacement decreased with a load increasing. For the horizontal

tension and shear with

TE D

direction, the material showed an elastic-linear plastic behavior like the one under combined

= 70° loading condition. It is interesting to find that the vertical

displacement decreased as the load was increasing. This resulted from the strong anisotropy of

EP

the unidirectional PMCs material. As the material went through increasing tensile and shear loads, matrix degraded faster than fiber. The specimen had to be rotated more so that the force

AC C

decomposed from fiber tension force can contribute more to the horizontal direction. In this way, fibers got stretched with the increasing horizontal displacement and therefore the vertical force increased with a decreasing vertical displacement. For this loading conditions, material did not totally fail during the tests. Delamination is the main degradation. The specimen can still sustain more loads under the same loading path.

Page 15 of 47

ACCEPTED MANUSCRIPT

2.2.7.Uniaxial tension Three specimens were tested under uniaxial tension along fiber direction. Good repeatability was

obtained. Material showed a small range of linear elasticity followed by large range of

RI PT

nonlinearity. The nonlinearity is brought in by delamination and sequential fiber pullout. The fiber pullout only happened close to total fracture. These severe delamination and fiber-matrix separation failure features are shown in Figure 7, which is similar to “ductile failure” in some

TE D

M AN U

SC

sense especially when the individual fibers are not simulated.

EP

Figure 7: (a) Severe delamination and (b) fiber-matrix separation observed during uniaxial

AC C

tension tests (a) side view (b) front view.

2.3.Summary of experimental results

A collection of force-displacement curves of all conducted tests are shown in Figure 4. The PMCs materials exhibit strong asymmetric behaviors between tension and compression. Brittle failure features like local buckling were shown in compressive dominated loading conditions while “ductile failure” features like delamination and fiber pullout were observed in tensile and shearing dominated loading conditions. Material strengths in tension and compression showed notable difference. Specimens can sustain up to 70 kN in tension while only about 30 kN in Page 16 of 47

ACCEPTED MANUSCRIPT

compression. With this great asymmetry in tension and compression, delamination became the main failure feature in tension dominated tests. In addition, an interesting phenomenon in elastic moduli of tension and compression along fiber direction was exhibited. One can clearly see that

RI PT

composites are stiffer in elastic range under tension loading conditions than that of compression loading conditions.

Tests under shear loading conditions exhibit a unique material behavior, which include linear

SC

elastic, material strain hardening and post-failure softening behaviors. Matrix takes the main loads under this loading condition at the beginning and then fibers gradually takes the main loads

M AN U

when the matrix goes to failure and fibers rotate. Tests of loading conditions combined with shear show plasticity in both vertical and horizontal directions for most of the cases. The material nonlinearity mainly comes from the matrix and progressive delamination of composites.

3. Theoretical modeling

TE D

This section presents the constitutive model formulation which comprises of elasticity, plasticity, damage accumulation and fracture propagation.

3.1.Elasticity

EP

In elastic range, the stress is a linear function of strain. An orthotropic elastic material model is used for this carbon fiber composites material, where there are nine engineering constants (three , three Poisson’s ratio , and three shear modulus

AC C

Young’s modulus

). If we define the

direction 1 as the fiber direction, a new stress state parameter describing arbitrary loading conditions for unidirectional fiber composites is defined as

where

=

!"#

,

(1)

is the stress along fiber direction and !"# is von-Mises equivalent stress. Note that the

von-Mises equivalent stress is used as a measure to normalize the

, rather than the composite

Page 17 of 47

ACCEPTED MANUSCRIPT

material strength. In this way,

can be used to distinguish different loading conditions including

tension dominated, compression dominated and shear dominated loading conditions, as shown in

Shear dominated

∞

Figure 8: Parameter

0

1 Uniaxial tension

+∞

M AN U

-1 Uniaxial compression

Tension dominated

SC

Compression dominated

RI PT

Figure 8.

describing different loading conditions of unidirectional fiber composite

As shown in the experimental section, tension and compression tests along the fiber direction have different Young’s modulus. This unsymmetrical elasticity brings difficulty in modeling the

follows.

TE D

behavior of PMCs. An elastic model considering the unsymmetrical elasticity is proposed as

( ) = %∘ +

(2)

is the Young’s modulus in fiber direction. % and %( are two parameters determining the

EP

where

%( , 1 + % *+, -

AC C

upper and lower bounds of

( ), and % is a parameter to fit the transition from tension to

compression loading conditions. The reference Young’s moduli on tension and compression can be obtained by converting the uniaxial tension and uniaxial compression force-displacement curves to stress-strain curves and calculating the slopes of the resultant curves. The reference Young’s moduli have been plotted in dots in Figure 9. With parameters % = 70521/01 ,

% = 21.42 and %( = 31160/01, an example of Young’s modulus (

) as a function of on

is

also illustrated in Figure 9. The proposed elastic model clearly distinguishes stiffness asymmetry between the tension and compression with a smooth transition through the shear dominated Page 18 of 47

ACCEPTED MANUSCRIPT

loading conditions. The parameters are modified to well fit all the loading conditions. Note that the stiffness asymmetries in other two transverse directions (

5

and

6

if exist) are not taken into

TE D

M AN U

SC

RI PT

account due to lack of experimental data.

parameter ( )

) and the stress state

EP

Figure 9: Relationship between Young’s modulus along fiber direction (

AC C

3.2.Yield locus and Failure initiation locus One can see that non-linear or unrecoverable deformation before total failure has been observed in almost all the experiments presented here except for uniaxial compression along fiber direction. This non-linear or unrecoverable deformation phenomenon was also reported by Lonetti, Barbero et al. (2004). This phenomenon is called “yield” in this paper as a concept borrowed from metal plasticity theories. An anisotropic yield function in the form of Tsai-Wu failure criterion (Tsai and Wu 1971) is adopted in this paper.

Page 19 of 47

ACCEPTED MANUSCRIPT 78 9

+766

:; < 5 6

=7

+ 75

+ 7==

5 =

5

+ 7>>

+ 76

6

+ 7=

+ 7??

5 >

+ 7>

=

>

+ 27 5

5 >

+ 7?

5

+7

?

+ 27 6

5

+ 755

+ 2756

6

5 5

(3)

5 6

where subscript 1 refers to the fiber direction while 2 and 3 refer to the transverse directions. 7:

RI PT

and 7:; are parameters to be determined from the yield or non-linear initiation stresses. The

interaction parameter 7 5 takes the form of Mises-Hencky criterion (Tsai and Hahn 1981) . If the

yield stress in tension (@A ), compression (@B ) and shear (@C ) from directions 1, 2, and 3 are known,

then the model coefficients can be determined as follows. 1 1 , 76 = @B5 @A6

1 , 7 = 7> = 7? = 0 @B6 =

(4)

1 1 1 1 1 1 , 755 = , 766 = , 7== = 5 , 7>> = 5 , 7?? = 5 @A @B @A5 @B5 @A6 @B6 @C56 @C6 @C 5

M AN U

7 =

1 1 , 75 = @B @A5

SC

1 @A

7 =

1

75=

2D@A @B @A5 @B5



Meanwhile, the Tsai-Wu criterion was also used to describe the failure initiation locus for PMCs

TE D

which takes the form of

E 9 :; <

and

:;

5 66 6

+

+

5 == =

EP

:

+

=

+

5 5

+

5 >> >

6 6

+

+

5 ?? >

= =

+2

+

5

> >

+

5

? ?

+2

+

5

6

6

+

5 55 5

(5)

56 5 6 .

+2

are free parameters to be determined from the fracture data points using Eq. (9). The

AC C

failure initiation stresses in tension (

A ),

B)

compression (

and shear ( C ) from directions 1, 2,

and 3 are needed to calibrate the model.

=

1

=

A

1

A

B

1 ,

B

55

=

,

5

1

=

A5 B5

1 ,

A5 66

5

=

1 =

B5

,

1

6

=

A6 B6

2D

A

,

1 B

1

A6

==

1 =

B6

1

,

=

=

5 ; >> C56

A5 B5



=

>

= 1

5 C6

?

;

= 0 ??

=

(6) 1

5 C 5

Page 20 of 47

ACCEPTED MANUSCRIPT

Based on the current test results, the initial yield and total fracture stresses compared with model prediction are shown in Figure 10. In the experimental data (both initial yield and total fracture), the nominal stress values are calculated based on the smallest cross section. The initial yield data

RI PT

points are usually calculated from the nonlinearity initiation points at the force-displacement curves. For those with very little or no material nonlinearity (because of brittleness under some loading conditions), initial yield and total fracture points are defined to be very close, i.e. the

SC

uniaxial compression. For other cases, the total fracture is defined as either peak load or clear failure observed from digital images.

M AN U

Regarding the model prediction, failure curves are fit with test data points, especially for those of compression, combined compression with shear, and shear loadings. It is noticed that the model predicted failure initiation locus is within the range between initial yield locus and total fracture data points since there are still a stage of fracture propagation. One may notice that in the

5-

stress space, the fracture data points in first quadrant are far outside the corresponding fracture

TE D

locus This can be explained as follows. For these shear combined with tension loading conditions, shear strength is enlarged due to the decomposed force component from fiber tension force. It is shown in Figure 11 that the shear strength is calculated from measured horizontal force which

EP

includes force component decomposed from total force along fiber direction. In this way, the yield and failure loci are somewhat conservative in the first quadrant of

5-

stress space in

AC C

order to better describe the material behavior. It should be noted that the uniaxial tension data points are “under predicted” in the failure initiation model, which is actually not true. This “early failure” is due to the delamination of the designed specimen notches rather than fiber break along the direction 1. The final yield and failure initiation loci will be determined through iterations in FE simulations.

Page 21 of 47

ACCEPTED MANUSCRIPT

Delamination induced failure

200

150 100 50

100

0

0

-200 -2000

-100 0

2000

4000

-150 -150

6000

-100

250 Shear strength enlarged Decomposed from fiber tension

200 100

0

50

100

In it ia l yield da t a poin t s

50

In it ia l yield m odel pr edict ion

0 -100 0

2000

M AN U

Tot a l fr a ct u r e da t a poin t s In it ia l fa ilu r e m odel pr edict ion

-50 -150 -2000

-50

SC

150

RI PT

-50 -100

4000

Delamination induced failure

TE D

Figure 10: Comparison between initial yield and total fracture data points from tests and model

AC C

EP

predicted initial yield locus and failure initiation loci

Figure 11: Force in fiber direction (G) can be decomposed into horizontal force component (GH) and vertical force component (GI )

Page 22 of 47

ACCEPTED MANUSCRIPT

3.3.Plastic flow rule An associated flow rule (AFR) is assumed for the PMCs materials undergoing inelastic deformation observed in the tests. An anisotropic plastic potential takes the following form.

:; <

:; <

= 78 9

:; <

J9K̅M < = !AN

JAN 9K̅M < = 0

RI PT

!AN = 78 9

79

(7)

is the Tsai-Wu yield function shown in Eq. (3) or called Tsai-Wu equivalent stress

hereinafter. In this way, the initial yield point of the material will naturally start from !AN = 1 :; < .

Material strain hardening is assumed to follow

SC

according to the 7:; coefficients in 78 9

isotropic Swift power hardening law, which reads JAN = O(K̅AN + K )P . Here JAN and K̅AN are M

M

M AN U

normalized Tsai-Wu yield stress and work conjugate Tsai-Wu equivalent plastic strain defined by the yield criteria. O , K and Q are determined by the stress-strain curve obtained from corresponding tests.

According to AFR, the incremental plastic strains can be expressed as

TE D

RSM = RT

where RT is the proportionality factor.

U7 , UV

(8)

EP

The increment of plastic work per unit volume can be obtained by RW M =

M :; RK:;

= !AN RK̅AN M

(9)

AC C

Therefore, the effective plastic strain increment can be expressed as follows: M RK̅AN

=

M :; RK:;

!AN



(10)

As observed from the experiments, vertical direction force-displacement curves show little plasticity but polymer matrix provides the major source of plasticity. So, the Swift power equation will be calibrated from the pure shear test. See Figure 12. Work conjugate stress and strain are used in the data processing. Therefore, the “equivalent stress” start from unity and the Page 23 of 47

ACCEPTED MANUSCRIPT

“equivalent strain” are relatively larger than traditional ways. One can find that the curve fitting does not exactly follow the experimental data. Here the experimental stress-strain curve is nominal because the stress is not very uniform on the notched area. The curve is provided as a

RI PT

benchmark for the initial hardening curve fitting. The illustrated curve in Figure 12 is an optimized result through iterative FE simulations to achieve the best correlation in force-

6

SC

displacement responses.

Failure initiation Nominal curve from test data

M AN U

5 4

Post-failure softening

Swift power hardening

3

1 0

20

40

60

80

100

AC C

0

EP

TE D

2

Figure 12: Curve fitting of power hardening law for the pure shear loading condition

3.4.Damage accumulation and material post-failure softening

When Tsai-Wu failure criterion is not reached, damage indicator (X) is set to be equal to the normalized Tsai-Wu failure stress (Eq. (5)). Therefore, the material damage evolution is defined as

Page 24 of 47

ACCEPTED MANUSCRIPT X= 9

:; <, Y7 X

≤ 1.

(11)

When damage indicator X < 1, the material is intact with no damage. When X = 1, it indicates that failure initiates as “equivalent stress” reaches the critical value 1. Typically the material

RI PT

exhibits both elastic and plastic behaviors before failure initiates (X ≤ 1). However, sometimes failure initiation could occur before material yielding and therefore material plastic strain hardening could vanish, under some loading conditions like uniaxial compression for this

SC

material.

After failure initiation, the damage accumulation is defined by an incremental form: \ ]AN RK̅AN RX = . ( )

(12)

M AN U

M

To sum up, the damage indicator follows the following equation. 9

X=^ \_ 1+

:; <

M ]AN RK̅AN

( )

,0 ≤ X ≤ 1 . ,1 < X ≤ 2

(13)

TE D

Here \ is the characteristic length of a finite element in the simulation. ]AN is Tsai-Wu equivalent stress that incorporates the effect of material post-failure softening, which is different from !AN , the equivalent stress from the hardening model. RK̅AN is the incremental work ( ) is the toughness or critical strain energy release rate at fracture

EP

conjugate plastic strain.

M

which is also a function of loading condition . R ]AN K̅AN is the plastic strain energy incremental M

AC C

per unit volume, and \R ]AN K̅AN represents strain energy incremental per unit surface. This M

incremental form is based on the strain energy accumulation. After damage initiation, the strain energy is accumulated in the form of \ ]AN RK̅AN . When it reaches the critical toughness, the material point is defined as fracture (X = 2).

M

As stated earlier, carbon fiber composite material shows brittle failure in compressive loading conditions and ductile failure in tensile and shear dominated loading conditions. This feature is Page 25 of 47

ACCEPTED MANUSCRIPT ( ), which is

characterized by applying a loading condition dependent fracture toughness defined by

`

,

`

,

` 5

and

a

`

+

`

1 + % ab,c (-d-ef )



(14)

RI PT

where

( )=

are model parameters for describing the brittle-ductile fracture

transition. Parameters were obtained by calibrating each tested loading condition and the results are shown in Figure 13. The negative range of <

1 corresponds to the brittle failure modes in > 1 corresponds to the

SC

the compression dominated loading conditions. The positive range of

ductile failure modes in the tension dominated loading conditions. The values in the range of ≤ 1 correspond to the shear dominated loading conditions which is also ductile. The

M AN U

1≤

exact transition area needs to be determined from iterations of simulations or more test data.

12

)

TE D

10

Tensile loadings

EP

6 4

Shear loadings

AC C

Toughness (

8

Compressive loadings

2 0

-3

-2

-1

0

1

2

3

Figure 13: Material fracture toughness versus loading condition parameter Page 26 of 47

ACCEPTED MANUSCRIPT It is assumed that material softening starts after failure initiation (X > 1). This is an important aspect which affects the prediction for failure stress, failure strain and failure modes (Matzenmiller, Lubliner et al. 1995, Donadon, De Almeida et al. 2009). A material post-failure

]AN = !AN h1

i%( )(X

RI PT

softening model is postulated in the following expression: 1)Cj k, (Y7 1 < X ≤ 2)

(15)

where !AN is the Tsai-Wu equivalent stress including strain hardening. Therefore post-failure equivalent stress takes the product of both hardening !AN and softening coefficient h1

1)Cj k. i%( ) is a parameter to control the ultimate failure stress which also depends

SC

i%( )(X

on loading condition , and i# is a parameter to determine stress softening rate with respect to i%( )(X

M AN U

damage accumulation indicator X . The material post-failure softening coefficient h1

1)Cj k is demonstrated in Figure 14. In order to distinguish the softening between

different loading conditions, the parameter i%( ) takes the following form. i%( ) = l∘

l( *m 1 + % , (|-|d-of )

TE D

The relationship between i%( ) and loading condition parameter shear dominated loading conditions (around

(16)

is illustrated in Figure 15. At

= 0), i%( ) = 1 which gives a sharp drop in the

softened equivalent stress. This was found in the pure shear tests. For other loading conditions,

EP

the drop is not that sharp and therefore i%( ) takes a smaller number around 0.65. Note that

AC C

these softening parameters will be calibrated through several iterations of FE simulations. The details of obtaining these parameters can be found in the model calibration procedure in the next section.

Page 27 of 47

ACCEPTED MANUSCRIPT

1.2 Material intact

RI PT

0.8 0.6 0.4

Material softening

SC

Softening Coefficient

1

0.2

0

0.5

M AN U

0 1

1.5

2

Figure 14: Material post-failure softening evolution curve (set i%( ) = 1, i# = 0.08)

1.2

TE D

Shear loadings

1

0.8

EP

Compressive loadings

0.6 0.4

AC C -3

-2

Tensile loadings

0.2 0 -1

0

1

2

3

Figure 15: Relationship between i%( ) and loading condition parameter Page 28 of 47

ACCEPTED MANUSCRIPT

3.5.Cohesive surface Delamination was observed in some of the loading conditions like simple tension and tension combined with shear tests. This special damage feature was due to the notch design in the

RI PT

specimens and it will be modeled as cohesive surface failure in finite element simulations using ABAQUS/Explicit (Shaw 2005). The surface-based cohesive behavior consists of a linear elastic traction-separation behavior, damage initiation, and damage evolution. An uncoupled tractionseparation feature is assigned to the linear elastic behavior. A maximum stress criterion is utilized

SC

to model damage initiation. After damage initiation, a linear damage evolution is utilized. The key parameters used in the model are the tangential stiffness pmm , maximum shear stress qm and induced delamination.

4. Numerical simulation

M AN U

maximum total separation r s . The parameters are critical in shear condition due to the shear

TE D

4.1.Model calibration procedure

Finite element simulations for all loading conditions were performed using ABAQUS/Explicit. The proposed material model was implemented as a user material subroutine (VUMAT). The

EP

phenomena of crack initiation and propagation were simulated using an element deletion technique for the continuum model and an ABAQUS built-in cohesive surface technique for

AC C

delamination. Two types of finite element models with or without cohesive surfaces are shown in Figure 16. Specimens were modeled using C3D8R solid elements. The attached aluminum plates on both shoulder sides of carbon fiber composite specimen are modelled as a linear elastic material.

For the single part model, the PMCs specimen is treated as a homogenized material with four aluminum plates attached to both sides of top and bottom. Boundary conditions are applied to the outer surfaces of aluminum plates. In the case of carbon fiber composite with cohesive surfaces

Page 29 of 47

ACCEPTED MANUSCRIPT

model, the specimen is treated as five parts with four aluminum plates attached to both sides. The five parts of PMCs are attached together using the cohesive surface feature. Boundary conditions

TE D

M AN U

SC

RI PT

are applied to the outer surfaces of aluminum plates, too.

EP

Figure 16: Finite element models for unidirectional carbon fiber composite specimen: (left) Single part model, (right) Multiple parts with cohesive surfaces

AC C

The model calibration procedures are summarized as follows. 1. The first step was to calibrate the elasticity model (

( ) and other constants in stiffness

tensor t). Tested Young’s moduli under both uniaxial tension and uniaxial compression were

correlated with the model ones by adjusting the parameter % and %( . The main consideration is to make sure the Young’s moduli under tension (

> 1 ) and compression (

< 1)

dominated conditions to fit with the corresponding values. Iterations would be required to fit

Page 30 of 47

ACCEPTED MANUSCRIPT

the other loading conditions like tension or compression combined with shear by modifying the transition parameter % .

2. Secondly, the yield and failure initiation loci ( 78 9

:; <

and

E ( :; ) )

were fitted through

RI PT

correlating force-displacement curves and images captured during tests. For the yield locus, it will be fitted to the material nonlinearity initiation point. For the ones without material nonlinearity, the material nonlinear initiation point will be the failure initiation point like the

SC

uniaxial compressive loading. For the failure initiation point, it is simple for compression and combined compression-shear conditions. These failure initiation points are also the total

M AN U

fracture points since they show linear elasticity with brittle failure modes. As for the shear loading condition, the maximum load point is set as the failure initiation point. For the loading conditions incorporating tensile loads, shear-induced failure or delamination will take place and hence total fracture data points will be taken as outer limit for failure initiation locus.

TE D

3. Thirdly, the strain hardening model based on Tsai-Wu equivalent stress ( JAN (K̅AN )) was M

fitted using tested stress-strain data under shear. The selection of data was based on the occurrence of plastic behavior under this loading condition. Usually, carbon fiber does not

EP

show plasticity and polymer matrix will show plasticity. Shear loading condition helped to reveal the matrix plasticity and therefore the curve fitting of material hardening was based on

AC C

it. Modifications on the hardening model would be required to fit the other loading conditions. 4. Fourthly, the toughness function (

( )) was calibrated based on areas under the tested

stress-strain (or force-displacement) curves under different loading conditions. This was to assure the correlation of the ultimate fracture displacement. The other purpose was to simulate the brittle failure in compression dominated loading conditions and ductile failure in tension dominated loading conditions.

Page 31 of 47

ACCEPTED MANUSCRIPT 5. The softening function ( ]AN ) was calibrated by using the tested shear loading stress-strain curve to achieve the correct post-failure softening mode. As shown in the experimental results, shear tests have a long range of softening without total failure which brought difficulty in the softening calibration. A sharp softening was assigned to the softening model

The softening function needs to be calibrated with toughness (

( ), step 4) since they work

SC

together in the material failure behavior.

< 1).

RI PT

by setting i%( ) to be around 1 in the shear dominated loading conditions ( 1 <

6. In addition, for the multiple parts model with cohesive surfaces, additional steps are needed

to calibrate the stiffness matrix u, critical stress vector v and maximum effective separation

M AN U

#wx r# . A simple uncoupled traction-separation feature is assigned to the cohesive surface.

Since shear-induced failure/delamination is shown in the experiments, a critical shear stress component is the key parameter.

A set of fully calibrated model parameters for the unidirectional IM7 graphite fiber with 8552

illustrated in Figure 17.

TE D

epoxy composite materials is listed in Table 2. A flow chart of the above calibration process is

EP

Table 2: Calibrated material model parameters for numerical simulations (unit in MPa or unitless)

(e )

AC C

Elastic property

Yield criterion & Material hardening

5

6

70521

8217

8217

3720

21.42

31160

1800

770

43

@A

6

%

@B

%(

@A5

5

56

0.28

0.28

@B5

@C

133

5

48

6

5

56

0.28

3720

3720

O

K

Q

2.55

0.002 Page 32 of 47

0.15

ACCEPTED MANUSCRIPT

A

B

4150

Cohesive surface

771

`

Toughness & Material softening

A5

`

10

43

133

12.7

0.78

` 5

8.0

B

a

Calibration start

qm

25000

99

0.015

0.04

10.7

0.3

0.6

i#

i

i(

Toughness fitting

Yes Yield and failure initiation loci fitting

Does F-D hardening fit?

Does failure mode and F-D fit?

Yes

Calibration completed

Yes

EP

No

Material softening fitting

No

TE D

Plasticity fitting for shear test

No

M AN U

Does tension & compression F-D slope fit?

rs

89

SC

Young’s modulus fitting

pmm

C 5

RI PT

Failure criterion &

AC C

Figure 17: A flow chart of model calibration procedure. F-D stands for force-displacement curve.

4.2.Simulation results

The FE simulation results will be presented in two subsections. Tests without delamination failure were simulated using single part model. Other tests with delamination failure were simulated using multiple parts model with cohesive surface.

Page 33 of 47

C

ACCEPTED MANUSCRIPT

4.2.1. Simulations using single part model (a) Uniaxial compression Figure 18 shows a photo of the uniaxial compression test of PMCs specimen compared to the FE

RI PT

simulation results. A brittle fracture due to local buckling failure was observed in both experiment and simulation. A straight failure surface is well predicted by the simulation. In both experiments and simulation, the failure happens in a short time and the whole cross section breaks immediately. The elements shown in red are the most critical and close to fracture. The predicted

AC C

EP

TE D

M AN U

elasticity slope and ultimate fracture limit.

SC

force-displacement curve from simulation well duplicated the experimental ones in terms of both

Page 34 of 47

ACCEPTED MANUSCRIPT

Figure 18: Comparison between FE simulations and test results of uniaxial compression test. Failure modes in (a) experiment (b) FE simulation; (c) Force–displacement curves of experiments and simulation

=

70°, FE simulation well reproduced

RI PT

(b) Compression + Shear with = 70° For the combined compression and shear loading with

the experiment in regards to force-displacement evolution, fracture limit and failure modes. See

SC

Figure 19. Both the vertical and horizontal force-displacement curves from simulation follow the experimental ones. In fiber direction the material mostly exhibits elasticity while in transverse

M AN U

direction it shows nonlinearity due to the polymer matrix plasticity. The predicted ultimate fracture forces in both directions fit the experiments with a reasonable accuracy. The brittle

AC C

EP

TE D

failure mode and skew failure surfaces due to combined loading are also duplicated in simulation.

Figure 19: Comparison between numerical simulation and test results of combined compression and shear with

=

70°. Total failure in (a) experiment (b) simulation. Force-displacement

Page 35 of 47

ACCEPTED MANUSCRIPT

curves of (c) vertical direction (d) horizontal direction. (Solid curve: experiment; Dashed curve: simulation)

=

45°, it is similar to the test of

=

70°.

RI PT

(c) Compression + Shear with = 45° For the combined compression-shear loading with

The FE simulation duplicates the experiments well. Both force-displacement curves from simulation correlate well with the experimental ones. See Figure 20. Similarly, in fiber direction

SC

the material exhibits elasticity while in transverse direction shows nonlinearity. The fracture limits in both directions well fit the experiments. The brittle failure mode and skew failure surface

M AN U

are reproduced by simulation. For the above three types of tests with compressive loading

AC C

EP

TE D

investigated, brittle failure modes play a major role.

Figure 20: Comparison between numerical simulation and test results of combined compression and shear with

=

45°. Total failure in (a) experiment (b) simulation. Force-displacement

Page 36 of 47

ACCEPTED MANUSCRIPT

curves of (c) vertical direction (d) horizontal direction. (Solid curve: experiment; Dashed curve: simulation)

RI PT

(d) Shear For the shear loading condition, a significant amount of matrix plasticity is revealed. A

comparison of the test and simulation results is shown in Figure 21. The matrix plasticity is well captured by the model, and the simulation results well duplicate this feature. The simulation used

SC

the same displacement control as the experimental one. The force-displacement curve fits well in the elastic and plastic hardening part. After the damage initiation point, the model exhibits a long

M AN U

range of material softening. Final failure is not shown in both experiment and simulation in the tests with displacement control. The material can still sustain some deformation and absorb more fracture energy. A deformation pattern comparison between experiment and simulation

AC C

EP

TE D

demonstrates the good predicting capability of the model in capturing the material plastic flow.

Page 37 of 47

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Figure 21: Comparison between numerical simulation and test results of the shear test. Final

EP

specimen configuration of (a) experiment (b) simulation; (c) Force-displacement curves of

AC C

experiments and simulation

4.2.2. Simulations using multiple parts model with cohesive surfaces

As delamination feature is not shown in the loading conditions incorporating compression loads in the fiber direction, simulations with cohesive surface model are done for the loading conditions incorporating tensile loads. Good delamination features have been shown in the simulations.

Page 38 of 47

ACCEPTED MANUSCRIPT (a) Tension + Shear with = 70° For the combined tension and shear loading with

= 70°, FE simulation with the cohesive

surface model well duplicates force-displacement curves. The key consideration lies in the failure

RI PT

features where delamination is clearly shown in the simulation as the loads gets larger and the matrix between fibers cannot sustain the shear stress. As the delamination propagates in the simulation, contacts between the newly developed surfaces decrease and therefore the loads decreases in the simulation. In the experiments, however, there are large clamping force on the

SC

aluminum tabs and friction between the newly developed surfaces which can further sustain the loads. Using the cohesive surface model, the simulation can well capture the elastic and material

AC C

EP

TE D

feature can be reproduced.

M AN U

strain hardening behavior before delamination propagation. In addition, the delamination failure

Page 39 of 47

TE D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

shear with

EP

Figure 22: Comparison between numerical simulation and test results of combined tension and = 70°. Specimen configuration before severe delamination propagation of (a)

AC C

experiment (b) simulation (contour showing damage indicator X , similarly hereinafter);

Delamination failure feature (c) remaining after experiment (d) during simulation. Forcedisplacement curves of (e) vertical direction (f) horizontal direction. (Solid curve: experiment; Dashed curve: simulation) (b) Tension + Shear with = 80° For the combined tension and shear loading with

= 80°, FE simulations correlate well with the

experiments in the force-displacement curve from both directions. In the vertical direction, the Page 40 of 47

ACCEPTED MANUSCRIPT

simulation reproduces the reverse curve from the experiment. In the horizontal direction, the force-displacement curve shows an elastic-plastic trend similar to the experiments. Experiments shows delamination as the force in the horizontal direction gets large. This failure feature can also

AC C

EP

TE D

M AN U

SC

RI PT

be reproduced by the model with cohesive surface.

Figure 23: Comparison between numerical simulation and test results of combined tension and shear with

= 80°. Specimen configuration before severe delamination propagation of (a)

experiment (b) simulation. Delamination failure feature (c) remaining after experiment (d) during simulation. Force-displacement curves of (e) vertical direction (f) horizontal direction. (Solid curve: experiment; Dashed curve: simulation)

Page 41 of 47

ACCEPTED MANUSCRIPT

(c) Uniaxial tension For the uniaxial tension loading condition, force-displacement curve from simulation correlates

well with experiments before severe delamination occurs. A linear elastic range and small range

RI PT

of delamination propagation range are well duplicated. The delamination and subsequent propagation are shown in the simulation as the experiments. The slope of the simulation curve is

EP

TE D

M AN U

SC

larger than the compression side as observed in the experiments.

Figure 24: Comparison between numerical simulation and test results of uniaxial tension test.

AC C

Specimen configuration before severe delamination propagation of (a) experiment (b) simulation. Delamination initiation during (c) experiment (d) simulation. (e) Force-displacement curves of experiments and simulation

To sum up, the multiple parts model with cohesive surfaces behave similar to the single part model before delamination takes place. Delamination and propagation failure feature are well reproduced by the cohesive surface method.

Page 42 of 47

ACCEPTED MANUSCRIPT

5. Conclusions In this paper, experimental investigation and numerical simulations have been conducted on unidirectional carbon fiber composites. Biaxial loading conditions including loads along fiber and

RI PT

transverse direction applied on the composites are investigated. Strength asymmetry is shown between tension and compression dominated tests. Compression dominated tests show brittle fracture mode like local buckling, while tension dominated tests show fracture mode like

material hardening and softening without total failure.

SC

delamination and progressive fiber breakage. Pure shear tests with displacement control show

is proposed to distinguish among

M AN U

In the modelling method, a new loading condition parameter

compression, shear and tension dominated conditions. Young’s modulus asymmetry is found in tension and compression tests and a model has been proposed to describe this phenomenon. A Tsai-Wu form yield criterion and associated plastic flow rule are applied to this material in modelling the material nonlinearity. The modified Tsai-Wu failure criterion is also utilized as the

TE D

failure initiation criterion. The combined effects of toughness and softening behavior lead to the two different failure modes: brittle fracture and ductile fracture. However, the model still cannot well capture the behavior after severe delamination. In general, good correlations are achieved for

EP

all the loading conditions up to failure initiation. Failure propagation like delamination is simulated using the cohesive surface method.

AC C

The FE simulation results validated the applicability of this stress based damage/plasticity coupling nonlinear model for unidirectional carbon fiber material. This model will have a good potential to predict fracture modes for a wide range of fiber reinforced composite materials. Validation of this model on laminates with different layups will be our next step study.

Page 43 of 47

ACCEPTED MANUSCRIPT

Acknowledgements Thanks due to Dr. Kai Wang and Dr. Meng Luo from MIT for assisting the biaxial tests. Thanks are also due to Prof. Tomasz Wierzbicki for providing the access to the loading frame. The

financial support from University of Central Florida is acknowledged.

SC

References

RI PT

authors appreciate Dassault Systèmes for providing software license of Abaqus (Simulia). Partial

AC C

EP

TE D

M AN U

Bandaru, A. K., V. V. Chavan, S. Ahmad, R. Alagirusamy and N. Bhatnagar (2016). "Ballistic impact response of Kevlar® reinforced thermoplastic composite armors." International Journal of Impact Engineering 89: 1-13. Barbero, E. J. and P. Lonetti (2002). "An inelastic damage model for fiber reinforced laminates." Journal of Composite Materials 36(8): 941-962. Benzeggagh, M. and M. Kenane (1996). "Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus." Composites science and technology 56(4): 439-449. Brewer, J. C. and P. A. Lagace (1988). "Quadratic stress criterion for initiation of delamination." Journal of composite materials 22(12): 1141-1155. Buitrago, B. L., S. K. García-Castillo and E. Barbero (2013). "Influence of shear plugging in the energy absorbed by thin carbon-fibre laminates subjected to high-velocity impacts." Composites Part B: Engineering 49: 86-92. Camanho, P. P., P. Maimí and C. Dávila (2007). "Prediction of size effects in notched laminates using continuum damage mechanics." Composites Science and Technology 67(13): 2715-2727. Car, E., S. Oller and E. Oñate (2000). "An anisotropic elastoplastic constitutive model for large strain analysis of fiber reinforced composite materials." Computer Methods in Applied Mechanics and Engineering 185(2): 245-277. Cherniaev, A. and I. Telichev (2015). "Meso-scale modeling of hypervelocity impact damage in composite laminates." Composites Part B: Engineering 74: 95-103. Chow, C. and F. Yang (1997). "Three-dimensional inelastic stress analysis of center notched composite laminates with damage." International Journal of Damage Mechanics 6(1): 23-50. Chowdhury, N. M., W. K. Chiu, J. Wang and P. Chang (2016). "Experimental and finite element studies of bolted, bonded and hybrid step lap joints of thick carbon fibre/epoxy panels used in aircraft structures." Composites Part B: Engineering 100: 68-77. Company, T. B. (2008). "COMPOSITES IN THE AIRFRAME AND PRIMARY STRUCTURE." from http://www.boeing.com/commercial/aeromagazine/articles/qtr_4_06/article_04_2.html. Djordjević, I. M., D. R. Sekulić, M. N. Mitrić and M. M. Stevanović (2010). "Non-Hookean elastic behavior and crystallite orientation in carbon fibers." Journal of Composite Materials. Donadon, M. V., S. F. M. De Almeida, M. A. Arbelo and A. R. de Faria (2009). "A threedimensional ply failure model for composite structures." International Journal of Aerospace Engineering 2009. Donaldson, S. (1985). "Fracture toughness testing of graphite/epoxy and graphite/PEEK composites." Composites 16(2): 103-112.

Page 44 of 47

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

Đorđević, I. M., D. R. Sekulić and M. M. Stevanović (2007). "Non-linear elastic behavior of carbon fibres of different structural and mechanical characteristic." Journal of the Serbian Chemical Society 72(5): 513-521. Ferry, L., D. Perreux, J. Rousseau and F. Richard (1998). "Interaction between plasticity and damage in the behaviour of [+ φ,− φ] n fibre reinforced composite pipes in biaxial loading (internal pressure and tension)." Composites Part B: Engineering 29(6): 715-723. Franklin, H. G. (1968). "Classic theories of failure of anisotropic materials." Fibre Science and Technology 1(2): 137-150. Furuyama, M., M. Higuchi, K. Kubomura, H. Sunago, H. Jiang and S. Kumar (1993). "Compressive properties of single-filament carbon fibres." Journal of materials science 28(6): 1611-1616. Goyal, V. K., E. R. Johnson and C. G. Davila (2004). "Irreversible constitutive law for modeling the delamination process using interfacial surface discontinuities." Composite Structures 65(3): 289-305. Guermazi, N., A. B. Tarjem, I. Ksouri and H. F. Ayedi (2016). "On the durability of FRP composites for aircraft structures in hygrothermal conditioning." Composites Part B: Engineering 85: 294-304. Hahn, H. (1983). "A mixed-mode fracture criterion for composite materials." Composites Technology Review 5(1): 26-29. Hahn, H. and T. Johannesson (1984). "A correlation between fracture energy and fracture morphology in mixed-mode fracture of composites." Mechanical behaviour of materials- IV: 431438. Hashemi, S., A. Kinloch and J. Williams (1990). "The effects of geometry, rate and temperature on the mode I, mode II and mixed-mode I/II interlaminar fracture of carbon-fibre/poly (etherether ketone) composites." Journal of Composite Materials 24(9): 918-956. Hashin, Z. (1980). "Failure criteria for unidirectional fiber composites." Journal of applied mechanics 47(2): 329-334. Hashin, Z. and A. Rotem (1973). "A fatigue failure criterion for fiber reinforced materials." Journal of composite materials 7(4): 448-464. Hayakawa, K., S. Murakami and Y. Liu (1998). "An irreversible thermodynamics theory for elastic-plastic-damage materials." European Journal of Mechanics-A/Solids 17(1): 13-32. Hinton, M. J., A. S. Kaddour and P. D. Soden (2004). Failure criteria in fibre reinforced polymer composites: the world-wide failure exercise, Elsevier. Hoffman, O. (1967). "The brittle strength of orthotropic materials." Journal of Composite Materials 1(2): 200-206. Huang, J. and J. Black (1996). "An evaluation of chip separation criteria for the FEM simulation of machining." Journal of Manufacturing Science and Engineering 118(4): 545-554. Kachanov, L. (1958). "Time of the rupture process under creep conditions." Isv. Akad. Nauk. SSR. Otd Tekh. Nauk 8: 26-31. Kwon, Y. and C. Liu (1997). "Study of damage evolution in composites using damage mechanics and micromechanics." Composite structures 38(1): 133-139. Lee, J. D. (1982). Three dimensional finite element analysis of damage accumulation in composite laminate. Fracture of Composite Materials, Springer: 291-306. Lin, W.-P. and H.-T. Hu (2002). "Nonlinear analysis of fiber-reinforced composite laminates subjected to uniaxial tensile load." Journal of Composite Materials 36(12): 1429-1450. Lin, Z.-C. and S.-P. Lo (2001). "2-D discontinuous chip cutting model by using strain energy density theory and elastic-plastic finite element method." International journal of mechanical sciences 43(2): 381-398. Liu, P. and J. Zheng (2010). "Recent developments on damage modeling and finite element analysis for composite laminates: a review." Materials & Design 31(8): 3825-3834.

Page 45 of 47

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

Lonetti, P., E. J. Barbero, R. Zinno and F. Greco (2004). "Interlaminar Damage Model for Polymer Matrix Composites." Journal of Composite Materials 38(9): 799-800. Luo, H., Y. Yan, X. Meng and C. Jin (2016). "Progressive failure analysis and energy-absorbing experiment of composite tubes under axial dynamic impact." Composites Part B: Engineering 87: 1-11. Maa, R.-H. and J.-H. Cheng (2002). "A CDM-based failure model for predicting strength of notched composite laminates." Composites Part B: Engineering 33(6): 479-489. Maimí, P., P. P. Camanho, J. Mayugo and C. Dávila (2007). "A continuum damage model for composite laminates: Part I–Constitutive model." Mechanics of Materials 39(10): 897-908. Matzenmiller, A., J. Lubliner and R. Taylor (1995). "A constitutive model for anisotropic damage in fiber-composites." Mechanics of materials 20(2): 125-152. Melanitis, N., P. Tetlow, C. Galiotis and S. Smith (1994). "Compressional behaviour of carbon fibres." Journal of materials science 29(3): 786-799. Murakami, S. and K. Kamiya (1997). "Constitutive and damage evolution equations of elasticbrittle materials based on irreversible thermodynamics." International Journal of Mechanical Sciences 39(4): 473-486. Obikawa, T. and E. Usui (1996). "Computational machining of titanium alloy—finite element modeling and a few results." Journal of Manufacturing Science and Engineering 118(2): 208-215. Olsson, M. and M. Ristinmaa (2003). "Damage evolution in elasto-plastic materials-material response due to different concepts." International Journal of Damage Mechanics 12(2): 115-139. Orifici, A., I. Herszberg and R. Thomson (2008). "Review of methodologies for composite material modelling incorporating failure." Composite Structures 86(1): 194-210. Pinho, S., L. Iannucci and P. Robinson (2006). "Physically based failure models and criteria for laminated fibre-reinforced composites with emphasis on fibre kinking. Part II: FE implementation." Composites Part A: Applied Science and Manufacturing 37(5): 766-777. Schapery, R. (1990). "A theory of mechanical behavior of elastic media with growing damage and other changes in structure." Journal of the Mechanics and Physics of Solids 38(2): 215-253. Schipperen, J. (2001). "An anisotropic damage model for the description of transverse matrix cracking in a graphite–epoxy laminate." Composite structures 53(3): 295-299. Shaw, M. C. (2005). Metal cutting principles, Oxford university press New York. Shih, A. J. and H. T. Yang (1993). "Experimental and finite element predictions of residual stresses due to orthogonal metal cutting." International Journal for Numerical Methods in Engineering 36(9): 1487-1507. Sun, C. and J. Chen (1989). "A simple flow rule for characterizing nonlinear behavior of fiber composites." Journal of Composite Materials 23(10): 1009-1020. Tekalur, S. A., K. Shivakumar and A. Shukla (2008). "Mechanical behavior and damage evolution in E-glass vinyl ester and carbon composites subjected to static and blast loads." Composites Part B: Engineering 39(1): 57-65. Tong, L. (1997). "An assessment of failure criteria to predict the strength of adhesively bonded composite double lap joints." Journal of Reinforced Plastics and Composites 16(8): 698-713. Tsai, S. W. (1968). "Strength theories of filamentary structures." Fundamental aspects of fiber reinforced plastic composites: 3-11. Tsai, S. W. and H. T. Hahn (1981). "Introduction to composite materials, 1980." Lancaster, Pennsylvania, Technomic 453. Tsai, S. W. and E. M. Wu (1971). "A general theory of strength for anisotropic materials." Journal of composite materials 5(1): 58-80. Usui, E. (1997). "Application of computational machining method to discontinuous chip formation." Journal of Manufacturing Science and Engineering 119: 667. van Dreumel, W. (1982). A short note on the compressive behaviour of aramid fibre reinforced plastics, Delft University of Technology.

Page 46 of 47

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

Van Dreumel, W. H. and J. L. Kamp (1977). Non Hookean behaviour in the fibre direction of carbonfibre composites and the influence of fibre waviness on the tensile properties, Delft University of Technology. Vaziri, R., M. Olson and D. Anderson (1991). "A plasticity-based constitutive model for fibrereinforced composite laminates." Journal of Composite Materials 25(5): 512-535. Vyas, G., S. Pinho and P. Robinson (2011). "Constitutive modelling of fibre-reinforced composites with unidirectional plies using a plasticity-based approach." Composites Science and Technology 71(8): 1068-1074. Wang, C., Z. Liu, B. Xia, S. Duan, X. Nie and Z. Zhuang (2015). "Development of a new constitutive model considering the shearing effect for anisotropic progressive damage in fiberreinforced composites." Composites Part B: Engineering 75: 288-297. Xie, M. and D. F. Adams (1995). "A plasticity model for unidirectional composite materials and its applications in modeling composites testing." Composites science and technology 54(1): 11-21. Yamada, S. and C. Sun (1978). "Analysis of laminate strength and its distribution." Journal of Composite Materials 12(3): 275-284.

Page 47 of 47

ACCEPTED MANUSCRIPT

Biaxial fracture testing data of carbon fiber polymer matrix composites are provided. Brittle and ductile fracture modes are found in different loading conditions of PMCs. The dependency of material post-failure softening on stress states is studied. Good force-displacement curve correlations are obtained using the proposed model. Both failure initiation and propagation are well predicted by the proposed model.

AC C

EP

TE D

M AN U

SC

RI PT

• • • • •