Axial quasi-static crushing of composite tubes: An investigation on the influence of the modeling techniques

Abstract

Carbon fiber-reinforced plastic (CFRP) composites perform remarkably as crashworthy structures in transport vehicles. They promise to improve the crashworthiness of vehicles in addition to weight savings thanks to their superior strength-to-weight ratio. However, designing composite crashworthy structures is challenging since predicting the energy absorption values and crushing behavior are not straightforward. Finite element modeling is a very effective numerical technique to predict energy absorption values and shorten the design process of composite structures, yet it is also quite hard to simulate the crushing of composite structures due to their complex damage mechanisms. This paper presents an investigation of the axial crushing behavior of composite tubes made of woven fabric CFRP and the influence of the different modeling techniques. Macro-scale stacked shell modeling approach is adopted to model the composite tubes. Various virtual debris wedge models are incorporated to obtain realistic failure modes in simulations. Simulated crushing morphologies and energy absorption values are compared with the experimental test results. Furthermore, the prediction capability of the models and the influence of mesh geometry used for rigid flat platen crushers are examined. It is found that the geometry of the virtual debris wedge and mesh geometry of the flat platen crusher affect the simulated crushing modes and level of the absorbed energy predicted.

Keywords

Carbon fiber-reinforced plastic composites crashworthiness crush plugs finite element modeling virtual debris wedge

Introduction

Carbon Fiber-Reinforced Plastic (CFRP) composites have excellent mechanical properties, making them a subject of interest for industrial applications. In automotive and aerospace industries, demands for improving the fuel efficiency and safety motivate manufacturers to improve designs and utilize materials with high strength-to-weight ratios.^1–3 Due to their lightweight and high energy absorption capacity, CFRP composites can perform better than conventional metallic materials in crash events, making them beneficial to be used in crashworthy structures.

Over the years, a vast number of works have been conducted on CFRP composite crashworthy structures and their crushing characteristics. During a crash incident, the crashworthy structures are expected to exhibit a progressive crushing behavior to absorb the kinetic energy in a stable and controlled way. The conventional metallic crash components absorb the kinetic energy through plastic deformation and friction mechanisms, whereas CFRP composites absorb the kinetic energy through a combination of different mechanisms: (1) intralaminar damage mechanisms, which involve fiber and matrix fractures, (2) interlaminar damage mechanism, referred as delamination, and (3) friction.^4–8 According to the literature, the progressive crushing modes of CFRP composites subjected to axial crush can be categorized into three modes. These modes are splaying, fragmentation, and local buckling.^4,5,9 Local buckling is the same progressive folding mode that occurs on metallic structures. Splaying is the mode that involves the formation of fronds combined with vertical tearing of the composite wall. Fragmentation is the mode in which the small fragments occurs in the crush zone and forced to be separated from the wall. These modes can be observed in tandem during crushing incidents.^4,5

Carbon fiber-reinforced plastic composite has great advantages over traditional materials as crashworthy components. However, it is very challenging to predict the energy absorption values due to the complexity of damage mechanisms. Numerous studies exist in the literature about the parameters that affect the crushing characteristics of CFRP composite tubes subjected to axial crush.^4,5,8–13 Different parameters such as composites microstructure, component geometry, trigger mechanisms, and crushing conditions affect the crushing characteristics of the crashworthy structures. Yet, there is not a complete agreement about which parameter has the most significant influence. Therefore, studies to investigate and understand the crushing characteristics of CFRP composites and the parameters that affect the crushing behavior are still critical.

Evaluation of crushing characteristics of crashworthy structures depends mostly on experimental testing. With the developments in numerical methods and computational techniques, the capability of predicting crushing characteristics and energy absorption values of composite structures are highly improved. The essential numerical method for investigating crashworthy structures is Finite Element Analysis (FEA). Researchers are interested in developing FE models since they shorten the design process through simulations rather than costly experimental tests. FEA is quite capable of simulating the crushing behavior of metallic components accurately and rapidly due to their simple plastic folding crushing behavior and well-defined constitutive models. However, due to the complex structure of composite materials, simulating CFRP composite structures requires more advanced constitutive models and it is more time consuming.

Various studies exist in the literature about modeling the axial crushing behavior of CFRP composite structures. Israr et al.¹⁴ conducted experimental quasi-static compression tests on cross-ply CFRP flat plates at low crush speed rates. They also developed a macro-mechanical model where each ply is meshed with 3D brick finite elements and cohesive elements for interlaminar behavior. Chiu et al.¹⁵ implemented a macro-scale approach composite damage model that addresses intralaminar and interlaminar failure behaviors on a unidirectional composite tube crushing modeled with 3D elements. The intralaminar damage model is based on continuum damage mechanics (CDM). It is implemented as a VUMAT subroutine to Abaqus/Explicit software and interlaminar behavior is defined with cohesive surfaces.

Alternative to modeling with 3D brick elements, there are studies in which structures are modeled using shell elements. Sokolinsky et al.¹⁶ examined the crushing behavior of a corrugated plate manufactured from carbon-epoxy fabric prepregs experimentally and numerically. They modeled each ply individually using continuum shell elements in a stacked shell approach. They also proposed a constitutive intralaminar behavior model for fabric-reinforced composites using cohesive surfaces. Numerical simulations are carried out on Abaqus/Explicit software. Zhu et al.¹⁷ investigated the crushing characteristics of square tubes both numerically and experimentally. They used the same method as Sokolinsky’s study to model tubes in Abaqus/Explicit software. Both Sokolinsky et al. and Zhou et al. validated the models with the results of experimental tests. Bussadori et al.¹⁸ also simulated the crushing behavior of a square tube by modeling the tube as a single layered shell and stacked shell. They developed a crushing zone formulation for single layer shell models in which the mechanical behavior changes when an element is assumed to go into crushing condition and they obtained very close load displacement curves to the experimental ones. Even though the simulated crushing modes did not match with the modes observed during tests, they developed a model that leads to faster results than stacked shell models that can be used for large crash structures. Chen et al.¹⁹ conducted experimental tests and numerical simulations on double hat shaped CFRP, GFRP, and CFRP/GFRP hybrid woven composite structures using conventional shell elements (S4R). They validated the simulation results with experimental results as well. Recently, Engül and Ersoy,²⁰ carried out experimental tests and numerical simulations on structures made of unidirectional carbon fiber epoxy prepregs on Abaqus/Explicit software using continuum shell elements. They modeled the composite flat plates by reducing the number of interfaces, which decreased run times and they validated the numerical results with experimental results.

During the axial crushing of composite structures, a combination of splaying and fragmentation modes can be observed.^4,5 In structures with this type of crushing behavior, debris usually occurs between the structure’s walls and the crusher part, generating a wedge effect. Accumulated debris wedge on the crushing zone has a critical effect on the crushing behavior and morphology since debris pushes the structure’s walls apart and creates fronds of composites. In 3D brick element models used by Israr et al.,¹³ accumulated debris and wedge effect can be observed. However, in shell element models, during axial crush, the deleted elements and the residual shell elements prevent the formation of a debris wedge. In completely closed geometries such as circular and square tubes, the simulations results in a nonrealistic crushing morphology and reduced energy absorption values. McGregor et al.,²¹ investigated the crashing of a braided CFRP square tubes numerically and experimentally. They modeled the tube using shell elements and carried out the simulations on LS-DYNA software. During simulations, they generated a predefined virtual debris wedge from rigid elements on the impact platen. They achieved very close results on both load displacement curves and crushing morphology. Liu et al.²² adopted the same virtual debris wedge approach to investigate double cell square tubes experimentally and numerically. They conducted their simulations on LS-DYNA software and achieved coherent results on crushing morphology and load-displacement curves. Recently, Engül and Ersoy²³ conducted a study on the axial crushing of various sinusoidal CFRP structures both numerically and experimentally. They used the debris wedge approach and obtained a good correlation between experimental and numerical load displacement curves and crushing morphology.

Although there are numerous studies in the literature about FE modeling of crushing of composite structures using shell elements, debris accumulation incident, and the debris effect on crushing morphology are not investigated in detail. The studies that compare the crushing morphology simulations with experimental results present the morphologies from a distant isometric point of view which only shows vertical tearing. Also, a detailed discussion on the effect of the assumptions adopted in FE models on energy absorption characteristics are still missing. In studies that adopt the virtual debris wedge approach, the influence of the wedge geometry is not fully comprehended as well.

In this study, the axial crushing behavior of composite tubes manufactured from carbon/epoxy plain woven fabric prepregs are investigated numerically and experimentally. Axial crushing behaviors of composite tubes are simulated using different rigid crushing platens. The simulated crushing morphology and energy absorption capabilities are investigated and compared to the experimental test results. The effects of the modeling techniques are also assessed by using different types of rigid crushing platens consisting of coarse and fine meshed square platens, coarse and fine meshed circular platens, and circular platens with different shapes of virtual debris wedges. Also, composite tubes crushed with inwards crusher and outwards crusher external plugs are investigated both numerically and experimentally to validate the FE models. FEA simulations are carried out on Abaqus/Explicit commercial software. By comparing the numerical results and experimental test data, it is aimed to achieve an accurate model that simulates energy absorption values and crushing morphologies realistically and accurately.

Experimental study

Manufacturing of specimens

Carbon fiber-reinforced plastic composite tubes are manufactured from KordSA OM10 carbon fiber/epoxy plain weave prepregs that contains carbon 3K carbon fiber tows and epoxy matrix with a nominal areal weight of 200 g/m². Six layers of prepregs are roll wrapped on an aluminum mandrel of diameter Ø30 mm. After roll wrapping, heat shrink tapes are wrapped to the surface of tubes. Heat shrink tapes are designed to bind and constrict the underlying layers when heat is applied. Tubes are then cured in a vacuum oven according to the recommended curing cycle provided by manufacturer²⁴ which consists of heating-up at 2°C/minute to 120°C, holding at 120°C for 60 min and cooling down at 2°C/minute. After the curing process, cured mandrels are ejected from the tubes. Using a mandrel with a high thermal expansion and applying a release agent plays critical roles to separate products easily. Tubes are then cut to 40 mm long specimens using a water-cooled diamond disc. Finally, 45° chamfer triggers are cut at the tip of the specimens by a lathe. Final state of the crush tube specimens and zoomed view of chamfer trigger are illustrated in Figure 1.

Figure 1.

Manufactured Ø30 mm crush tube specimens; (a) whole specimen (b) trigger section.

Test procedure

Crush tube specimens are tested by performing quasi-static compression tests. A Zwick Roell Z100 Universal Testing Machine with 100 kN load cell is used for the experimental testing. Since tubular specimens are self-supporting due to their geometry, no test fixtures are used. Specimens are positioned as the triggered tip faces the moving upper rig and flat end of the tubes faces the bottom rig. Test setup for quasi-static tests are illustrated in Figure 2.

Figure 2.

Test setup for quasi-static compression tests.

Since quasi-static testing is not standardized, a recommended value of loading rate does not exist to crush the structures. However, most of the studies in the literature perform quasi-static tests with crushing speeds varying between 0.5 mm/min and 15 mm/min.^{9,10,14,16,23,25} Farley²⁶ stated that the crushing speed indeed affects the energy absorption characteristics. Yet, Farley²⁶ also stated that, at lower speeds, the energy absorption does not change significantly in carbon/epoxy specimens. During testing, a constant crushing rate of 2 mm/min is used.

Quasi-static tests are performed by crushing the composite tubes using an inwards crusher plug, an outwards crusher plug and crushing by only flat platens of the test rig. Both inwards and outwards crusher plugs have a radius of 2 mm on the inside corner of the plugs. Crush plugs used in the present work are shown in Figure 3.

Figure 3.

External plugs used for testing; (a) outwards crusher plug, (b) inwards crusher plug.

External crusher plugs used in the tests by placing the plugs of the top end of the tubes. Both outwards and inwards crusher plugs used in the test setup is shown in Figure 4.

Figure 4.

External plugs used for testing illustrated in the test setup; (a) outwards crusher plug, (b) inwards crusher plug.

Final crushing displacement during the tests are set as 20 mm. The displacement is measured from the crosshead movement of the testing machine, and the load is measured with a 100 kN capacity load cell. Load displacement curves are obtained from quasi-static tests and specific energy absorption (SEA) values are evaluated from the curves. Tested specimens are investigated in terms of crushing morphology, load displacement curve characteristics, SEA, peak force (PF), and mean crush force (MCF) values. SEA values are calculated by considering the crushed part of the specimen. Average mass of the manufactured specimen with 40 mm length is 8.04 g. While investigating the 20 mm crush displacement tests, half of the tubes mass is accounted.

Finite element analysis

The axial crushing behaviors of composite tubes are modeled by using Abaqus/Explicit software with shell elements according to the stacked shell model approach. FE models are generated considering the intralaminar and interlaminar damage mechanisms. For modeling the intralaminar damage behavior, built-in ABQ_PLY_FABRIC subroutine is used. ABQ_PLY_FABRIC contains the same constitutive equations used in Sokolinsky et al.’s¹⁶ and Zhu et al.’s¹⁷ studies. Cohesive Zone Method (CZM) is used to model the interlaminar damage behavior by defining cohesive interaction between plies surfaces.

Constitutive model for intralaminar damage

ABQ_PLY_FABRIC subroutine provides a continuum damage material model for fabric reinforced composites where each ply is modeled as homogeneous orthotropic material that shows progressive stiffness degradation due to fiber/matrix damage.²⁷ Elastic stress–strain relations of the model are governed by orthotropic damaged elasticity. Elastic stress-strain relations of the material model²⁷ can be expressed as

(\begin{array}{l} ε_{11} \\ ε_{22} \\ ε_{12}^{e l} \end{array}) = (\begin{array}{l} \frac{1}{(1 - d_{1}) E_{1}} & \frac{- v_{12}}{E_{1}} & 0 \\ \frac{- v_{21}}{E_{2}} & \frac{1}{(1 - d_{2}) E_{2}} & 0 \\ 0 & 0 & \frac{1}{(1 - d_{12}) 2 G_{12}} \end{array}) (\begin{array}{l} σ_{11} \\ σ_{22} \\ σ_{12} \end{array}),

(1)

where the subscripts 1 and 2 represent warp and weft directions of the fabric,

ε_{11}

ε_{22}

ε_{12}^{e l}

are the elastic strain components;

σ_{11}

σ_{22}

σ_{12}

are the stress components;

E_{1}

and

E_{2}

are the Young’s moduli along the 1 and 2 directions;

G_{12}

is the in-plane shear modulus;

v_{12}

and

v_{21}

are the principal Poisson’s ratios. The variables

d_{1}

and

d_{2}

represent the fiber damage along directions 1 and 2 directions,

d_{1 +}

and

d_{2 +}

refer to the tensile fiber damage along 1 and 2, respectively, and

d_{1 -}

and

d_{2 -}

refers to the compressive damages. Variable

d_{12}

represents the damage variable related to matrix micro-cracking due to shear deformation. Damage variables on local fiber directions must differ, corresponding to the stress state. Thus, the damage variables can be defined in the material model as

d_{1} = d_{1 +} \frac{〈 σ_{11} 〉}{| σ_{11} |} + d_{1 -} \frac{〈 {- σ}_{11} 〉}{| σ_{11} |}; d_{2} = d_{2 +} \frac{〈 σ_{22} 〉}{| σ_{22} |} + d_{2 -} \frac{〈 {- σ}_{22} 〉}{| σ_{22} |}; d_{12} = d_{21}

(2)

where the angle brackets, “

〈 〉

,” signifies the Macaulay operator. Fiber damage variables are assumed to be a function of corresponding effective stress in the model.²⁷ The relation between the effective stress and the damage variables can be formulated as

{\tilde{σ}}_{1 +} = \frac{〈 σ_{11} 〉}{(1 - d_{1 +})}, {\tilde{σ}}_{1 -} = \frac{〈 - σ_{11} 〉}{(1 - d_{1 -})}, {\tilde{σ}}_{2 +} = \frac{〈 σ_{22} 〉}{(1 - d_{2 +})}, {\tilde{σ}}_{2 -} = \frac{〈 - σ_{22} 〉}{(1 - d_{2 -})}

(3)

Damage initiation of the material model in fiber directions²⁷ can be demonstrated as

ϕ_{α} = \frac{{\tilde{σ}}_{α}}{X_{α}}; (α = 1 +, 1 -, 2 +, 2 -),

(4)

where

X_{α}

represents the strength along the fiber direction. Damage evolution of the material model in fiber directions²⁷ can be formulated as

d_{α} = 1 - \frac{1}{r_{α}} \exp [- \frac{2 g_{0}^{α} L_{c}}{G_{f}^{α} - g_{0}^{α} L_{c}} (r_{α} - 1)]; (α = 1 +, 1 -, 2 +, 2 -),

(5)

where

L_{c}

is the characteristic length of the element,

G_{f}^{α}

is the fracture energy per unit area of composite material under uniaxial tensile or compressive loading along fiber directions,

g_{0}^{α}

is the elastic energy per unit volume under uniaxial tensile or compressive loading along fiber directions at damage initiation, and

r_{α}

is the damage threshold for fiber damage. Once damage initiation is reached, the damage threshold will be the maximum value of the failure criteria function that exceeds the value of 1²⁷ as it is stated on

r_{α} = \max (1, ϕ_{α}) .

(6)

Elastic energy density at the point of damage initiation²⁷ can be formulated as

g_{0}^{α} = \frac{X_{α}^{2}}{2 E_{α}} .

(7)

Since elastic energy density only includes the energy values in the elastic region, the fracture energy per unit area must be higher than the elastic energy per unit area. This provides the maximum value of $L_{c}$ ²⁷ as it is formulated as

G_{f}^{α} - g_{0}^{α} L_{c} > 0 \Leftrightarrow L_{c} < L_{\max} = \frac{G_{f}^{α}}{g_{0}^{α}} .

(8)

Unlike the fiber direction loading response, the shear response is dominated by non-linear plastic behavior. The elastic stress–strain equations for shear loading²⁷ can be expressed as

{\tilde{σ}}_{12} = \frac{σ_{12}}{1 - d_{12}} = 2 G_{12} ε_{12}^{e l} = 2 G_{12} (ε_{12} - ε_{12}^{p l}),

(9)

The yield function of the material model²⁷ can be demonstrated as

F = | {\tilde{σ}}_{12} | - {\tilde{σ}}_{0} (ε^{p l}) \leq 0,

(10)

The hardening law of the material model²⁷ can be demonstrated as

{\tilde{σ}}_{0} (ε^{p l}) = {\tilde{σ}}_{y 0} + C {(ε^{p l})}^{p},

(11)

where

{\tilde{σ}}_{y 0}

, denotes the initial effective shear yield stress,

C

and

p

are the constants of the hardening equation.

ε^{p l}

, is the equivalent plastic strain. After reaching the yield point, the relationship between effective stress and strain is defined according to yield function and hardening law equations. Shear damage initiation criterion for matrix²⁷ can be expressed as

ϕ_{12} = | \frac{{\tilde{σ}}_{12}}{S} |,

(12)

where

S

is the in-plane shear strength. Damage threshold for shear loading²⁷ can be formulated as

r_{12} = \max (1, ϕ_{12}) .

(13)

Damage evolution for shear loading²⁷ can be demonstrated as

d_{12} = \min (α_{12} \ln (r_{12}), d_{12}^{\max}), α_{12} > 0, d_{12}^{\max} \leq 1,

(14)

where

α_{12}

, represents the shear damage variable coefficient. Since ABQ_PLY_FABRIC constitutive model is a built-in user subroutine for Abaqus/Explicit, it is required to activate it before inputting the material properties. VUMAT for Fabric Reinforced Composites document²⁷ provides the necessary information to use the subroutine.

The composite material used in this study is very similar to the one used by Zhu et al.,¹⁷ hence the properties used in this study are taken from the cited reference and listed in Table 1. Plain weave prepregs are woven from T300 carbon fiber yarns manufactured by TORAYCA with a TEX number of 3K and the resin system is epoxy resin with a mass fraction of 42%.

Table 1.

Intralaminar properties of the composite tube.¹⁷

Description	Symbol	Value	Unit
Density	$ρ$	1.56	g/cm³
Young modulus along fiber direction 1	$E_{1}$	65.1	GPa
Young modulus along fiber direction 2	$E_{2}$	64.4	GPa
Shear modulus	$G_{12}$	4.5	GPa
Principal Poisson ratio	$v_{12}$	0.037	—
Tensile strength along fiber direction 1	$X_{1 +}$	776	MPa
Compressive strength along fiber direction 1	$X_{1 -}$	704	MPa
Tensile strength along fiber direction 2	$X_{2 +}$	760	MPa
Compressive strength along fiber direction 2	$X_{2 -}$	698	MPa
Shear stress at the initiation of shear damage	$S$	95	MPa
Tensile fracture energy per unit area along fiber direction 1	$G_{f}^{1 +}$	125	kJ/m²
Compressive fracture energy per unit area along fiber direction 1	$G_{f}^{1 -}$	250	kJ/m²
Tensile fracture energy per unit area along fiber direction 2	$G_{f}^{2 +}$	95	kJ/m²
Compressive fracture energy per unit area along fiber direction 1	$G_{f}^{2 -}$	245	kJ/m²
Shear damage equation parameter	$α_{12}$	0.18	—
Max shear damage	$d_{12}^{\max}$	0.99	—
Initial effective shear yield stress	${\tilde{σ}}_{y 0}$	185	MPa
Hardening equation coefficient	$C$	1053	—
Hardening equation power term	$p$	0.41	—
Element deletion flag	1DelFlag	1	—
Maximum value of damage variables	$d_{\max}$	0.99	—

Constitutive model for interlaminar damage

Interlaminar damage behavior refers to the delamination of plies. It is the other predominant failure mode of axially crushed tubes. In composite structures subjected to axial crushing, delamination occurs through a mixed mode delamination consisting of normal opening mode (Mode I), first shear mode (Mode II) and second shear mode (Mode III). As it is proposed in referenced studies,^16,17,28 Cohesive Zone Model (CZM), which defines the interlaminar damage behavior by generating cohesive interactions between plies, is a very convenient method to model delamination failure.

In this work, interlaminar damage is modeled by generating cohesive contacts based on the traction-separation law between surfaces. The interlaminar behavior is considered as linear elastic until the damage initiation point. Normal and shear traction components are assumed as uncoupled in elastic range. The elastic behavior of cohesive contacts can be demonstrated as^17,29

t = {\begin{cases} t_{n} \\ t_{s} \\ t_{t} \end{cases}} = [\begin{array}{l} K_{n} & 0 & 0 \\ 0 & K_{s} & 0 \\ 0 & 0 & K_{t} \end{array}] {\begin{cases} δ_{n} \\ δ_{s} \\ δ_{t} \end{cases}} = K δ,

(15)

where

t

is the traction,

K

is the stiffness coefficient, and

δ

is the separation between plies. The subscripts

n

s

, and

t

refers to the normal opening (Mode I), first shear (Mode II), and second shear (Mode III) delamination modes, respectively. Damage initiation behavior is defined with quadratic stress damage initiation criterion as it is formulated as

{(\frac{〈 t_{n} 〉}{t_{n}^{\max}})}^{2} + {(\frac{t_{s}}{t_{s}^{\max}})}^{2} + {(\frac{t_{t}}{t_{t}^{\max}})}^{2} = 1 .

(16)

where

t_{n}^{\max}

t_{s}^{\max}

, and

t_{t}^{\max}

refers to the interfacial strength values of the corresponding delamination mode. When the condition in the quadratic stress equation is confirmed, delamination initiates. Damage evolution of the mixed delamination is characterized by using the energy based Benzeggagh–Kenane criterion. Benzeggagh–Kenane damage evolution law is illustrated as

G^{c} = G_{n}^{c} + (G_{s}^{c} - G_{n}^{c}) {(\frac{(G_{s} + G_{t})}{G_{n} + G_{s} + G_{t}})}^{η},

(17)

where

G

refers to the fracture energy, superscript

c

denotes critical fracture energy, and

η

refers to the Benzeggagh–Kenane cohesive property. Softening of the fracture energy is selected as linear in the model.

Inter-laminar failure properties used as input data to define the delamination behavior in the simulations are listed in Table 2. Penalty stiffness values are selected as it is suggested in the studies of Turon et al. and Camanho et al.^28,29 Detailed information about the damage initiation strength and fracture toughness values can be found in Sokolinsky et al.’s study.¹⁶

Table 2.

Interlaminar properties of the composite tube.¹⁶

	Mode I	Mode II	Mode III	Unit
Penalty stiffness	2 × 10⁶	1 × 10⁶	1 × 10⁶	[N/mm³]
Damage initiation	54	70	70	[MPa]
Fracture energy	0.504	1.566	1.566	[kJ/mm²]

Construction of the models

FE models are constructed on Abaqus/Explict software since explicit solver is more efficient for crushing simulations that involve large deformations and contacts.³⁰ At the beginning of the construction of the model, each ply is generated individually as solid layers and meshed with continuum shell elements (SC8R) which have the geometry of 3D elements yet have the constitutive behavior similar to the conventional shell elements. Material properties are defined by assigning homogeneous continuum shell sections to the plies and defining ply orientations.

Composite tube shown in Figure 5 is subjected to axial crushing between a fixed rigid bottom platen and a moving rigid crushing surface at the top. The composite tube model consists of six plies with a thickness of 0.2 mm each. Plies are meshed with approximately 1.0 × 1.0 mm sized elements. The size of the elements of plies chosen considering the Turon et al.’s,²⁸ Sokolinsky et al.’s¹⁶ and Zhou et al.’s¹⁷ studies. According to Turon et al.’s study, a minimum number of 4–5 elements should be in the cohesive zone which is related to interlaminar properties. The interlaminar properties are taken from Sokolinsky et al. and Zhou et al. studies, therefore the mesh size implemented in these studies are used. The tube is connected to the bottom rigid platen with tie contacts between the tube’s bottom nodes and rigid surface. Bevel trigger on the top of the tube is modeled in step form by adjusting the height of each ply with an offset of 0.2 mm. The trigger model is also illustrated in Figure 5 in a detailed view.

Figure 5.

Abaqus CAE model of the composite tubes and rigid bottom platen.

Boundary conditions for upper rigid surfaces are set in such a way that only motion in z axis (U3) is permitted. Crushing rate of the rigid upper surface is set to the value of 1 m/s in simulations. The velocity of the rigid upper surfaces is defined with a loading rate higher than the experimental tests in order to shorten the required time for simulation. According to the studies in the literature that adopts the same approach, dynamic effects become significant after 1 m/s, so the increased loading rate up to 1 m/s in simulations does not change the quasi-static nature of the analysis and does not affect the accuracy, but shortens the analysis run time.^14–23

The friction coefficient between composite tubes and rigs is defined as 0.2 according to Zhu et al.’s study¹⁷ and the friction coefficient between plies is defined as 0.3, according to Pinho et al.’s study³¹ by generating “Tangential behavior” contact with penalty friction formulation. Also, “Normal behavior” contact is defined between composite tubes and rigid surfaces also in between plies. Mass scaling method is applied to the whole model with the target increment of 1e − 7. Semi-automatic mass scaling method that scales the elements’ mass if a time increment below the target is used in the model. To model delamination correctly and achieve splaying behavior, CZM was not applied to the first two elements’ surfaces in the trigger tip.

Figure 6 illustrates various flat rigid upper surface geometries used in the simulations which involves fine and coarse meshed square platens and circular rigid platens. All rigid surfaces in the models are meshed with rigid elements (R3D4). Square platen’s dimensions are 60 × 60 mm, coarse meshed square platen is meshed with 10 × 10 mm sized elements, whereas fine meshed rectangular platen is meshed with 0.5 × 0.5 mm sized elements. Circular platens are generated as Ø30 mm flat disks and meshed with triangular slice-shaped elements in hoop direction. Circular coarse meshed platens have an element size of 1 × 15 mm (hoop, axial), whereas fine meshed circular platens have a size of 0.25 × 0.25 mm (hoop, axial).

Figure 6.

Flat rigid upper surfaces, (a) coarse mesh square platen (CMSP) (b) fine meshed square platen (FMSP) (c) coarse meshed circular platen (CMCP) (d) fine meshed circular platen (FMCP).

In most of the previous studies about FE modeling of composite structures crushing, upper crusher part is generally modeled as a square platen with a coarse mesh. Rigid elements do not deform during the simulation so the mesh density and geometry of the rigid platen are not expected to cause much difference in the composite tube crushing simulations in the first place. However, during the numerical studies it is found out that the mesh density of rigid platens alters the crushing morphology, yet the local buckling mode always occurred in flat platen simulations. So, a very fine mesh and coarse mesh sizes for flat platens are chosen for comparison with debris wedge simulations to demonstrate that local buckling occurs in flat platens regardless of mesh size and geometry, demonstrating the necessity to include a virtual debris wedge to promote delaminations and suppress local buckling.

Rigid upper surfaces with various virtual debris wedges are demonstrated in Figure 7 with section view of the crush zone and the schematics of the debris wedges. The surfaces of the virtual debris and crusher plugs in crushing zone are meshed with 0.5 × 0.5 mm sized elements, in order to model the curved crush surfaces smoothly in a piecewise manner. Using a coarser mesh size, would result the curved surfaces of VDR2 radius wedge to look like triangular, similar to the triangular debris wedge surfaces VD45, VD75, VD45NN, and does not provide enough contact nodes in the most critical part of the crush zone and does not simulate the observed crush morphologies accurately.

Figure 7.

Cross-sectional views (left) and schematics (right) of various rigid surfaces with virtual debris wedge (all dimensions are in mm), (a) radius debris wedge (VDR2) (b) 45° angle debris wedge (VD45) (c) 75° angle debris wedge (VD75) (d) 45° angle debris wedge with no needle (VD45NN).

Sharp edges used in the debris wedge models are used to separate the layers that create inwards and outwards fronds. So, it is used to prevent local buckling during crushing simulations. The sharp edge goes through the interfaces of middle plies.

The specific wedge geometries investigated in the study are inspired from the studies in the literature that adopt virtual debris wedge approach.^21–23 In these studies, there is not a consensus about the wedge geometry. However, most of them have reasonable geometric shapes which mitigate local buckling and promotes delaminations in accord with experimentally observed crushing morphologies. In all these studies, there is only one type of wedge geometry adopted. So, in this study the intention is to investigate the effect of wedge geometry on simulated crushing morphology and SEA.

Section views and schematics of the inwards and outwards crusher plugs are shown in Figure 8. In these models, it is expected that the tubes will slip through the surfaces of the plugs, so virtual debris wedge is not required since the geometry of the plugs are directing the debris.

Figure 8.

Crusher plug meshed models cross-sectional view (left) and schematics (right) (all dimensions are in mm), (a) inwards crusher plug (ICP) (b) outwards crusher plug (OCP).

In all of the simulations, crushing displacement is set as 10 mm since it is sufficient to investigate the crushing characteristics in a reasonable time. All simulations are carried out on an 8-core computer with approximately 150 min of run time each.

Results and discussion

Experimental test results

Load-displacement curves, SEA-displacement curves and crushing morphology of the tubes crushed with flat test rigs are illustrated in Figure 9. During axial crushing, composite tubes exhibited a splaying crushing mode that can be characterized with transverse tears, inwards and outwards fronds. Accumulated debris of material caused by matrix and fiber cracks can be seen in the middle layer of the crushed tube. SEA values and Peak Forces (PF) obtained from the load displacement curves are shown in Table 3. Average SEA value for three tests is calculated as 73.65 J/g. From the load displacement curves, it is observed that mean crush force is approximately 15 kN and the highest peak force obtained from test results is around 19.5 kN.

Figure 9.

Test results of the tubes crushed with flat test rigs, (a) load displacement curves, (b) crushing morphology.

Table 3.

Crushing characteristic values for crushing with flat test rigs.

Test number	Crushed mass (g)	PF (kN)	MCF (kN)	SEA (J/g)	Average SEA (J/g)
Test 1	3.94	19.99	14.27	71.48	73.65
Test 2	4.05	19.45	14.76	73.87
Test 3	3.99	18.19	15.09	75.60

Load-displacement curves, SEA-displacement of composite tubes crushed with outwards crusher plug and the crush morphology are given in Figure 10. Crushing modes observed in the tests include outwards fronds and transverse tears on the tube wall, resembling the shape of a palm tree. SEA values and the PF values for tubes crushed with outwards crusher plug are given in Table 4. Average SEA value for tubes crushed with outwards crusher plug is 34.58 J/g, highest peak force observed in the tests is 8.63 kN and mean crush force is approximately 8 kN. Apparently, SEA values for outwards crushed tubes are lowered about half of the value of the tubes crushed with flat test platens. This can be explained by referring to the nature of the crush morphology with a dominant splaying mode, which involves more material fragmented during tests, while in the tubes crushed with outwards crushing plugs have the shape of a palm tree that involves lesser material being fragmented. It must be noted that some of the fronds are damaged while removing the crush plug from crushed specimen. So, some of the fronds occurred during crushing cannot be seen in crushing morphology image in Figure 10.

Figure 10.

Test results of the tubes crushed with outwards crusher plug, (a) load displacement curves, (b) crushing morphology.

Table 4.

Crushing characteristic values for crushing with outwards crusher plugs.

Test number	Crushed mass (g)	PF (kN)	MCF (kN)	SEA (J/g)	Average SEA (J/g)
Test 1	4.05	8.63	7.05	35.55	34.58
Test 2	4.03	8.09	6.52	33.43
Test 3	4.01	8.48	6.86	34.75

Load-displacement curves, SEA-displacement curves, and crush morphology of the crushed tubes with inwards crusher plug are demonstrated in Figure 11. PF and SEA values for three tests are given in Table 5. In the crushing tests, it is observed that the crushed tube wall moves inwards and fronds are compressed inside the tube squeezing each other. That increases the rate of the crushed material during tests. Average SEA value of the tests carried out is 68.83 J/g, highest PF is 15.79 kN and mean crush force is approximately 14–15 kN. During the tests, it is also observed that the material debris accumulated in the center of the tube during crushing could not escape. Thus, it will lead to increase of the crush force exponentially after some point is reached and the debris trapped in the center starts to be crushed by the plug.

Figure 11.

Test results of the tubes crushed with inwards crusher plug, (a) load displacement curves, (b) crushing morphology.

Table 5.

Crushing characteristic values for crushing with inwards crusher plugs.

Test number	Crushed mass (g)	PF (kN)	MCF (kN)	SEA (J/g)	Average SEA (J/g)
Test 1	4.04	15.79	13.76	69.36	68.83
Test 2	3.97	15.63	13.38	67.57
Test 3	4.06	16.05	13.86	69.58

Numerical simulations results

In order to verify the accuracy of the constructed models and explore the suitability of the modeling approaches, simulation results are compared to experimental results in terms of load displacement curves and crush morphologies. The load values in load-displacement curves are the reaction force exerted by the composite tube during crushing. Numerical SEA values are calculated by considering the average mass of tubes. First, results of the square and circular crusher platen models with fine and coarse mesh structures are compared with the experimental results. Here, although there are three repeats for each test, only one representative load-displacement curve is presented for clarity. In Figure 12, load-displacement curves and SEA-displacement curves of specimens crushed by flat platens, obtained numerically and experimentally are compared. Numerical results are filtered by using SAE 100 filter. SAE filtering is a technique applied to X-Y graphics to remove the noise on curves. It is a built-in visualization tool in Abaqus software. SAE100 filtering is the class of filtering with cutoff frequency multiplier of 165.³¹

Figure 12.

Experimental and numerical load displacement curves comparisons for flat rigid platen simulations, (a) coarse meshed square platen (b) fine meshed square platen (c) coarse meshed circular platen (d) fine meshed circular platen.

As it is observed in Figure 12, load-displacement curves obtained from simulations with various platen mesh geometries are quite close to the experimental test curves. However, it is also observed that mesh structure of the upper rigid surfaces changes the crushing behavior as it affects the curves. Fluctuations in the beginning of the curves are caused by the free surfaces at first two elements in the trigger zone. The delamination behavior is not defined in first two elements to obtain realistic crush characteristics. Influence of the mesh structure can also be seen in the simulated crushing modes as can be seen in Figure 13.

Figure 13.

Crushing modes of the flat rigid platen simulations isometric (left), top (middle), and section (right) views (red elements are damaged); (a) experimental crushing morphology (b) coarse meshed square platen (c) fine meshed square platen (d) coarse meshed circular platen (e) fine meshed circular platen.

Even though the flat platen simulations did not capture exact crushing morphology due to local buckling, the load-displacement curves exhibited very close results to the experimental tests. Although, local wall buckling occurred during crushing simulations, the tubes showed a progressive damage behavior. Furthermore, although the experimental and simulated SEAs are very close to each other, there are large fluctuations in the simulated load-displacement curves possibly due to local buckling during the delamination growth.

Crushing morphologies of simulations using flat rigid platens of various mesh geometries proves that mesh structure of the top surface has a considerable effect on the simulated crushing behavior of the composite tubes. Local buckling behavior observed in the tube walls, in addition to splaying modes and vertical tears. It can be stated that, the accumulation of debris cannot be represented in this type of modeling. Considering the load displacement curves in Figure 12, using flat rigid platens with this modeling construction is acceptable. Yet better modeling techniques required to get accurate crushing morphology results. Comparing various mesh structures used for the flat rigid platens, it can be stated that fine meshed circular platen has given the closest load displacement curve to the experimental one. This may be due to the fact that the tubes fail due to splaying in the radial direction, and the points on the tube and circular rigid platen get in contact successively in the radial direction more smoothly in the fine meshed circular platen.

Load-displacement curves and SEA-displacement curves for crushing simulations using virtual debris wedge approach are given in Figure 14. The simulation results are compared with experimental test results of the tubes crushed with test rigs.

Figure 14.

Experimental and numerical load displacement curves comparisons for virtual debris wedge simulations, (a) radius debris wedge (VDR2) (b) 45° angle debris wedge (VD45) (c) 75° angle debris wedge (VD75) (d) 45° angle debris wedge with no needle (VD45NN).

Apparently, the geometry of the debris wedge has significant influence on the crushing characteristics of the simulated structures. In simulations using radius debris wedge (VDR2) and 45° angle debris wedge with no needle (VD45NN), the trends are close to the experimental curve values. In 75° angle debris wedge (VD75), PF value is much higher than expected value and MCF value is slightly higher than the experimental tests. In 45° angle debris wedge (VD45), simulation PF and MCF are slightly lower than experimental curve values. Crushing morphologies of the four virtual debris wedge simulations are given in Figure 15.

Figure 15.

Crushing modes of the virtual debris wedge simulations isometric (left), top (middle) and section (right) views (red elements are damaged inactive elements); (a) experimental crushing morphology (b) radius debris wedge (VDR2) (c) 45° angle debris wedge (VD45) (d) 75° angle debris wedge (VD75) (e) 45° angle debris wedge with no needle (VD45NN).

From crushing modes illustrated in Figure 15, the load-displacement curve results are found reasonable and accurate. In virtual debris wedge simulations, the tube walls’ lower part maintains its integrity during crushing. So, it puts the required reactionary force close to the experimental values. It is suggested this is the reason for all MCF values are close to the experimental results. The only significant difference in virtual debris wedge simulations is the high PF in VD75 simulations. During the initiation of crushing process, the needle generated on VD75 debris wedge part enters between the middle layers of tube. In VD75 debris wedge simulation, the ply elements face the planar section of the rigid part in a more vertical angle than other simulations so it is thought that the third and fourth plies in the model showed higher reactionary force while tightened between outer plies and rigid needle part which prevents the buckling behavior. After crusher part moves downwards, passing the peak force displacement, the plies are relaxed a little bit and the results converged to the experimental tests.

The influence of the debris wedge geometry on crushing modes can be seen in the simulation results. Clearly, crushing modes of the simulations using virtual debris wedge reflects the splaying morphology better than the simulations using rigid flat platen. Local buckling behavior observed in simulations using the rigid flat platen is suppressed in simulations using virtual debris wedge resulting in vertical transverse tears and splaying fronds. Similar crushing behavior including vertical tears and splaying fronds can be observed in all crushing morphology results using the virtual debris wedge simulations with slight differences.

Needle shaped geometry in the VDR2, VD45, and VD75 debris wedge simulations separates the layers in the tube wall and prevents local buckling behavior by acting as a horizontal support that constricts layers. Even though local buckling behavior didn’t occur on VD45NN simulations, it can be seen that the tube wall slightly tends to move outwards in the section view.

Load-displacement curves and SEA-displacement curves for the simulations of crushing tubes with outwards and inwards crusher plugs are demonstrated and compared to experimental test curves in Figure 16.

Figure 16.

Experimental and numerical load displacement curves comparisons for crush plug simulations, (a) outwards crusher plug (OCP) (b) inwards crusher plug (ICP).

In Figure 16, it is clearly seen that simulation load displacement curves exhibits close results to experimental test curves for tubes crushed with plugs. There is only a slight difference in the curves for outwards crusher plug simulation during the first 4 mm. Inwards crusher plug simulations have shown very good correlation with experimental tests for almost the whole range. In Figure 17, crushing modes obtained from crush plug simulations are shown and compared with experimental results. Crush morphology obtained from simulations shows also good correlation with experimental crushing modes.

Figure 17.

Crushing modes of the tubes crushed with plugs; simulation results (left), experimental results (right), (a) outwards crusher plug (OCP) (b) inwards crusher plug (ICP).

Clearly, outwards and inwards fronds in plug crush simulations are very similar to the modes observed in experimental test morphologies. Only, in inwards crushing, the complete closing of the tubes towards the center with fronds did not occur on simulation morphologies due to the limited crushing displacement in simulations. No local buckling behavior occurred on crush plug simulations as expected.

During crushing, plugs direct the tube inwards or outwards. Since tubes are directed while crushing, the effect of the debris on crushing behavior is minimized. In outwards crusher plug simulations, the tube wall is pushed outwards. The dominant damage mechanism is the tensile damage in the hoop direction that leads to axial tearing of the tube. In inwards crusher plug simulations, the tube wall is pushed inwards, which will cause the tube wall to intertwine in the center of the tube. This will result in the compression of tube material inside the center, and reaction loads will increase. Thus, higher load values achieved in inwards crusher plug simulations.

Commenting together on numerical and experimental load displacement curves and crushing morphologies, it is clearly observed that virtual debris wedge simulations demonstrate the crushing behavior more accurately. Load displacement curves show that energy absorption values are also predicted accurately with virtual debris wedge simulations. Rigid flat platen simulations also give load displacement curves close to the experimental ones. Even though the exact crushing modes couldn’t be achieved with rigid flat platen simulations due to local buckling behavior, the axial tearing of the tubes and splaying behavior can be observed. Crush plug simulations give accurate results in accordance with the experimental tests in terms of both load displacement curves and crushing morphology patterns. This also validates the statement that in the case that the damaged section is directed by plugs, using continuum shell elements to simulate crushing behavior of composite structures gives accurate results without the need of using virtual debris wedge, since the effect of accumulated debris is much smaller when compared to the cases where crushing occurs under rigid flat platen.

In Table 6, crushing characteristic values of the FE simulations are given. In Figure 18, overall comparison of the average experimental SEA values and FE simulations values are illustrated. In Table 6, simulation results are given together with the deviations from experimental data. In virtual debris wedge simulations, VDR2 gave the closest results to the experimental test results. VD45NN also give remarkably close results. Flat platen simulations except fine mesh square platen simulations (FMSP) gave close values as well. Simulations using fine meshed circular platen (FMCP) gave the best results amongst the flat platens in terms of crushing morphologies and SEA predictions. It can be stated that the flat platens can estimate the energy absorption values closely, yet it cannot simulate the exact crushing morphology due to local buckling. If a virtual debris is utilized, numerical SEA values are very close to experimental values and the crushing morphology is very accurately simulated.

Table 6.

Crushing characteristics values of crushing simulations compared with experimental data.

	Flat test rig exp. (Avg.)	CM SP	FM SP	CM CP	FM CP	VD R2	VD 45	VD75	VD45NN	OCP exp. (Avg.)	OCP	ICP exp. (Avg.)	ICP
SEA (J/g)	73.65	74.33 + 0.68	64.99 − 8.66	74.48 + 0.82	71.65 − 2.00	71.47 − 2.18	63.05 − 10.60	88.56 + 14.91	71.14 − 2.51	34.58	38.06 + 3.48	68.83	65.00 − 3.83
PF (kN)	19.21	20.46 + 1.25	22.29 + 3.08	19.70 + 0.41	20.73 + 1.52	22.30 + 3.09	17.52 − 1.69	34.93 + 15.72	24.37 + 5.16	8.40	11.10 + 2.70	15.82	22.29 + 6.47
MCF (kN)	14.71	15.70 + 0.99	13.38 − 1.33	15.39 + 0.68	14.99 + 0.28	15.26 + 0.55	13.21 − 1.50	18.96 + 4.25	14.51 − 0.20	6.81	8.24 + 1.43	13.67	13.38 − 0.29

Figure 18.

Overall comparison of SEA values of rigid flat platen and virtual debris wedge simulations (maximum and minimum values obtained from tests are given as error bars).

Finite element models to simulate crushing of the composite tubes can be improved by incorporating debris accumulation behavior by not deleting the failed elements in the progressive failure model and letting them to accumulate between the delaminated plies and act as debris to exert wedge action, however, this is not possible by using the available capabilities of ABAQUS software, since the failed elements are either deleted or their stiffnesses are reduced close to nil and they lose their integrity. More elaborate user defined subroutines should be developed to capture this type of behavior.

Conclusion

In this study, numerical and experimental investigations of crushing process of wound composite tubes made of plain weave prepregs are performed and a detailed research on FE modeling techniques, considering the effects of the geometry and mesh of the rigid crusher part, is conducted. This study also aims to examine the virtual debris wedge modeling approach. All numerical modeling and experimental test results are evaluated considering the SEA values, load-displacement curves, crushing behavior, and crushing morphology.

It can be asserted that the mesh geometry of the crusher enormously influences the crushing characteristics and energy absorption values. The results of the crushed tube with flat test platens showed that the tubes show the brittle fracturing crushing mode, the combined form of splaying and fragmentation damage modes. Debris accumulation occurs due to fragmentation and debris trapped inside the tube wall acts as a wedge. The tubes are also tested using two different crusher plugs that direct the crushed tube wall toward the inside and outside of the tube during the crushing progress. A palm tree-like crushing morphology with axial tearing is observed in experimental tests using the outwards crusher plug. The crushing morphology of the tubes crushed with the inwards crusher plug showed that the tube wall directed to move inwards and generated inwards fronds are intertwined in the center of the tube. This caused fragmentation behavior and compression of the tube wall inside the tube, resulting in higher reaction forces against crushing. Since most severe damage occurs in tubes crushed with rigid flat platens, energy absorption values are the highest in these tests. However, peak forces are also higher with flat platens and this can result in more severe decelerations as compared to tubes with plugs which are more advantageous for a crash box design.

In numerical studies, it has been concluded that the mesh structure of the rigid surface influences the crushing mode and energy absorption prediction. From the simulations performed, the fine meshed circular platen showed the best prediction among rigid flat platen simulations regarding crushing morphology and energy absorption.

Then, virtual debris wedge simulations are carried out by employing four different debris wedge geometries. The simulation results conclude that the geometry of the virtual debris wedge influenced predictions obtained from simulations. VDR2 simulation predicted the experimental test crush behavior and energy absorption values most accurately.

The composite tubes crushed with inwards and outwards crusher plugs are modeled and investigated in the final two simulations. These two simulations correlated well with experimental test results regarding crushing morphology and energy absorption values. No buckling behavior is observed in simulation results.

The main inferences derived from investigations are that the crusher has a quite important effect on the simulated crush behavior of tubes. The virtual debris wedge approach indeed gives stable crushing behavior in simulations. Also, virtual debris wedge geometry has a considerable effect on crushing characteristics, with reasonable geometries gives very accurate results.

Footnotes

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Serdar Demir

Mehmet C Engül

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