Crushing response of circular thin-walled tube with non-propagating crack subjected to dynamic oblique impact loading

Abstract

The dynamic oblique crushing of circular thin-walled tubes with the presence of non-propagating crack was investigated numerically. The material considered was strain rate sensitive with crack located at the distal end of the tube. Major crashworthiness parameters were obtained and the analysis of the structural response for idealized and finite element crushed thin-walled tubes was also carried out. The study shows that crack initiation on energy absorbing tubes increase their crushing force efficiency under oblique impact, decrease their crushing force efficiency under axial impact and reduce their crashworthiness performance such as the energy absorption capacity and specific energy absorption under axial and oblique impact. Results of the crashworthiness parameters, deformation modes, damage morphology, stress–strain relations, absorption energy characteristics and crushing force-displacement history were obtained. Furthermore, the numerical study reveals both the desirable and undesirable consequence of crack on the overall crashworthiness performance of energy absorbing circular thin-walled tubes.

Keywords

Crashworthiness performance non-propagating crack dynamic oblique crushing deformation mode energy absorption crush force efficiency

Introduction

The protection of passengers and cargoes during high impact is of utmost importance. Thin-walled tubes are designed for such crashworthiness purpose to reduce the high incidence of injuries, damage and fatalities that could result from such collision. During the deformation of these tubes, structural failure could occur. A major cause of such failure is the presence of crack or discontinuity. The effect of this failure is the reduction in the overall crashworthiness performance of such energy absorbing tubes. Crack initiation on the tube affects the deformation mode and the efficiency of the tube to optimally absorb energy. Moreover, the type of material used as energy absorbers can contribute to the efficiency and energy absorption capacity of the thin-walled structure. Materials that are brittle-like may fracture in a catastrophic way as they reach their maximum tensile strength. These materials could dissipate energy through some fracture mechanisms such as delamination, fibre breakage and matrix cracking (Mamalis et al., 1997). Many composites materials used for energy absorbers are in this category. However, ductile-like materials such as mild steel and aluminium alloy used in crashworthiness application first undergo some form of yielding and the dissipation of energy is through progressive plastic deformation. Despite the advantage of these ductile-like materials, the composite materials have higher specific energy absorption (SEA) than their metallic counterpart (Ramakrishna and Hamada, 1998).

Recent advances in the design of energy absorbing thin-walled tubes show that circular tubes are most efficient. Reid (1993) showed that for a circular thin-walled tube, more than one-half of its length undergo plastic deformation. Also, the type of loading can affect the energy absorption capacity of thin-walled tubes. Oblique impact loading reduces the energy absorption capability of the tube than axial impact loading. This is because during oblique loading condition, the tube is susceptible to bending and the consequence is that not all the tube lengths contribute to plastic deformation. However, under oblique loading, the energy absorption can be improved by filling the tube with foams (Hussein et al., 2017; Li et al., 2015a). A number of researches have been made on the energy absorption of thin-walled tubes under axial impact (e.g. Costas et al., 2016; Hao et al., 2017; Qiang et al., 2017), oblique impact (e.g. Gao et al., 2016; Li et al., 2015b) and a combination of axial and oblique impact loading conditions (Fang et al., 2015b; Pirmohammad and Marzdashti, 2016; Tarlochan et al., 2013). Moreover, the velocity of impact can also affect the deformation time. Two forms of impact speed have been adopted in the crushing behaviour of thin-walled tubes by contemporary researchers. They include the quasi-static and the dynamic forms of impacts (Fan et al., 2013; Fang et al., 2015a; Xu et al., 2014). During the quasi-static or low velocity impact, regular and deformation pattern were observed (Altin et al., 2017; Han et al., 2007). However, most crash scenarios are due to dynamic impact. In this case, the deformation of the tube is erratic and unstable which can be observed from the crushing force-displacement history curve (Al Galib and Limam, 2004; Duarte et al., 2014; Isaac and Oluwole, 2016). The finite element integration schemes that have been used to analyse these two forms of impacts are the implicit and explicit time integration schemes (Kazanci and Bathe, 2012; Noels et al., 2004). The explicit integration scheme has gained popularity than their implicit counterpart because it requires lesser time step for the crushed tubes to completely deform.

Crushing response of energy absorbing tubes has been performed analytically (Enakoutsa, 2014; Hao et al., 2017), numerically (Isaac and Oluwole, 2015; Wu et al., 2017) and experimentally (Eyvazian et al., 2014; Mirzaei et al., 2012). During experimental investigations, a number of crushed tubes develop cracks at their distal end. Severe stress concentration and insufficient time for microscopic movement to take place could lead to crack. Depending on the type of the tube material used, the fracture could be brittle-cracking or ductile-cracking. Very few authors have investigated the crashworthiness performance and damaging effect of cracks on these thin-walled tubes. Crack initiation on the tube predisposes it to a very high plastic behaviour and prolongs the deformation process. Some authors, such as Zavattieri (2006), Zheng (2005), Falzon et al. (2015) and Yang et al. (2015), respectively, modelled crack propagation with a cohesive model for shell element, predicted the formation of crack by the use of s-rails, predicted the impact of damage due to compression in composite and performed fracture analysis of cracks growing from surface to fully through-wall crack in tubular members. Javidruzi et al. (2004) performed finite element study of the behaviour of cracked cylinder shell. In their results, the crack size, crack orientation and loading parameters affect the stability and behaviour of thin-walled cylinder subjected to compressive and tensile loading conditions. Gu et al. (2012) performed a theoretical and finite element analysis of the fracture possibility of squared thin-walled tubes under axial crushing. They found out fracture was more likely to occur with increased tube thickness and the peak value of strains in the tube decreased with wall thickness and circumference. A number of authors have investigated the effect of holes (Gupta and Gupta, 1993), grooves (Xu et al., 2016) and imperfections (Yuen and Nurick, 2008) on energy absorbing structures. In their investigations, the effect of geometric modifications on the thin-walled tubes was that it decreased the initial peak force. In recent years, Isaac and Oluwole (2017) investigated the effect of crack on thin-walled tubes under axial impact. Their findings showed that the crashworthiness performance of the cracked tubes was lowered when compared with tubes without crack initiation.

The limited information on the subject of crashworthiness effect of crack in thin-walled tubes necessitated this study. Many authors who investigated geometric modifications only considered axial crushing of thin-walled tubes. In this study, the tube was modelled with a horizontal non-propagating centre through crack at its distal end and was subjected to dynamic axial and oblique impact loading conditions. The extended finite element method (FEM) was adopted for modelling the crack and the major crashworthiness parameters, such as the energy absorption, SEA, crush force efficiency (CFE), mean and peak crushing forces, were evaluated.

Structure, materials and method

Structural description

The modelled structure consists of a striker plate, a circular thin-walled tube and a fixed wall at the distal end of the tube. The striker plate typically represents a moving object colliding with a vehicle. This plate is modelled to impact the tube at different angles with a speed of 15.56 m/s. The thin-walled tube is a surface-like structure used as an energy absorber and could be attached at the front or rear of a vehicle. It is called thin-walled because the thickness $t_{i}$ is significantly smaller than the length and diameter of the tube. In this study, the length of the modelled circular tube is 350 mm with an inner diameter of 75 mm. During crash scenario, the thin-walled tube converts kinetic energy into plastic deformation energy. The choice of using circular geometry for the thin-walled tube is due to its excellent energy absorbing properties and structural response during the plastic deformation process. The tube is subjected to both axial impact (i.e. at 0°) and oblique impact (i.e. at 10°, 20° and 30°) by the striker plate. In some cases, during this dynamic impact loading condition, the thin-walled tube is predisposed to ductile-cracking. To take into account the crack initiation, a horizontal centre through crack is modelled on the tube at its distal end. A fixed and rigid wall is attached firmly to the end of the tube to allow complete and accurate deformation process to take place.

Problem description

An incident striker plate of mass $m = 350 kg$ crushes a circular thin-walled tube of thicknesses $t_{i}$ and cross-sectional perimeter of 235.6 mm with various impacting angles $θ_{i}$ . Table 1 shows a summary of the dimensions of the structural model. The striker plate is given an initial velocity $v_{0} = 15.56 m/s$ . This velocity is according to the impact speed requirement of the National Highway Traffic Safety Administration (NHTSA) New Car Assessment Programme (NCAP) (Witteman, 1999). The mass of the incident plate is 31.82% of 1100 kg mass of a compact car and is significantly greater than the mass of the crushed tube. The selected mass ensured a negligible effect of inertia, which causes deformation to develop over the entire length of the tube and allows the tube to absorb more energy during its initial deformation. For the cracked thin-walled tube, a horizontal centre through crack (i.e. $2 a = 15 mm$ ) was modelled at the distal end of the tube as shown in Figure 1. To compare crashworthiness characteristics, a thin-walled tube without crack (TOC) but with the same mass and loading conditions of the thin-walled tube with crack (TWC) was modelled as represented in Figure 2.

Table 1.

Designation, dimensions and angle of impacts used for study.

Tube profile	Tube length (mm)	Tube perimeter (mm)	Model identification	Impacting angle (°)				Tube thickness (mm)
Tube profile	Tube length (mm)	Tube perimeter (mm)	Model identification	$θ_{1}$	$θ_{2}$	$θ_{3}$	$θ_{4}$	$t_{1}$	$t_{2}$	$t_{3}$	$t_{4}$
Circular	350	235.6	TOC	0	10	20	30	1.8	2.0	2.2	2.5
Circular	350	235.6	TWC	0	10	20	30	1.8	2.0	2.2	2.5

TOC: thin-walled tube without crack; TWC: thin-walled tube with crack.

Figure 1.

Schematic of modelled circular thin-walled tube set-up with crack at the distal end and tube subjected to oblique impact loading.

Figure 2.

Representation of the two modelled thin-walled tubes used for study.

Material property

The material used for this study is made from A36 steel hot rolled carbon and it is strain rate sensitive with a dynamic flow stress $σ_{F}$ dependent of the strain rate. It has a yield strength of 250 MPa and an ultimate tensile strength of 550 MPa. To fully take into account strain hardening and thermal softening process, the Johnson-Cook constitutive isotropic hardening model (Johnson and Cook, 1983) was used for the material model and it is given as

σ_{F} = [A + B {(ε_{e f}^{p})}^{N}] (1 + C I n \frac{{\dot{ε}}_{e f}^{p}}{{\dot{ε}}_{0}}) [1 - {(\frac{T - T_{0}}{T_{m} - T_{0}})}^{M}]

(1)

The Johnson-Cook constitutive model introduces the reference strain rate $\overset{•}{ε_{0}}$ , the effective plastic strain $ε_{e f}^{p}$ and the effective plastic strain rate ${\dot{ε}}_{e f}^{p}$ . The terms $A$ , $B$ and $C$ are the Johnson-Cook material parameters whose values are given in Table 2 and were obtained from Lacy (2010). Also, $N$ and $M$ are the strain and power coefficient, respectively, while $T$ , $T_{0}$ and $T_{m}$ are the room, absolute and melting temperatures, respectively. A summary of the material properties of the modelled thin-walled tubes used in this study is presented in Table 2.

Table 2.

Material properties for the A36 steel hot rolled carbon.

$C_{p} (J/kg K)$	$T_{m} (K)$	$T_{0} (K)$	$E (GPa)$	${\dot{ε}}_{0} (s^{- 1})$	$ρ ({kg/m}^{3})$	$ν$	Material constant (Lacy, 2010)
$C_{p} (J/kg K)$	$T_{m} (K)$	$T_{0} (K)$	$E (GPa)$	${\dot{ε}}_{0} (s^{- 1})$	$ρ ({kg/m}^{3})$	$ν$	$A (MPa)$	$B (MPa)$	$C$	$N$	$M$
486	1773	295	200	1	7850	0.26	146.7	896.9	0.033	0.320	0.323

Extended finite element method (X-FEM) formulation of cracked thin-walled tube

In the traditional FEM, the modelling of crack can be computationally burdensome and generally inaccurate. This is because both the crack and the boundary of the mesh surrounding the crack must conform to each other. To achieve this, the material has to be continuously re-meshed until all the crack paths on the material are completely conformed to the boundaries of the finite element mesh. However, the X-FEM has become popular as an accurate and less burdensome numerical method for modelling cracks in two-dimensional (2D) and three-dimensional (3D) structures. In this method, the finite element meshes does not necessarily conform to the path of the crack (Möes et al., 1999). Both the mesh and the crack are completely independent of each other. Figure 3(a) shows the boundary of the unstructured meshes conforming to the crack path using the traditional FEM. In Figure 3(b), the X-FEM solution was applied and it can be observed that the boundary of the meshes does not necessarily conform to the path of the centre through crack. Only one simulation is run to generate the mesh and obtain accurate, approximated solution. The X-FEM solution enriches the finite element or mesh free approximation by adding new functions through the concept of partition of unity (Duarte and Oden, 1996; Melenk and Babuska, 1996). Moreover, X-FEM has been used to analyse the interaction of main crack and microcracks (Wang et al., 2016). In this study, the centre through crack which is located at the distal end of the circular thin-walled tube was modelled using the X-FEM as shown in Figure 4. The displacement approximation for this particular stationary crack is generally expressed as (Fleming et al., 1997)

u_{T} = \sum_{I ϵ N_{a}} Φ_{I} (X) [u_{I} + \underset{I ϵ N_{h}}{\underset{︸}{(H (X) α_{I})}} + \underset{I ϵ N_{c}}{\underset{︸}{(\sum_{m = 1}^{4} ψ_{m} (X) β_{I}^{m})}}]

(2)

where $N_{a}$ is the set of all nodes in the mesh, $N_{h}$ is the set of all nodes whose support is cut by the Heaviside and $N_{c}$ is the set of nodes whose support is cut by the crack. Also, $X$ is the position vector, $Φ_{I} (X)$ is the shape function, $u_{I}$ is the nodal displacement vector, $H (X)$ is the jump function which represents the gap between the surfaces of the crack. The crack tip function $ψ_{m} (X)$ represents the tip asymptotic singularity. Also, $α_{I}$ is the nodal enriched degree of freedom vector associated with the jump function and $β_{I}^{m}$ is the nodal enriched degree of freedom vector associated with the crack tip function. At the crack interior, a generalized Heaviside enrichment jump function is defined as

H (X) = {\begin{matrix} 1 if (x - x^{*}) n \geq 0 \\ - 1 otherwise \end{matrix}

(3)

Figure 3.

Unstructured meshes with centre through crack (a) finite element method – conforming meshes and (b) extended finite element method – nonconforming meshes.

Figure 4.

Three-dimensional X-FEM and geometric model of a centre through crack at the distal end of circular thin-walled tube.

The first term on the right-hand side of equation (2) is all nodes, the second term applies to nodes whose shape function support is cut by the crack interior and the third term is applicable to nodes whose shape function support is cut by the crack tip. Figure 5 represents the crack portion of the circular thin-walled tube domain. At the crack tip $c_{t}$ , as represented in Figure 5(b), the isotropic elastic property by Fleming et al. (1997) is written as

[ψ_{m} (X), m = 1 - 4] = [\sqrt{r} \sin \frac{θ_{c}}{2}, \sqrt{r} \cos \frac{θ_{c}}{2}, \sqrt{r} \sin θ_{c} \sin \frac{θ_{c}}{2}, \sqrt{r} \sin θ_{c} \cos \frac{θ_{c}}{2}]

(4)

where $r$ and $θ_{c}$ are the polar coordinates in the local crack tip coordinate system.

Figure 5.

Body of the circular thin-walled tube showing the (a) crack domain Ωc and (b) coordinate system at the crack tip $c_{t}$ .

In this study, the X-FEM model was applied and specified on the crack location with a crack length $2 a = 15 mm$ . At the crack domain, the X-FEM enrichment features were added to the FE approximation to give the displacement solution of equation (2).

Structural crashworthiness indicators

To assess the crushing performance of the two modelled thin-walled tubes, certain crashworthiness indicators must be defined. First, during the dynamic crushing of the tubes, the energy absorption capacity is calculated by simply integrating the crushing force $F (x)$ with respect to the displacement $x$ over the crushed tube distance $d_{t}$ given as

E_{a} = \int_{0}^{d_{t}} F (x) d x

(5)

The maximum load $F_{m x}$ gives the peak crushing force required to cause a complete deformation or permanent change. During quasi-static or dynamic axial impact loading, the maximum load corresponds to the initial peak force. This initial peak force is a very useful indicator that should be observed during the design of thin-walled tubes used as energy absorbers and should also be kept below their allowable value. This is because the occupant survival rate during crash incidence is highly dependent on this peak value. However, during oblique impact loading, investigation shows that the maximum load does not necessarily correspond to the initial peak force (Børvik et al., 2003).

Another useful indicator which is in relation with the energy absorption capacity is the mean crushing force (MCF). It is defined as the total energy absorption divided by the crushing displacement of the tube given as

M C F = \frac{1}{d_{t}} \int_{0}^{d_{t}} F (x) d x

(6)

The crashworthiness indicator that relates both the peak force and the MCF is the CFE. It is defined as the ratio of the MCF to the peak crushing force and expressed as

C F E = \frac{M C F}{F_{m x}}

(7)

A number of researchers have shown that a high CFE results in better performance of the energy absorbing tubes (Alia et al., 2014; Fang et al., 2015b; Taştan et al., 2016). Lower CFE value depicts high peak crushing force and greater possibilities of potential damages during crash incidence. It is therefore of utmost importance to keep it high during the design of thin-walled tubes used as energy absorbers.

In the analysis of the crashworthiness performance, the measure of the capability of the energy absorbed material is taken into consideration. This is in relation to the crashworthiness indicator called the SEA. It is the energy absorption per unit mass of the crushed tube $m_{c}$ and is given as

S E A = \frac{E_{a}}{m_{c}}

(8)

The absorber mass of the tube is a significant factor that affects the SEA and should be decreased so as to obtain higher values of SEA.

In this study, the numerical analysis of the major crashworthiness indicators for idealized tube without crack, finite element tube without crack and finite element tube with crack was performed. The exact solution of the idealized tube is compared with the approximate solutions of the finite element tubes with and without crack.

Numerical modelling and validation

Crushing parameters for idealized thin-walled tube

Consider the one complete folding process of the circular thin-walled tube section as shown in Figure 6. A striking plate moving with an initial velocity $v_{0}$ is impacted on the tube of wall thickness $t$ (see Table 1. for the different tube thickness), radius $r_{t}$ and folding length $l_{i f}$ . It is assumed that folding of the tube commences at the point where the tube is attached to the rigid wall. The type of deformation mode formed is non-axisymmetrical due to the dynamic impact and loading conditions. In Figure 6(a), the striking plate is closer to the tube section and it is assumed that wrinkle is formed on the outside of the thin-walled tube. The unfolded axial length at this left-hand section is represented by $L_{y}$ . However, it is expected that the unfolded axial length would increase with the angle about the tube longitudinal axis as depicted in Figure 6(b). For uniform evaluation of $L_{y}$ at both the left- and right-hand sections of the thin-walled tube, the small increase in length $(Δ)$ (∆) at the right-hand section is assumed negligible and $l_{i f}$ of the left-hand section is approximately equal to that at the right-hand section. A relationship between the axial unfolding length, original length and number of complete folds was deduced by Isaac and Oluwole (2018) and it is written as

L_{y} = \frac{L_{0}}{1 + 0.1162 n_{f}}

(9)

where $L_{0}$ is the original length of the tube and $n_{f}$ is the total number of complete folds. Equation (9) assumes that folding around the entire tube during impact is identical. The deformed length is the difference between $L_{0}$ and $L_{y}$ . Dividing the deformed length by $n_{f}$ gives the folding length $l_{i f}$ . When the thin-walled tube is subjected to oblique impact loading conditions, the folding length can be expressed as (Isaac and Oluwole, 2018)

l_{i f} = \frac{L_{0} - L_{y}}{n_{f}} \cos (θ_{i})

(10)

Figure 6.

Idealized circular thin-walled tube sections depicting non-axisymmetrical mode for one complete folding process when subjected to oblique impact loading condition (a) axial length $(L_{y})$ , left-hand side and (b) increased axial length $(L_{y} + Δ)$ , right-hand side of the deformed tube.

For $L_{0} = 350 mm$ (i.e. same value of the finite element model) and $n_{f} = 1$ , equation (9) can be evaluated and the value substituted into equation (10) to obtain $l_{1 f}$ , $l_{2 f}$ , $l_{3 f}$ and $l_{4 f}$ at their respective impacting angles $θ_{1}$ , $θ_{2}$ , $θ_{3}$ and $θ_{4}$ as seen in Table 1.

In Figure 6, during the collapsed process, the folding angle $φ$ increases to $φ + d φ$ to make a complete fold. The kinetic and internal energy released is converted to plastic deformation energy. The total energy absorption $(E_{a T})$ can be obtained by summing the plastic energy absorbed, energy due to incremental change and energy absorbed due to circumferential stretching. For the idealized circular thin-walled tube, the total energy absorbed is given by (Isaac and Oluwole, 2016)

E_{a T} = 4 π σ_{F} t_{i} (4 l_{i f}^{2} + 0.58 t_{i} (π r_{t} + 2 l_{i f}))

(11)

where $r_{t}$ is the radius of the thin-walled tube (i.e. $r_{t} = 37.5 mm$ ) and the parameter $σ_{F}$ is the dynamic flow stress. For a strain rate sensitive material, the dynamic yield stress is obtained from the Cowper–Symonds dynamic constitutive equation (Hu and Zhao, 2001) expressed as

σ_{d y} = σ_{s t} (1 + {(\frac{\dot{ε}}{D})}^{1 / q})

(12)

where $σ_{s t}$ is the static yield stress (i.e. 250 MPa) which can be replaced by the dynamic flow stress $σ_{F}$ . The mean strain rate $\dot{ε} = (v_{0} / 4 r_{t})$ . The material constants $D$ and $q$ are taken from Jones (2011 [1989]) and their values are 40.4 s^–1 and 5, respectively. By substituting the dynamic flow stress, mean strain rate parameters and material constants into equation (12), an approximation of 552 MPa of the dynamic yield stress $(σ_{d y})$ is obtained.

To obtain the MCF, from Figure 6, under axial impact, the force due to the plastic collapsed moment is given as

P_{m} = M_{p} \times 5.26 \sqrt{\frac{π^{3} r_{t}}{t}}

(13)

where $M_{p}$ is the plastic collapsed moment for the circular tube which is expressed as

M_{p} = \frac{σ_{F}}{2 \sqrt{3}} t^{2}

(14)

Substituting equation (14) into equation (13) gives

P_{m} = \frac{σ_{F}}{2 \sqrt{3}} t^{2} \times 5.26 \sqrt{\frac{π^{3} r_{t}}{t}}

(15a)

P_{m} = 1.52 σ_{F} \sqrt{{(π t)}^{3} r_{t}}

(15b)

Since the material is strain rate sensitive, the dynamic speed impact behaviour of the metallic material is modelled by the Cowper–Symond constitutive equation which is obtained from equation (12) as

S = 1 + {(\frac{\dot{ε}}{D})}^{1 / q}

(16)

Multiplying equation (15b) by equation (16) gives the $MCF$ for a strain rate sensitive material written as

M C F = 1.52 σ_{F} \sqrt{{(π t)}^{3} r_{t}} (1 + {(\frac{\dot{ε}}{D})}^{1 / q})

(17)

For reasonable approximation, equation (17) is multiplied by a factor of 1.45 to give

M C F = 2.2 σ_{F} \sqrt{{(π t)}^{3} r_{t}} (1 + {(\frac{v_{0}}{4 r D})}^{1 / q})

(18)

From the plastic strain rate, the axial impact velocity at the end of the circular thin-walled tube is influenced by the angle of impact the striker plate makes with the thin-walled tube. Therefore, under various oblique loading conditions, the MCF can reasonably be approximated as (Isaac and Oluwole, 2018)

M C F = 2.2 σ_{F} \sqrt{{(π t_{i})}^{3} r_{t}} (1 + {(\frac{v_{0} \cos (θ_{i})}{4 r_{t} D})}^{1 / q})

(19)

The value of the dynamic flow stress for mild steel is taken as 250 MPa (i.e. $σ_{F} = σ_{s t}$ ).

Finite element modelling and mesh convergence

The crushing response of the modelled circular thin-walled tube was predicted using the non-linear finite element code Abaqus/Explicit. Dynamic crushing of the tube with or without crack results in non-linear effects of large deformation and displacement. The finite element type used for the thin-walled tube is the S4R element. This is a four-node continuum shell element used to model the tube thickness with a Simpson integration rule and thickness integration points of five along the element thickness of direction. Convergence study to obtain suitable mesh sizes was carried out for both tubes with and without crack. Figure 7 shows the effect of mesh refinement on the deformation mode of TWC and TOC configurations under 30° impacting angle. Generally, mesh refinement produces better energy absorption results. However, a point is reached where further mesh refinement gives no significant increase in the energy absorption result. In Figure 8, eight mesh sizes ranging from 0.3 to 1.0 mm at intervals of 0.1 mm were investigated for the two different configurations under 30° impacting angle as represented in Table 3. Convergence commenced from 5497 up to 9703 number of elements for TWC configuration while it commenced from 3294 up to 9670 number of elements for TOC configuration. The mesh sizes in the range where convergence occurred for the two configurations were from 0.3 to 0.6 mm. Therefore, performing mesh convergence study ensures correct and suitable mesh sizes that produce accurate results within acceptable and satisfactory computational time.

Figure 7.

Effect of mesh refinement on the collapsed mode. The mesh size for the upper tubes is 0.3 mm while that for the lower tubes is 0.5 mm of tube thickness $t_{1}$ , subjected to 30° impacting angle (a) TWC configuration and (b) TOC configuration.

Figure 8.

Eight-point convergence curves for TWC and TOC configurations of tube thickness $t_{1}$ , subjected to 30° impacting angle.

Table 3.

Relationship between mesh size and number of elements for two different configurations in this study.

Mesh size (mm)	Number of elements
Mesh size (mm)	TWC	TOC
0.3	9703	9670
0.4	6403	5869
0.5	5871	3815
0.6	5497	3294
0.7	4120	3004
0.8	3457	2746
0.9	3107	2418
1.0	2689	2296

TWC: thin-walled tube with crack; TOC: thin-walled tube without crack.

The more number of elements generated by the tube with crack is due to finer elements generated around the crack tips. These finer elements around the crack tip help to effectively capture the crack behaviour during the deformation process. A 3D discrete rigid shell planer was used to model both the striker plate and the fixed wall and therefore does not require to be meshed to perform the finite element simulation. The deformation process requires a general contact interaction which allows contact between the Lagrangian surface and Eulerian material. A self-contact is also assigned to the tube so as to account for all possible self-interactions and avoid interpenetration of folding of the tube during the collapsed process. The self-contact interactions describe different contact between the various parts of the thin-walled surface such as the interaction between the entire tube surface and the crack surface. The self-contact was modelled by ‘Hard’ contact pressure over-closure relationship. To make the thin-walled tube, striker plate and fixed wall a continuous interaction system, a tie constrain was used. The system interaction was defined by surface-to-surface contact interactions and it describes contact between the thin-walled tube and the rigid surfaces such as the rigid wall and the striker plate. Moreover, the thin-walled tube was modelled as finite sliding penalty for the contact surfaces. The value of the Coulomb friction penalty coefficient is 0.2 (Ahmad and Thambiratnam, 2009).

Finite element model validation

While the centre through crack introduced at the distal end of the thin-walled tube was successfully initiated using the X-FEM approach, the centre through crack on the thin-walled tube was intractable, making it a major drawback to the analytical solution of the idealized thin-walled tube. Hence, the exact solution of the idealized crushed TOC (i.e. the theoretical formulation in section ‘Crushing parameters for idealized thin-walled tube’) is used to compare the finite element solution without crack. Figure 9 represents a comparison of the MCF and their corresponding energy absorption of both the idealized and finite element solutions without crack initiation and under different impacting angles. The exact solutions of the idealized circular tube at different tube thickness subjected to different impacting angles can be seen to verify the finite element approximated solutions of circular tubes without crack initiation. The results of the MCF and energy absorption obtained from these two solutions are further used to compare the approximated solutions of the finite element crushed TWC.

Figure 9.

Analytical and finite element comparison of the mean crushing force and total energy absorption at different impacting angles $θ_{i}$ .

Numerical results and discussion

Deformation mode and force-displacement history

Figure 10 shows the deformation modes for TOC, damage morphologies for TWC and their corresponding force-displacement curves under various impacting angles. At impacting angle of 0°, the collapsed mode of TOC is axisymmetric or concertina. For this type of collapsed mode, regular pattern of peak forces can be observed in the force-displacement history. However, the collapsed mode of TWC is diamond or non-axisymmetric and their subsequent pair of peak forces are erratic or irregular. Moreover, at this 0° angle of impact, it is evident that folding commences at the distal end of both tubes. It is also seen that the initial peak force is higher than the subsequent pair of peak forces. At increasing impacting angles, the initial peak force is seen to be lowered than their subsequent pair of peak forces. On the collapsed tube, folding is seen to start at the impacting surface. The tube without crack is susceptible to bending with increase in impacting angles and the force-displacement diagram becomes erratic. Also, the initial peak force is lower than the subsequent pair of peak forces. For the tubes with crack, one observes very little bending during the deformation process. The bending reduction of TWC tube profile gives an advantage of a higher crushing force efficiency over TOC tube profile. However, the highly non-linear behaviour due to the initiation of the crack on the tube produces erratic responses and consequently lowers both the initial peak force and energy absorption capacity.

Figure 10.

Collapsed mode of thin-walled tubes with and without crack of thickness $t_{4}$ at various impacting angles and showing their corresponding crushing force-displacement characteristics.

Stress–strain relationship

Figure 11 compares the finite element prediction of the stress–strain curves for TOC and TWC tube profiles of thickness $t_{3}$ when subjected to an initial velocity of 15.56 m/s at various impacting angles. The stress–strain relation was measured at the point where the striker plate impacted the thin-walled tube. Moreover, the deformation of the tube produces plastic strain. At 0° impacting angle, large plastic strain develop for both TOC and TWC profiles. It can be seen that with the cracked thin-walled tube, subjected to high dynamic axial impact loading, a point is reached where the tube fails catastrophically as shown in Figure 11(a). This situation could arise when there is a decrease in temperature. At this point, little or no energy is being absorbed by the thin-walled tube. This catastrophic behaviour is unlikely to occur for ductile energy absorbing tubes subjected to oblique impact loading. In all dynamic crash cases, it is observed that the plastic strain decreases rapidly with increasing impacting angles especially for the cracked tube profile. This reveals the effect of crack to the plastic behaviour of thin-walled tubes used as energy absorbers. Moreover, it is evident that the flow stress decreases with impacting oblique angle during the deformation process.

Figure 11.

Stress–strain relationship of deformed circular thin-walled tubes with and without crack at different impacting angles (a) 0°, (b) 10°, (c) 20° and (d) 30°.

Effect of tube thickness and crack initiation

Thickness of the thin-walled tube plays a vital role to obtaining the initial peak force. In Figure 12, it is evident that with higher tube thickness, the initial peak crushing force increases. Moreover, more serrations develop with higher tube thickness. With fewer serrations, better crushing performance are obtained. A careful study shows that crack initiation on the thin-walled tube causes the initial peak force to increase as shown in Figure 12(b). At an approximate displacement of 2 mm for the different tube thickness, the initial peak forces for the tube with crack are seen to be higher than the initial peak forces of tubes without crack. The increase in initial peak force for the tubes with crack contributes to why their energy absorption capacities are lower than those without crack initiation. Moreover, one of the possible reasons for the increased initial peak force of TWC is the uneven distribution of stress around the tube during initial impact. Stress tends to concentrate more at the crack tip locations. It is therefore important to minimize the initial peak force during design. This can be achieved by introducing initiators or triggering mechanisms which help to raise the initial stress during loading and also reduce the effect of the transfer of load to the entire structure. Also, it is observed that more serrations are found for tubes with crack. The more the number of serrations the less stable the collapsed structure. The effect of crack initiation on TWC tube profiles, therefore, makes them to have less stable collapsed mode than the TOC tube profiles. Tube thickness at $t_{1}$ gives repeated pattern of peak forces than those at $t_{4}$ . The repeated pattern obtained at lesser tube thickness indicates more stability of the deformed system during the entire plastic deformation process. The crushing length during the deformation process increases with increasing thickness. However, for the tube with crack, the crushing length is reduced when compared with those without crack because the presence of crack in the TWC tube profile does not allow the entire tube lengths to contribute to the whole deformation process. Also, for the TWC tube profile, folding did not commence at the cracked portion of the tube because the range of tube thickness used in this study is relatively high. However, it was shown by Gu et al. (2012) that with lower tube thickness, initial folding of the tube begins at the cracked portion of the tube when subjected to axial impact.

Figure 12.

Crushing force versus displacement history showing the effect of tube thickness for (a) TOC tube profile and (b) TWC tube profile.

Energy absorption characteristics

The energy absorbed by the thin-walled tube along its length of deformation during and after impact is represented in Figure 13. These diagrams depict the energy-displacement comparison between TOC and TWC of tube thickness $t_{4}$ at different impacting angles $θ_{i}$ . At an impacting angle of $θ_{1} = 0 °$ , the energy absorption is greatest and the crushing length/displacement is largest. This result shows that higher energy absorption is obtained during axial impact loading condition. However, for different oblique angle of impacts, the energy absorption and crushing length decreases with increase in the impacting angles. This is because, during oblique impact loading conditions, the tube is susceptible to bending and some of the tube lengths do not contribute in the plastic deformation process. For both axial and oblique impacts, it is evident that the TOC gives higher energy absorption than those with crack (TWC). Therefore, this numerical investigation shows that the effects of crack on the thin-walled tube is that it reduces the energy absorption capacity and produces an upwards dispersion of the energy absorption curves against the tube without crack initiation.

Figure 13.

Energy absorption versus displacement characteristics of TOC and TWC profiles of tube thickness $t_{4}$ and at different impacting angles (a) 0°, (b) 10°, (c) 20° and (d) 30°.

Effect of crack size and location on tube

A further study is carried out to investigate the influence of crack size and location on the TWC tube profile. The crack is first located at the distal end of the tube and is designated as ‘Bottom’. Crack placed at the middle position of the tube is designated as ‘Centre’ while the crack placed at the top end of the tube is designated as ‘Top’ as represented in Figure 14(a). As the position of the crack moves from the distal end to the top end of the tube where the moving plate strikes it, the total energy absorption capacity slightly reduces. The reason for the energy reduction is because of the less contribution of the tube length to plastic deformation as the crack approaches the top of the tube. In Figure 14(b), the total energy absorption of TWC profile with crack at bottom, centre and top locations are 23.91, 23.01 and 22.17 kJ, respectively. Moreover, their initial peak forces were observed to be approximately the same. However, with increasing crack size and at the same crack location, the initial peak forces of these tube profiles varied. The longer the crack, the higher the initial peak force and lower the energy absorption capacity.

Figure 14.

Influence of crack location for TWC profile of tube thickness $t_{1}$ and impacting angle of 30° (a) deformation modes and (b) energy absorption versus displacement characteristics.

SEA

In all cases, the SEA of TOC tube profile is higher than TWC tube profile as represented in Figure 15. This is because after the crushing process, it was observed that the crushed mass of the tube with crack was slightly higher than those without crack. However, the mass of the tube before and after the plastic deformation is negligible as compared with the mass of the striker plate. It is evident that at impacting angle of 0°, the SEA increases with wall thickness for both TOC and TWC. However, as the impacting angle increases, the SEA shows strong likelihood to decrease. The SEA of the TOC of tube thickness $t_{3} = 2.2 mm$ at 0°, 10°, 20° and 30° are 26.55, 23.88, 22.64 and 24.33 kJ/kg, respectively, while that of TWC are 23.46, 22.37, 22.11 and 21.89 kJ/kg, respectively.

Figure 15.

Specific energy absorption comparison between TOC and TWC profiles at different impacting angles (a) 0°, (b) 10°, (c) 20° and (d) 30°.

Summary of crashworthiness parameters

Major crashworthiness parameters such as the total energy absorption $(E_{a T})$ ,MCF, maximum load $(F_{m x})$ , initial peak force $(F_{i p})$ and CFE are represented in Table 4 for TOC and TWC tube profiles. For all parameter cases $(t_{i}, θ_{i})$ , investigation shows that $E_{a T}$ of the TOC profile are greater than the $E_{a T}$ of the TWC profile. This crashworthiness parameter of TOC gives it advantage over the TWC tube profile. At impacting angle of 0°, the $F_{i p}$ of TOC profile are higher than the TWC profile. However, at increasing impacting angles, most of the $F_{i p}$ of TOC profile are lower than TWC profile. During axial loading condition (i.e. at 0°), the $F_{m x}$ correspond to the $F_{i p}$ for both profiles. However, under oblique impact loading, the $F_{m x}$ of TOC profile is higher than the TWC profile. The presence of crack on the tube during oblique impact lowers the maximum load and also reduces the susceptibility of bending of the thin-walled tube during the entire deformation process. Thus, the CFE of the TWC profile under oblique loading condition is higher than the CFE of the TOC profile under the same loading condition.

Table 4.

Summary of the crashworthiness parameters comparison between TOC and TWC tube configurations.

Parameter (mm, °)	TOC profile					TWC profile
Parameter (mm, °)	$E_{a T} (kJ)$	MCF (kN)	$F_{m x} (kN)$	$F_{i p} (kN)$	CFE (%)	$E_{a T} (kJ)$	MCF (kN)	$F_{m x} (kN)$	$F_{i p} (kN)$	CFE (%)
$(t_{1}, θ_{1})$	30.25	114.07	211.33	211.33	53.98	26.73	106.05	208.21	208.21	50.93
$(t_{2}, θ_{1})$	33.57	131.41	240.66	240.66	54.60	29.74	121.57	235.87	235.87	51.54
$(t_{3}, θ_{1})$	37.82	159.06	273.51	273.51	58.16	33.41	140.55	269.98	269.98	52.06
$(t_{4}, θ_{1})$	45.92	212.39	321.40	321.40	66.08	38.78	175.92	317.00	317.00	55.50
$(t_{1}, θ_{2})$	30.58	110.68	220.01	103.31	50.31	25.01	96.78	123.96	119.04	78.07
$(t_{2}, θ_{2})$	32.29	120.87	169.18	118.08	71.44	28.58	113.55	154.04	120.35	73.71
$(t_{3}, θ_{2})$	34.00	131.49	192.65	143.32	68.25	31.85	130.27	174.90	137.45	74.48
$(t_{4}, θ_{2})$	40.64	167.09	229.45	186.57	72.82	36.34	157.15	214.23	161.26	73.36
$(t_{1}, θ_{3})$	24.95	88.75	142.23	79.75	62.40	24.07	91.89	129.92	90.19	70.73
$(t_{2}, θ_{3})$	29.95	110.65	178.76	93.92	61.90	27.97	109.69	155.79	101.35	70.41
$(t_{3}, θ_{3})$	32.23	130.73	196.49	108.38	66.53	31.47	127.65	193.05	112.45	66.12
$(t_{4}, θ_{3})$	37.97	153.23	247.74	129.46	61.85	35.42	150.59	202.36	135.40	74.42
$(t_{1}, θ_{4})$	25.70	97.48	140.54	70.42	69.36	23.91	90.65	131.46	80.28	68.96
$(t_{2}, θ_{4})$	29.85	108.69	167.29	83.46	64.97	27.49	106.34	153.62	90.41	69.22
$(t_{3}, θ_{4})$	34.63	131.19	196.98	96.62	66.60	31.17	125.07	174.65	107.17	71.61
$(t_{4}, θ_{4})$	36.35	143.19	247.06	112.77	57.96	35.04	146.80	218.98	122.90	67.04

TOC: thin-walled tube without crack; TWC: thin-walled tube with crack; MCF: mean crushing force; CFE: crush force efficiency.

Analytical and finite element results comparison

In section ‘Finite element model validation’, the analytical solution of idealized circular thin-walled without crack was used to verify the finite element approximations of the TOC. In this section, a comparison of the MCF and the energy absorption of the finite element approximations of cracked thin-walled tube (TWC-FE) with both the tube without crack (TOC-FE) and the idealized tube without crack (Analytical) is performed. Figure 16 shows the energy absorption versus thickness of the thin-walled tube at all impacting angles used in this study. It is clear that the energy absorption capacities of TWC-FE are lower than TOC-FE while TOC-FE and Analytical show very close relationship with each other. Also, the MCFs for TOC-FE are much greater than TWC-FE at 0° impacting angle as depicted in Figure 17(a). With increasing impacting angles, good agreement of the MCF for the three numerical modelled circular thin-walled tubes is evident as represented in Figure 17(b) to (d). The numerical predictions of these major crashworthiness parameters reveal the undesirable effects of crack on energy absorbing circular tubes and also offer a guide for the design of an efficient circular thin-walled tube used as energy absorber for crashworthiness application.

Figure 16.

Summary of energy absorption comparison between analytical and finite element models for all tube thicknesses $t_{i}$ and at different impacting angles (a) 0°, (b) 10°, (c) 20° and (d) 30°.

Figure 17.

Summary of mean crushing force comparison between analytical and finite element models for all tube thicknesses $t_{i}$ and at different impacting angles (a) 0°, (b) 10°, (c) 20° and (d) 30°.

Reliability and accuracy of numerical solution

For the credibility of numerical solution, it is necessary that the material used for modelling is of the same material property, loading and parametric conditions of the material used for the experiment. These conditions influence the crashworthiness performance of the thin-walled tube significantly. Experimental result obtained under quasi-static loading produces a different result from numerical model when the tube is subjected to dynamic loading. In the latter case, higher energy absorption, MCF and other crashworthiness parameters are obtained. Where no experimental data are available to validate the finite element solution, it behoves to use analytical solution to verify the finite element approximation. In this study, the analytical solution of the total energy absorption of the circular thin-walled tube (TOC) was predicted in section ‘Crushing parameters for idealized thin-walled tube’. A summary of the percentage error of the total energy absorption $(E_{a T})$ of both TOC and Analytical thin-walled tube configurations is shown in Table 5. It should be noted that both the finite element and analytical models are made from the same strain rate sensitive material and subjected to the same loading and parametric conditions. At all impacting angles, it is evident that the errors between the analytical solution and finite element approximation are below 10%. This gives the credibility of the finite element solutions of the TOC configuration. The verified solution of TOC configuration can therefore be reliably used as a basis for comparison of the finite element approximation of the TWC initiation.

Table 5.

Percentage error of the total energy absorption comparison between analytical and finite element solutions.

Parameter (mm, °)	$E_{a T} (kJ)$
Parameter (mm, °)	TOC	Analytical	Error (%)
$(t_{1}, θ_{1})$	30.25	31.17	3.04
$(t_{2}, θ_{1})$	33.57	34.78	3.60
$(t_{3}, θ_{1})$	37.82	38.41	1.56
$(t_{4}, θ_{1})$	45.92	43.90	−4.40
$(t_{1}, θ_{2})$	30.58	30.25	−1.08
$(t_{2}, θ_{2})$	32.29	33.75	4.52
$(t_{3}, θ_{2})$	34.00	37.29	9.68
$(t_{4}, θ_{2})$	40.64	42.20	3.84
$(t_{1}, θ_{3})$	24.95	27.30	9.42
$(t_{2}, θ_{3})$	29.95	30.84	2.97
$(t_{3}, θ_{3})$	32.23	34.07	5.71
$(t_{4}, θ_{3})$	37.97	38.97	2.63
$(t_{1}, θ_{4})$	25.70	23.59	−8.21
$(t_{2}, θ_{4})$	29.85	26.35	−11.73
$(t_{3}, θ_{4})$	34.63	29.13	−15.88
$(t_{4}, θ_{4})$	36.35	33.35	−8.25

TOC: thin-walled tube without crack.

Summary and conclusion

In this study, the effect of crack initiation on the crashworthiness performance of circular thin-walled tubes subjected to various loading conditions was investigated numerically. First, an analytical solution for circular tubes without crack was established and the results were used to verify the finite element approximations of circular TOC under the same parametric and loading conditions. The analytical solutions show good agreement with the finite element approximations. Crack was then initiated at the distal end of the circular thin-wall tube (TWC) using the X-FEM. The results of TOC and TWC profiles were compared with each other.

Based on the findings of this study, the following conclusions can be drawn:

During oblique crash case, the crushing force efficiency of circular TWC initiation is higher than those without crack. For tube thickness of 2.5 mm and impacting angles of 10°, 20° and 30°, the CFE of TWC are 73.36%, 74.42% and 67.04%, respectively, while that of TOC are 72.82%, 61.85% and 57.96%, respectively. The increased CFE of the TWC is due to the ability of the tube to withstand bending during the deformation process and this also causes a reduction in the MCF.

However, in terms of the energy absorption, the tube without crack gives higher results than the tube with crack. Also, better results in the SEA were obtained by the tube without crack. The simulation results show that for tube thickness of 2.5 mm and at impacting angles of 0°, 10°, 20° and 30°, the SEA for TOC are 28.37, 25.10, 23.47 and 22.47 kJ/kg, respectively, while the SEA for TWC are 23.96, 22.45, 21.89 and 21.66 kJ/kg, respectively.

In spite of the improved CFE of cracked thin-walled tubes during oblique loading, the reduction in energy absorption and SEA make them less attractive for energy absorbing structures. During fabrication of circular thin-walled tubes, discontinuities may be introduced. Therefore, the need to test these tubes for any defect or small crack is of utmost importance. This will help to minimize uncertainty of the energy absorbing structure during crash operation and ensure maximum crashworthiness performance.

The higher crashworthiness performance as observed by the TOC profile gives it desirable advantage as protective structure over the TWC profile.

The study therefore shows how the presence of cracks in energy absorbing thin-walled tubes contributes to their overall crashworthiness performance.

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 iD

Chukwuemeke William Isaac

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