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This is the published version of a paper published in Composite structures.

Citation for the original published paper (version of record):

Costa, S., Gutkin, R., Olsson, R. (2017)

Mesh objective implementation of a fibre kinking model for damage growth with

friction

Composite structures, 168: 384-391

https://doi.org/10.1016/j.compstruct.2017.02.057

Access to the published version may require subscription.

N.B. When citing this work, cite the original published paper.

Permanent link to this version:

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Mesh objective implementation of a fibre kinking model for damage

growth with friction

Sérgio Costa

a,b,⇑

, Renaud Gutkin

a

, Robin Olsson

a

a

Swerea SICOMP AB, SE-431 22 Mölndal, Sweden

b

Department of Applied Mechanics, Division of Material and Computational Mechanics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden

a r t i c l e i n f o

Article history:

Received 12 October 2016 Revised 23 January 2017 Accepted 10 February 2017 Available online 13 February 2017 Keywords: Crushing Fibre kinking Friction Damage mechanics Mesh objectivity

a b s t r a c t

A newly developed physically based model for the longitudinal response of laminated fibre-reinforced composites during compressive damage growth is implemented in a Finite Element (FE) software. It is a mesoscale model able to capture the physics of kink-band formation by shear instability, the influence of the matrix in supporting the fibres and the rotation of the fibres during compression, resulting in more abrupt failure for smaller misalignments. The fibre kinking response is obtained by solving simultane-ously for stress equilibrium and strain compatibility in an FE framework.

Strain softening creates pathological sensitivity when the mesh is refined. To make the model mesh objective, a methodology based on scaling the strain with the kink-band width is developed. The FE implementation of the current model is detailed with focus on mesh objectivity, and generalized to irreg-ular meshes. The results show that the current model can be used to predict the whole kinking response in a 3D framework and thus account for the correct energy dissipation.

1. Introduction

Fibre kink-band formation is one of the most studied intralam-inar failure modes[1]. This particular mode has received interest because of its influence on stiffness, strength and energy absorp-tion. There are reasonably well-established models to predict stiff-ness and strength[2–4], but not energy absorption, i.e. the post-peak response has not been fully addressed. Modelling kink-band formation is, however, fundamental for predicting the longitudinal compressive response, such as for bolted joints, impact and crash situations.

Fibre kinking is a very complex phenomenon due to interacting mechanisms and instabilities created at failure. Experimental[5– 7], analytical [8,9] and micromechanical finite element [10,11]

studies show that the main mechanism occurring during kinking is the interaction between fibre rotation and a degraded surround-ing matrix. Thus, kink-band formation is strongly influenced by ini-tial misalignment of the fibres as well as by other in-plane stress components, such as transverse normal and shear stress. Analytical models are, however, not suitable to represent the behaviour of complex-shaped structures in multiaxial loading situations. The

high level of detail of the micromechanical models makes them prohibitively expensive for big structures. Therefore, it is required to combine the properties of the matrix and individual fibres, i.e. homogenization at the ply level/mesoscale. In this paper we use a mesoscale model developed by the authors in[12], based on Con-tinuum Damage Mechanics (CDM) and suitable for analysis of large structures. Since fibre kinking is a matrix dominated failure mode

[13], an accurate description of the matrix response becomes nec-essary to predict the fibre kinking behaviour. Thus, the matrix behaviour is accounted for in the transverse and shear responses that are based on a physical coupling of damage growth and fric-tion created at microcrack closure in associafric-tion with compressive load, as described in[14]. As a result, a non-zero frictional stress acts on crack faces when the material is partially or fully damaged. This modelling approach is important to accurately account for the energy dissipation at different strain levels, e.g. during progressive crushing and crash situations.

Constitutive FE models with strain-softening cause convergence difficulties especially for damage in the longitudinal (fibre) direc-tion[15]. A detailed review on this issue is given in[16]. During kink-band formation, softening occurs in the physical material and this is also captured in the current numerical model. This leads to a non-convergent solution as the mesh is refined, as observed in

[17]. In order to account for the correct energy absorption it is nec-essary to have a mesh objective model.

http://dx.doi.org/10.1016/j.compstruct.2017.02.057 0263-8223/Ó 2017 Elsevier Ltd. All rights reserved.

⇑Corresponding author at: Swerea SICOMP AB, SE-431 22 Mölndal, Sweden. E-mail address:sergio.costa@swerea.se(S. Costa).

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Other well-known models [18]took advantage of the critical strain energy release rate Gcto define the final strain (for the fully

damaged material) that is smeared over the element characteristic length. The drawbacks are the cumbersome tests required to mea-sure Gc, and their highly scattered results[19]. A major advantage

of the model developed in [12], and implemented here, is the reduced material characterisation, since it does not require Gc.

However, it requires new approaches to obtain a mesh objective implementation.

To deal with the localization phenomena, one possible way is to use regularization techniques, commonly referred to as localiza-tion limiters. They are usually based on continuum enrichments, which typically introduce a parameter defining a characteristic length of the material. Several alternatives have been exploited as a localization limiter in[20] and in nonlocal damage theory

[16,21]. The crack band model is one of the most widely used to avoid the pathological sensitivity to mesh refinement. However, only the global response is objective and mesh-induced directional bias still occurs.

For typical CDM problems, such as the cohesive zone models, or traction-separation laws, the damage starts when the peak stress is reached and the stress can decrease either by increasing strain (softening) or decreasing strain (elastic unloading). In the current model the material is already partially damaged when the peak stress is reached., resulting in a slightly nonlinear response before the peak stress. This is physically correct and observed in[22], but numerically it causes more difficulties since it requires the formu-lation of an nonlinear unloading behaviour, which is also detailed here.

The longitudinal response is obtained with a model able to pre-dict the response after the peak load, as developed by the authors in[12]. The model is an extension of fibre kinking theory, i.e. non-perfectly straight fibres rotating under an applied compressive load induce a loss of stability in the resin that eventually leads to kink-band formation. The model is also able to predict the whole kinking response under a 3D stress state. The peak stress results from shear instability due to the highly nonlinear shear response

[18]. The resulting compressive-softening behaviour differs signif-icantly from the most commonly used bilinear approaches.

The current work details the FE implementation of the referred kinking model[12], with focus on solving the mesh objectivity issues arising from strain-softening behaviour during kinking for-mation. This is a significant contribution towards the crash simula-tion of composite structures.

2. Overview of the constitutive model

In this section, we present a succinct summary of the model. Further details about the development of the current 3D model are explained in detail in[12]. The fibre kinking response is found from the actual rotation of the fibres in the kink-band. To obtain both kinking stress and fibre rotation it is necessary to solve simultaneously:

I. The strain compatibility;

II. The stress equilibrium between applied global stresses and nonlinear local stresses;

III. The nonlinear constitutive response of the material in the kink-band.

In summary, the kink-band is influenced by the compressive longi-tudinal strains, the transverse and shear stresses as well as the ini-tial misalignment hiof the fibres in the composite.

The stress-strain relation predicted by the current model was validated by comparison against two experimental test cases as

well as against analytical and numerical models in[12]. In partic-ular, the model is able to predict the measured peak stress, and corresponding strain, for typical initial fibre misalignments observed experimentally. The stress predicted for large compres-sive strains is in good agreement with the measured average crush stress[23].

2.1. Geometrical perspective

Due to the 3D nature of kinking formation, a 3D framework is required for an accurate prediction. However, the focus of the cur-rent paper is mesh objectivity, and hence we only consider the rotations in the kink-band plane (r).

Fig. 1(a) shows a micrograph of a typical kink-band. Two coor-dinate systems are introduced: the first one rotated to the orienta-tion of the kink-band through the thickness is denoted r, and the second (misaligned) coordinate system, m, is associated with the rotating fibres, and is characterised by the angle h. The main factors contributing to fibre rotation in the kink-band are represented in

Fig. 1(b). Fig. 1(c) defines the initial and current configurations and associated fibre rotations. The numerical subscripts (11, 22, 12) refer to the rotated global coordinate system r, and the sub-scripts using m (11m, 22m, 12m) refer to the misaligned frame, m (also referred to as local frame or kink-band frame later on).

Based onFig. 1(c) the following relation can be established

h ¼

c12m

þ hi ð1Þ

2.2. Model formulation

The model predicts the longitudinal compressive response for an arbitrary loading. The three aforementioned conditions, I–III, are summarized in the equations that follow. Starting with strain compatibility (I), expressed as a function of the shear angle it becomes

c12m

Þ ¼

e11m

cos2

h þ

e22m

sin2h 

c12m

cos h sin h 

e11

¼ 0:5½

e11m

ð1 þ cos 2hÞ þ

e22m

ð1  cos 2hÞ 

c12m

sin 2h 

e11

¼ 0 where h ¼

c12m

þ hi

ð2Þ Numerical methods are required in order to find the shear angle,

c

12m, as it is also present inside the trigonometric functions. For the remaining unknowns, the strains

e

11mand

e

22min the

mis-aligned frame, we can take advantage of the elastic assumptions. Thus, the strains in the kink-band coordinate system can be easily solved using transverse isotropy[12].

The second line in Eq.(2)is a reformulation used only to reduce the computational costs. Using double angle formulation does not require the use of more expensive operations like the square of the trigonometric functions. This will be more notorious in the upcom-ing equations.

The two unknowns from transverse isotropy are the stresses,

r

11mand

r

22mobtained from the stress equilibrium (II), which will

constitute the second set of equations as follows:

r

11m¼

r

11cos2h þ

r

22sin2h þ 2

s

12sin h cos h ¼ 0:5½

r

11ð1 þ cos2hÞ

þ

r

22ð1  cos2hÞ þ

s

12sin 2h

r

22m¼

r

11sin2h þ

r

22cos2h  2

s

12sin h cos h ¼ 0:5½

r

11ð1  cos2hÞ

þ

r

22ð1 þ cos2hÞ 

s

12sin 2h

s

12m¼ 

r

11sin h cos h þ

r

22sin hcos h þ

s

12ðcos2h  sin 2

hÞ where h ¼

c12m

þ hi

(4)

The stresses without subscript are in the r coordinate frame. The stress component of interest is the kinking stress,

r

11obtained from

rearranging the local in-plane shear equation in Eq.(3)as follows:

r

11¼ ½

r

22sin h cos h þ

s

12ðcos2h  sin 2

hÞ 

s

12m=ðsinh coshÞ

¼

r

22þ 2ð

s

12cos 2h 

s

12mÞ= sin 2h

where h ¼

c12m

þ hi

ð4Þ The transverse and the shear stress,

r

22 and

s

12 respectively,

need to be calculated from the nonlinear response of the matrix (III), detailed in[14]. The shear response in the kink-band plane accounts for nonlinearity by combining damage and friction in a physical way according to[14]as follows

s

12m¼ G12

c12m

ð1  dÞ þ d

s

friction where

s

friction

¼ G12ð

c12m



cs

Þ if no sliding

lL

h

r

22m p0Li if sliding



ð5Þ where

s

friction represents the in-plane frictional stress. The

dissipa-tive mechanisms are represented by damage, d and the sliding strain,

c

s, as detailed in[14]. Solving the nonlinear Eqs.(2)-(5)using

numerical methods, we obtain the fibre kinking stress

r

11; and the

shear angle

c

12m.

For a more tangible understanding, Eq.(4)can be simplified, assuming uniaxial stress state (

r

22=

s

12= 0) and small rotations.

This results in a variant of Budiansky’s equation [13] in which the shear and compressive strengths are replaced by the in-plane shear and kinking stresses respectively

r

11 ¼



s

12m

c12m

þ hi

ð6Þ

2.3. Material properties

The material data used in this manuscript were taken from[22], which provides further information about the characterisation of

the mechanical properties. The necessary parameters for the model were gathered inTable 1. Xcis the peak stress during compression

in the fibre direction, while p is a shape parameter for the shear stress-strain curve, as described in[12]. STand SLare the shear

strengths in the material 1-3 and 2-3 planes, and

l

Land

l

Tare

the friction coefficients on the corresponding fracture planes. Note that the shear strengths are the values at the onset of shear nonlin-earity. The parameter p0Laccounts for the apparent internal

pres-sure built up during manufacturing. The kink-band width, w in

Fig. 1a, works as a physical process zone required for the mesh objective formulation.

3. Mesh objectivity

Mesh dependency is an issue with damage models due to their softening behaviour. Localization occurs only after the maximum stress has been reached, and creates the mesh size dependencies. For the longitudinal compression of composites, fibres with lower misalignment not only contribute to a higher strength but also cre-ate a more catastrophic failure. The ply undergoes a higher load drop accompanied by sudden fibre rotation, Fig. 2. This effect, where small misalignment angles may cause a snap-back, has pre-viously been demonstrated in numerical experiments by e.g. Kyri-akides et al.[5]. The current model is able to capture the existence of snapback behaviour for fibres with lower misalignment angles than 1°. This constitutes a plus for the model, but it can also cause numerical difficulties. The abrupt changes in stress may lead to divergence when solving

r

11and

c

12m; and/or instability problems

in the explicit simulations.

The simplest way to verify the behaviour of the model is to carry out elementary studies. A cube of side L = 1 mm was meshed with one element ‘‘1 1  1”, and successively refined until ‘‘5 5  5” elements. When more elements are added, the soften-ing response becomes mesh dependent,Fig. 3. In the current study we consider hi= 3° for all the simulations. The reason for this Table 1

Mechanical properties of the uni-weave NCF composite HTS45/LY556,[22].

Elastic properties

ModulusðGPaÞ Poisson’s ratios

E11¼ 136 E22¼ 9:15 G12¼ 4:9 m12¼ 0:28 v23¼ 0:43

Strength propertiesðMPaÞ Damage and kinking

Xc¼ 626 ST¼ 47:44 SL¼ 20 p¼ 0:7 w = 0.2 mm

Friction properties

Internal pressureðMPaÞ Coefficient of friction

p0L¼ 60 lL¼ 0:34 lT¼ 0:4

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choice is that 3° results in the measured compressive strength, Xc¼ 626 MPa for the chosen material, i.e. we are assuming that

the fibres are misaligned 3 degrees. Interestingly, this value is in between the mean misalignment angle and the maximum angle measured in[24].

When more elements are added in the loading direction, then more elements undergo partially elastic unloading resulting in a lower load for the same deformation. Thus, one possible approach is to scale the strain of each element with a physical process zone. For kink-band formation that zone is the kink-band width (w). This width can be interpreted as a material parameter that scales lin-early with fibre diameter and could be measured[25]or easily cal-culated as in[13,26]. Note that the uncertainty in Gcdirectly affects

the underlying CDM model and has a profound influence on the behaviour after peak stress, while w only affects the influence of mesh size in a moderate fashion. This width is significantly larger than the localization limiters representing cracks, thus allowing higher minimum critical element length. It is out of the scope of this paper to obtain the actual width of the kink-band since it will not influence neither the implementation nor the convergence

studies. In the following analyses, two methods will be considered for mesh objectivity.

3.1. Method 1 – strain decomposition

As shown inFig. 4, Method 1 decomposes the strain into a bulk/ elastic component

e

eland a kink-band component

e

k. The total

dis-placement, d is expressed as

d ¼ dkþ del) 

e

Le¼

e

kwþ

e

elðLe wÞ ð7Þ

where

e

is the average strain applied to the element with length Le.

The previous equation can be rearranged and the kinking strain becomes

ek

¼ ½

e

 ð1  wÞ

r

11=E11= w; where w ¼ w=Le ð8Þ

and

r

11is the fibre kinking stress that must be recovered from a

previous iteration. The whole iteration process stops when a toler-ance criterion is met. These iterations add computational costs.

Fig. 2. Analytical response showing the correlation between compressive stresses and fibre rotation for 1°, 2° and 4° and 15° of initial misalignment.

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3.2. Method 2 – smear the kink-band over the element

In Method 2, the strain in the kink-band is distributed (smeared) over the whole element,Fig. 5. In this method the longi-tudinal strain that goes into the element must be split into the strain before peak stress (

e

o) and the strain during softening

(

e

soft), i.e. prelocalization and postlocalization parts. The reasoning

for this splitting approach is because the unobjective response (Fig. 3) starts with of softening, which coincides with the peak stress. Thus, the longitudinal strain must be reformulated accord-ing to

e

e

e

soft)

e

e

oþ ð

e



e

oÞLe=w ð9Þ

4. FE implementation

The fibre kinking model was implemented into a user subrou-tine, VUMAT, in the commercial FE code ABAQUS. The lack of mesh objectivity due to the softening behaviour the main challenges. In addition, the numerical difficulties to solve the nonlinear Eqs. (2)-(5)are also discussed.

4.1. Numerical method to solve conditions I–III

In order to solve the stress equilibrium and the strain compat-ibility, it is necessary to use a root-finding algorithm. In the current approach, the bisection method was chosen. The reasons to choose bisection over other methods with faster convergence are its sim-plicity and robustness in particular close to the peak load. Taking advantage of the fact that there is a known interval with a root, using bisection, we can directly input all the equations as described into the model formulation. Using other methods with higher con-vergence will create additional difficulties due to the coupling of

Eqs.(2)-(5)with the equation governing damage growth. Damage growth is modelled through a damage variable, d, degrading only the shear components and regulating the contribution of friction. Irreversible damage is assumed for structural unloading as well as for the elements undergoing decreasing strain. Thus, the damage variable at a given increment n is defined as follows:

dn¼ maxfdn

; dn1g where 0 6 d 6 1 ð10Þ

The bisection method makes it easy to handle the irreversibility of Eq. (10). More efficient methods, such as Newton-Raphson, could be used, but were left for future investigation, as computa-tional efficiency was not the main focus of the current paper. 4.2. Unloading effects

In a model with multiple elements, some undergo partially elas-tic unloading behaviour when the peak stress has been reached, i.e. the absolute value of the longitudinal strain, |

e

11| decreases,

repre-senting an unloading situation. The unloading behaviour for two examples of unloading-loading cycles, shown in Fig. 6, is based on the sliding-sticking response of the friction, as described in

[14]. The shear response shown in Fig. 6(b) is the local shear response defined in Eqs.(3) and (5). It governs the response of the material in the kink-band and therefore the global response

r

11 shown in the same figure. The shear instability has been

defined according to [18], i.e. there is no equilibrium point between the compressive stress and the nonlinear shear. Further-more, the maximum kinking stress does not result from matrix failure, but is rather determined from instability due to shear nonlinearity.

During unloading the damage variable remains constant follow-ing Eq.(10). It is important to point out that the hysteresis loops in the kinking response are simply driven by the cyclic behaviour in

Fig. 4. Representation of the strain decomposition used in Method 1.

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shear. The increase of the permanent strain from the first to the second loop is evident, and constitutes an asset of the model. The shear strength represented, SLinTable 1, dictates the initiation of

the damage variable. Shear instability occurs at higher shear strains, resulting in the load drop in the kinking response, i.e. the kinking strength is determined by shear instability.

5. Results 5.1. Method 1

Using Method 1 we obtained mesh independent results. How-ever, this method has two drawbacks. One is that it requires more iterations, making it more computationally expensive. Another

important drawback with this method is the snap-back behaviour observed for ratios ofw smaller than one-fourth, observed in Fig. 7

for 1 element.

The results are shown inFig. 7(a) and the elements with higher damage are shown in red (or lighter colour) inFig. 7(b). The solu-tion ‘‘5 5  5” corresponds to the case when the element size coincides with the kink-band width, i.e.w = 1, which allows simu- lation of the kink band localisation in the real material.

5.2. Method 2

This method is computationally cheaper and more stable than Method 1, since there is no need to iterate on the fibre kinking stresses again. The spurious snap-back behaviour observed for one element in Method 1 for ratios ofw smaller than one-fourth is absent for Method 2, as shown inFig. 8. This method allows a very high critical maximum element size contributing to more effi-cient simulations. The maximum element size is limited by the predefined lock-up angle, i.e. when the required shear angle ð

c

12mÞ to make the model mesh objective is greater than the

lock-up angle.

5.3. Method 2 – irregular mesh

Distorted meshes could add additional difficulties for any mesh objective method. Thus, we tried the method with best results, method 2, on irregular meshes. Using badly shaped meshes may affect the localization of the fracture into a kink-band as well as the measure of the effective size of the element and consequently affect the objectivity of the results. In spite of this, the results are converging quite well considering the highly distorted mesh and the amount of softening,Fig. 9.

The fact that the damaged area tends to follow the element path is visible in ‘‘3 3  3” but is somehow lost in ‘‘5  5  5”. This cannot be judged as a deficiency of the method because the load-displacement diagram and overall direction of the cracking band is captured, regardless of the mesh size and orientation.

6. Discussion

Understanding fibre kinking of composite materials and mesh dependency in damaged models are two important and linked problems. Both have been extensively studied in the past decades with significant progress in both understanding of the mechanisms and creation of mesh objective models. However, none of the fibre kinking models implemented in an FE framework gives the whole

Fig. 6. Cyclic response: (a) cyclic kinking stress and associated damage, (b) cyclic shear stress.

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response in a homogenized/mesoscale model. Solving simultane-ously for stress equilibrium, strain compatibility and nonlinear shear response of the matrix results in a physically sound response for the whole compressive behaviour. Thus, the model also cap-tures the influence on stiffness and strength for varying misalign-ment angles. This physically meaningful behaviour comes at a cost, as it requires the use of numerical methods to solve the referred equations. Since iterations inside the VUMAT cannot be avoided, the Eqs.(2)-(4)were simplified by using double angle for-mulas to avoid more expensive calculations with quadratic trigonometric terms. This results in an increased speed of approx-imately 25% for the whole simulation, which is meaningful for the typically long crash simulations.

Models with a strain-softening response develop strain localiza-tion, as discussed in[16]. In the current model, the stress distribu-tion is uniform until the maximum stress is reached. Immediately after this, the deformation localizes into a kink-band and the cor-responding load drop occurs. A similar behaviour was obtained in [27]. Strain localization can be interpreted as formation and propagation of the kink-band. The drawback is that this spurious localization results in pathological mesh sensitivity in numerical simulations. A proper scaling of the longitudinal strain with the introduction of a localization limiter, corresponding to the width of the kink-band, gives mesh objective response. The use of a local-ization limiter implied that there is a critical minimum length of the element given by the kink-band width, and a critical maximum length. Method 1 indicates the presence of snap-back behaviour for elements four times larger than the kink-band width, but this may

cause numerical problems in the FE solutions. Method 2 allows for significantly larger element size, in the order of 25 times larger than the kink-band width.

The implemented model is able to predict the interaction with splitting and the appearance of transverse stresses during kinking. The raise of tensile strains in the transverse direction is in accor-dance with[28]. Introducing a maximum transverse strain crite-rion during the kinking would allow the detection of splitting. 7. Conclusions

An FE implementation of a validated material model is detailed with focus on mesh objectivity. The bisection root finding method was used with satisfactory results for solving the stress-equilibrium and the strain-compatibility robustly and simply. The mesh objectivity, a major issue of the models with softening behaviour, is resolved for different refinement levels, without tak-ing advantage of the fracture toughness. It is shown that a method based on strain decomposition gives mesh objective results, but is limited in element size and that elastic snap-back otherwise occurs. Another method based on smeared strains is proposed (Method 2), and is shown to be computationally more efficient and able to avoid snapback for higher ratios of element size/ kink-band width than Method 1, while still being mesh objective. Thus, a model that requires less material characterization has been implemented in an FE framework that allows crash simulations to become more predictive. This results in designs at lower cost and further use of composites into lower-end applications, e.g.

main-Fig. 8. Method 2: (a) kinking responses, (b) respective mesh refinements.

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stream cars. For future work the current FE model will be validated against experimental tests.

Acknowledgment

The funding for this research from Fordonsstrategisk Forskning och Innovation (FFI) via VINNOVA is gratefully acknowledged. References

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References

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