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Abstract

This article numerically investigated the curing temperature effect on matrix shrinkage crack and the effect of matrix shrinkage cracking on delamination in composite T-piece specimens using a modified interface cohesive model with thermal effects accounted. Thermal relative coefficient was introduced to produce the relative thermal displacement in the formulation of interface cohesive elements. The thermal shrinkage crack in the deltoid region of T-piece was simulated. Effect of this thermal initial cracking on the prediction of dominated delamination [Chen, J., Ravey, E., Hallett, S., Wisnom, M. and Grassi, M. (2009). Prediction of Delamination in Braided Composite T-Piece Specimen, Composites Science and Technology, 69(14): 2363–2367; Chen, J. (2011) Simulation of Multi-directional Crack in Braided Composite T-Piece Specimens Using Cohesive Models, Fatigue & Fracture of Engineering Materials & Structures, 34(2): 123–130.] of T-piece under T-pull loading case was also studied. The investigation indicated that some improper restraints to T-piece specimens during the curing process will induce so called thermal shrinkage cracks in the deltoid region of T-piece. This sort of thermal related matrix shrinkage crack has limited effect on the capacity of T-piece to resist T-pull loading. Radius laminates are the main load carrier in the T-pull loading case. This modeling investigation supplied considerable information for the design and manufacture of T-piece related composite components under pulling condition. Further investigation considering loading cases such as bending and combination with bending and T-pull is suggested in the future work to explore general effects of thermal related matrix shrinkage cracking on delaminating.

Keywords

thermal matrix shrinkage crack thermal initial damage delamination composite T-piece specimen modified interface cohesive model

INTRODUCTION

Study of composite T-piece specimens has received sound attention as it is an important component in aerospace structures and wind turbine blades. Thermal effects from manufacture will remain in components as residual stresses or strains, and possibly incur thermal shrinkage cracking in the matrix. There was evidence of this in manufacturing some T-piece specimens. Investigation of thermal effects on matrix cracking and the effect of thermal related matrix initial crack on delamination mechanisms will benefit the design and manufacture of such composite components. Figure 1 shows a composite T-piece specimen under pull load together with local failure pattern in the area of deltoid of T-piece, which presents a thermal shrinkage crack in matrix, initial delamination point, and dominated delamination. Dominated delamination was recognised at the site between radius laminates and deltoid region by experimental and numerical work reported in [1]. The dominated delamination is actually a main crack growth in the failure pattern of T-piece under a pull load, which associates a significant reduction of structural stiffness.

Figure 1.

(a) T-piece specimen under pull loading, the grey region is the deformed element; (b) failure modes in deltoid region.

Over the last decade there have been a large number of applications using interface cohesive models to predict delamination of fiber composites [1–12]. However, there have only been a few publications studying the failure mechanism by prediction of delamination of T-piece specimens using interface cohesive elements. Previous work [7] initially investigated the potential of interface elements in prediction of delamination of T-piece specimens. And previous investigations [1,13] studied the basic failure mechanism of T-piece under T-pull loading using interface cohesive models. This article aims to use a modified interface cohesive model with thermal effects added to investigate the thermal related matrix shrinkage crack in the deltoid region, and the effect of thermal initial cracking on dominated delamination along the interface between radius laminates and deltoid of T-piece. Thermal loading from curing laminates and mechanical T-pull loading were considered in this investigation. Basic materials of laminates are given in the Tables 1 and 2 [1], geometry of T-piece and T-pull loading were also taken from [1], and clamped restraints on the foot of T-piece is shown in Figure 2. The temperature change 175°C from curing process was used to investigate the thermal shrinkage cracks in the deltoid of T-piece.

Figure 2.

A half 2D model.

Table 1.

Materials properties.

	E₁₁ (GPa)	E₂₂ (GPa)	E₃₃ (GPa)	G₁₂ (GPa)	G₁₃ (GPa)	G₂₃ (GPa)	ν ₁₂	ν ₁₃	ν ₂₃	α ₁₁	α ₂₂	α ₃₃
Outer braided wrap	59.7	60.1	9.7	21.95	4.7	4.7	0.279	0.28	0.28	1.94e-6	2.22e-6	2.8e-5
Braided UD layer	160	9.7	9.7	5.9	5.9	4.7	0.33	0.33	0.28	-1.1e-7	2.8e-5	2.8e-5
[0°] layer	152	9.7	9.7	5.9	5.9	4.7	0.33	0.33	0.28	-1.1e-7	2.8e-5	2.8e-5
[90°] layer and Deltoid	9.7	152.0	9.7	5.9	4.7	5.9	0.021	0.28	0.33	2.8e-5	-1.0e-7	2.8e-5
Platform braids	65.8	46.1	9.7	25.8	4.7	4.7	0.421	0.28	0.28	2.9e-6	1.2e-6	2.8e-5

Table 2.

Interlaminar material strength and fracture energy.

σ ₃₃ _c	σ ₁₃ _c	σ ₂₃ _c	G_Ic	G_Iic	G_IIIc
45 MPa	35 MPa	35 MPa	300 J/m²	1000 J/m²	1000 J/m²

A MODIFIED INTERFACE COHESIVE MODEL WITH ADDED THERMAL EFFECTS

The numerical approach in this work is a modified interface cohesive model which adds thermal effects in the formulation of interface cohesive elements. Thermal and mechanical relative displacements are proposed to be applied together for the study of cohesive interfacial behavior. Firstly, the damage initiation state variable F(ϵ₀) is proposed to be assessed by a quadratic formula given in Equation (1):

(1)

where ϵ_I₀, ϵ_II₀, and ϵ_III₀ are the interface initial damage relative displacements corresponding to mode I, mode II, and mode III fracture, respectively. Equation (1) is a combined initial damage mode considering the coupling effects, which determine the point of damage initiation when F(ϵ₀) = 1. This can be referred to the Figure 3 in which ϵ_j = ϵ_j₀ (j = I, II, III) when F(ϵ₀) = 1 in single fracture case. Corresponding interface strength state σ_j = σ_jt, where, σ_jt (j = I, II, III) is interface normal or shear strength or yield traction. Thus, F(ϵ₀) = 1 in Equation (1) generally presents interface strength state in which interface material begins softening in mixed fracture case.

Figure 3.

A general bilinear softening damage law.

Each relative displacement ϵ_j (j = I, II, III) in Equation (1) can be accounted by a simple superposition as:

(2)

for each fracture mode. In Equation (2), ϵ_jm is usual mechanical relative displacement, ϵ_jt is recognized as a thermal relative displacement which can be accounted by Equation (3):

(3)

where, Δα_j defined as a relative thermal coefficient is the difference between the value of thermal coefficient α_j at upper face and lower face of interface. Thermal coefficient α_j varies with composite material directions, which can be seen in the Table 1. ΔT is the temperature change taken from the manufacturing conditions in curing composite components. h is the thickness of interface, which is usually taken as a unit when zero thickness is proposed in the interface cohesive element. This actually makes the value of thermal relative displacement equal to the value of thermal strain at interface. A temperature change as a thermal load through the defined relative thermal coefficient forms the thermal relative displacement at interface, which consists of, together with relative mechanical displacement, the total relative displacement.

Delamination propagation is proposed to be assessed by the fracture state variable F(ϵ_c), which can be accounted by Equation (4). Figure 3 shows a single fracture case, when fracture occurs F(ϵ_c) = 1, this leads ϵ_j = ϵ_jc (j = I, II, III), the critical relative displacement:

(4)

Actually, Equation (4) presents a mixed mode fracture case in which all contribution from three fracture modes are included. Corresponding fracture energy state variable F(G_c) accounted by Equation (5) becomes a unit state when mixed fracture occurs. In single fracture case shown in Figure 3, when F(G_c) = 1, G_j = G_jc (j = I, II, III):

(5)

Using Equation (2), Equation (4) can be rewritten as Equation (6) at fracture occurred:

(6)

where, each single critical relative displacement ϵ_jc (j = I, II, III) in Equation (6) can be accounted by 2G_jc/σ_jt (j = I, II, III), which is determined by the geometrical triangle relationship between the area G_jc and ϵ_jc, together with yield traction σ_jt, shown in Figure 3. G_Ic, G_IIc, and G_IIIc are the fracture toughness corresponding to mode I, mode II, and mode III crack, respectively. In each single fracture mode the fracture toughness G_jc determines the area under bilinear lines, called a material softening damage law, shown in Figure 3. Corresponding strain energy release rate G_j (j = I, II, III) can be accounted by Equation (7) when fracture occurs:

(7)

This integral needs the relationship between tractions and relative displacements shown in Figure 3, which can be expressed by the Equation (8) [2–4,8]:

(8)

where, the initial stiffness K_j₀ (j = I, II, III) in Equation (8) is determined by the interlaminar strength σ_jt and initial damage relative displacement ϵ_j₀ as K_j₀ = σ_jt/ϵ_j₀ (j = I, II, III). The value of each σ_jt as a yield traction corresponding to different fracture mode can be seen in Table 2, and each ϵ_j₀ used in this investigation of composites was taken as 10⁻³ mm, which is less important than the fracture toughness G_jc and yield traction σ_jt because they determine the critical relative displacement ϵ_jc [2,5,6,8].

The damage scale d in Equation (8) is proposed to be expressed by a quadratic relationship:

(9)

where, each d_j (j = I, II, III) in Equation (9) is used to measure the reduction of stiffness in the material softening stage shown in Figure 2, and is determined by the reduction of stiffness from elastic stage to softening stage. This can be expressed by Equation (10):

(10)

Using the geometrical triangle relationship shown in Figure 3, d_j can be defined by Equation (11) [8]:

(11)

Replacing ϵ_j by ϵ_jt + ϵ_jm, d_j can be expressed as follows:

(12)

The damage coupling factor γ_j in Equation (9) is determined as 0 ≤ γ_j ≤ 1.0, j = I, II, III.

For the single fracture mode, for example, in opening mode, j = I, γ_I = 1.0 and all others are zero. Therefore, the damage scale d = d_I. For the mixed fracture mode in this investigation of plain strain case, the value of γ_j is derived as follows. First, assume the amount of total damage scale in mixed facture mode equals the value of damage scale in the pure single mode I case. This can be expressed as follows:

(13)

γ_I and γ_II were simply treated to have a same value as γ_I₌ γ_II₌ γ in this investigation, and assume the ratio between d_II and d_I was taken as β = d_II / d_I = ¾. Using Equation (13), γ can be worked out by Equation (14):

(14)

It should be noticed that γ_j in 3D mixed fracture mode can be determined in the same way as that used in this 2D case.

An Interface Cohesive Element

Using Equations (1)–(14) given in the last section, the constitution of 2D interface cohesive element shown in Figure 4 can be expressed as a same format as that in references [2–4], and partly repeating as follows:

Figure 4.

A linear 2D interface cohesive element.

Using virtual work to compute the internal nodal forces, q, via:

(15)

where, σ are the tractions mentioned above, t is the thickness, and a is the half-length of the element. The tangent stiffness relationships follow as:

(16)

where:

(17)

In Equation (17), B is the matrix of nodal shape functions of interface cohesive element that relates the nodal relative displacement, p, of interface area, which can be seen as detailed in [2]. B is given as follows, with sub-matrices:

(18)

where, sub-matrices of shape function are given as:

(19)

where ζ is the isoparametric coordinate of the interface element which is shown in Figure 4. D _t in Equation (17) is the cohesive material constitutive relationship, which is given by the derivation of traction as follows:

(20)

In Equation (20), the normal and tangential components of the traction acting on the interface in the cohesive process zone can be derived by the relationship between tractions and relative displacements given in the Equation (8), and the damage scale d with coupled three fracture modes. It should be noticed that the nodal relative displacements, p, (strain producing) are the differences in displacements between the top and bottom surface:

(21)

where, it should be noted that ϵ defines a set of relative displacements and not strains. Nodal relative displacements p are defined as a set of degree of freedom for the top and bottom surfaces in the 2D element shown in Figure 3:

(22)

where, the displacement fields, u ^T= (v, u) would then (the order is v, u because of use of mode I, mode II) be as follows.

(23)

Equations (1)–(23) were used to develop a linear 2D interface cohesive element shown in Figure 4 through a user subroutine in ABAQUS [14], which was used in the investigation of a plain strain case presented in this article. However, the function to account all three fracture modes mentioned in last section are available in the developed user subroutines.

A Compression Model in the Material Softening Damage Law

Figure 3 also shows a proposed compression model which deals with the compressive failure at interface as an opposite failure case against opening fracture. $G'_{c}$ , $ɛ'_{0}$ , $ɛ'_{c}$ , and $σ_{t}^{'}$ shown in Figure 3 are the fracture energy, damage initiation relative displacement, critical relative displacement, and interface strength in compression case. In general, they are similar to that given in the opening failure mode and should be determined by relevant experimental tests. However, they were simply treated to have same values as that in opening mode in this investigation. Thus, Figure 3 shows a symmetric softening damage law to treat both opening fracture and compressive failure. And this symmetric model was also used to treat two-way sliding case in shear fracture.

Modeling Thermal Shrinkage Cracks

A half plane strain model of T-piece is shown in Figure 2. Except the symmetric condition (horizontal movement restrained in the middle plane of web), the middle point on the top of web of T-piece was supported vertically. Four-noded quadrilateral plane strain elements CPE4I from ABAQUS were employed in this modeling. Interface cohesive user elements were inserted in the potential crack area, that is, deltoid region, and the interface between laminate radius and deltoid. Mesh density in crack potential area was limited by the critical length 0.36 mm calculated from the linear fracture mechanics based on formula as Equation (24):

(24)

where, fracture energy G_c was taken as G_Ic, material property E was taken as E₃₃ of UD ply, and interlaminar strength σ_c was taken as σ₃₃_c. ψ is cracking shape parameter and was taken the same value 1.12 used in [1]. The actual modelled crack length should be less than this value, and 0.125 mm was therefore used as element length along potential crack path in this model, which determines the length of each cohesive element and hence ensures that modeling of the crack can be captured below its critical length.

A released restraint condition on the foot of T-piece referred to Figure 2 was applied firstly to study the effect of temperature change from curing laminates on the behavior of T-piece. The temperature change 175°C as a thermal loading was applied on the entire mesh of T-piece. General material properties and fracture toughness are given in the Tables 1 and 2 respectively, which were taken from [1]. Figure 5(a) shows the deformed deltoid region in a half mesh which shows that there was no significant cracking simulated by interface cohesive user elements. This is because the relative thermal coefficients Δα_j, worked out from Table 1, are small which have not produced enough thermal relative displacements at interface under given temperature change. The foot of T-piece was freely rotating in this case, thus deltoid region was not deformed much. When restraint was given by clamping the foot of T-piece, significant compression in the deltoid was observed, and material compressive failure simulated by interface cohesive user elements is shown in Figure 5(b). This implicated that materials would fail in deltoid when improper restraints are applied in curing composite T-piece, and failed materials would incur cracks in the deltoid when specimens cool down and restraints are released after curing. This was observed in manufacturing some T-piece components.

Figure 5.

(a) Deformed deltoid region in a half mesh due to thermal effects without restraints; (b) deformed deltoid region in a half mesh due to thermal effects with restraints.

Predicting Delamination of T-Piece with Thermal Initial Crack

Based on the modeling work presented in last section, investigation of the effect of thermal initial crack on delamination was carried out in which failed materials in deltoid due to curing temperature as a thermal initial crack was existing. Figure 6(a) shows delamination and thermal shrinkage crack in deltoid region. Delamination without thermal initial crack is shown in Figure 6(b). Load–displacement curves for two cases, pull and pull plus thermal initial crack, are given in Figure 7, which shows two curves are mostly in superposition, a slight difference can be seen at the point of failure load. Figure 8 shows predicted failure loads together with experimental solutions marked by recorded numbers 4 to 9. In Figure 8, the first recorded failure load gives the average tested value 49.4 N/mm, which was worked out from six specimens with all different manufacturing quality levels [15]. It should be noticed that the tested mean 55 N/mm reported in [1] was obtained as an averaged value from specimens with good quality only. The second and third recorded numbers in Figure 8 give the predicted failure load 54.9 and 54.1 N/mm for T-pull loading case and T-pull plus thermal initial crack, respectively. Both predicted failure loads are higher than the tested average failure load 49.4 N/mm by 10%. Comparing to the pure T-pull loading case, the reduced percentage of predicted failure load in the loading case of T-pull plus thermal initial crack is less than 1.5%. This indicates that the effect of thermal initial crack on delamination is not significant in this T-pull loading case.

Figure 6.

(a) Delamination due to pull plus thermal initial crack; (b) delamination due to pull with no thermal initial crack.

Figure 7.

Load–displacement curve.

Figure 8.

Failure response.

DISCUSSION AND FUTURE WORK

Development and application of a modified interface cohesive model with added thermal effects are presented in this article. This model was used to simulate the matrix thermal shrinkage crack and delamiantion propagation in deltoid of T-piece under pull loading. Investigation indicated that some improper restraints to T-piece specimens in curing process will bring so called thermal shrinkage cracks in deltoid region. However, this sort of thermal related matrix initial crack had limited effects on the capacity of T-piece to resist T-pull loading. Radius laminates were the main load carrier in the pull loading case. It should be noticed that above comments were given based on the comparison using the average tested value of failure loads [1,15]. Present experimental work has not yet supplied exact individual failure mechanism of T-piece with or without thermal initial crack in deltoid region. It is the modeling simulation that filled the gap in investigation of such failure mechanism. Although this investigation was single pulling case only, supplied information is considerable for some T-piece related composite components which are mainly subjected to pulling loads.

Future work is expected to investigate thermal effects in more loading cases to study general effects of thermal matrix initial crack on the delamiantion of T-piece specimens under bending, and mixed loading case with pulling and bending. The later one is actually a real loading condition for some components such as aero engine blades. Future work will entirely assess the general capacity of T-piece specimen under combined loading case. This would be valuable in the progressive failure analysis of practical composite blades which are doubly curved bodies. A study of fiber orientation in this sort of curved blades was given in author's previous work [16]. Finally, this modified interface cohesive model with added thermal effects can be applied in a wide range of investigation of composite components under the service load with temperature change in the future.

Footnotes

Figures 1, 2 and 5–7 appear in color online

Nomenclature

F(ϵ₀)

= Damage initiation state variable

ϵ ₀

= Initial damage relative displacement

= Relative displacement

ϵ_c

= Critical relative displacement

σ_t

= Interface strength

= Interface traction

ϵ_t

= Thermal relative displacement

ϵ_m

= Mechanical relative displacement

Δα_j

= Relative thermal coefficient

ΔT

= Temperature change

F(ϵ_c)

= Fracture state variable

G_c

= Fracture toughness

F(G_c)

= Fracture energy state variable

K ₀

= Interface initial stiffness

= Interface reduced stiffness

= Damage scale

= Damage coupling factor

= Nodal forces of interface cohesive element

= Nodal relative displacements of interface cohesive element

K_t

= Tangent stiffness matrix of interface cohesive element

= Nodal shape function of interface cohesive element

D_t

= Constitute relationship of cohesive materials

= Opening relative displacement relating to mode I crack

= Sliding relative displacement relating to mode II crack

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A Numerical Investigation of Thermal-related Matrix Shrinkage Crack and Delamination in Composite T-Piece Specimens Using a Modified Interface Cohesive Model

Abstract

Keywords

INTRODUCTION

A MODIFIED INTERFACE COHESIVE MODEL WITH ADDED THERMAL EFFECTS

An Interface Cohesive Element

A Compression Model in the Material Softening Damage Law

Modeling Thermal Shrinkage Cracks

Predicting Delamination of T-Piece with Thermal Initial Crack

DISCUSSION AND FUTURE WORK

Footnotes

Nomenclature

References