FAILURE PREDICTION AND DAMAGE MODELLING OF MATRIX CRACKING IN - - PDF document

18TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS

1 Introduction Matrix cracking under quasi-static loading is an important failure mode in quasi-isotropic laminates. Cracking can result in a significant loss of structural stiffness and lead to other forms of damage, such as

• delamination. Early attempts for the modelling of

matrix cracking based on shear-lag theories over predicted the saturation crack density. With regards to crack opening/sliding approaches, although direct finite element modelling of matrix cracking is feasible at the micro-scale, it can be a costly approach in terms of time and computational effort, as the exact geometry of the laminate must be

• modelled. Furthermore, the measured crack opening

and/or sliding displacements are then input into a macro-scale model which must be run separately. At the ply level, continuum damage models are preferred as the fibres and matrix are not individually modelled, allowing results to be more readily obtained. Models using variational mechanics [1-3] are more conservative, but until recently have been limited to (S°/90°)s layups with either in-plane uniaxial or biaxial loading. Other analytical and numerical models in the literature require the use of iteration to

• btain the current crack density and cannot therefore

be readily implemented into explicit finite element codes. 2 Proposed Model A model for matrix cracking based on the concepts previously used in variational and stress transfer approaches [4] is refined and combined with the LaRC05 failure criteria [5] to detect the onset of matrix cracking and subsequently degrade properties for generic in-plane loading in S°/90° laminates. Furthermore, the saturation crack density is calculated by comparing the energy release rates for matrix cracking and delamination. Unlike earlier models, to allow for the prediction of crack formation in non-90° plies the effects of shear stress are included and the ply of interest is assumed to be supported by two different sublaminates. The model is able to predict the stiffness reduction of the laminate as matrix cracks accumulate and accurately calculate the saturation crack density, after which delamination is induced. The combined failure criteria and damage model are coded as a user defined material subroutine in an explicit commercial finite element package (Abaqus\Explicit VUMAT) and are shown to compare favourably with the experimental data available in the literature. 2.1 Failure detection The model uses the failure indices previously

• utlined by Pinho et al. [5]. The failure index for

matrix failure is a modified version of the Mohr- Coulomb criterion, adapted for UD composites in a similar manner to Puck et al. [6]: (1) where and are the transverse and longitudinal shear components of the traction,

is the in-situ

in-plane transverse tensile shear strength,

and

are the transverse and longitudinal in-situ transverse shear strengths. is the normal component of the traction and and are the slope coefficients for transverse and longitudinal shear strength, respectively.

FAILURE PREDICTION AND DAMAGE MODELLING OF MATRIX CRACKING IN QUASI-ISOTROPIC LAMINATES AT THE PLY LEVEL

G.M. Vyas1*, S.T. Pinho1, P. Robinson1

1 Department of Aeronautics, Imperial College London, South Kensington Campus, London,

SW7 2AZ, UK * Corresponding author (gaurav.vyas04@imperial.ac.uk)

Keywords: Matrix cracking, Stiffness reduction, Stress field

2.2 Stress and Displacement Fields The determination of the stress and displacement fields follows the derivation

• utlined

by Farrokhabadi et al. [4]. The dimensions of the representative volume element of the lamina shown in Fig. 1 are given by , and , where is the width of the lamina. Generalised plane strain is assumed for the displacement field, which implies the displacement field is: (2) where

denotes the average strain along the fibre

direction of the cracked lamina, which is assumed to be equivalent to that of the uncracked lamina. It is assumed that after cracks have formed, two perturbation stresses appear in the -direction at the interface between the thick and thin sub-laminae of the 90° lamina, and . The through the thickness shear stress components of the cracked lamina are assumed to be piecewise linear for each sub lamina and are independent of the -direction due to the generalised plane strain assumption: (3) (4) where the superscripts (1) and (2) denote the thick and thin sub-lamina respectively and 90 indicates a remote stress applied on the undamaged lamina. From equilibrium with the above equations, the following stresses are derived: (5) (6) The derivation of the normal through the thickness stress components assumes that continuity of is enforced across the interface between the thick and thin sub-laminae due to perfect bonding at the interface , with the remote stress,

, as a

boundary condition, as well as satisfying the equilibrium equations: (7) In the following derivation, and are defined as: (8) Using the relation

and

the stresses for each sub-lamina, the through the thickness displacements are derived as: (9) The transverse displacements are derived from the through thickness displacements using

. Integrating in the cracked and

uncracked regions with respect to z and simplifying:

3 PAPER TITLE

(10) The in-plane displacements in the fibre direction are derived using the relation

. Integrating in both regions gives:

(11) where ,

and are the displacements at the

interface and are determined as follows. By symmetry, the vertical displacement at can be set to zero, which prevents rigid body displacements in z and rigid body rotations in x and y. This allows for the determination of

. At ,

continuity of displacements implies that

, allowing for the

determination of

:

(12) Similarly, an expression for

can be obtained

by using the horizontal displacement at , and rearranging the resultant expression. As before,

is found using continuity:

(13) The determination

• f

requires the displacement in the x-direction at , . Using the x-symmetry, rigid body displacements in x and rigid body rotations in z can be prevented imposing . As with the

• ther

displacement components, continuity infers and allows for the calculation of : (14) Using averaged values for all displacements and stresses, the in-plane stress-strain relations are: (15) The averaged values for

and in each sub-

lamina in equation are then subtracted from each

• ther, leading to the differential equations:

(16) The constants are material dependent and given by: (17) The boundary conditions required for the solution of the differential equations are: (18) 2.3 Damaged compliance matrix and saturation crack density The saturation crack density is calculated by comparing the energy release rates for matrix cracking and delamination. The intersection of the two curves is defined as the saturation value, after which delamination will occur instead of additional crack accumulation. The modelling of the reduction in laminate stiffness after damage requires the definition of two functions

• f the damage parameter, , given by:

(19) The evolution of the variables in equation 19 is given by the compliance relationships used in the variational models by Nairn & Hu [1] and Hashin [2]. Following the work of McCartney [7], the damaged compliance matrix is then given by: (20) Where k and k’ are given by: (21) 3 Results An example comparing the proposed variational approach to a shear lag approach for the calculation

• f saturation crack density is shown in Fig. 2.

Predictions for the change in modulus of cross-ply laminates of glass/epoxy [8] and AS4/3502 [9] are

5 PAPER TITLE

shown in Fig. 3 and Fig. 4 respectively and compared with experimental data.

• Fig. 5, Fig. 6 and Fig. 7 show predictions of stresses

and displacements made by the model compared with FE simulations. 4 Discussion The proposed approach produces predictions which are close to the experimentally measured of saturation crack density, as shown in Fig. 2.

• Fig. 3 and Fig. 4 indicate the model is able to predict

the modulus change of laminates up to the predicted saturation crack density. The predictions are conservative and in good agreement with the experimental data. The stress and displacement fields are very well reproduced by the model as shown in Fig. 5 to Fig. 7 5 Conclusions A model for the prediction of matrix cracking is combined with the existing LaRC05 failure criteria and damage model. The new model shows improved predictions for damage accumulation and is able to predict the stiffness change of laminates undergoing matrix cracking. The model allows for generic in- plane loading and further examples will be presented at the conference. 6 Acknowledgements The authors wish to acknowledge the support of the EPSRC and Airbus through CASE award number 08000674.

• Fig. 1. A 90° lamina divided into a cracked thick sub

lamina and uncracked thin sub lamina

• Fig. 2. Comparison of saturation crack densities for

various laminates and layups using a shear lag approach and a variational approach

• Fig. 3. Predicted normalised modulus up to

saturation crack density for a cross-ply glass/epoxy laminate

• Fig. 4. Predicted normalised modulus up to

saturation crack density for cross-ply AS4/3502 laminates

z y L z2 z1 h2 h1

Thin sub lamina (2) Thick sub lamina (1) Cracked surface

h

• 50

50 100 150 200 0.2 0.4 0.6 0.8 1 1.2 1.4

% Difference from experimental Ply Thickness (mm) Shear Lag Current Variational Model

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Normalised Modulus Crack Density (/mm) Predicted Experimental Data (Varna et al. 2001) Glass/Epoxy (02/904)s

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 0.5 1 1.5 2 2.5

Normalised Modulus Crack Density (/mm) AS4/3502 (0/90)s (02/902)s (02/902)s (02/903)s Predicted Experimental (Lee and Hong 1992)

• Fig. 5. Comparison of predicted σy in cracked

region of a (0°/90°/0°) glass/epoxy laminate with a finite element simulation

• Fig. 6. Comparison of predicted τyz in cracked region
• f a (0°/90°/0°) glass/epoxy laminate with a finite

element simulation

• Fig. 7. Comparison of predicted through thickness

displacement in cracked region of a (0°/90°/0°) glass/epoxy laminate with a finite element simulation References

[1] J.A. Nairn and S. Hu. “The initiation and growth of delaminations induced by matrix microcracks in laminated composites”. International Journal of Fracture, Vol. 57, No. 1, pp 1-24, 1992. [2] Z. Hashin, “Analysis of stiffness reduction of cracked cross-ply laminates”. Engineering Fracture Mechanics, Vol. 25, pp 771-778, 1986 [3] J.L Rebiere et al. “Initiation and growth of transverse and longitudinal cracks in composite cross-ply laminates”, Composite Structures, Vol. 53, pp 173- 187, 2001. [4] A. Farrokhabadi, H. Hosseini-Toudeshky and B. Mohammadi. “A generalized micromechanical approach for the analysis of transverse crack and induced delamination in composite laminates”. Composite Structures, Vol. 93, pp 443-455, 2011. [5] S.T. Pinho and R. Darvizeh and P. Robinson and C. Schuecker and P.P. Camanho. “Material and structural response

• f

polymer-matrix fibre- reinforced composites”. Composites Science and Technology, 2008; Accepted for publication (Special issue on Second World Wide Failure Exercise). [6] A. Puck and H. Schürmann. “Failure analysis of FRP laminates by means

• f

physically based phenomenological models”, Composites Science and Technology, Vol. 62, pp 1633-1662, 2002. [7] L. McCartney. “Physically based damage models for laminated composites”, Proceedings of the Institution

• f Mechanical Engineers, Part L: Journal of

Materials: Design and Applications, Prof Eng Publishing, pp 163-199, 2003 [8] J. Varna, R. Joffe and R. Talreja. “Mixed micromechanics and continuum damage mechanics approach to transverse cracking in [S, 90n]s laminates”. Mechanics of Composite Materials, Vol. 37, No. 2, pp 115-126, 2001 [9] J.H, Lee. and C.S. Hong. “Refined two-dimensional analysis of cross-ply laminates with transverse cracks based on the assumed crack opening deformation”, Composites Science and Technology, Vol. 46, No. 2, pp 157-166, 1993

• 5

5 10 15 20 25 30 35 40 0.1 0.2 0.3 0.4 0.5

Stress (MPa) y distance (mm) Predicted FE results

• 2

2 4 6 8 10 12 14 16 0.1 0.2 0.3 0.4 0.5

Stress (MPa) y distance (mm) Predicted FE results

• 0.00045
• 0.0004
• 0.00035
• 0.0003
• 0.00025
• 0.0002
• 0.00015
• 0.0001
• 0.00005

0.00005 0.1 0.2 0.3 0.4 0.5 Displacement (mm) y distance (mm) Predicted FE Results