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Microplane model for composite laminates

G. Cusatis1, Z.P. Bazant2, A. Beghini3

1Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute

4048 Johnsson Engineering Center, 110 Eighth St, Troy, NY, USA

cusatg@rpi.edu

2Department of Civil and Environmental Engineering, Northwestern University

2145 Sheridan Rd., Evanston, IL, USA

z-bazant@northwestern.edu

3Skidmore, Owings & Merril LLP

224 S. Michigan Av. Chicago, IL, USA

alessandro.beghini@som.com

Summary: This paper presents the spectral stiffness microplane (SSM) model, which is a general constitutive model for compos-

ite laminates, able to simulate the orthotropic stiffness, pre-peak nonlinearity, failure envelopes, post-peak softening and fracture.The model is verified by comparisons with experimental data for uniaxial and biaxial tests of unidirectional and multidirectional

laminates

Introduction

Various theories can be found in the literature for the descrip-

tion of the mechanical behavior of fiber-polymer composites

[1]. These theories, however, generally neglect the quasibrittle

character of these materials. In quasibrittle fracture, the crack

tip is surrounded by a nonlinear zone (fracture process zone)

that is not negligible compared to the cross section dimension

of the structures. The fracture process zone (FPZ) at crack tip

occupies almost the entire nonlinear zone and undergoes soft-

ening damage instead of plastic deformation typical of ductile

behavior. The stress along the FPZ is nonuniform and the stress

decreases with crack opening gradually, due to discontinuous

cracking in the FPZ, crack bridging by fibers, and frictional

pullout of inhomogeneities.

The present paper summarizes the outcome of a recent research

effort [2, 3] in which a complete theory for the mechani-

cal behavior of fiber-polymer laminates has been formulated

in the framework of the microplane model. A constitutive law

is first developed for laminates with unidirectional reinforce-

ment. Subsequently, general laminates with multidirectional re-inforcement are modeled as an overlay of lamina with unidirec-

tional reinforcements of different orientations.

Microplane Model Formulation with Spectral De-

composition

By exploiting the spectral decomposition theorem, the material

stiffness matrix can be decomposed [4] as follows:

E =

I

IEI (1)

where I are the eigenvalues of the stiffness matrix and EI de-

fine a set of matrices constructed through the diadic products

of the eigenvectors of the materials stiffness matrix. The matri-

ces EI also decompose the stress and strain vectors into ener-

getically orthogonal modes, which are called eigenstresses and

eigenstrains. In the case of isotropic materials, these orthogo-

nal modes represents the volumetric and deviatoric deformation

modes.

By projecting the eigenstrains on a generic microplane of a

kinematically constrained microplane model [5], it is possi-

ble to decompose the microplane strain vector into orthogonal

components (PI, microplane eigenstrains) that can be used to

drive the constitutive behavior at the microplane level. From themicroplane eigenstrains, the microplane eigenstresses, PI,

can be calculated according to suitable constitutive relations

for the normal and shear components of each eigenmode. The

macroscopic stress tensor may then be computed from the prin-

ciple of virtual work, which reads:

=3

2

I

EIPTPId (2)

where is the surface of a unit hemisphere, is the contraction

of the stress tensor into a six-dimensional vector, and P is the

macro-micro projection operator.

In [2] and [3], the approach highlighted above has been

fully developed with reference to unidirectional (transversely

isotropic) composite laminates. In this case, four orthogonal

modes exist and each mode can be approximately associated

with a specific failure mode. This observation greatly simplifies

the formulation of the constitutive behavior at the microplane

level.

Calibration and Validation

An extensive calibration and rigorous validation of the pre-sented model is still under way. Nevertheless, preliminary re-

sults [2, 3] are promising and the developed theory seems to

be able to capture the most relevant aspects of the behavior of

composite laminates. Fig. 1 shows the comparison between the

microplane simulation, the well known Tsai-Wu criterion [7],

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Figure 1: Comparison between numerical simulations (solid

line), Tsai-Wu criterion (dashed line), and experimental results

(points) from [6] for the biaxial failure envelope ofa unidirec-

tional laminate.

and the experimental results (published in [6]) for the biaxial

failure envelope of a unidirectional laminate.

The developed and calibrated microplane model for unidirec-

tional laminates can be then used for the simulation of multidi-

rectional laminates. A widely used laminate lay-up is (90/+45/-

45/0)S , which is quasiisotropic. The behavior of this multi-

directional laminate is here simulated assuming each ply to be

governed by the microplane model for unidirectional laminates.

Fig. 2 shows the comparison between the experiments, the mi-

croplane model prediction and the prediction of the Tsai-Wu

criterion. The microplane model theory agrees very well with

the experimental data in the tension-tension quadrant of the en-

velope. For the tension-compression quadrant, the prediction is

less accurate but still satisfactory. However, marked disagree-

ment is found in the compression-compression quadrant, in

which both the microplane model and the Tsai-Wu criterion

severely overestimate the laminate strength.

Conclusion

1. The spectral decomposition theorem, applied to the mate-rial stiffness matrix, is a powerful tool to analyze generally

anisotropic materials.

2. The present SSM (spectral stiffness microplane) model

describes well the experimentally observed behavior of

fiber composites, not only for uniaxial stress-strain curves,

but also for multiaxial failure envelopes.

3. The main advantage of the SSM model is that one and

the same model can simulate the orthotropic stiffness, fail-

ure envelopes and the post-peak behavior, which include

strain-softening damage and fracture mechanics aspects.

This further implies that the SSM model must be able toautomatically predict the energetic size effect.

4. The SSM model can be implemented as a material sub-

routine in finite element codes, either implicit or explicit.

From experience with microplane models for concrete, the

Figure 2: Comparison between numerical simulations (solid

line), Tsai-Wu criterion (dashed line), and experimental results

(points) from [6] for multiaxial failure envelope fora multidi-

rectional laminate.

kinematically constrained formulation is known to be very

stable in finite element analysis.

References

[1] I. M. Daniel, and O. Ishai (1994): Engineering Mechan-

ics of Composite Materials. New York: Oxford University

Press.

[2] G. Cusatis, A. Beghini, and Z. P. Bazant (2007): Spec-

tral Stiffness Microplane Model for QuasiBrittle Compos-

ite Laminates: I. Theory Journal of Applied Mechanics,

ASME, in press.

[3] A. Beghini, G. Cusatis, and Z. P. Bazant (2007): Spectral

Stiffness Microplane Model for QuasiBrittle Composite

Laminates: II. Calibration and Validation. Journal of Ap-

plied Mechanics, ASME, in press.

[4] P. S. Theocaris, and D. P. Sokolis (1999): Spectral decom-

position of the linear elastic tensor for monoclinic sym-

metry. Acta Crystallographica, A55, 635647.

[5] Z. P. Bazant, and B. H. Oh (1985): Microplane model for

progressive fracture of concrete and rock. Journal of En-

gineering Mechanics, Trans. ASCE 111, 559582.

[6] P. D. Soden, M. J. Hinton, and A. S. Kaddour (2002): Bi-

axial test results for strength and deformation of a range of

E-glass and carbon fibre reinforced composite laminates:

failure exercise benchmark data. Composites Science and

Technology 62, 1489-1514.

[7] S. W. Tsai, and E. M. Wu (1972): A general theory of

strength for anisotropic materials. Journal of Composite

Materials, 5, 5880.