A finite element based micromechanical model to simulate fatigue damage development in open-hole laminated composites

A micromechanical model is implemented to indicate the progressive fatigue problem of a laminated composite with a central circular hole under fatigue loading based on a finite element model. The mechanical properties of the composite lamina are represented based on the characteristics of the fiber and the matrix through a micromechanics model. An appropriate algorithm is then adopted to simulate fatigue damage development in the composite lamina. According to this algorithm the stress field of the composite subjected to fatigue load is initially obtained using finite element method. Finally, the predicted results of the stresses in the constituents i.e. fiber and matrix are determined according to the micromechanical bridging model. Finally applying proper damage driving relations leads to damage degree in each element. The proposed model is proven to be successful in the observation of the fatigue behavior with stiffness degradation in each elements of the composite in each cycle. Results are reported and validated using those micromechanical model and experimental data available in the literature.


Introduction
In recent years advanced composite materials [1,2] are being used increasingly in many engineering applications. The high stiffness to weight ratio [3,4] coupled with the flexibility in the selection of lamination scheme make the laminated composite shells attractive structural components for aerospace, automobile and many other vehicles [5,6].
Employing laminated composites in various structures has motivated researchers to propose innovative algorithms and/or methods to simulate their behaviors durin g design courses [7][8][9]. On e of these main challenges and interests is predicating fatigue life by modeling damage which is created by fatigue as a critical mechanisms of failure for composite materials [10][11][12][13][14] .
Predicting the behavior of composite materials is more complex than homogeneous materials under fatigue load and is controlled by the types of the fiber and matrix, directions and number of fiber layers, and also stress ratio [15]. They are anisotropic and the damage mechanisms like delaminat ion , deboning and breakage of fibers, and matrix cracking [16][17][18][19] are highlighted as the main reason of failures in these materials. So, the life assessments models based on the fatigue failure in composite materials can be listed in different classifications likes phenomenological residual stiffness and strength models [20] .
Shokrieh and Lessard [21] predicted the composite materials fatigue life under the cyclic loading using their proposed a progressive damage model. So, the laminate fatigue life was represented as number of cycle's relationships. Indeed, the approach is emerged from three major components, thes e components are based on stress calculations, failure assessments and the rules which specified the degradation of material property. This method will be capable to simulate the residual strength and stiffness and life of arbitrary geometry composite under every cyclic loading. Using the same progressive model, Naderi and Maligno [11] adopted an ABAQUS finite element simulation to estimate the fatigue life of carbon/epoxy laminates with different layup sequences. Later on, Lian and Yao [22] developed a MSC Patran/Nastran finite element analysis to simulate the fatigue damage in composite laminates with an own stiffness degradation model to reduce the elastic constants based on the failure mechanism.
Macro mechanical modeling is common approaches for modeling the failure behavior of composite materials [23,24] .These models and its governing effective parameters which related to the material properties can be extracted through unidirectional tests on the fibrous lamina. Because of the fiber and matrix materials stresses and the internal stresses causes, it is not possible to study the constituent fiber or matrix failure causes.
In order to overcome on the latter issue, few researchers have proposed employing fiber and matrix properties to determine the internal stresses in the constituents of the composite subjected to fatigue load in micromechanical scale. Zheng et al. [25] proposed the bridging micromechanical model to estimate fatigue strength of unidirectional composites. Later on, Zabihpour et al. [26]   According to this algorithm after set on initial data, the given geometry, material properties of constituents (i.e. the fiber and the matrix) along with fatigue loading is firstly defined. The geometry data like number of layers, thicknesses and fiber directions together with geometry dimensions is employed to model and discretize the composite structure to proper sizes of the elemen ts. Then using fiber volume fraction and properties of the constituents, equivalent property for each element in discretized composite structure is calculated based on rule of mixtures. Now finite element analysis is carried out to calculated element stresses for given cycle of loading. Some transformation matrices should be employed here to transform stresses from global coordinate systems to on-axis coordina t es for all elements. Now using micromechanical relations, the average stresses in the fiber and matrix can be calculated regarding to the average stress of lamina. Then using damage evolution law for fiber and matrix the amount of the damage increment in constituent and consequently total damage degrees (failure indices) are calculated for current cycle of loading. In this stage a couple of states may happens.
 Both failure indices are smaller than 1.0. In this situation the solution continues after calculating new fiber and matrix properties.
 Failure index for fiber is still smaller than 1.0 but failure index for matrix exceeds the value 1.0. In this situation we have matrix cracking. 3) Evaluate the new value of the damage indexes in matrix and fiber for cycle N i+1 as follows: This purposed model that defines the damage status at each load cycles is explained by means of a flowchart ( Figure 1) which was discussed above and implemented in ABAQUS software as a UMAT subroutine.

Micromechanical model
Indeed, the micromechanical models which are represented here want to extract the composite materials properties based on their constituent properties. Some of these models are as follows: Mixed Role (ROM), Modified Mixed Role and Bridging Model. The bridging model p roposed by Huang [25] is based on establishing strain relationships between fibers and matrix. The elastic properties of the composite layer can be obtained in terms of the elastic properties of the fiber and matrix as below. (2) where where  is the angle between the natural coordinate system ( , ) xy and the material coordinate system '' ( , ) xy which is shown in Figure 2.

Figure 2. Plane stress transformation
The constitutive equation correlating the stress and str ain in a lamina is expressed as It is assumed that the fibers are transversely isotropic thus, the stress and strain relationship are defined as if matrix is isotropic, then In the strain-stress equation of the lamina assuming plane stress condition is reasonable and can represented as The incremental stresses in the fiber are related to those associated w ith the matrix as bellow The relation between stress increments in the fiber and matrix and overall applied stress increment,   d is considered via equation (10).  a a a a a   a a a a a   a a a a  A a a a aa zero a If the problem is taken as two dimensional the matrix form of A can be represented as follow (13)  

Material Property Degradation
Under the first cycles of fatigue loading, the strength of the plies can be considered as higher value of the fiber and matrix as following:  is the medium mass density. During the isothermal process the following relation is obtained as: (24) , In last equation

Model verification
In order to validate the results of the present model, experimental data reported by Shi et al. [27] for E-glass/ polyester GT200 are considered.Tthe mechanical properties of fiber and matrix are show n in Table 1 and Damage parameters of evolution equation are listed in Table 2Table 1. The fatigue life of the lamina for fiber angle 0 o is compared to the experimental ones in Figure 3.   The behavior of elastic modulus E11 of composite lamina verses number of cycle is shown in Figure   10, in the damaged element B and C, E11 decreases gradually as the number of cycles increases.