Multi Scale Modeling and Failure Analysis of Laminated Composites

In present study a multi scale modeling and failure analysis of laminated composites is performed. for micro level study Rule of Mixtures and Halphin-Tsai equations are used to determine lamina properties. Off-axis failure strength of lamina for different volume fractions are calculated using Finite Element software ANSYS. Finite element analysis results are compared with analytical results and published experimental results. In macro level study of laminates first ply failure load of laminates is calculated using ANSYS and compared with analytical results. Various failure theories i.e. maximum stress theory, maximum strain theory, Tsai-Wu, Tsai-Hill and Puck failure criteria are implemented. First ply failure load for different lamination schemes are calculated for uni-axial and Bi-axial loading conditions.


Introductıon
Laminated composites are made by stacking various layers of unidirectional lamina at different angles to provide required stiffness and strength in particular direction. Each lamina is made up of unidirectional fibers arranged in a matrix. Hence study of laminated composites can be performed at different scales. In micromechanical analysis of lamina properties of individual constituents, interaction between fiber and matrix, distribution of fibers in matrix all these factors are considered. In macromechanical analysis lamina is considered as a homogenous and orthotropic body. In macromechanical analysis of laminates entire stack of lamina is considered as a single body and integration of individual properties of lamina are used for analysis. Micromechanics of lamina is helpful while developing new fiber matrix system. Macro level analysis of lamina can be used when determine stress and strains in lamina. Further off axis strengths can be calculated when loading is not along the direction of fibers. Macromechanical analysis of laminates is capable to predict behavior of laminates under different loading conditions. Experimental study of laminated composites is expensive and time consuming work. Hence so many theoretical models have been proposed to determine properties of laminated composites. For micromechanical analysis of lamina Halphin and Tsai proposed model to predict elastic properties of lamina. Huang developed a formulae based on micromechanics to predict strength properties of lamina [1]. Off-axis strength of lamina is important when direction of loading is not along the direction of fibers. Pipes performed a benchmark experimental study on boron-epoxy, aramid-epoxy and graphite epoxy lamina to determine off-axis failure strength [2]. First ply failure load is important parameter while designing laminated composite structure. Theoretical and finite element analysis procedures have been developed to determine first ply failure of laminated composites. Reddy et al. performed a benchmark study to determine first ply failure load of laminated composites using finite element method [3]. Kam et al. determined first ply failure load of laminated composites using analytical and experimental methods [4]. Rahimi et al. determined first ply failure and last ply failure loads for [Ө 4 /0 4 /-Ө 4 ] s lamination scheme using finite element software ANSYS [5,6].
In present work modeling and failure analysis of laminated composites is performed using finite element software ANSYS. Finite element results are compared with theoretical results. Theoretical procedure is coded in MATLAB program.

Methodology
In present study investigation is carried out at three different scales.

Maximum stress failure theory
This theory is based on maximum stress theory of Rankine and maximum shear stress theory of Tresca. According to this theory failure occurs when any one of stress in material axis exceeded the failure value of stress.
Lamina is considered to be failed if any one of the following conditions violates- Lamina is considered to be failed if any one of the following conditions violates-

Tsai-Wu failure theory
This interactive failure theory is based on strain energy theory for isotropic materials.
Tsai-Wu failure theory when applied to a lamina states that, a lamina is considered to be safe if:  For an angle lamina it required to define a different co-ordinate system which is known as local co-ordinate system. Axis along the direction of fibers is known as longitudinal local axis whereas axis perpendicular to the fiber is known as transverse local axis.
The hook's law for a 2D angle lamina can be written as: Above equation can be written as-

Tsai-Hill failure theory
Based on distortion energy theory for isotropic materials. Lamina is considered fail when following condition violates-

Modeling and Finite Element Analysis of Laminates in ANSYS Steps involve in finite element analysis (ANSYS)
Finite element analysis involves three stages of activity: preprocessing, processing and post processing. A complete finite element analysis is a logical interaction of the three stages.
Steps of finite element analysis in ANSYS-

Micromechanical analysis of lamina
In this section three different fibers Graphite, Boron and Aramid fibers with epoxy as a matrix system has been analyzed. Elastic properties such as Young's modulus, Poission'ratio, shear modulus and strength properties has been calculated and compared at different fiber volume fractions. Volume fraction of fibers is varied from 30% to 65%.

Macromechanical analysis of lamina
In this section strength of lamina is calculated when orientation of fibers is not along the direction of loading. This strength is also known as off-axis strength of lamina. Off axis strength of lamina is calculated using FEA software ANSYS and stress-strain relationship and failure theory are   Table 3: Properties of aramid and epoxy [15].     coded in MATLAB. Results obtained using ANSYS and MATLAB are compared with available experimental results. Finally variation of off-axis strength with fiber orientation angle is plotted for different volume fraction of boron fiber in epoxy matrix (Figures 2 and 3).

Validation of FEA model (ANSYS)
Comparison of FEM results with Experimental Results [7] shown in Table 5. In Figure 4 comparison between FEA (ANSYS) results and theoretical (MATLAB) results is presented. A very good agreement is found between ANSYS and MATLAB results. Failure strength at different fiber orientation angles and local stress results are calculated using ANSYS and MATLAB. From above observations it can be concluded that analysis of lamina can be performed accurately using ANSYS.

Variation of off-axis strength with fiber orientation angle for different volume fractions
It has been observed from Figure 5 that failure strength decreases with increase in fiber orientation angle. Maximum strength is obtained when loading is along the direction of fibers and minimum when loading is perpendicular to the direction of fibers. Higher strength values are associated with higher fiber volume fractions. It is also noticeable that effect of fiber volume fraction is only significant for only low fiber orientation angles. Beyond 10 ̊ fiber orientation angles failure strength of lamina is almost same for all volume fractions. Since cost is directly associated with fiber volume fractions. Higher the fiber volume fraction, higher will be cost of lamina. Hence it is suggested to use lamina with lower fiber volume fraction when off axis angles are higher than 10 .
Failure strength drastically decreases from 0 ̊ to 15 ̊ which is from 1260MPa to 760.75MPa according to maximum stress theory. Results obtained using Tsai-Wu and Puck failure criteria are very close whereas maximum stress theory predicts higher failure load in range of 15 ̊ to 60 .