Investigation of Minimum RVE Size for CFRP Laminates

Therefore, to determine the minimum RVE size, the RVE element was studied by varying the ratio of the fiber diameter (D) to the RVE size (H). Nβ Number of subcells iny2 towards Nγ Number of subcells iny3 towards u(βγ) Displacement of (βγ) subcell. Subcell local coordinate system (βγ). βγ) Total tension in subcell (βγ) σ(βγ) Total tension in subcell (βγ).

Overview of Composite Materials

Among the above, fibrous composites have been selected for further studies as it uses long fibers that have stronger and stronger properties compared to the same bulk composite material [4]. Some of the common fiber materials and their strength-to-density and stiffness-to-density ratios are described in Table 1.1. It can be observed from the values given in Table 1.1 that glass fibers exhibit greater strength to density ratio as shown in Tab.

Figure 1.2: Percentage distribution of composites in various applications [1]

Damage Mechanics in Composite Materials

Analysis of Composite Materials

Macro-mechanical analysis

The micromechanics approach enables the physics of deformation and damage to be captured at a more fundamental scale [8]. The simplest and most basic failure criteria are sufficient for the micromechanical analysis of composite materials.

Micro-mechanical analysis

Analytical and Semi-analytical Methods for Micro-mechanical Analysis

This method is only valid for composites with an infinite body, an isotropic microstructure and the materials exhibiting an isotropic effective response. This method does not provide accurate results for the micromechanical analysis of composites, as they generally exhibit an orthotropic microstructure. This method is based on an averaging scheme used to model the elastic and plastic response of the composites.

Numerical Methods for Micro-mechanical Analysis

Introduction to Representative Volume Element (RVE)

Literature Review

3] have used the high-fidelity generalized cell method to study the behavior of a nonlinear elastic, elastic-plastic, and viscous-plastic component. 2] studied the modeling of micromechanical damage in composite materials using HFGMC and the failure mechanism in a heterogeneous material. In the first step, the generalized high-fidelity cell method is used to obtain the actual laminate stress state.

Motivation, Scope and Objectives

Thesis Layout

This framework is used for micromechanical modeling of multiphase composites with an improved computational efficiency that also enables multi-scale analysis. HFGMC is a powerful analytical form used to solve nonlinear component equations for estimating stress and strain components developed at finite intervals of applied load. For the calculation of stress-strain variables under finite increases in strain, the high fidelity generalized method for cells has therefore been used.

General Formulation

It is a non-local, micromechanical analysis method that can also capture the shear coupling effect. The stress influence matrix P(βγ) can be calculated using the following expression, which is necessary for calculating the effective stiffness matrix for the RUC element. 2.6, ¯σ(βγ) is an average voltage for the subcell and can be obtained using the following mathematical equation:

Figure 2.1: Micro-structure of CFRP composite with (x 2 , x 3 ) global co-ordinate system

Equilibrium and Continuity Conditions

Equilibrium condition

Displacement continuity conditions

Traction boundary condition:-

Boundary Conditions Applied on Repeating Unit Cell (RUC):-

Global Residual Vector

The final system of tangential equations is obtained by composing the contributions of all subcells (Nβ, Nγ). The global TG and DG matrices are obtained by constructing T and D matrices for all subcells, as shown in Figure 2.4. The last two terms in the residual vector represent a displacement residual that is zero, since these equations do not take stiffness variables into account.

The voltages needed to calculate the residual vector are the total voltages at the voltage integration points shown in Fig. If the absolute value of the global residual vector is within the tolerance, there is no need to find the correction variables, otherwise the Jacobian matrix is estimated to correct these variables using the following expression:. where i denotes the number of iterations and J is the Jacobian matrix given by the following expression: .

Newton - Raphson Iteration Scheme

Summary

Introduction to PDM

Damage Detection

Multi-continuum theory

In this theory, fibers are assumed to be linear elastic in nature and matrix material exhibits non-linear elastic behavior. It is a component-based failure criterion used to construct a nonlinear progressive failure algorithm for investigating the material fracture strengths of composite laminates. The homogenized value used to characterize the stress tensor at a point in a single continuum material is obtained by taking a volume average of all stresses in the specified region.

Damage Modeling

Sudden material property degradation rule (MPDM)

Summary

Input Parameters Required for PDM of CFRP Laminate

Numerical Implementation

The benefits of the sparse implementation are the finer discretization of the unit cells as well as the significant improvement in computational efficiency. By solving the equations of the global system, the displacement, strain, and stress components are obtained for each subcell. Further, if the solution is in the convergence limit, MCT is used to find the damage condition of the subcell.

Newton-Raphson iteration scheme described by Figure 2.6 is used to obtain the solution in convergence limit. Once the damage is detected, material properties of the failed subcells are degraded using sudden material property degradation rule. Once the global system is solved, output variables such as homogenized mechanical properties, stress concentration tensors, and failure indices are plotted from the micromechanical failure criteria.

During this PDM analysis described using Figure 4.2, the residual limit is given as 10−10 and ultimate failure is considered when one-third of the total number of subcells fail.

Figure 4.2: Flowchart for HFGMC method [2]

Validation of Numerical Results using Experimental and Analytical Approach

Analytical approach

Comparison between Numerical and FEM Results

Finite element modeling

A 2D finite element model is developed for a CFRP laminate using a SOLID 183 element with a plane stress state. The dimensions and load condition for the RUC are shown in Figure 4.1. a) Finite element method (b) Average HFGMC solution.

Stress-strain Behavior of CFRP Composite Laminate

In the figures, the black dot represents the damage initiation point at the strain level. Simulations are stopped and considered the failure endpoint when 35% of the total volume is occupied by damaged subcells. It can be seen from the figures that up to the point of damage initiation, the CFRP composite material shows a linear behavior, but once the damage starts, softening is observed as shown in Figure 4.5.

Therefore, the strength of the CFRP laminate at the damaged location decreases abruptly and softening is observed. Therefore, to capture the size effect, the subcell size was varied and the behavior of CFRP composite material after damage onset was observed.

Figure 4.5: Stress-strain curve for CFRP composite

Stress-strain components in the CFRP laminate

Damage Propagation in CFRP Laminate

Ultimate failure is observed when 35% of the material volume is damaged. g) Variation of the refractive index from 0 to 1. In this figure, the variation of the refractive index for different deformation values is presented for a single RUC element. As the refractive index reaches 1 for any RUC subcell, that subcell is assumed to be failed.

Figure 4.10c shows that failure starts at = 0.00221 value of applied load and that four subcells fail.

Figure 4.10: Damage propagation in CFRP composite material with increasing strain

Stiffness Properties of CFRP Composite Laminate

From Figure 4.12, the nature of degradation of stiffness properties beyond the initial point of failure can be observed, in which the longitudinal elastic modulus E11 gradually decreases until the final failure point. It is observed from this figure that E22 and E33 maintain the same value up to the initial point of failure where both the values weaken differently up to the final point of failure. CFRP laminate has only five independent constants in the effective constitutive matrix up to failure initiation point.

Therefore, it can be clearly seen that the (y2, y3) plane exhibits transversely isotropic behavior up to the point where damage begins. In this plane, the stiffness properties are constant when calculated in different directions up to the damage origin. Therefore, the stiffness matrix is at the point where the stiffness properties begin to deteriorate, which can be called the operating point.

The effective stiffness matrix at = 3×10−3, which is in the region between damage initiation and ultimate failure, is given by.

Figure 4.12: Properties of CFRP composite with increasing applied strain

Summary

For simplicity, the microstructure of the CFRP composite has been assumed as doubly periodic, shown in Figure 5.1, which is defined in the (x2, x3) coordinate system. In such cases, mean-field homogenization is used to achieve the homogeneous properties of the CFRP composites. Figure 5.1 shows (blue) RVE without interaction, (orange) with near-neighbor interaction and (green) with next-near-neighbor interaction.

In the current analysis, the near-neighbor interaction effect was taken into account when calculating the minimum RVE size and assumed that the following near-neighbor effect is absent.

Figure 5.1: Micro-structure of CFRP composite

Fiber Interaction Effect

Different Types of RVE Configurations

A single RUC element was studied in the 1I RVE configuration, as shown in Figure 5.3. Using the diameter and volume fraction of the fiber shown in Figure 5.5, the dimensions of the 1I RVE are calculated for each (D/H) ratio. Figure 5.5 shows that the volume fraction of the RVE increases as the (D/H) ratio increases.

The microstructure of the 1I RVE configuration for said (D/H) ratios is shown in Figure 5.6. 2IT RVE configuration is shown in the Figure 5.3 in which two fiber elements are placed in the direction perpendicular to the loading. Therefore, estimated minimum RVE size for 2IT RVE configuration is for (D/H) = 0.3 which is stated in the Table 5.2.

The 2IL RVE configuration is shown in Figure 5.3, in which two fiber elements are placed in the loading direction. The microstructure of the RVE configuration for increasing (D/H) ratios is shown in Figure 5.10, in which the diameter of the fiber has been kept constant. The variation in the stiffness properties calculated using the HFGMC method and Halphin-Tsai model with increasing (D/H) ratio is shown in figure 5.11.

Minimum RVE size for 2IL configuration considering 5% variation in stiffness properties obtained using HFGMC results w.r.t.

Figure 5.5: Volume fraction of fiber vs (D/H)

Comparison Between Different RUC Configurations

From Figures 5.15a and 5.15b, it can be easily concluded that the results obtained for configuration 1I and 2I are in perfect agreement, while configuration 2IT gives the highest values for the effective stiffness properties. The shear modulus for the 2IL configuration is the lowest among all configurations, while the transverse elastic properties of the CFRP composite are the lowest for the 1I and 2I configurations.

Figure 5.15a and 5.15b explains the behavior of transverse elastic modulus (E 22 ) and shear modulus (G 12 ) respectively w.r.t

Summary

Conclusions

Scope for Future Work

Equation of stress

Displacement and traction continuity conditions

Residual vector

The generalized method for cells and high-fidelity generalized method for cells micromechanical models a review. Mechanics of advanced materials and structures. A micromechanics based nonlocal constitutive equation and representative volume element size estimates for elastic composites. Micromechanics-based variational estimates for a higher-order nonlocal constitutive equation and optimal selection of effective moduli for elastic composites.

Strength prediction and progressive failure analysis of carbon fiber reinforced polymer laminates with multiple interacting holes involving three-dimensional finite element analysis and digital image correlation.