Composite materials are specially used in aircraft structures, automobiles, and sport goods.
Composite material models can be modelled with layer elements . After creating a model using layered elements, structural analysis including large deflation and stress stiffening can be performed [34].
Composites materials are more challenging to model than isotropic materials due to each layer may have different orthotropic material property, orientation angle and layer thickness.
Depending upon the application and type of results required, SHELL181, SHELL281, SOLSH190, SOLID185 and SOLID186 (Layered Solid) are types of element available for modeling composite materials [35]. Shell elements allow to define layered composite of thin-walled structures which are common in aircraft structure, boat hulls and racing cars analysis [38].
There are several ways of modelling composite materials in Ansys software. It can be modelled in Workbench or in Mechanical Parametric Design Language (APDL) Ansys software [35].
The FE models in this study were modeled and analyzed in ANSYS Mechanical APDL 17.0 . Finite Element Analysis (FEA) of composite materials can be performed in micromechanical, lamina, and laminate level of approach [34]. One method to achieve a lamina or mesoscale level analysis is using shell element, which were used in this study [35]. To analyses a composite laminate, input parameters including number of layers, orthotropic material properties and orientation angle of the reinforcing material must be provided [36, 37]. An isotropic material requires minimum two material properties ; Youngs modulus and Poisson’s ratio in x-direction. Whereas an orthotropic material requires nine elastic constants, three Young’s modulus, three Poisson’s ratio and three shear modulus values [37]. The nine elastic constants are used in the strain-stress relation matrix of Hook’s law. The properties of the materials are stored in the material stiffness matrix [D]-1 as in equation 2.4.
From orthotropic form of Hook’s law:
{𝜀} = [𝐷−1]{𝜎} =
Laminate beam theory can be used to construct finite element for analyzing of composite structures [36]. In static analysis of a simple loaded composite beam, the deflection and stress at the beam can be derived by differentiation. The thin beam theory considers that normal-to-the- beam- mid-surface remain straight and normal after deformation. Furthermore, if rotation and shear are ignored then the strain and curvature equations can be expressed as follows [38].
𝜀0 = 𝜕𝑢
𝜕𝑥 , 𝑘 = −𝜕2𝑤
𝜕𝑥2 (2.5)
Where ε, κ, represents strain and curvature, while u and w are displacements in x and z-directions, respectively. Normal strain at any point can be then expressed as 𝜀 = 𝜀0+ 𝑧𝑘.
The stress in the axial direction is determined by
𝜎𝑥 = 𝑄11(𝜀0+ 𝑧𝑘) (2.6)
For simply supported beam zero displacement in the z-direction and zero moment are defined at the supports. Furthermore, the equation for the Hook’s law can be stated as in equation 2.7.
[ 𝑁
𝑀 ] = [ 𝐴 𝐵 𝐵 𝐷 ] [ԑ°
к] (2.7)
Where A ,B and D are 3x3 matrix. A matrix represents the in-plane stiffness properties [36], B matrix is coupling that arise between the bending and the membrane action and B is zero in case of symmetric laminate. D-matrix is the bending stiffness properties. The mid-plane strain, curvature, in-plane loads and moment are represented by ε, κ, N and M, respectively.
In equation 2.7 the stiffness matrix relates the stress results to strains. In case of non-symmetric laminate of composite materials, an out-of-plane bending can be occurred [36]. The non-symmetric lay-up of fiber give non-zero value to the B-matrix of the laminate stiffness and that coupled the in-plane and bending response. Therefore, the material lay-up in this study were modeled symmetry to avoid the out of plane movements [10] . Furthermore, it is important to remember the FE results are an approximate result and their occupancy depends among others on the choice of element type and mesh density [35].
Rule of Mixture
For a continuous and aligned fiber reinforced composite, modulus of elasticity in the longitudinal direction is described by the ‘Rule of Mixtures’ [10, 11]. The rule shows that the stiffness of the composite material is the sum of the individual volume fraction and their corresponding material property . The ROM simply depends on the volume fraction of fibers [2, 10].
If we assume there is no interfacial gliding between the layers of the matrix and reinforcement during loading along the fiber axis, then there will be equal strain for both plies [2, 10].
𝜀11 = 𝜀11𝑓 =𝜎11𝑓
𝐸𝑓 = 𝜀11𝑚= 𝜎11𝑚
𝐸𝑚 (2.8)
Where f refers for fiber and m is for matrix and the numbers represents the direction with respect to the fiber axis, σ11f and σ11m are strength of the fiber and matrix parallel to the fiber, respectively.
Since the reinforcing fiber is much stiffer than the matrix phase and the fiber will be subjected to higher stress. The overall stress is then the sum of the stresses from both materials with the factor of their strength[2, 13].
𝜎11 = (1 − 𝑉𝑓)𝜎11𝑚+ 𝑉𝑓𝜎11𝑓 (2.9) The Vf represents volume percent of fiber and (1-Vf ) is matrix volume fraction and σ11 is the tensile strength of composite parallel to the fiber . Furthermore, the elastic modulus of the composite along the fiber axis will be formulated as [10]:
𝐸11 = (1 − 𝑉𝑓)𝐸𝑚+ 𝑉𝑓𝐸𝑓 (2.10) Where E11 is the Young’s modulus of the composite material along the fiber. Ef and Vf are elastic modulus and volume fraction of the fiber respectively, while the Em and Vm represents the elastic modulus of the matrix and its volume fraction.
If parts have uniform cross-sectional area, the volume fraction of the fiber can be estimated by the area ratio:
𝑉𝑓 = 𝐴𝑓
𝐴𝑡𝑜𝑡𝑎𝑙, 𝑉𝑚 = 𝐴𝑚
𝐴𝑡𝑜𝑡𝑎𝑙 = (1 − 𝑉𝑓) (2.11) If no void presents and the material is assumed to have transversely isotropic mechanical property, then E22= E33, v12=v13, and G12=G13.
E22 = EfEm
Ef− Vf(Ef− Em) (2.12) The in-plane shear modulus, G12, can be estimated from:
G12 = GfGm
Gf− Vf(Gf− Vm) (2.13) 𝐺23= 𝐸3
2(1 + 𝑣23) The Poisson’s ratio coefficient can be obtained from:
𝜈c = νmVm+ νfVf (2.14) Based on the knowledge of the fiber elastic modulus Ef , matrix modulus Em , and fiber volume fraction Vf , the rule-of-matrix approach allows prediction of E11, E22=E33, v12=v13, and G12=G13 of a composite [2, 15, 34].
The ROM (equations 1.2-1.6) predicts the mechanical properties of a part fabricated from composite materials based on the mechanical properties of the individual.
CHAPTER 3
3 METHODOLOGY
To characterize the mechanical properties of 3D printed composite materials, tensile and flexural experimental tests was performed. Furthermore, a Finite Element Model (FEM) was developed.
The 3D printer used in this thesis was Markforged® Mark-Two, continuous fiber 3D printer[32]. Test samples were fabricated from carbon fiber filament imbedded in a thermoset plastic matrix named “Onyx” by its producer Markforged®. Both materials were delivered by the 3D printer manufacturer Markforged®. “Onyx” is the matrix phase material and it is used in synonym to the matrix material or matrix phase in this report[39].
First, a 3D model of the specimen was modeled in Autodesk inventor 2018 and the file was converted into STL-file. STL-file is type of file commonly used by additive manufacturing machines. The STL-file contains a triangular mesh patterns of the 3D CAD model. This 3D model was then sliced into thin layers in parallel to the working plane by a slicing software called “Eiger”. The working plane lays in XY-plane as illustrated in Figure 3-1. “Eiger” is an online freely available cloud-based software, developed by Markforged® 3D printer manufacturer. In this study, this slicing software was used to cut the 3D model provided as STL-file into several thin layers.
During slicing, the slicing software generate information about the tool path, layer thickness, fiber orientation angle, number of fiber layers, infill density, type of pattern, number of total layers and type of material in each layer according to the settings provided. This information is used as a command by the 3D printer during printing.