Analysis of the Material Behavior of 3D Printed Laminates Via FFF

Abstract

A comprehensive understanding of process–structure–property relationship of 3D printed parts is currently limited. In the present study, we investigate the influence of the mesostructure on the overall mechanical behavior of the parts synthesized via fused filament fabrication. In particular, characterization of anisotropic behavior is carefully studied by performing mechanical testing on the printed parts. The printed parts are treated as laminates and are characterized using laminate mechanics. Test coupons of thick layered and also thin layered unidirectional as well as bidirectional laminates are printed with polymeric material for tensile and bending tests. Test results revealed that the process parameters govern the mesostructure and therefore the material behavior of the parts. Mechanical behavior of the bidirectional printed laminates is studied in detail. The properties are significantly influenced by the layer thickness and layup order of the printed parts. Mechanical behavior of the printed parts can be characterized using laminate theory. The effect of lamina layup and layer thickness on the flexural properties of the laminates is significant. Furthermore, the first ply failure theory is employed for the finite element failure analysis of the printed parts. The results provide insights in the relationship between mesostructure–mechanical properties of the printed parts.

Appendix A

Classical Laminate Theory

The constitutive relation of a lamina is written as

$$ \left\{\begin{array}{c}{\sigma}_{11}\\ {}{\sigma}_{22}\\ {}{\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{Q}_{11}& {Q}_{12}& 0\\ {}{Q}_{12}& {Q}_{22}& 0\\ {}0& 0& {Q}_{66}\end{array}\right]\left\{\begin{array}{c}{\varepsilon}_{11}\\ {}{\varepsilon}_{22}\\ {}{\gamma}_{12}\end{array}\right\}\ \mathrm{in}\ \mathrm{short}\ \left\{\sigma \right\}=\left[Q\right]\left\{\varepsilon \right\} $$

(1)

where

$$ {Q}_{11}=\frac{E_1}{1-{\nu}_{12}{\nu}_{21}},{Q}_{12}=\frac{\nu_{12}{E}_1}{1-{\nu}_{12}{\nu}_{21}},{Q}_{22}=\frac{E_2}{1-{\nu}_{12}{\nu}_{21}},{Q}_{66}={G}_{12} $$

(2)

Strains of the laminate is written as

$$ \left\{\begin{array}{c}{\varepsilon}_{xx}\\ {}{\varepsilon}_{yy}\\ {}{\gamma}_{xy}\end{array}\right\}=\left\{\begin{array}{c}{\varepsilon}_{xx}^0\\ {}{\varepsilon}_{yy}^0\\ {}{\gamma}_{xy}^0\end{array}\right\}+z\left\{\begin{array}{c}{k}_{xx}\\ {}{k}_{yy}\\ {}{k}_{xy}\end{array}\right\},\mathrm{in}\ \mathrm{short}\ \left\{\varepsilon \right\}=\left\{{\varepsilon}^0\right\}+z\left\{k\right\} $$

(3)

where {ε⁰} are in-plane strains; {k}are curvatures of the laminate; z is distance from mid plane in the thickness direction. The constitutive matrix for a lamina in global coordinate system is given as

$$ \left\{\sigma \right\}=\left[\overline{Q}\right]\left\{\varepsilon \right\} $$

(4)

where ${\overline{Q}}_{ij}$ is written as$\left[\overline{Q}\right]={\left[T\right]}^{-1}\left[Q\right]{\left[T\right]}^{-T}$and [T] is a transformation matrix. The resultant force and moment per unit width for a laminate with N number of layers are expressed as

$$ \left\{N\right\}=\sum \limits_{k=1}^N\underset{h_k}{\overset{h_{k+1}}{\int }}\left\{\sigma \right\}\; dz $$

(5)

$$ \left\{M\right\}=\sum \limits_{k=1}^N\underset{h_k}{\overset{h_{k+1}}{\int }}\left\{\sigma \right\}\;z\; dz $$

(6)

Using (equation (3), (4)), and (equation (5), (6)) become

$$ \left\{N\right\}=\left[A\right]\left\{{\varepsilon}^0\right\}+\left[B\right]\left\{k\right\} $$

(7)

$$ \left\{M\right\}=\left[B\right]\left\{{\varepsilon}^0\right\}+\left[D\right]\left\{k\right\} $$

(8)

where $\left[A\right]=\sum \limits_{k=1}^N{\left[\overline{Q}\right]}_k\left({z}_k-{z}_{k-1}\right)$, $\left[B\right]=\frac{1}{2}\sum \limits_{k=1}^N{\left[\overline{Q}\right]}_k\left({z}_k^2-{z}_{k-1}^2\right)$, $\left[D\right]=\frac{1}{3}\sum \limits_{k=1}^N{\left[\overline{Q}\right]}_k\left({z}_k^3-{z}_{k-1}^3\right)$.

The [A], [B] and [D] are stiffness matrices for the laminate. The [B] = [0] for a symmetric laminate. The mid-plane strains and curvatures can be calculated from (equation (7), (8)), once we know the normal force and moment acting on a lamina. Strains for a symmetric laminate subjected to only in-plane forces are given from (equation (7)) as

$$ \left\{\begin{array}{c}{\varepsilon}_{xx}^0\\ {}{\varepsilon}_{yy}^0\\ {}{\gamma}_{xy}^0\end{array}\right\}={\left[A\right]}^{-1}\left\{\begin{array}{c}{N}_{xx}\\ {}{N}_{yy}\\ {}{N}_{xy}\end{array}\right\} $$

(9)

Strains for a symmetric laminate subjected to only transverse loads are given from (equation (8)) as

$$ \left\{\begin{array}{c}{k}_{xx}\\ {}{k}_{yy}\\ {}{k}_{xy}\end{array}\right\}={\left[D\right]}^{-1}\left\{\begin{array}{c}{M}_{xx}\\ {}{M}_{yy}\\ {}{M}_{xy}\end{array}\right\} $$

(10)

Uniaxial tensile loading along x-axis: In the uniaxial tensile test, the load is applied in the x direction and for laminate thickness h, N_xx = hσ_xxN_yy = 0 and N_xy = 0. The stress-strain relation for uniaxial tensile test is ${\sigma}_{xx}={E}_{xx}{\varepsilon}_{xx}^0$, using the relation (equation (9)), the modulus of elasticity along the x direction of the laminate is calculated as follows

$$ {E}_{xx}=\frac{1}{{\left[{A}^{-1}\right]}_{11}h} $$

(11)

Flexural loads: In the 3-point bending test, the load is applied in the z direction and for laminate thickness h, M_xx ≠ 0, M_yy = 0 and M_xy = 0. The relationship between flexural stress and stiffness is written as ${E}_x^f={\sigma}_{xx}^f/{\varepsilon}_{xx}^f$, using the (equation (10)), the flexural modulus of elasticity of the laminate along the x direction is given as follows

$$ {E}_x^f=\frac{12}{{\left[{D}^{-1}\right]}_{11}{h}^3} $$

(12)

The elastic moduli such as E₁, E₂, G₁₂, ν₁₂ of the lamina found from the experimental tensile test results are used for the calculation of matrices [A], [B] and [D]. Then, E_xx and ${E}_x^f$of the laminate can be calculated using (equation (11), (12)), respectively. More details about the laminate theory available in [35].

Tsai-Hill Failure Criterion

The failure criterion for a planar stress is written as

$$ \frac{\sigma_1^2}{X^2}-\frac{\sigma_1{\sigma}_2}{X^2}+\frac{\sigma_2^2}{Y^2}+\frac{\tau_{12}^2}{S^2}=1 $$

(13)

The lamina properties available in Table 3 are useful in the failure analysis of the printed laminates. The nonlinear quasi-static finite element failure analysis of the laminates was done in Hyperworks. The first ply failure stresses and corresponding strains are reported in the results when the laminate just met the failure criterion. More details about the nonlinear quasi static analysis and finite element modeling of composites laminates can be found in the Hyperworks manual.