As mentioned before, the complex curvilinear fibre trajectories of 3Dprinted continuous fibre composites, such as those shown in Figure 1(a) and (b), lead to variable mechanical properties which are difficult to be accurately modelled using a conventional progressive failure (CPF) model based on approximated fibre orientations with continuum damage mechanics [9, 15]. In this study, a novel numerical modelling method, i.e., a PDPF analysis model, is proposed and developed to predict and analyse the progressive failure response of 3Dprinted CCF composites.
First, the actual fibre deposition trajectories are straightforwardly extracted from the Markforged system. Then, the basic width of consolidated CCF/PA filaments is experimentally measured and determined as shown in Figure 2(a), leading to a width value of around 0.72 mm for the 3Dprinted CCF/PA in the fibre rich areas. The fibre rich area is where the fibre filaments are deposited back and forth without any gap and within which there are no other material parts.
Taking the CCF composites with negative Poisson’s ratio as an example, based on the basic fibre width of 0.72 mm and fibre trajectories of Figure 1(a), the areas of 3D printed CCF/PA parts with their fibre placement directions can be determined as presented in Figure 2(a). Besides, the mechanical properties in the rich area are the same as those of 3Dprinted CCF/PA standard tensile specimens that can be experimentally characterised, as reported in [16, 17]. The curved areas of 3Dprinted CCF composites may possess a lower fibre volume fraction due to fibre loss during printing [18], however the effect is not considered in the PDPF model due to its marginal impact in this study.
3.1 Progressive failure of 3Dprinted CCF/PA composite
The damage behaviours of CCF composites are complicated. They comprise intralaminar damage e.g., continuous fibre fracture/rupture, matrix cracking/crushing and interlaminar damage/delamination. [19]. To date, continuum damage mechanics is of great advantage in calculating the intralaminar damage of CCF composites in a damage accumulation manner based on the damage variables ranging from 0 to 1, for the description of a damage initial state and the progressive development process. In the PDPF model, the orthogonal material constitutive equations were developed based on continuum elements in an Abaqus simulation platform, and the corresponding failure criteria were utilised for calculating the damage initiation and propagation. Among several failure criteria [20–24], one of the mostly utilised method for unidirectional CCF composites, namely, Hashin criterion [21, 22], was employed to predict the failure responses of the 3Dprinted CCF composites.
Details for the constitutive equations and failure criteria for 3Dprinted CCF/PA composites are presented in Appendix A. The basic mechanical properties were obtained from previous studies [9, 17] for the same 3Dprinted CCF/PA composites, with a nominal CCF volume fraction of 31.4% and a void content of 7.5% [17]. With these previous characterisations, the basic mechanical properties are summarised in Table 1. In this study, the interlaminar damage/delamination was observed as an insignificant behaviour [14], hence it has not been taken into account.
As mentioned, in the PDPF model, the 3Dprinted CCF/PA filaments in composite layers were calculated using anisotropic material constitutive properties, where the fibre longitudinal direction was defined along the CCF filament trajectories. The fibre transverse direction was then automatically set as perpendicular to the longitudinal direction, as exhibited in Figure 2(a).
Table 1
Mechanical properties of 3Dprinted CCF/PA composites [9, 17].
Properties

Variable

Value

Density (kg/m3)

ρ

1250

Fibre volume (%)

V

31.4

Longitudinal Young’s modulus (GPa)

E11

69.4

Transverse Young’s modulus (GPa)

E22

3.5

Principal Poisson’s ratio ()

v12

0.33

Shear modulus (GPa)

G12

1.9

Longitudinal tensile strength (MPa)

XT

905.3

Longitudinal compressive strength (MPa)

XC

426

Transverse tensile strength (MPa)

YT

17.9

Transverse compressive strength (MPa)

YC

66

Inplane shear strength (MPa)

S12

43.4

Longitudinal traction fracture energy (KJ/m2)

GCft

91.6

Longitudinal compression fracture energy (KJ/m2)

GCfc

79.9

Transverse traction fracture energy (KJ/m2)

GCmt

0.22

Transverse compression fracture energy (KJ/m2)

GCmc

1.1

3.2 Elasticplastic behaviour of SCF/PA composite
As previously mentioned, SCF/PA was utilised as additional material components filling the space where CCF/PA placement was difficult in continuous printing. SCF/PA was selected because of its better performance than the PA counterpart [10]. As introduced in Section 2, the SCF/PA filament was placed in ±45° configurations in all printing, and the nominal volume fraction was ~10% [9], leading to an increment of elastic modulus of PA but providing no big improvement in tensile strength. Hence, for modelling purposes, it was treated as an isotropic material for numerical analysis, though it has minor orthotropy [9].
Instead of using the testing results presented in [16], the mechanical properties of 3Dprinted SCF/PA composites were specifically characterised in this study. According to ASTM D638, tensile tests were conducted for type IV dog bone SCF/PA specimens (deposited with ±45° path configurations), using a universal testing machine with a quasistatic constant velocity of 5.0 mm/min and a strain gauge with a gauge length of 10 mm to obtain the effective or engineering stressstrain relation.
The mean results of the experimental stressstrain curves of standard specimens produced by repetitive testing were extracted and converted into the true stressstrain curve for modelling [25], as shown in Figure 3. Then, an elasticplastic model was established using the true stressstrain curve. A summary of its basic properties is depicted: the density is 1160 kg/m3, tensile modulus is 898 MPa, tensile strength is 50 MPa and failure strain is 0.4. In failure analysis, the equivalent plastic strain (PEEQ) describing an accumulation of the plastic deformation during the process was utilised for characterising the plastic deformation. This was done because of PEEQ’s wide applications with benefits for comprehensive and effective evaluation using a scalar rather than a complex strain tensor [26, 27]. In addition, a ductile criterion in Abaqus was applied to determine the final failure of SCF/PA materials using a predefined failure strain (=0.4) based on the elasticplastic model, to exhibit the materials’ fracture behaviour.
3.3 Numerical definitions and boundary conditions
As shown in Figure 2(b), the throughthickness distribution of sixtytwo layers of 3Dprinted CCF/PA with additional SCF/PA parts and two layers of complete SCF/PA was numerically built, with two SCF/PA layers of the upper and bottom surfaces being compulsory options because of using the Markforged printer. Hence, the specimens consisted of sixtyfour plies and each ply had the same thickness of 0.125 mm, based on the printing setup. In each layer, the continuous fibre longitudinal directions, as mentioned, were defined to be the same as those of printing trajectories as shown in Figure 1(a). With the Markforged printer, there is one single wall of SCF/PA, as the casing around a printed profile, which was ignored in the simulation, due to its very limited volume fraction and insignificant effects on characterisation.
The numerical models were developed using Abaqus/Explicit to calculate the structural responses and materials failure. Under compression, 8node quadrilateral continuum shell elements (SC8R) were applied to guarantee accurate modelling and to improve the calculation’s efficiency [2]. The moving and fixed plates were modelled with conventional shell elements (S4R). After a convergence study on mesh and velocity, a constant displacement rate of 100 mm/s was applied for the moving plate and a uniform mesh with the element size of 0.9 mm was used for modelling the 3Dprinted specimens [9].
The meshwork of 3Dprinted CCF/PA auxetic composite and specific boundary conditions are exhibited in Figure 2(c). A general contact algorithm was deployed to calculate and simulate the general interaction between the elements [25]. The friction coefficients for the contact interfaces were generally defined as 0.3. During the simulation, in accordance with the failure criteria for CCF/PA and the failure strain for SCF/PA, the totally failed elements were removed for further calculation. The same numerical definitions were performed for modelling the high stiffness 3Dprinted CCF composites.