Swinburne has been undertaking extensive research in development of new composite materials involving acrylonitrile-butadiene-styrene ABS and other materials including metals. In order to predict the behaviour of new ABS based composite materials in the course of FDM process, it is necessary to investigate the flow of the composite material in liquefier head. No such study is available considering the geometry of the liquefier head. Main flow parameters including temperature, velocity, and pressure drop have been investigated. Filaments of the filled ABS have been fabricated and characterized to verify the possibility of prototyping using the new material on the current FDM machine.

Results provide promising information in developing the melt flow modelling of metal-plastic composites and in optimising the FDM parameters for better part quality with such composites.

Analytical investigations for composite materials under low to high impact velocities have been carried out and several theoretical models have been also proposed by many researchers Wen, ; Billon and Robinson, ; Naik and Doshi, Experimental study is another significant method because it is not only closer to the real physical process but also is the base of the other methods. Many experimental studies are reported in the literatures Zhu et al. Numerous intervening parameters as mentioned above, make the impact issue of composite materials are rather complicated.

Theoretical models usually depend on many assumptions and simplified conditions, which seriously hinder their utility for general engineering problems. Moreover, experiment methods are usually subjected to the technical conditions, and the internal impact process is difficult to observe.

Therefore, designing composite armors or shields solely based on the theoretical models and experimental data is uneconomical and impracticable Silva et al.

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Fortunately, recent advances of computer science and numerical algorithms, offer a possibility of using numerical simulation combined with slight experiment tests for evaluating composite materials. The material model and modeling method for composite play a key role in numerical simulation. Hoof developed a numerical model for composite laminate. In this model, composite laminates were constructed ply by ply and a tie-break contact algorithm was introduced to simulate the interlaminar failure. Tabiei and Ivanov presented a micromechanical model for flexible woven fabric.

It could simulate the dual behaviors of the flexible fabric, i. Using this fiber-level model, the influence, of fiber transverse properties and the friction among fibers, on the ballistic behavior of fabric have been investigated. Anderson presented a constitutive relationship for anisotropic materials, in which the coupling problem of the volumetric and deviatoric had been overcome. Hayhurst , Clegg and Riedel et al. Composite laminate is fabricated by several single fabrics, and fabric is assembled by a number of yarns in specific patterns such as weaving or stacking.

While the yarns are usually made by hundreds of the high-strength fibers Grujicic et al. However, it will make the computational scale increase rapidly, and thus result in a dramatic decline in computational efficiency.

On the other hand, for the engineering issues, e. The organization of the paper is as follows. The theoretical framework of this model is discussed and an inverse flyer plate simulation is conducted to demonstrate the nonlinear shock response of this developed model in Section 2. The main conclusions are summarized in Section 5. As mentioned above, composite materials are generally synthesized by two or more materials through chemical method and possess themselves hierarchical structure.

To enhance the engineering practicability and improve the computational efficiency, the composite material model developed in this paper is a computational macro-mechanical model, which equates the whole composite laminate into an orthotropic homogeneous material as shown in Figure 1.

This model accounting for the nonlinear shock effects, aims to capture the main mechanical behaviors and evaluate the macroscopic parameters of composite laminates under high-velocity impact loading. Comparing with isotropic materials e. The phenomena observed in the impact tests for composite laminates are as follows Hayhurst et al. Material anisotropy. Different to isotropic materials, composite laminate is usually orthotropic and the strength effects constitutive model and pressure effects EOS are strongly coupled together.

Therefore, the stress-strain relationship of composite laminate can be expressed as:. Pressure is defined as a third of the trace of the stresses, i. Therefore, expanding Eq. From the Eq. For the composite materials, a significant conclusion can be drawn that the volumetric and deviatoric response are strongly coupled, as that deviatoric strain can result in spherical stress, while volumetric strain can lead to deviatoric stress. The first term, on the right side of Eq. However, for orthotropic materials, it represents the volumetric thermodynamic response.

To involve the nonlinear shock effects, the first term of Eq.

## Mathematical modeling and numerical analysis of reinforced composite beams

Meanwhile, the contribution to pressure from the deviatoric strain is remained as a correction. Thus the expression of pressure can be rewritten as:. The failure model developed in this paper is an orthotropic failure model, which includes two stages, i. Subsequent to failure initiation, stiffness and strength properties of the failed material will be updated based on the direction or modes of the failure Silva et al.

In this study, the through-thickness direction of composite laminate is defined as the direction, while the in-plane principal directions are and directions. Delamination will be caused by the excessive stresses or strains in the or plane direction. If failure is initiated in either of these two modes, the stress in the direction and the corresponding orthotropic stiffness coefficients C ij is instantaneously set to zero Tham et al. The and directions are assumed to be in the plane of the composite.

If failure is initiated in and directions, the post-failure response is similar to direction,. If all the three material directions fail simultaneously, the material stiffness and strength will become isotropic with no stress deviators and tensile material stresses, indicating that the material can withstand only hydrostatic pressure. Due to the orthogonality of composite laminates, some elements in the impact region will be more easily distorted during the simulations, which may lead to numerical instabilities or even calculation failure Deka et al.

In this study, element effective strain is used as the erosion criterion, similar approach has been adopted in the references Grujicic et al. During the impact process, the composite material will undergo non-failure and failure states and the corresponding cell strain will also undergo this two stages. At this point, the meshes will be extremely distorted and the deletion mechanism will be triggered. The layered projectile, carried by a polymer sabot, is accelerated by means of a single stage gas gun.

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The corresponding simulation has been implemented in this study, which can be used to verify the shock response of the developed material model Figure 4. The numerical and experimental results are compared in Figure 5. Figure 5 shows that the two simulation results present a dramatic difference.

It is encouraging and means that the developed material model is able to simulate the nonlinear shock response, i. The theoretical framework of the developed model was discussed and its nonlinear shock response has been also demonstrated in Section 2. In this section, the performance of this model will be further verified in high-velocity impact condition. The projectile and its sabot are launched together by the gun and the impact velocity is controlled by the amount of gun powder. The classic ballistic impact case is selected form the literature Hoof et al.

This is a 1.

## Numerical Material Model for Composite Laminates in High-Velocity Impact Simulation

It is used for streamlining the engineering design process and to cut time and cost while obtaining improved results. It has a workflow-based environment and several multi-objective optimization algorithms. The half of the plate was firstly cured, then a non-adhesive insert and the not-cured half of the plate were placed on the first part, after which a final curing phase was applied.

A non-adhesive insert was introduced at the midplane of the laminates for inducing the delamination onset [ 18 ].

### Bibliographic Information

The length of the film was 45 mm, while its thickness was less than 0. The specimens were cut from the plates; their sections had a rectangular shape, were 25 mm wide and mm long, and had a uniform thickness of 3 mm Figure 1. In Figure 2 the specimen gripped in the testing machine is reported. Before each test, a digital grid was realized and calibrated on the specimen.

It substituted the manual reference, suggested by the standard, for live monitoring the crack growth. The entire tests were recorded by placing the optical system in front of each sample; the images were acquired in continuous mode. The blue grid reported in Figure 3 , whose length is named a 0 in the standard calculation, represents the preliminary delamination length included between the pin of the hinge and the end of the non-adhesive insert. The red grid was used for estimating the real crack opening during the test.

The total length obtained by combining a 0 and the real crack opening is defined as a in all the calculation methods. Different fracture criterions exist for identifying the damage in the materials, and each of them is based on some specific hypothesis. A failure index is evaluated by the stress-based criterions; its critical value is the same as that of the energy-based criterions. They were able to distinguish specific failure modes, but their application is not defined in the proximity of crack tips where a singular stress field exists. Griffith was the first to quantitatively connect crack size and strength.

Later, Irwin and Rice also seriously contributed to the fracture mechanics [ 32 ]. The Griffith—Irwin approach is based on linear elastic fracture mechanics. The stress intensity factor or the critical strain energy release rate ruled both the crack initiation and propagation. Four types of data reduction methods were calculated in all the tests for estimating G Ic values.

The BT expression for the strain energy release rate of a perfectly built-in double cantilever beam is as follows:. Equation 2 considers a built-in condition in correspondence with the double cantilever beam clamping to the delamination front. The possibility of rotation at the end of the beam, owing to the real clamping conditions, leads to a correction of the previous formulation in favor of the MBT, which provides a new calculation of the strain energy release rate, as follows:.

Then, the best least-squares fit was drawn through the data. Finally, the exponent n representing the slope of this line was used to calculate the mode I interlaminar fracture toughness as follows:. Lastly, the MCC method was applied. The slope of this line, indicated as A 1 , was used for evaluating the mode I interlaminar fracture toughness as follows:. The CZM approach consisted of introducing fracture mechanisms by adopting softening relationships between tractions and separations, which in turn introduced a critical fracture energy that was also the energy required to break apart the interface surfaces [ 36 ].

In particular, the CZM model was based on contact elements, which means that the interfacial separation was defined in terms of contact gap or penetration and tangential slip distance. In particular, cohesive zone modeling with contact elements included two different traction separation laws; namely a bilinear traction separation law and an exponential traction separation law.

For this study, the bilinear behaviour with linear softening characterized by maximum traction and critical energy release rate was chosen. According to this behaviour, as described by Alfano [ 22 ], mode I debonding defined a mode of separation of the interface surfaces in which the separation normal to the interface dominated the slip tangent to the interface. The normal contact stress tension and contact gap behavior are plotted in Figure 4. Figure 4 shows linear elastic loading OA followed by linear softening AC.

The maximum normal contact stress is achieved at point A. Debonding begins at point A and is completed at point C, when the normal contact stress reaches zero value; any further separation occurs without any normal contact stress. The area under curve OAC represents the energy released due to debonding, and is called critical fracture energy. The slope of the line OA determines the contact gap at the maximum normal contact stress and, hence, it characterizes how the normal contact stress decreases with the contact gap, that is, whether the fracture is brittle or ductile.

After debonding is initiated, it is assumed to be cumulative, and any unloading and subsequent reloading occurs in a linear elastic way along line OB at a more gradual slope. As far the actual tests are concerned, the numerical model see Figure 5 and Figure 6 was representative of the flat hinges method [ 18 ] and the hinge rotations were taken into account by means of remote points pilot nodes , connected to the inner edges of the plates, representing the part of the hinges glued to the specimen, in which the proper rotational degree of freedom was left free.

Applied load in the numerical model of the double cantilever beam DCB delamination tests. The aim of the FEA modeling was to calibrate the constitutive model by identifying the set of constants related to the CZM method e. In order to achieve this result, the optimization tool i. The mesh sensitivity analysis showed a behavior quite independent from the in-plane mesh size, because of the fact that both parts of the model below and upside the delamination plane had homologous meshes. This means that the positions of the nodes of both parts matched each other, nodes shared by the contact elements on which the CZM method was activated Figure 8.

During the analyses, no convergence issues arose, thus the default settings L2-Norm were mantained. The performance comparison is reported in Figure 9. It demonstrates that, from the coarse model on, the results are pretty overlapped. The analyses were carried out on a workstation esacore with 64 Gb of RAM. In Figure 10 , the debonding area is highlighted.

Following the recommended procedure for fracture mechanics, after the bonded contact set up, all cohesive properties were applied using fracture mechanics options in the highlighted area. It can be seen that deformation energy is accumulated in the sample in a load range between 98 N and N. In correspondence with those values, the accumulated energy G I reaches the critical value G IC so that the delamination starts.

After that point, the material response becomes non-linear. In Table 1 , a synopsis of G I energy values is reported for the different computation approaches [ 18 ]. Synoptic of the experimental G I energy values for delamination propagation. The curve related to specimen number 3 was chosen as a target. To take into account the possible influence of the adhesion force of the teflon insert representing the initial defect as trigger of delamination , the FEM model was made parametric.

CZM—cohesive zone material. Two different types of DoEs designs of experiment and one optimization algorithm were used during the numerical experimental calibration with modeFRONTIER, in order to target the calibration in a fast and efficient way. In particular, in different steps, the utilized DoEs are as follows [ 37 ]:. Uniform Latin hypercube ULH : this is a stochastic DoE algorithm that generates random numbers conforming to a uniform distribution.

It is particularly suited for optimization with genetic algorithms and Response Surface Methodology training. ULH is an advanced form of Monte Carlo sampling; more precisely, it is constrained Monte Carlo sampling in which the constraint refer to the way each variable is sampled. The uniform statistical distribution is split in n intervals with the same probability and a random value is selected in each interval. ULH also tries to minimize correlations between input variables and to maximize the distance between the generated designs. In this way, the points are relatively uniformly distributed over the variable range.

This algorithm is particularly suitable for generating the initial dataset for the optimization with a genetic algorithm and RSM training. Incremental space filler ISF : this is an augmenting algorithm thta sequentially adds new design configurations to a database by maximizing the minimum distance from the existing points optimization of the maximin criterion. This algorithm is particularly suited for generating a dataset for response surface training as it improves both the RSM approximation quality and reliability, as well as the numerical stability of the training.

In general, new points are added in such way to uniformly fill the input space. However, if the zone filling option is enabled, the new points will be added in hyperspheres centered around the marked designs from the existing database and with a defined radius, expressed as a percentage of the variable ranges.

It is the ESTECO proprietary version of the multi-objective genetic algorithm that uses a smart and efficient multi-search elitism, which is able to preserve excellent Pareto or non-dominated solutions without converging prematurely to a local optimum. ESTECO is an independent software provider, highly specialized in numerical optimization and simulation data management with a sound scientific foundation and a flexible approach to customer needs.