### Project focus since last update

As the printing for our project was all completed before the first update, our progress since then has mostly been focused on data analysis and consideration of future work. Additionally, we focused on understanding the causes of some issues we had with testing, mostly stress concentrations causing premature failure.

### Data analysis

The primary analysis in this project is to apply the maximum stress failure criterion to the tensile test result and examine how well the criterion predicts FFF part failure. The experiment previously conducted consisted of FFF coupons printed at seven different angles (0, 15, 30, 45, 60, 75, and 90 degrees); five samples were tested for each printing direction. Thirty five FFF coupons in total were loaded with a unidirectional tensile force, and the maximum stresses were recorded. The data was summarized and shown in Figure 1 where the blue dots and orange bars represent the mean and standard deviation. Note that the 15-degrees data point has a smaller maximum tensile stress than expected due to excessive stress concentrations from FFF printing (which will be discussed later); in theory, it should lie between the maximum stresses of 0- and 30-degrees samples.

With this data, the maximum stress failure criterion is then applied. The key stress transformation formula is shown below. σ_{xx}, σ_{yy}, and τ_{xy} are the stresses being loaded to the coupon which can be transformed by the matrix multiplication to the equivalent stress state in the bead direction: σ_{11}, σ_{22}, and τ_{12}. The letters “s” and “c” in the transformation matrix indicate sine and cosine respectively. In the maximum stress failure criterion, part failure is predicted when at least one stress component in the transformed stress state (σ_{11}, σ_{22}, and τ_{12}) exceeds the maximum, predetermined values that the part can withstand. This failure criterion assumes that there is no interaction between each stress component and, therefore, treats stress in each direction as a pure load.

In this experiment, the test specimens were only loaded with a tensile force in axial direction (σ_{xx} > 0) meaning that σ_{yy} and τ_{xy} are equal to zero. With such condition, the matrix multiplication yields three equations.

The previous test done on similar tensile specimens found that under pure-load condition the maximum values of σ_{11}, σ_{22}, and τ_{12} (designated by X_{T}, Y_{T}, and S) are 40.29 kPa, 31.13 kPa, and 23.38 kPa [1]. Knowing that the coupon will fail when at least one of the three stress values is reached, equations (1), (2), and (3) can be rewritten to define the failure region as following. The negative sign in (3) is dropped out since shear stress is symmetric.

At a given printing angle (θ), the coupon is predicted to fail when any of the equations above is satisfied. Hence, the failure region of σ_{xx} at a given printing angle θ can be constructed, and the result is shown in Figure 2. The orange, purple, and yellow dotted lines represent the predicted stress limit resulted from σ_{11}, σ_{22}, and τ_{12} respectively. For instance, at θ equals 45 degrees, the part is predicted to break at σ_{xx} = 46 kPa as τ_{12} reaches its maximum value.

From Figure 2, it can be seen that the maximum stress failure criterion successfully predicts part failure at the printing angles of 0, 75, and 90 degrees (data points fell in the failure region). This observation is as expected since at the angle near 0 and 90, the equivalent stress state in the bead direction is close to a pure load. In other words, from equations (1) – (3), at θ close to 0 degree the filament experiences σ_{11} ≃ σ_{xx} (and σ_{22} ≃ τ_{12} ≃ 0) while at ? near 90 degrees, σ_{22} ≃ σ_{xx} (and σ_{11} ≃ τ_{12} ≃ 0).

However, at the angles from 15 to 60 degrees, the data points fell far below the predicted failure region. This is because at these printing angles, the equivalent stress in the bead direction is a combination of σ_{11}, σ_{22}, and τ_{12}. As previously mentioned, the maximum stress criterion ignores stress interactions and treats the loads as independent components, but in reality the interactions do exist. As a result, the failure region predicted by the maximum stress failure criterion is not accurate for the FFF part.

This leads to a need for failure criteria that take into account stress interactions which play a significant role in FFF part failure. Among several models, Osswald-Osswald is one criterion that shows potential to be used with FFF. In fact, such a model has been applied to the data in this experiment (Figure 3) by Gerardo. The Osswald-Osswald model successfully predicts all the failure observed in this experiment (except the 15-degrees data point which is dropped out due to a significant effect from stress concentrations). In addition to the Osswald-Osswald criterion, Tsai-Hill and Tsai-Wu, which are widely used for composite materials, are also possible options. All in all, the development of failure criteria for FFF materials is a new field of study, and there are still many questions to be answered. This experiment only covers one type of material in a specific shape under one loading condition. More work needs to be done in order to fully understand the behavior of the FFF material.

### Solidworks modeling of stress concentrations

Stress concentration simulation was conducted for each degree print orientation do identify what stress concentrations existed among the different coupons. The models were generated based on the coupon dimensions used to 3D print the coupon, the critical area being the neck, was dimensioned at 13 mm wide and 3.2 mm thick. The models and simulation were both conducted in SolidWorks. One end of the model was fixed, the other end experienced a force of 1500[N] acting along the axial direction providing tensile stress. From the simulation the maximum Von Mises Stress was obtained within the neck region of the coupon model.

#### Modeling Process:

Models were generated making the assumption that no max stress concentrations would exist outside of the neck area, this allowed for only modeling features on the neck area of the coupon and not extending the features beyond that into the area that would be held with clamps. Bead paths were established by creating 3 initial circles which would then be aligned with appropriate degree oriented lines (Figure 4). The circles had a diameter of 0.5 [mm], the same as the print settings used. The travel path was assumed to start at the top circle then travel vertically down to the middle circle, then travel up and to the right following the print degree lines. The bottom circle represents where the next “down and to the left” print bead would be placed thus providing the bounds for profile of bead path which would be cut from a normal coupon profile (Figure 5). To establish the opposite side profile, the print degree lines were extended across the coupon and the path was mapped to determine where the profile would exist (Figure 6). Once the profiles were cut from both sides of the coupon, a linear pattern was applied to project the profile onto the rest of the neck area of the coupon. The linear pattern was repeated at intervals of twice the distance between the circles established by the bead path. For the 15 degree model seen in Figure 4, the value would be approximately 2*1.93 or 3.86mm.

#### Results:

The stress used for the 0 degree print was simply stress across a cross sectional area equal to that of the coupon (stress = 1500N/(3.2mm * 13mm)). For all others the Von Mises stress was taken from SolidWorks simulation. For every simulation except for the 75 degree orientation the stress concentrations were consistently located at those features at the end of their “row” of concentrations (Figure 7). Figure 8 is a chart outlining all maximum stress vs print orientation.

#### Considerations for future work:

All of the simulations used the same force of 1500N and the resultant pattern of max stress seems to loosely fit the general shape provided by the maximum stress failure criterion. Simulations at the different angles should be repeated with at the very least averages of their maximum stress induced during testing, this would then be compared with the maximum stress failure criterion to determine if it fits better or worse. This may also reveal other patterns of maximum stress and print bead orientation. Lastly, to explore differences in print bead width, the models could be repeated for a 0.25mm bead width to determine what changes to modeled maximum stress would exist.

### Reference

[1] G. A. Mazzei Capote, “Defining a failure surface for Fused Filament Fabrication parts using a novel failure criterion,” Sep. 2018.