Summary

 In fiber-reinforced composites, composites exhibit complex internal deformation behavior even under simple external loading conditions due to the interactive load transfer between matrix and reinforcement. Composite components usually experience much more complex loading regimes than simple uniaxial loading under actual service conditions, so computational methods must be used to examine the multiaxial mechanical behavior at different local material points.

 In 2023, the journal "Composite Structures" published a research work by the Department of Mechanical Engineering of Incheon National University to characterize the biaxial compression deformation behavior of epoxy polymers through cross experiments and finite element analysis.

The purpose of this study was to develop an accurate and systematic method for evaluating the initial yield surface of epoxy polymers using existing cross-type specimen types and testing methods, and to directly compare the resulting surface with the third quadrant of the σ3=0 plane Various yield functions for comparison.

Test materials and methods

The cross-shaped sample was prepared by the research team. The geometric shape of the sample refers to the research results of Smits et al. The loading arm adopts a transition method that gradually narrows to the central area, and the central area is milled and thinned.

Fig. 1 (a) Geometry of cross- shaped specimen ( top view ) and (b) central area of three different thicknesses ; (c) Prepared cross- shaped specimen; (d) MTS planar biaxial and torsion test system ( all units in millimeters ) .

The focus of this study is the deformation behavior of cross-shaped specimens under biaxial compression loading path, and the compression load ratios are to be set as: -4:-10, -6:-10, -8:-10 and -10:-10 , due to a certain degree of buckling due to specimen compression, the out-of-plane displacement of the measurement region due to specimen buckling was measured using a laser distance tracker with a frequency of 2 Hz to determine the onset of buckling (Fig. 2(a) and 2(b )). 

Fig . 2 (a) Image of a bent cruciform specimen ruptured in the central standard region by biaxial compressive loading; ( b) Schematic illustration of deflection measurement using a laser distance tracker; (c) Quarter finite element of a cruciform specimen Model.

Finite Element Analysis

Through finite element modeling analysis, unlike 1D deformation tests, for some biaxial tests with cruciform geometry, the biaxial stress state cannot be calculated from the applied load and the area of the measurement region. In fact, for the present specimen geometry, the applied load cannot be fully transferred to the central strain region, resulting in an estimate of the strain that is lower than that determined by dividing the applied load by the cross-sectional area, the numerically obtained strain The zone stress is about 35% lower than that obtained by simply dividing the applied load by the cross-sectional area.

Fig. 3 (a) Comparison of the stress in the measured area between the analytical results and the finite element results; (b) the von Mises stress distribution in the entire cross-shaped specimen under equal biaxial compressive loads ; (c) under equal biaxial compressive loads Von  Mises stress distribution in the canonical region .

In order to solve this problem, several researchers established a force-stress coupling relationship corresponding to the geometric characteristics of the cross-shaped sample. In the in-plane stress state, the relationship between the stress in the strain zone and the applied arm load can be described as follows:

Constants a and b are constants that describe the degree of coupling. In the same way, there is also a certain linear conversion law between the strain in the measurement area and the displacement applied by the loading arm, that is, the displacement-strain coupling equation:

 In order to evaluate the effect of buckling displacement on the results, three samples with different center thickness were subjected to equal load compression. The results showed that the initial yield of the sample with smaller center thickness was earlier, and the center thickness was proportional to the load, and would not Affects the corresponding displacement. For all thickness ratios considered, the specimen buckled near the end of loading after reaching the maximum force. The occurrence of buckling is related to the instability caused by the strain softening mechanism after the continuous application of external loads.

Fig . 4 (a) Force - displacement curves and (b) buckling-deflection curves under different specification area thicknesses (3 , 4 and 5 mm) .

 The force-displacement curves obtained from biaxial compression tests of cross-shaped specimens subjected to different load ratios are shown in Fig. 5. Almost all biaxial force profiles exhibit nearly linear elastic deformation, plastic hardening, and softening states, except for the 4:-10 loading ratio, which shows a large continuous increase in force on the more heavily loaded axis, whereas The linear increase in force in the other axis is small.

Fig.5 The force - displacement curve 

The force - displacement curve measured by the experiment in Fig . 5  , the load ratio is : (a)-10:-10 , (b)-8:-10 , (c)-6:-10 , (d)-4:-10 . Blue dots represent force - displacement curves obtained from a finite element model using the linear elastic material law ; the inset shows a magnification near the determined yield point.

Fig.6 Schematic diagram of  offset-based yield criterion applied to biaxial test results

Figure 6 is a schematic diagram of the proposed yield criterion, using the offset method to determine the yield point:

1. Determine the one-dimensional yield point A through the slope of the offset line, the offset strain and the offset strain range.

2. Determine the corresponding effective stress and strain ( points B and C ) in the central specification region based on the results of the FE simulation using the linear elastic material model .

3.The determination of points B and C provides a basis for estimating the displacement of each loading arm at each moment by the displacement-strain coupling equation at points B' and C' in Figure 6.

4.Draw a linear offset line from the biaxial test structure, the slope of which is from zero point to point D, and determine the yield point E at this time by the obtained offset displacement and offset displacement range.

5. Finally, the biaxial yield stress is obtained through the force-stress coupling equation.

The article goes on to illustrate the effect of different sums on the initial yield surface, and the results show that the yield surfaces of all considered load ratios are slightly enlarged as , due to the non-linear attenuation of the slope of the offset line (Fig. 7(a)). Whereas when increasing to 6e-4 and 7e-4, the yield stress of the specimen increases suddenly, the increased stress state results in a considerable deviation from the smooth surface, and is judged to be unlikely to belong to the appropriate yield point of the standard region (as Figure 7(b)).

Fig.7 The yield criterion based on offset under different offset strains and offset strain ranges 

Comparison of Several Yield Criteria

The article compares the differences between parabolic, MD data-driven and conical , and P MvM yield functions. The results show that the biaxial compression yield response of epoxy polymers deviates from the predictions of parabolic and conical functions, and is slightly smaller than that of MD data-driven functions. predicted (Fig. 8(a)), the fitted PMvM function shows relatively good agreement with the biaxial test data (Fig. 8(b)), and the comparison of equibiaxial compressive stress and one-dimensional stress shows that M D The data-driven and fitted PMvM function is a reasonable choice for the initial yield surface of epoxy polymers (Fig. 8(c)).

Fig.8.  Comparison of experimentally obtained yield points with initial yield surface predictions for (a) parabolic, conical, and multidimensional data-driven yield functions, and (b) PMvM and multidimensional data-driven yield functions; (c) equibiaxial compression Stress Magnitude Compared to 1D Stress Magnitude

A series of finite element simulations were performed under equibiaxial compression loading using a material plasticity model based on parabolic and MD data-driven yield functions . Figure 9 shows that the starting point of plastic deformation is the rounded corner of the sample, and extends from here to the central test area. A, B, C, and D in Figure 9 (b) represent the beginning of plastic deformation of the sample and the beginning of plastic deformation in the central area. , complete plastic deformation in the central area, and the deformation state of the specimen under maximum load.

Fig. 9 (a) Comparison of the force - displacement curves of the equiproportional loading biaxial test and the finite element analysis results , (b) the corresponding equivalent plastic strain distribution under different displacement conditions. The yield range in (a) represents the displacement range determined by the offset yield criterion; A and B represent the time of the first plastic deformation in the cross-shaped specimen and the measurement area, respectively; C represents the plastic deformation in the entire measurement range Fully developed; D indicates the deformation of the specimen under maximum load.

Conclusion

The initial yield surface of the epoxy polymer was characterized using biaxial testing and finite element analysis. It has been observed that until the epoxy polymer reaches its maximum stress state, the effect of buckling is negligible. The initial yield surface exhibits a wide range of stress states and is not described by the well-known parabolic or conical yield functions. Under the biaxial compressive load path, the multiaxial yield response was slower but closer to that predicted by the yield function driven by MD data. The equiaxed compression behavior after yielding is better described by the yield function driven by MD data. However, as the applied load ratio deviates from the isometric condition, all considered yield functions become increasingly unable to correctly describe the multiaxial deformation response.