- RJST - Theory of Elastic-Plastic Shells
- Phase-field analysis of finite-strain plates and shells including element subdivision
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The formation mechanisms of Al 3 Ti through the reaction of Al and Ti have been investigated systematically by D. Harach and K. Vecchio [ 9 ]. Intermetallic Al 3 Ti formation involved interfacial diffusion behavior between solid Ti and liquid Al. The volume fraction of residual Ti f Ti was calculated by the equation below. Figure 18 b displays the interfacial characteristics between the SiC fiber and the Al 3 Ti intermetallic alloy.
Note that the circular interface surrounding the SiC fiber is not complete and there are interstices and gaps in some parts of the interface. This observation indicates that bonding types between the fiber and Al 3 Ti include both the metallurgical reaction and the mechanical joint. The results of mechanical tests are summarized in Table 4. This result is extracted from the compressive stress-strain curves 3 and 4 plotted in Figure 19 a. Together, these two results suggest the strength of the fiber reinforced laminated composite is remarkably enhanced and the ductility is well maintained, which provides a feasible approach for solving the contradictory issue of strength and ductility.
Similar studies were also formerly presented by Han et al. This indicates that the reinforcement effect of SiC fibers on strength are almost the same under the sphere-plane and the plane-plane contact mode loading. The introduction of the concept of average compressive strength is meant to better understand the mechanical property of the materials under the sphere-plane contact mode. This does not have the same meaning as the compressive strength mentioned under a plane-plane contact mode.
Moreover, the damaged specimen after material failure under the sphere-plane contact load is shown in Figure 19 b.
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It can be observed that crack initials from the center of the contact surface owes to the loading of the spherical indenter and then extends to the side boundaries of the specimen, which causes the final failure of the material. This suggests that the failure behaviors of the CCFR-MIL composite are not only related to the microstructure but also to the loading contact mode. It is noted that under the sphere-plane mode, the compressive strengths of the two composites with and without fiber reinforcement are respectively lower than those under plane-plane mode.
This is because the sphere indenter would apply a concentrated force and induce higher plastic strain concentration in the loaded material, which leads to earlier crack initiation and earlier material failure. This shows that the SiC reinforcement effect for strength is still apparent while the effect for ductility is not as clear as that under plane-plane contact load. It can be inferred that the reinforcement effect of SiC fibers on the ductility of the composite is significant when under a uniform normal contact load.
RJST - Theory of Elastic-Plastic Shells
While under a sphere-plane contact load, the material around the indentation is subject to a tangential force from the spherical indenter. This force pushing the material away horizontally expends the crack at the contact point and leads to the final material failure. The SiC fibers can only prevent this kind of failure behavior in one horizontal direction since they are continuous in one horizontal direction and discretely distributed in another. Accordingly, the SiC reinforcement effect on mechanical properties of the material under the sphere-plane loading contact mode is more complicated, which is further studied in the following sections.
The equivalent plastic strain, which can represent the magnitude of plastic deformation concentration within the material, is expressed in Equation 9. The comparison between plastic strain fields with and without SiC inhomogeneities also reveals the reinforcement effects of SiC fibers. Considering that the value of total strain includes a large proportion of elastic strain, the plastic strain is contoured to display the difference of the strain status among each part in the Al 3 Ti matrix material more clearly.
It is known that the maximum plastic strain concentration MPSC area is also where cracks may initiate and extend from, which leads to the final material failure.
As the load increases, it is substituted by the closest location to the upper boundary of the central SiC fiber point A and then extends along depth around the fiber as the load continues increasing. By indicating the location where the cracks may initially form, extend from, and lead to the final material failure under continuous increasing load, also reveals the failure mechanism at the early stage of bullet penetration process. The variations of the equivalent plastic strains at point A and B with different loads are plotted in Figure Apparently, the equivalent plastic strain PEEQ of point A increases slowly when compared to that of point B at the beginning of loading while, after the plastic strain concentration area moves down to the SiC fiber string with the increasing load, the PEEQ of point A grows faster than that of point B and exceeds this point at the load of This occurs because, at the beginning of the contact loading process, most of the concentrated load is absorbed by the soft Ti layer to generate deformation within this layer.
Therefore, the plastic strain only concentrates on the shallow layer of the Al 3 Ti layer around point B and cannot extend to the deeper layer around point A. With the increasing amount of load, the Ti layer deforms to the max but will not fracture due to the high ductility. Therefore, a more concentrated load is imposed on the Al 3 Ti layer. It should be noted that the matrix material Al 3 Ti at point A is subject to not only the normal stress from the load above but also to the tangential stress induced by SiC fibers as foreign bodies block the material from moving deeper.
The continuous fibers are subject to a non-uniform force under a sphere-plane contact mode of loading, the middle part of the fiber bear the load while the two ends do not.
Therefore, the resilience of the high-strength SiC fiber would create considerable tangential stress at the middle part of fiber to separate the material at point A. Since the diameter of fiber is much smaller than that of the spherical indenter, the tangential stress at point A is much higher than that at point B. Therefore, the PEEQ of point A is higher than that of point B under the same normal load as long as the concentrated loading force would propagate to the deeper layer around point A, which explains the phenomenon in Figure 18 beyond By comparing the images of Figure 20 a,b, it was noted that, at the beginning of loading, the plastic strain distributions of the materials with and without the reinforced SiC fiber are almost the same.
Then, as the load increases, which is shown in Figure 20 c, the plastic strain concentration area moves deeper along the loading direction and is getting closer to the SiC fiber string. This indicated that the SiC fibers do make a difference in improving the strength of the Ti-Al 3 Ti MIL composite by effectively preventing the plastic strain from extending to the deeper layer. In addition, this improving effect is more obvious at large load than at small ones. Correspondingly, when the load is removed, the SiC fibers with elastic deformation can provide more of a restoring force than the matrix material of Al 3 Ti.
This explains the phenomenon in Figure 17 , which showed how the SiC fibers improve both strength and resilience of the laminated composite. The comparison between the equivalent plastic strain fields with and without SiC fiber reinforcement indicated that the SiC fiber string is also a local plastic strain concentration raiser even though it can lower the global strain.
Phase-field analysis of finite-strain plates and shells including element subdivision
This explains the phenomenon in Figure 19 a and the reason why the SiC fiber string can lower the failure strain while alsostrengthening the material. When the fracture or breakage occur on SiC fibers, the high plastic strain concentration around the fibers would extend to other parts of the Al 3 Ti layer immediately.
The strengthening effect of the SiC fiber string is connected with the geometric parameters of SiC fibers and the buffer effect of the Ti layer on the load is related to the volume fraction of Ti. Accordingly, in-depth parametric studies are necessary. These are carried out in the following section. It is well known that the appropriate distribution of SiC fibers plays an important role in improving the mechanical properties of the fiber reinforced composites.
The magnitude of the load is fixed at The equivalent plastic strain fields in cross section XOZ with varying ratios between the center distance of adjacent SiC fibers d 0 and fiber diameter d f is shown in Figure Note that the PEEQ in the center of the interface between the Ti and Al 3 Ti layer point B is barely affected by changing the ratio, which means the analysis focuses on the maximum PEEQ around the central SiC fiber at point A, the closest location to upper boundary of the fiber and of the residual indentation.
Indicating that, when the SiC fibers are distributed more densely, the effect of fiber reinforcement on compressive strength is more visible. These observations can also be viewed from another scenario, which is shown in Figure 23 a,b. Accordingly, it can be inferred that the sparsely distributed SiC fibers would cause higher plastic strain concentration at point A, which may lead to an earlier material failure around the fibers. However, when the center distance gets too short, the plastic strain concentration induced by the stiff SiC fiber, which concentrated around the fiber before, extends along the horizontal center line of fiber string This is shown in Figure 22 d.
As shown in Figure 24 , it can be observed that there are void strings along the horizontal center line of the SiC fiber string between adjacent fibers. Therefore, the strength of this layer is weakened dramatically. So theoretically, the material can be identified as failing when the plastic strain concentration comes into being in this layer [ 96 , 99 ]. Accordingly, the optimal ratio between the center distance of adjacent SiC fibers d 0 and fiber diameter d f is 4 in which the plastic strain ratio of the global material and the local position in the Al 3 Ti layer both drop significantly while the plastic strain concentration along the horizontal center line of the SiC fiber string between adjacent fibers has not yet come into being.
The ductile Ti layers provided sufficient bridging tractions to enhance fatigue resistance by almost an order of magnitude over the monolithic intermetallic known as Al 3 Ti [ ]. Such a buffer effect on the load is closely related to the volume fraction of the Ti layer [ ]. The magnitude of load is fixed at The SiC fiber strings in the cases of this section are always located in the center layer of the Al 3 Ti layer.
The equivalent plastic strain fields in cross section XOZ with varying volume fractions of Ti is shown in Figure Considering that the variation in volume fraction of Ti affects the global plastic strain field much more dramatically than local fields, the analysis focuses on the maximum PEEQ at the interface between Ti and Al 3 Ti layers at point B and the residual indentation, which can better reveal the plasticity behaviors of the global field. Comparing Figure 25 a—d, it can be observed that the plastic strain concentration region at the interface between two layers reduces both size and magnitude since the increasing thickness of the compliant Ti layer can provide a better buffer effect to the contact load.
These observations can also be viewed from another scenario, which is shown in Figure 26 a. The maximum PEEQ at the interface between two layers at point B and the residual indentation both drop with the growth of the Ti volume fraction by Implying that the maximum PEEQ of residual indentation is more vulnerable for the change of the Ti volume fraction than that at the interface between two layers.
Plot of a the maximum plastic strain of residual indentation and the equivalent plastic strain of point B and b the average equivalent plastic strain of the Al 3 Ti layer with different volume fractions of Ti. It can be observed from Figure 25 d that the plastic strain concentration due to the bending of the whole unit has come into being at the bottom of the layer and the bending effect is highly possible. It amplifies the plastic strain concentration at the center of interface between Ti and Al 3 Ti layers.
This shows that the plastic deformation of the whole composite material has expanded to a considerable degree. To quantitatively analyze the global plasticity behaviors, the definition of the average equivalent plastic strain is introduced [ 31 ]. This corresponds to the trend of the fatigue threshold [ ] and stress intensity [ ] with the increase of Ti volume fraction in the former references, which also fluctuate. The investigation focused on microscopic strengthening, penetration, and failure mechanisms at the early stage of penetration when the laminated material along the loading direction was not damaged and the material was continuous.
The parameters of magnitude of load, distribution of SiC fibers, and volume fraction of Ti were investigated for their effects on the plastic strain fields within the material measured by the equivalent plastic strains. The results indicated following major findings:. SiC fibers distribute in the horizontal center layer of the Al 3 Ti intermetallic layers and react with the intermetallic during the heat-treating process. Both are lower than the values under the plane-plane contact load. In addition, the SiC fiber reinforcing effect for strength is still apparent while the effect for ductility is not as clear as that under the plane-plane contact load.
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Under the sphere-plane contact loading, the maximum plastic strain concentration in the Al 3 Ti layer is closest to the upper boundary of the central SiC fiber. It then extends along depth as the load increases, which are the locations where cracks may initiate and extend from. The optimal ratio between center distance of adjacent SiC fibers d 0 and fiber diameter d f under sphere-plane contact loading is 4 at which ratio the plastic strain of the global material and the local position in Al 3 Ti layer both drop significantly.
When this occurs, the plastic strain concentration along the horizontal center line of the SiC fiber string between adjacent fibers has not yet come into being. The authors gratefully acknowledge the financial support of this research by the National Natural Science Foundation of China No. Conceptualization, J. National Center for Biotechnology Information , U.