In order to investigate if the anisotropy of the spacetime may induce futuresingularities at a finite value of the cosmic time on the evolution ofcosmological models, we study vacuum and non-vacuum Bianchi type I spacetimesexhibiting such future singularities. We show that in the case of Kasner vacuumcosmologies the spacetime may rip itself apart in a finite time, and only inthe direction corresponding to the unique scale factor which diverges at thisfinite value of the cosmic time. The other two directional scale factors andthe average scale factor do not diverge and tend to zero at this time, whilethe directional and average expansion rates also become infinite. Due to theabsence of the matter content this anisotropic future singularity is induced bythe shear scalar, which also blows up at this time. We call such a singularity"Vacuum Rip". For non-vacuum solutions we discuss fully anisotropic Bianchitype I spacetimes filled with a stiff fluid and ellipsoidal (axisymmetric)cosmological models filled with matter with isotropic and anisotropicbarotropic pressure, and characterized by $\sigma/\theta=const$, where $\sigma$and $\theta$ are the shear scalar and the expansion scalar respectively.
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