Eigenvalues of the 3D critical point equation (∇u)ν = λν are normally computed numerically. In the letter, we present analytic solutions for 3D swirling strength in both compressible and incompressible flows. The solutions expose functional dependencies that cannot be seen in numerical solutions. To illustrate, we study the difference between using fluctuating and total velocity gradient tensors for vortex identification. Results show that mean shear influences vortex detection and that distortion can occur, depending on the strength of mean shear relative to the vorticity at the vortex center.

In this paper, we present a Bayesian analysis for the Weibull proportional hazard (PH) model used in step-stress accelerated life testings. The key mathematical and graphical difference between the Weibull cumulative exposure (CE) model and the PH model is illustrated. Compared with the CE model, the PH model provides more flexibility in fitting step-stress testing data and has the attractive mathematical properties of being desirable in the Bayesian framework. A Markov chain Monte Carlo algorithm with adaptive rejection sampling technique is used for posterior inference. We demonstrate the performance of this method on both simulated and real datasets.

We describe mechanical metamaterials created by folding flat sheets in the tradition of origami, the art of paper folding, and study them in terms of their basic geometric and stiffness properties, as well as load bearing capability. A periodic Miura-ori pattern and a non-periodic Ron Resch pattern were studied. Unexceptional coexistence of positive and negative Poisson's ratio was reported for Miura-ori pattern, which are consistent with the interesting shear behavior and infinity bulk modulus of the same pattern. Unusually strong load bearing capability of the Ron Resch pattern was found and attributed to the unique way of folding. This work paves the way to the study of intriguing properties of origami structures as mechanical metamaterials.