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Geometry
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
A prism whose base is a parallelogram is a parallelepiped. The volume of a parallelepiped with one vertex at the origin and adjacent vertices at (x1,y1,z1) $ (x_{1} , y_{1} , z_{1} ) $ , (x2,y2,z2) $ (x_{2} , y_{2} , z_{2} ) $ , and (x3,y3,z3) $ (x_{3} , y_{3} , z_{3} ) $ is given by volume=x1y1z1x2y2z2x3y3z3. $$ {\text{volume}} = \left| {\begin{array}{*{20}c} {x_{1} } & {y_{1} } & {z_{1} } \\ {x_{2} } & {y_{2} } & {z_{2} } \\ {x_{3} } & {y_{3} } & {z_{3} } \\ \end{array} } \right|. $$
Microstructural model of the behavior of a ferroalloy with shape memory in a magnetic field
Published in Mechanics of Advanced Materials and Structures, 2022
The symmetry of the crystal is characterized by the orthogonal tensors proper2 forming a group of rotations in which the shape of the crystal remains unchanged: a cube is transformed into a cube, a parallelepiped into exactly the same parallelepiped, etc. During the phase transition of the first kind, a cubic cell of austenite (high-temperature state) ferroalloy with shape memory (FSMA) Ni2MnGa is transformed into a tetragonal cell of martensite (low-temperature state). The rotation group for a cell in the austenitic state (for a cube) consists of 24 orthogonal tensors, The rotation group for a cell in the martensitic state (for a tetragonal cell) consists of 8 orthogonal tensors, (the orthogonal tensors corresponding to these states are given, for example, by Hane and Shield [18]). The number of independent variants of martensite during the phase transition is determined by the relation and for Ni2MnGa is 3.
Modelling of asphalt mixes based on X-ray computed tomography and random heterogeneous generation
Published in International Journal of Pavement Engineering, 2020
Yassine El Haloui, Fateh Fakhari Tehrani, Joseph Absi, Fabien Courreges, Mohamed El Omari, Fatima Allou, Christophe Petit
As a next step, a Boolean operation is performed between the inclusions and the container (rectangle, cylinder or parallelepiped shape) in order to obtain the matrix. Moreover, an interaction between aggregates and the outer surface of the container can be introduced. This option presents the heterogeneous surface of sawn laboratory samples from an HMA plate. To obtain models with air voids, another Boolean operation is required between the matrix and air void distribution. Lastly, both mortar (with or without voids) and aggregate models are ready to be run. A fully cohesive state at the mortar/aggregate interfaces is imposed on the entire aggregate surface through use of the ‘tied contact’ function, which has been preloaded in the FEM code. This function relies on the principle of full cohesion between the master and slave surfaces. For each node of the slave surface (i.e. first surface), the finite element code captures the nearest point of the master surface (second surface) and then projects perpendicularly from the master to the slave surface. Figure 6 illustrates the Boolean operations required to produce an example of the final heterogeneous numerical model.
Microbes induced biofabrication of nanoparticles: a review
Published in Inorganic and Nano-Metal Chemistry, 2020
Devendra Kumar Golhani, Ayush Khare, Gopal Krishna Burra, Vikas Kumar Jain, Jagannadha Rao Mokka
Many Bacterial species are used for synthesis of magnetite nanoparticles namely A. magnetotacticum,[70]Magnetotactic bacterium MV-1,[71]Sulfate reducing bacteria,[72]M. magnetotacticum,[73]M. magnetotacticum (MS-1)[74] and M. gryphiswaldense.[75] They produced NPs of size 2–200 nm with various morphologies, such as parallelepiped, octahedral prism, hexagonal prism, and cubo-octahedral. Similarly, non-magnetotactic bacterium Geobacter metallireducens GS-15 in anaerobic growth conditions is found to produce magnetite particles in the size range 10–50 nm by extracellular routes.[76] Bazylinski et al.[77] reported the formation of Fe3S4 by uncultured magnetotactic bacteria. In this study, sulfate-reducing bacteria were found capable of producing magnetic FeS nanoparticles.[78]