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arbitrary coordinates

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Series: ASM Handbook
Volume: 14A
Publisher: ASM International
Published: 01 January 2005
DOI: 10.31399/asm.hb.v14a.a0004030
EISBN: 978-1-62708-185-6
... Abstract This article contains nine tables that present useful formulas for deformation analysis and workability testing. The tables present formulas for effective stress, strain, and strain rate in arbitrary coordinates, principal, compression and tension testing of isotropic material. The...
Book Chapter

Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005452
EISBN: 978-1-62708-196-2
... has the coordinates: x C = ∫ a b x y   d x S y C = 1 2 ∫ a b y 2   d x S where S is the area of the trapezoid, and y = f ( x ) is the equation of the curve AB . For an arbitrary plane figure, as shown in Fig...
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005233
EISBN: 978-1-62708-187-0
... computational region into a mesh of cells, but finite-volume cells can be arbitrary quadrilaterals in two dimensions, hexahedra in three dimensions, or indeed any shape enclosed by a set of corner points. In contrast, FDMs are defined on grids that are obtained using orthogonal curvilinear coordinate systems...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005453
EISBN: 978-1-62708-196-2
... , …, c n are arbitrary constants. If Eq 6 is differentiated n times with respect to x , and if the n constants c i that appear in the resulting n equations and Eq 6 are eliminated, the resulting differential equation is Eq 5 . Equation 6 is the primitive of Eq 5 . A particular...
Book Chapter

Series: ASM Handbook
Volume: 3
Publisher: ASM International
Published: 27 April 2016
DOI: 10.31399/asm.hb.v03.a0006292
EISBN: 978-1-62708-163-4
... of atomic arrangement. A crystal consists of unit cells stacked tightly together, each identical in size, shape, and orientation with all others. The choice of the boundaries of a unit cell is somewhat arbitrary, being conditioned by symmetry considerations and by convenience. Crystallography uses...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005400
EISBN: 978-1-62708-196-2
... independent slip systems to accommodate an arbitrary strain state in an fcc crystal was first pointed out by von Mises ( Ref 8 ). Since one can find more than one set of five independent shear strains to accommodate a given imposed strain state, the solution for ε ij in terms of a i ( Eq 4(b) ) is not...
Series: ASM Handbook
Volume: 12
Publisher: ASM International
Published: 01 January 1987
DOI: 10.31399/asm.hb.v12.a0001837
EISBN: 978-1-62708-181-8
... surface. For example, at point p in Fig. 5 , the magnification is proportional to ss′/sm, but at point q, it is proportional to ss′/sn. The coordinates of the points in the fracture surface are usually measured by stereo SEM pairs, that is, two photographs of the same field taken at small tilt angles...
Series: ASM Handbook
Volume: 17
Publisher: ASM International
Published: 01 August 2018
DOI: 10.31399/asm.hb.v17.a0006439
EISBN: 978-1-62708-190-0
..., microfinished end diameter. (b) Screen display of dimension A-A. (c) Monitor image of dimension B-B. (d) Screen display of dimension C-C. Courtesy of Industrial Systems Division, Ball Corporation One technique applicable to the inspection of simple geometries is polar coordinate vision ( Fig. 11 ). This...
Series: ASM Handbook
Volume: 10
Publisher: ASM International
Published: 15 December 2019
DOI: 10.31399/asm.hb.v10.a0006662
EISBN: 978-1-62708-213-6
... proportional to concentration must be used. The molecular geometry, atomic masses, and a complete description of the forces between the atoms (force field) are required to calculate the vibrational frequencies and describe the fundamental vibrations of a molecule. This is normal-coordinate analysis ( Ref 1...
Series: ASM Handbook
Volume: 10
Publisher: ASM International
Published: 15 December 2019
DOI: 10.31399/asm.hb.v10.a0006631
EISBN: 978-1-62708-213-6
... The main application of single-crystal x-ray diffraction (XRD) is to characterize the crystal structure, including determining symmetry, unit cell parameters, atomic coordinates and thermal displacement parameters, bond lengths and angles between the atoms, and structural motive (or topology). During...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005426
EISBN: 978-1-62708-196-2
... These equations are written in Cartesian tensor notation ( Ref 15 ), according to which the subscripts i and j take the values 1, 2, or 3 corresponding to the three Cartesian coordinate directions. A subscript that appears just once in a term takes on one of the three values 1, 2, or 3; repeated...
Series: ASM Handbook
Volume: 11
Publisher: ASM International
Published: 15 January 2021
DOI: 10.31399/asm.hb.v11.a0006767
EISBN: 978-1-62708-295-2
.... Convergence techniques are generally employed to ensure that a given solution is “correct.” Program output is the stress state in a given component or assembly for a given combination of loading and boundary conditions. Although this stress state is unique to a specific coordinate system, most commercially...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005431
EISBN: 978-1-62708-196-2
... described by Lagrangian, Eulerian, and Arbitrary Lagrangian-Eulerian (ALE) formulations. In the Lagrangian method, the finite-element mesh is attached to the material and moves through space along with the material. In the Eulerian formulation, the finite-element mesh is fixed in space, and the material...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005450
EISBN: 978-1-62708-196-2
... vorticity vector for Cartesian coordinates is: (Eq 37) Ω = ( ∂ w ∂ y − ∂ v ∂ z ) i + ( ∂ u ∂ z − ∂ w ∂ x ) j + ( ∂ v ∂ x − ∂ u ∂ y ) k     = 2 ⋅ ω = 2 ( ∇ × V ) For incompressible...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005422
EISBN: 978-1-62708-196-2
... allows the partitioning of space by considering the size associated with a site, and thus, it is capable of mimicking an arbitrary size distribution. This mosaic differs from a Voronoi tessellation in the definition of the planes delimiting the regions of the tessellation. It can be noticed from Eq 1...
Series: ASM Handbook
Volume: 4B
Publisher: ASM International
Published: 30 September 2014
DOI: 10.31399/asm.hb.v04b.a0005993
EISBN: 978-1-62708-166-5
... Radiation spectrum The basic law of heat conduction was first presented by a French physicist, Fourier, and is referred to as Fourier's law of heat conduction. Table 1 presents the general form related to various coordinate systems. From Fourier's law and the first law of...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005449
EISBN: 978-1-62708-196-2
... coordinate systems. From Fourier's law and the first law of thermodynamics, the differential equations for heat conduction in solids can be derived ( Table 2 ). To determine temperature distribution due to heat conduction, boundary conditions must be known. Frequently encountered boundary conditions in...
Series: ASM Handbook
Volume: 17
Publisher: ASM International
Published: 01 August 2018
DOI: 10.31399/asm.hb.v17.a0006452
EISBN: 978-1-62708-190-0
... A-scans, w k ( t ), are generated and used to form scan lines in an image. In a B-scan, the t in each A-scan always is associated with a range coordinate equal to ct /2, where c is the speed of sound, and the focal law, k , defines the other coordinate. Examples for swept-aperture linear...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005415
EISBN: 978-1-62708-196-2
... microstructure involves a large number of variables (one or several for each spatial coordinate, r , with each one regarded as a degree of freedom). An effective way to formulate their time evolution (the kinetics of microstructure change) is to define a total energy of the system and derive the kinetics along...
Series: ASM Handbook
Volume: 17
Publisher: ASM International
Published: 01 August 2018
DOI: 10.31399/asm.hb.v17.a0006456
EISBN: 978-1-62708-190-0
... nondestructive tests and dimensional metrology, because it is an ideal technique to carry out the defect analysis and the measurement of the outer as well as the inner geometry of a specimen without the need to cut it through (therefore preventing a complete nondestructive coordinate measurement). Thanks to CT...