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A Review on Plasticity and the Finite Element Method
Published in Uday S Dixit, Seikh Mustafa Kamal, Rajkumar Shufen, Autofrettage Processes, 2019
Uday S. Dixit, Seikh Mustafa Kamal, Rajkumar Shufen
The field variables in the governing equations of elastic–plastic deformation use operations on scalars, vectors, and tensors. For the convenience of representation of various equations, the index notation along with Einstein’s summation convention is used. In index notation, a tensor component is represented by using indices. An index is designated by a small letter, say i, which can take on the values 1, 2, and 3 for the x, y, and z coordinates of a Cartesian system, respectively. For example, the components of a typical vector a (which is a first-order tensor) are represented by ai that denotes 3 components formed by the index i (1, 2, 3). Thus, ai≡a1a2a3.
Mathematical preliminaries
Published in J.L. Meek, Computer Methods in Structural Analysis, 2017
The use of the index notation is greatly enhanced by the Einstein summation convention. This simple idea states that a repeated index in an expression implies a sum over the range of the index. For example, the expression AiBi for two three–dimensional vectors {A}, {B} is AiBi=A1B1+A2B2+A3B3
Clarifying the definition of ‘transonic’ screw dislocations
Published in Philosophical Magazine, 2021
Daniel N. Blaschke, Jie Chen, Saryu Fensin, Benjamin A. Szajewski
Both sound waves and moving dislocations may be quantified in terms of spatiotemporal displacement fields applied to atoms with respect to a perfect lattice configuration. For sound waves, these displacement fields are generally small and smooth; this is not the case for line defects (dislocations). Independent of motion, dislocations are most readily described by a non-vanishing Burgers vector which introduces a spatial displacement discontinuity across the slip plane. In the continuum limit, the displacement field satisfies the balance of linear momentum and the leading order stress-strain relations known as Hooke's law: Within these equations, denotes stress, is the infinitesimal strain tensor (i.e. the symmetrised displacement gradient field), ρ the material density, and we have introduced the common shorthand notation for partial derivatives. denotes the components of the fourth rank tensor of second order elastic constants (SOEC), also known as the 'stiffness tensor'. Within these expressions, we employ index notation and Einstein's summation convention where the sum over repeated indices is implied. In addition, a dot denotes differentiation with respect to time.
Fast accurate seakeeping predictions
Published in Ship Technology Research, 2020
Equations (46) and (47) for translational and rotational motion are combined to one matrix equation for the six accelerations. To that end, the vector product involving in (46) is written in index notation: for arbitrary three-dimensional vectors and their vector product can be written (using Einstein summation) aswhere are the elements of the Levi–Civita tensor: is 1 if ijk constitute an even permutation of (1,2,3); for an odd permutation; and 0 if i,j,k are not all different. Using this notation, (46) and (47) can be combined into the matrix equationHere E is the 3 by 3 unit matrix, 0 the 3 by 3 zero matrix, and A the 6 by 6 added mass matrix. The rigid mass matrix on the left-hand side (in parentheses) may appear unusual. The reason is: here moments and moments of inertia refer to the centre of gravity G, but the translation refers to the origin of the body-fixed coordinate system, which usually differs from G. If both coincide, we have , causing the upper right 3 by 3 submatrix to vanish.
Exergetic port-Hamiltonian systems: modelling basics
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Markus Lohmayer, Paul Kotyczka, Sigrid Leyendecker
For tensorial quantities, we use (abstract) index notation with Einstein’s convention: Indices of contravariant slots are written as superscript and indices of covariant slots are written as subscript. Repeated indices (up-down pairs) imply contraction. With a smooth manifold, denotes the tangent bundle and the cotangent bundle over . We write for a general vector bundle with total space and base space . When the latter is clear from the context, we just write . For vector bundles and , is the vector bundle over where . A section of a bundle is a function satisfying where is the bundle projection. The set of all sections of is denoted by . Given a contravariant -tensor field , the sharp map is the (curried) function defined by . Dually, the flat map corresponding to a covariant -tensor field is a bundle map from the tangent to the cotangent bundle. Its name derives from the fact that in index notation, it lowers the up-index of a tangent vector into the down-index of the covector .