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Fracture Mechanics
Published in Cameron Coates, Valmiki Sooklal, Modern Applied Fracture Mechanics, 2022
Cameron Coates, Valmiki Sooklal
Note that stress is NOT a vector, it is a second-order tensor. The term “tensor” describes any physical quantity that can be represented by a scalar, vector, or a matrix. Physical properties such as mass, temperature, or density, which are scalars, would be considered zero-order tensors. Physical properties that may be represented as vectors such as force, velocity, or acceleration are first-order tensors. Stress is a second-order tensor as the stress state at a point is represented by a matrix. Moment equilibrium requires that σij = σji (i ≠ j), therefore the six independent shear stress components reduce to three. The total number of independent stress components will therefore be six; three independent shear stresses and three independent normal stresses.
Introduction
Published in Georgios A. Drosopoulos, Georgios E. Stavroulakis, Nonlinear Mechanics for Composite Heterogeneous Structures, 2022
Georgios A. Drosopoulos, Georgios E. Stavroulakis
A tensor is a matrix with physical meaning, with its components being dependent on a given coordinate system. A tensor A is of order n when each of its components Aijk…m has n indices and obeys certain transformation rules from an initial to a new coordinate system.
Vector and Tensor Calculus
Published in Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou, ViscousFluid Flow, 2021
Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou
Tensors encountered in fluid mechanics are of second order, i.e., they are characterized by an ordered pair of coordinate directions. Tensors are often denoted by uppercase, boldface type or lower case, boldface Greek letters, such as τ for the stress tensor. A tensor field is a tensor-valued function that associates a tensor with each point of a given region in space. Tensor addition and multiplication of a tensor by a scalar are commutative and associative. If R, S, and T are tensors of the same type, and m and n are scalars, then
Re-examining urban region and inferring regional function based on spatial–temporal interaction
Published in International Journal of Digital Earth, 2019
Haiyan Tao, Keli Wang, Li Zhuo, Xuliang Li
Tensor is the generalization of vector and matrix, which is actually multi-dimensional array or matrix. Its order corresponds to the number of dimensions, usually called the modes (Zdunek 2009). Tensor decomposition extends the decomposition of two-dimensional matrices to high-dimensional data sets. Large complex data sets are usually composed of interrelated variables and potential factors between them. With rich dimensional information, it is usually necessary to decompose the data set into the corresponding element composition to excavate the structure and implicit information (Zdunek 2009). The tensor decomposition has the advantage of retrieving information from large data sets, and we can use the simple structure with relatively few parameters to depict the interaction between different modes in multi-dimensional data (Sun and Axhausen 2016). Therefore, the tensor decomposition has become one popular choice to deal with large and complex multi-dimensional data, which is widely applied to the traffic field and urban computing (Han and Moutarde 2016; Ma et al. 2013; Peng et al. 2012; Wang et al. 2014).
Tensor calculus: unlearning vector calculus
Published in International Journal of Mathematical Education in Science and Technology, 2018
Wha-Suck Lee, Johann Engelbrecht, Rita Moller
A typical course in vector calculus sometimes leaves some students with concepts that appear to be scattered and unrelated. Much of this confusion results from an inability to move away from using Cartesian coordinates as a reference point to judge all other coordinate systems. Since high school all the way into the first year of mathematics, students are exposed to Cartesian coordinates. Furthermore, the distinction between the actual physical point in space and its Cartesian coordinates has not been emphasized. This addiction to the Cartesian coordinate system when applied to the ubiquitous geometric concept of the gradient manifests in the form of the algebraic formulation of the gradient (2.1). Many students perceive this as the geometric definition of the gradient. In Section 2, we show the pitfall of this misconception in order to break away from the addiction to Cartesian coordinates. More precisely, we show (1) that the algebraic undergraduate definition of the gradient works in exactly one coordinate system, the Cartesian coordinate system and (2) a case at where the algebraic formulation of the gradient (2.1) gives the wrong gradient in a slightly modified Cartesian coordinate system. Tensor calculus is an upgraded vector calculus with improved algebraic formulations which are not tied down to any single coordinate system. We use the tensor calculus notation of [1].
The role of phosphorus in pore rounding of sintered steels
Published in Cogent Engineering, 2018
Walid Khraisat, Wisam Abu Jadayil, Nathir Rawashdeh, Henrik Borgström
This expression is known as total derivative. Mathematically surface tension τ, which is referred to as surface stress in solid surfaces, and surface energy σ are different. This means that the solid free surface sustains a state of stress leading to strain energy stretching the free surface without adding new atoms to the surface. The surface stress, which is a second order tensor, is associated with the work against surface deformation, it is the result of forces acting at the material surface. The surface stress can be thought of as an excess of bulk stress. Tensors are quantities which possess both magnitude and direction. The order of the tensor is the number of physical directions associated with the tensor. A second order means that there are two directions associated with the stress, (1) the direction of the force and (2) the direction of the unit normal to the surface which gives the orientation of the surface. Surface stresses are ⊥ to the surface they act upon thus the surface is in a plane stress condition. The components of the surface stress are shown in Figure 8.