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Mathematical Preliminaries
Published in Michael R. Gosz, Finite Element Method, 2017
A vector is a physical quantity that has both magnitude and direction. Force is an example of a vector quantity. Both magnitude and direction are needed to completely describe it. A vector u is illustrated in Figure 2.1. The X1, X2, X3 axes labeled in the figure form what is called a Cartesian coordinate system. There are three other vectors shown in the figure. These are labeled e1, e2, and e3. They represent unit vectors (otherwise known as base vectors) pointing in the X1, X2, and X3 directions, respectively. It turns out that any vector u can be expressed in terms of the base vectors as follows: u=u1e1+u2e2+u3e3
Fundamentals of Elasto-Plastic Mechanics
Published in Yichun Zhou, Li Yang, Yongli Huang, Micro- and MacroMechanical Properties of Materials, 2013
Yichun Zhou, Li Yang, Yongli Huang
Einstein said that any theory that cannot be described by a mathematical language is not yet a science. The state of a nonisolated material must be described by a mathematical language. Therefore, the state of a material is described by a series of physical quantities. Because of the number and variety of these quantities, it is easy to become confused. However, if the physical quantities are expressed by a mathematical language, the problem is very easy. After careful analysis, it is discovered that some physical quantities-such as mass, density, volume, and the kinetic energy of an object-can be completely described by one value. A physical quantity having only magnitude without direction is called a scalar. However, some physical quantities, such as velocity, acceleration, and force, cannot be represented by a single value. They not only have magnitude, but also direction. We have to set up a coordinate system to describe these quantities. For example, as shown in Figure 1.3, a description of the geometric position of point A in a space requires three independent coordinates (x,y,z) in a reference system.
The importance of equation η = μn 2 in dimensional analysis and scaled vehicle experiments in vehicle dynamics
Published in Vehicle System Dynamics, 2022
Sina Milani, Hormoz Marzbani, Nasser Lashgarian Azad, William Melek, Reza N. Jazar
Maxwell assumed a set of independent abstract base physical dimensional quantities, and expressed every other dependant physical quantity based on them. Therefore, every dependant physical quantity can be derived as a product of the base quantities, each with an exponent. The exponent indicates the dimension of that quantity. Dimensional homogeneity requires the dimension of each base quantity to be equal in every term, and on both sides of a physical equation. A dimension indicates how the numerical value of a quantity changes when the basic units of measurement change. The abstract base physical quantities in the Newtonian mechanics are: Length , Time , Mass , Temperature . A physical quantity is anything that is expressible in certain units, and completely characterised by its numerical value. Dimensions of physical quantities are all human-made concepts, as there is no such thing as a dimension in the nature. However, these concepts help science and engineering to interpret and model the nature. The concept of ‘dimensional homogeneity’ clearly differentiated ‘variable,’ ‘parameter,’ and ‘constant.’ It clarified that any term of an equation is made of a multiplication of three items: variable, parameter, constant, [3].