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Spatial Displacement Gradient Tensor
In mechanics, a displacement field is the assignment of displacement vectors for all points in a region or body that are displaced from one state to another. A displacement vector specifies the position of a point or a particle in reference to an origin or to a previous position. For example, a displacement field may be used to describe the effects of deformation on a solid body. Formulation Before considering displacement, the state before deformation must be defined. It is a state in which the coordinates of all points are known and described by the function: \vec_0: \Omega \to P where *\vec_0 is a placement vector *\Omega are all the points of the body *P are all the points in the space in which the body is present Most often it is a state of the body in which no forces are applied. Then given any other state of this body in which coordinates of all its points are described as \vec_1 the displacement field is the difference between two body states: \vec = \vec_1 - \vec_0 wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mechanics
Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of an object's position relative to its environment. Theoretical expositions of this branch of physics has its origins in Ancient Greece, for instance, in the writings of Aristotle and Archimedes (see History of classical mechanics and Timeline of classical mechanics). During the early modern period, scientists such as Galileo Galilei, Johannes Kepler, Christiaan Huygens, and Isaac Newton laid the foundation for what is now known as classical mechanics. As a branch of classical physics, mechanics deals with bodies that are either at rest or are moving with velocities significantly less than the speed of light. It can also be defined as the physical science that deals with the motion of and forces on bodies not in the quantum realm. History ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function f(x, y, \dots) with respect to the variable x is variously denoted by It can be thought of as the rate of change of the function in the x-direction. Sometimes, for the partial derivative of z with respect to x is denoted as \tfrac. Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: f'_x(x, y, \ldots), \frac (x, y, \ldots). The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strain (mechanics)
In mechanics, strain is defined as relative deformation, compared to a position configuration. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the metric tensor or its dual is considered. Strain has dimension of a length ratio, with SI base units of meter per meter (m/m). Hence strains are dimensionless and are usually expressed as a decimal fraction or a percentage. Parts-per notation is also used, e.g., parts per million or parts per billion (sometimes called "microstrains" and "nanostrains", respectively), corresponding to μm/m and nm/m. Strain can be formulated as the spatial derivative of displacement: \boldsymbol \doteq \cfrac\left(\mathbf - \mathbf\right) = \boldsymbol'- \boldsymbol, where is the identity tensor. The displacement of a body may be expressed in the form , where is the reference position of material ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stress (mechanics)
In continuum mechanics, stress is a physical quantity that describes forces present during deformation. For example, an object being pulled apart, such as a stretched elastic band, is subject to ''tensile'' stress and may undergo elongation. An object being pushed together, such as a crumpled sponge, is subject to ''compressive'' stress and may undergo shortening. The greater the force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Stress has dimension of force per area, with SI units of newtons per square meter (N/m2) or pascal (Pa). Stress expresses the internal forces that neighbouring particles of a continuous material exert on each other, while ''strain'' is the measure of the relative deformation of the material. For example, when a solid vertical bar is supporting an overhead weight, each particle in the bar pushes on the particles immediately below it. When a liquid is in a closed container under pressure, each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\text i=j. \end or with use of Iverson brackets: \delta_ = =j, For example, \delta_ = 0 because 1 \ne 2, whereas \delta_ = 1 because 3 = 3. The Kronecker delta appears naturally in many areas of mathematics, physics, engineering and computer science, as a means of compactly expressing its definition above. Generalized versions of the Kronecker delta have found applications in differential geometry and modern tensor calculus, particularly in formulations of gauge theory and topological field models. In linear algebra, the n\times n identity matrix \mathbf has entries equal to the Kronecker delta: I_ = \delta_ where i and j take the values 1,2,\cdots,n, and the inner product of vectors can be written as \mathbf\cdot\mathbf = \sum_^n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direction Cosine
In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three positive coordinate axes. Equivalently, they are the contributions of each component of the basis to a unit vector in that direction. Three-dimensional Cartesian coordinates If is a Euclidean vector in three-dimensional Euclidean space, \mathbf v = v_x \mathbf e_x + v_y \mathbf e_y + v_z \mathbf e_z, where are the standard basis in Cartesian notation, then the direction cosines are \begin \alpha &= \cos a = \frac &&= \frac,\\ \beta &= \cos b = \frac &&= \frac,\\ \gamma &= \cos c = \frac &&= \frac. \end It follows that by squaring each equation and adding the results \cos^2 a + \cos^2 b + \cos^2 c = \alpha^ + \beta^ + \gamma^ = 1. Here are the direction cosines and the Cartesian coordinates of the unit vector \tfrac, and are the direction angles of the vector . The direction angles are acute or obtuse angles, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spatial Deformation Gradient Tensor
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically deforming materials and other fluids and biological soft tissue. Displacement field Deformation gradient tensor The deformation gradient tensor \mathbf F(\mathbf X,t) = F_ \mathbf e_j \otimes \mathbf I_K is related to both the reference and current configuration, as seen by the unit vectors \mathbf e_j and \mathbf I_K\,\!, therefore it is a '' two-point tensor''. Two types of deformation gradient tensor may be defined. Due to the assumption of continuity of \chi(\mathbf X,t)\,\!, \mathbf F has the inverse \mathbf H = \mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spatial Derivative
A spatial gradient is a gradient whose components are spatial derivatives, i.e., rate of change of a given scalar physical quantity with respect to the position coordinates in physical space. Homogeneous regions have spatial gradient vector norm equal to zero. When evaluated over vertical position (altitude or depth), it is called ''vertical derivative'' or ''vertical gradient''; the remainder is called ''horizontal gradient'' component, the vector projection of the full gradient onto the horizontal plane. Examples: ;Biology * Concentration gradient, the ratio of solute concentration between two adjoining regions * Potential gradient, the difference in electric charge between two adjoining regions ;Fluid dynamics and earth science * Density gradient * Pressure gradient * Temperature gradient A temperature gradient is a physical quantity that describes in which direction and at what rate the temperature changes the most rapidly around a particular location. The temperatu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
