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Pure Shear
In mechanics and geology, pure shear is a three-dimensional homogeneous flattening of a body. It is an example of irrotational strain in which body is elongated in one direction while being shortened perpendicularly. For soft materials, such as rubber, a strain state of pure shear is often used for characterizing hyperelastic and fracture mechanical behaviour. Pure shear is differentiated from simple shear in that pure shear involves no rigid body rotation. The deformation gradient for pure shear is given by: F = \begin1&\gamma&0 \\\gamma&1&0\\0&0&1\end Note that this gives a Green-Lagrange strain of: E = \frac\begin\gamma^2&2\gamma&0\\2\gamma&\gamma^2&0\\0&0&0\end Here there is no rotation occurring, which can be seen from the equal off-diagonal components of the strain tensor. The linear approximation to the Green-Lagrange strain shows that the small strain tensor is: \epsilon = \frac\begin0&2\gamma&0\\2\gamma&0&0\\0&0&0\end which has only shearing components. See ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects result in displacements, or 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, Kepler, Huygens, and 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 r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geology
Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Earth sciences, including hydrology, and so is treated as one major aspect of integrated Earth system science and planetary science. Geology describes the structure of the Earth on and beneath its surface, and the processes that have shaped that structure. It also provides tools to determine the relative and absolute ages of rocks found in a given location, and also to describe the histories of those rocks. By combining these tools, geologists are able to chronicle the geological history of the Earth as a whole, and also to demonstrate the age of the Earth. Geology provides the primary evidence for plate tectonics, the evolutionary history of life, and the Earth's past climates. Geologists broadly study the properties and processes of E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three-dimensional Space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal meaning of the term dimension. In mathematics, a tuple of numbers can be understood as the Cartesian coordinates of a location in a -dimensional Euclidean space. The set of these -tuples is commonly denoted \R^n, and can be identified to the -dimensional Euclidean space. When , this space is called three-dimensional Euclidean space (or simply Euclidean space when the context is clear). It serves as a model of the physical universe (when relativity theory is not considered), in which all known matter exists. While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
San Diego State University
San Diego State University (SDSU) is a public research university in San Diego, California. Founded in 1897 as San Diego Normal School, it is the third-oldest university and southernmost in the 23-member California State University (CSU) system. In Fall 2022, SDSU hit an all time high enrollment record student body of nearly 37,000 and an alumni base of more than 300,000. It is classified among "R2: Doctoral Universities – High research activity". In the 2015–16 fiscal year, the university obtained $130 million in public and private funding—a total of 707 awards—up from $120.6 million the previous fiscal year. As reported by the Faculty Scholarly Productivity Index released by the Academic Analytics organization of Stony Brook, New York, SDSU had the highest research output of any small research university in the United States in 2006 and 2007. SDSU sponsors the second-highest number of Fulbright Scholars in the State of California, just behind UC Berkeley. Since 2005, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deformation (mechanics)
In physics, deformation is the continuum mechanics transformation of a body from a ''reference'' configuration to a ''current'' configuration. A configuration is a set containing the positions of all particles of the body. A deformation can occur because of external loads, intrinsic activity (e.g. muscle contraction), body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc. Strain is related to deformation in terms of ''relative'' displacement of particles in the body that excludes rigid-body motions. 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. In a continuous body, a deformation field results from a stress field due to applied forces or because of some changes in the temperature field of the body. The relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendicularly
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can be defined between two lines (or two line segments), between a line and a plane, and between two planes. Perpendicularity is one particular instance of the more general mathematical concept of '' orthogonality''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its '' normal vector''. Definitions A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperelastic Material
A hyperelastic or Green elastic materialR.W. Ogden, 1984, ''Non-Linear Elastic Deformations'', , Dover. is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density function. The hyperelastic material is a special case of a Cauchy elastic material. For many materials, linear elastic models do not accurately describe the observed material behaviour. The most common example of this kind of material is rubber, whose stress- strain relationship can be defined as non-linearly elastic, isotropic and incompressible. Hyperelasticity provides a means of modeling the stress–strain behavior of such materials. The behavior of unfilled, vulcanized elastomers often conforms closely to the hyperelastic ideal. Filled elastomers and biological tissues are also often modeled via the hyperelastic idealization. Ronald Rivlin and Melvin Mooney developed the first hyperelastic models, the Neo-Hookean and Mooney–R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fracture Mechanics
Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture. Theoretically, the stress ahead of a sharp crack tip becomes infinite and cannot be used to describe the state around a crack. Fracture mechanics is used to characterise the loads on a crack, typically using a single parameter to describe the complete loading state at the crack tip. A number of different parameters have been developed. When the plastic zone at the tip of the crack is small relative to the crack length the stress state at the crack tip is the result of elastic forces within the material and is termed linear elastic fracture mechanics (LEFM) and can be characterised using the stress intensity factor K. Although the load on a crack can be arbitrary, in 1957 G. Irwin fou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Shear
Simple shear is a deformation in which parallel planes in a material remain parallel and maintain a constant distance, while translating relative to each other. In fluid mechanics In fluid mechanics, simple shear is a special case of deformation where only one component of velocity vectors has a non-zero value: :V_x=f(x,y) :V_y=V_z=0 And the gradient of velocity is constant and perpendicular to the velocity itself: :\frac = \dot \gamma , where \dot \gamma is the shear rate and: :\frac = \frac = 0 The displacement gradient tensor Γ for this deformation has only one nonzero term: :\Gamma = \begin 0 & & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end Simple shear with the rate \dot \gamma is the combination of pure shear strain with the rate of \dot \gamma and rotation with the rate of \dot \gamma: :\Gamma = \begin \underbrace \begin 0 & & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end \\ \mbox\end = \begin \underbrace \begin 0 & & 0 \\ & 0 & 0 \\ 0 & 0 & 0 \end \\ \mbox \end + \b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green-Lagrange Strain
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 The displacement of a body has two components: a rigid-body displacement and a deformation. * A rigid-body displacement consists of a simultaneous translation (physics) and rotation of the body without changing its shape or size. * Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration \kappa_0(\mathcal B) to a current or deformed configuration \kappa_t(\mathcal B) (Figure 1). A change in the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Shear
Simple shear is a deformation in which parallel planes in a material remain parallel and maintain a constant distance, while translating relative to each other. In fluid mechanics In fluid mechanics, simple shear is a special case of deformation where only one component of velocity vectors has a non-zero value: :V_x=f(x,y) :V_y=V_z=0 And the gradient of velocity is constant and perpendicular to the velocity itself: :\frac = \dot \gamma , where \dot \gamma is the shear rate and: :\frac = \frac = 0 The displacement gradient tensor Γ for this deformation has only one nonzero term: :\Gamma = \begin 0 & & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end Simple shear with the rate \dot \gamma is the combination of pure shear strain with the rate of \dot \gamma and rotation with the rate of \dot \gamma: :\Gamma = \begin \underbrace \begin 0 & & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end \\ \mbox\end = \begin \underbrace \begin 0 & & 0 \\ & 0 & 0 \\ 0 & 0 & 0 \end \\ \mbox \end + \b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squeeze Mapping
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping. For a fixed positive real number , the mapping :(x, y) \mapsto (ax, y/a) is the ''squeeze mapping'' with parameter . Since :\ is a hyperbola, if and , then and the points of the image of the squeeze mapping are on the same hyperbola as is. For this reason it is natural to think of the squeeze mapping as a hyperbolic rotation, as did Émile Borel in 1914, by analogy with ''circular rotations'', which preserve circles. Logarithm and hyperbolic angle The squeeze mapping sets the stage for development of the concept of logarithms. The problem of finding the area bounded by a hyperbola (such as is one of quadrature. The solution, found by Grégoire de Saint-Vincent and Alphonse Antonio de Sarasa in 1647, required the natural logarithm function, a new concept. Some insight ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
