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Crumpling
In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a random network of ridges and facets with variable density. The geometry of crumpled structures is the subject of some interest the mathematical community within the discipline of topology. Crumpled paper balls have been studied and found to exhibit surprisingly complex structures with compressive strength resulting from frictional interactions at locally flat facets between folds. The unusually high compressive strength of crumpled structures relative to their density is of interest in the disciplines of materials science and mechanical engineering. Significance The packing of a sheet by crumpling is a complex phenomenon that depends on material parameters and the packing protocol. Thus the crumpling behaviour of foil, paper and poly-membranes differs significantly and can be interp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mechanical Engineering
Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and maintain mechanical systems. It is one of the oldest and broadest of the engineering branches. Mechanical engineering requires an understanding of core areas including mechanics, dynamics, thermodynamics, materials science, structural analysis, and electricity. In addition to these core principles, mechanical engineers use tools such as computer-aided design (CAD), computer-aided manufacturing (CAM), and product lifecycle management to design and analyze manufacturing plants, industrial equipment and machinery, heating and cooling systems, transport systems, aircraft, watercraft, robotics, medical devices, weapons, and others. Mechanical engineering emerged as a field during the Industrial Revolution in Europe in the 18th century; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structural Analysis
Structural analysis is a branch of Solid Mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on the physical structures and their components. In contrast to theory of elasticity, the models used in structure analysis are often differential equations in one spatial variable. Structures subject to this type of analysis include all that must withstand loads, such as buildings, bridges, aircraft and ships. Structural analysis uses ideas from applied mechanics, materials science and applied mathematics to compute a structure's deformations, internal forces, stresses, support reactions, velocity, accelerations, and stability. The results of the analysis are used to verify a structure's fitness for use, often precluding physical tests. Structural analysis is thus a key part of the engineering design of structures. [...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 rel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Neighbourhood (mathematics), neighborhood that is homeomorphic to an open (topology), open subset of n-dimensional Euclidean space. One-dimensional manifolds include Line (geometry), lines and circles, but not Lemniscate, lemniscates. Two-dimensional manifolds are also called Surface (topology), surfaces. Examples include the Plane (geometry), plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as Graph of a function, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cat Play And Toys
Cat play and toys incorporates predatory games of "play aggression". Cats' behaviors when playing are similar to hunting behaviors. These activities allow kittens and younger cats to grow and acquire cognitive and motor skills, and to socialize with other cats. Cat play behavior can be either solitary (with toys or other objects) or social (with animals and people). They can play with a multitude of toys ranging from strings, to small furry toys resembling prey (e.g. mice), to plastic bags. Defining object play Object play for cats is the use of inanimate objects by the animal to express play behaviour. In the case of pet domestic cats, humans normally provide them with purchased, human-made toys such as toy mice, bird or feather toys, or toy insects. These may be suspended from a string attached to a wooden or fishing-style rod designed to simulate lifelike activity in the toy, triggering the cat's predatory insticts – this game is known as catfishing. Cat play can be enric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paper Recycling
The recycling of paper is the process by which waste paper is turned into new paper products. It has a number of important benefits: It saves waste paper from occupying homes of people and producing methane as it breaks down. Because paper fibre contains carbon (originally absorbed by the tree from which it was produced), recycling keeps the carbon locked up for longer and out of the atmosphere. Around two-thirds of all paper products in the US are now recovered and recycled, although it does not all become new paper. After repeated processing the fibres become too short for the production of new paper, which is why virgin fibre (from sustainably farmed trees) is frequently added to the pulp recipe. There are three categories of paper that can be used as feedstocks for making ''recycled paper'': mill broke, pre-consumer waste, and post-consumer waste. ''Mill broke'' is paper trimmings and other paper scrap from the manufacture of paper, and is recycled in a paper mill. ''Pre-c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cellulose
Cellulose is an organic compound with the formula , a polysaccharide consisting of a linear chain of several hundred to many thousands of β(1→4) linked D-glucose units. Cellulose is an important structural component of the primary cell wall of green plants, many forms of algae and the oomycetes. Some species of bacteria secrete it to form biofilms. Cellulose is the most abundant organic polymer on Earth. The cellulose content of cotton fiber is 90%, that of wood is 40–50%, and that of dried hemp is approximately 57%. Cellulose is mainly used to produce paperboard and paper. Smaller quantities are converted into a wide variety of derivative products such as cellophane and rayon. Conversion of cellulose from energy crops into biofuels such as cellulosic ethanol is under development as a renewable fuel source. Cellulose for industrial use is mainly obtained from wood pulp and cotton. Some animals, particularly ruminants and termites, can digest cellulose with the help of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Property
In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). Properties of a point on a function Perhaps the best-known example of the idea of locality lies in the concept of local minimum (or local maximum), which is a point in a function whose functional value is the smallest (resp., largest) within an immediate neighborhood of points. This is to be contrasted with the idea of global minimum (or global maximum), which corresponds to the minimum (resp., maximum) of the function across its entire domain. Properties of a single space A topological space is sometimes said to exhibit a property locally, if the property is exhibited "near" each point in one of the following ways: # Each point has a neighborhood exhibiting the property; # Each point has a neighborhood base of sets exhibiting the propert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
