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HRR Singularity
The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material. The theoretical concept of J-integral was developed in 1967 by G. P. Cherepanov and independently in 1968 by James R. Rice,J. R. Rice, ''A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks'', Journal of Applied Mechanics, 35, 1968, pp. 379–386. who showed that an energetic contour path integral (called ''J'') was independent of the path around a crack. Experimental methods were developed using the integral that allowed the measurement of critical fracture properties in sample sizes that are too small for Linear Elastic Fracture Mechanics (LEFM) to be valid.Lee, R. F., & Donovan, J. A. (1987). J-integral and crack opening displacement as crack initiation criteria in natural rubber in pure shear and tensile specimens. Rubber chemistry and technology, 60(4), 674–688/ref> These experiments ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Mechanics
Solid mechanics (also known as mechanics of solids) is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation (mechanics), deformation under the action of forces, temperature changes, phase (chemistry), phase changes, and other external or internal agents. Solid mechanics is fundamental for civil engineering, civil, Aerospace engineering, aerospace, nuclear engineering, nuclear, Biomedical engineering, biomedical and mechanical engineering, for geology, and for many branches of physics and chemistry such as materials science. It has specific applications in many other areas, such as understanding the anatomy of living beings, and the design of dental prosthesis, dental prostheses and surgical implants. One of the most common practical applications of solid mechanics is the Euler–Bernoulli beam theory, Euler–Bernoulli beam equation. Solid mechanics extensively uses tensors to describe stresses, strains, and the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Displacement Vector
In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a straight line from the initial position to the final position of the point trajectory. A displacement may be identified with the translation that maps the initial position to the final position. Displacement is the shift in location when an object in motion changes from one position to another. For motion over a given interval of time, the displacement divided by the length of the time interval defines the average velocity (a vector), whose magnitude is the average speed (a scalar quantity). Formulation A displacement may be formulated as a '' relative position'' (resulting from the motion), that is, as the final position of a point relative to its initial position . The corresponding displacement vector can be defined as the differen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young's Modulus
Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Young's modulus is defined as the ratio of the stress (force per unit area) applied to the object and the resulting axial strain (displacement or deformation) in the linear elastic region of the material. Although Young's modulus is named after the 19th-century British scientist Thomas Young, the concept was developed in 1727 by Leonhard Euler. The first experiments that used the concept of Young's modulus in its modern form were performed by the Italian scientist Giordano Riccati in 1782, pre-dating Young's work by 25 years. The term modulus is derived from the Latin root term '' modus'', which means ''measure''. Definition Young's modulus, E, quantifies the relationship between tensile or compressive stress \sigma (force per unit ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
J IntegralPathIndep
J, or j, is the tenth letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its usual name in English is ''jay'' (pronounced ), with a now-uncommon variant ''jy'' ."J", ''Oxford English Dictionary,'' 2nd edition (1989) When used in the International Phonetic Alphabet for the voiced palatal approximant (the sound of "y" in "yes") it may be called ''yod'' or ''jod'' (pronounced or ). History The letter ''J'' used to be used as the swash letter ''I'', used for the letter I at the end of Roman numerals when following another I, as in XXIIJ or xxiij instead of XXIII or xxiii for the Roman numeral twenty-three. A distinctive usage emerged in Middle High German. Gian Giorgio Trissino (1478–1550) was the first to explicitly distinguish I and J as representing separate sounds, in his ''Ɛpistola del Trissino de le lettere nuωvamente aggiunte ne la lingua italiana'' ("Trissino's epistle ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain Rule
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , then the chain rule is, in Lagrange's notation, h'(x) = f'(g(x)) g'(x). or, equivalently, h'=(f\circ g)'=(f'\circ g)\cdot g'. The chain rule may also be expressed in Leibniz's notation. If a variable depends on the variable , which itself depends on the variable (that is, and are dependent variables), then depends on as well, via the intermediate variable . In this case, the chain rule is expressed as \frac = \frac \cdot \frac, and \left.\frac\_ = \left.\frac\_ \cdot \left. \frac\_ , for indicating at which points the derivatives have to be evaluated. In integral, integration, the counterpart to the chain rule is the substitution rule. Intuitive explanation Intuitively, the chain rule states that knowing t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divergence Theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem relating the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the volume enclosed. More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region enclosed by the surface. Intuitively, it states that "the sum of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region". The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plasticity (physics)
In physics and materials science, plasticity (also known as plastic deformation) is the ability of a solid material to undergo permanent Deformation (engineering), deformation, a non-reversible change of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the material itself. In engineering, the transition from Elasticity (physics), elastic behavior to plastic behavior is known as Yield (engineering), yielding. Plastic deformation is observed in most materials, particularly metals, soils, Rock (geology), rocks, concrete, and foams. However, the physical mechanisms that cause plastic deformation can vary widely. At a crystalline scale, plasticity in metals is usually a consequence of dislocations. Such defects are relatively rare in most crystalline materials, but are numerous in some and part of their crystal structure; in such cases, plastic crystallinity can resul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elasticity (physics)
In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to ''plasticity'', in which the object fails to do so and instead remains in its deformed state. The physical reasons for elastic behavior can be quite different for different materials. In metals, the Crystal structure, atomic lattice changes size and shape when forces are applied (energy is added to the system). When forces are removed, the lattice goes back to the original lower energy state. For rubber elasticity, rubbers and other polymers, elasticity is caused by the stretching of polymer chains when forces are applied. Hooke's law states that the force required to deform elastic objects should be Prop ... [...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] |
Path Independence
Path dependence is a concept in the social sciences, referring to processes where past events or decisions constrain later events or decisions. It can be used to refer to outcomes at a single point in time or to long-run equilibria of a process. Path dependence has been used to describe institutions, technical standards, patterns of economic or social development, organizational behavior, and more. In common usage, the phrase can imply two types of claims. The first is the broad concept that "history matters", often articulated to challenge explanations that pay insufficient attention to historical factors. This claim can be formulated simply as "the future development of an economic system is affected by the path it has traced out in the past" or "particular events in the past can have crucial effects in the future." The second is a more specific claim about how past events or decisions affect future events or decisions in significant or disproportionate ways, through mechanisms s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Traction
Traction, traction force or tractive force is a force used to generate motion between a body and a tangential surface, through the use of either dry friction or shear force. It has important applications in vehicles, as in ''tractive effort''. ''Traction'' can also refer to the ''maximum'' tractive force between a body and a surface, as limited by available friction; when this is the case, traction is often expressed as the ratio of the maximum tractive force to the normal force and is termed the ''coefficient of traction'' (similar to coefficient of friction). It is the force which makes an object move over the surface by overcoming all the resisting forces like friction, normal loads (load acting on the tiers in negative ''Z'' axis), air resistance, rolling resistance, etc. Definitions Traction can be defined as: In vehicle dynamics, tractive force is closely related to the terms tractive effort and drawbar pull, though all three terms have different definitions. Coeffici ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no holes that go completely through it, because two paths going around different sides of such a hole cannot be continuously transformed into each other. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any Loop (topology), loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
