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A single-displacement reaction, also known as single replacement reaction or exchange reaction, is an archaic concept in chemistry. It describes the stoichiometry of some chemical reactions in which one element or ligand is replaced by atom or group. [1] [2] [3] It can be represented generically as:
Forming limit curves (FLC) for four steel sheet grades are displayed in the attached figure. All forming limit curves have essentially the same shape. A minimum of the curve exists at the intercept with the major strain axis or close thereby, the plane strain forming limit.
Examining the density formula, we see that the mass of a beam depends directly on the density. Thus if a beam's cross-sectional dimensions are constrained and weight reduction is the primary goal, performance of the beam will depend on Young's modulus divided by density .
In this case, the equation governing the beam's deflection can be approximated as: = () where the second derivative of its deflected shape with respect to (being the horizontal position along the length of the beam) is interpreted as its curvature, is the Young's modulus, is the area moment of inertia of the cross-section, and is the internal ...
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.
For 3D displacement fields it is expressed as derivatives of displacement functions in terms of a second-order tensor (with 6 independent elements). Deflection is a term to describe the magnitude to which a structural element is displaced when subject to an applied load.
The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in the generalized Hooke's law: . Young's modulus E describes the material's strain response to uniaxial stress in the direction of this stress (like pulling on the ends of a wire or putting a weight on top of a column, with the wire getting longer and the column losing height),
The equations are written only for the small domain of individual elements of the structure rather than a single equation that describes the response of the system as a whole (a continuum). The latter would result in an intractable problem, hence the utility of the finite element method.