Young's modulusYoung's modulus , the Young modulus, or the modulus of elasticity in tension or axial compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied lengthwise. It quantifies the relationship between tensile/compressive stress (force per unit area) and axial strain (proportional deformation) in the linear elastic region of a material and is determined using the formula: Young's moduli are typically so large that they are expressed not in pascals but in gigapascals (GPa).
Poisson's ratioIn materials science and solid mechanics, Poisson's ratio (nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Poisson's ratio is the negative of the ratio of transverse strain to axial strain. For small values of these changes, is the amount of transversal elongation divided by the amount of axial compression. Most materials have Poisson's ratio values ranging between 0.0 and 0.5.
Shear modulusIn materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain: where = shear stress is the force which acts is the area on which the force acts = shear strain. In engineering , elsewhere is the transverse displacement is the initial length of the area. The derived SI unit of shear modulus is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousand pounds per square inch (ksi).
Elastic modulusAn elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is defined as the slope of its stress–strain curve in the elastic deformation region: A stiffer material will have a higher elastic modulus. An elastic modulus has the form: where stress is the force causing the deformation divided by the area to which the force is applied and strain is the ratio of the change in some parameter caused by the deformation to the original value of the parameter.
BendingIn applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically 1/10 or less, of the other two. When the length is considerably longer than the width and the thickness, the element is called a beam.
Finite strain theoryIn 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.
Stress (mechanics)In continuum mechanics, stress is a physical quantity that describes forces present during deformation. 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 units of force per area, such as newtons per square meter (N/m2) or pascal (Pa).
I-beamI-beam is a generic lay term for a variety of structural members with an or -shaped cross-section. Technical terms for similar items include H-beam (for universal column, UC), w-beam (for "wide flange"), universal beam (UB), rolled steel joist (RSJ), or double-T (especially in Polish, Bulgarian, Spanish, Italian and German). I-beams are typically made of structural steel and serve a wide variety of construction uses. The horizontal elements of the are called flanges, and the vertical element is known as the "web".
Beam (structure)A beam is a structural element that primarily resists loads applied laterally to the beam's axis (an element designed to carry primarily axial load would be a strut or column). Its mode of deflection is primarily by bending. The loads applied to the beam result in reaction forces at the beam's support points. The total effect of all the forces acting on the beam is to produce shear forces and bending moments within the beams, that in turn induce internal stresses, strains and deflections of the beam.
Infinitesimal strain theoryIn continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as density and stiffness) at each point of space can be assumed to be unchanged by the deformation.
Bending momentIn solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending moments is the beam. The diagram shows a beam which is simply supported (free to rotate and therefore lacking bending moments) at both ends; the ends can only react to the shear loads. Other beams can have both ends fixed (known as encastre beam); therefore each end support has both bending moments and shear reaction loads.
Elasticity tensorThe elasticity tensor is a fourth-rank tensor describing the stress-strain relation in a linear elastic material. Other names are elastic modulus tensor and stiffness tensor. Common symbols include and . The defining equation can be written as where and are the components of the Cauchy stress tensor and infinitesimal strain tensor, and are the components of the elasticity tensor. Summation over repeated indices is implied. This relationship can be interpreted as a generalization of Hooke's law to a 3D continuum.
Shear stressShear stress (often denoted by τ (Greek: tau)) is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts. The formula to calculate average shear stress is force per unit area.: where: τ = the shear stress; F = the force applied; A = the cross-sectional area of material with area parallel to the applied force vector.
Linear elasticityLinear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics. The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations (or strains) and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding.
T-beamA T-beam (or tee beam), used in construction, is a load-bearing structure of reinforced concrete, wood or metal, with a -shaped cross section. The top of the -shaped cross section serves as a flange or compression member in resisting compressive stresses. The web (vertical section) of the beam below the compression flange serves to resist shear stress. When used for highway bridges the beam incorporates reinforcing bars in the bottom of the beam to resist the tensile stresses which occur during bending.
Stress–strain analysisStress–strain analysis (or stress analysis) is an engineering discipline that uses many methods to determine the stresses and strains in materials and structures subjected to forces. In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material. In simple terms we can define stress as the force of resistance per unit area, offered by a body against deformation.
Strain-rate tensorIn continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. It can be defined as the derivative of the strain tensor with respect to time, or as the symmetric component of the Jacobian matrix (derivative with respect to position) of the flow velocity. In fluid mechanics it also can be described as the velocity gradient, a measure of how the velocity of a fluid changes between different points within the fluid.
Price elasticity of demandA good's price elasticity of demand (, PED) is a measure of how sensitive the quantity demanded is to its price. When the price rises, quantity demanded falls for almost any good, but it falls more for some than for others. The price elasticity gives the percentage change in quantity demanded when there is a one percent increase in price, holding everything else constant. If the elasticity is −2, that means a one percent price rise leads to a two percent decline in quantity demanded.
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.
TensorIn mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product.
