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Föppl–von Kármán Equations
The Föppl–von Kármán equations, named after August Föppl and Theodore von Kármán, are a set of nonlinear partial differential equations describing the large deflections of thin flat plates. With applications ranging from the design of submarine hulls to the mechanical properties of cell wall, the equations are notoriously difficult to solve, and take the following form: "Theory of Elasticity". L. D. Landau, E. M. Lifshitz, (3rd ed. ) : \begin (1) \qquad & \frac\nabla^4 w-h\frac\left(\sigma_\frac\right)=P \\ (2) \qquad & \frac=0 \end where is the Young's modulus of the plate material (assumed homogeneous and isotropic), is the Poisson's ratio, is the thickness of the plate, is the out–of–plane deflection of the plate, is the external normal force per unit area of the plate, is the Cauchy stress tensor, and are indices that take values of 1 and 2 (the two orthogonal in-plane directions). The 2-dimensional biharmonic operator is defined as : \n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
August Föppl
August Otto Föppl (25 January 1854 – 12 August 1924) was a professor of Technical Mechanics and Graphical Statics at the Technical University of Munich, Germany. He is credited with introducing the Föppl–Klammer theory and the Föppl–von Kármán equations (large deflection of elastic plates). Life His doctoral advisor was Gustav Heinrich Wiedemann and one of Föppl's first doctoral students was Ludwig Prandtl, his future son-in-law. He had two sons Ludwig Föppl and Otto Föppl. Ludwig Föppl who was a mechanical engineer and Professor of Technical Mechanics at the Technical University of Munich. Otto Föppl who was an engineer and Professor of Applied Mechanics at the Technical University of Braunschweig for 30 years. Career In 1894, Föppl wrote a widely read introductory book on Maxwell's theory of electricity, titled ''Einführung in die Maxwellsche Theorie der Elektrizität''. (This was the first German-language textbook on Maxwell's theory of electrodynamics an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stress Functions
In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces (or body forces that could be expressed as potentials) on the boundary are (using index notation) the equilibrium equation: :\sigma_=0\, where \sigma is the stress tensor, and the Beltrami-Michell compatibility equations: :\sigma_+\frac\sigma_=0 A general solution of these equations may be expressed in terms of the Beltrami stress tensor. Stress functions are derived as special cases of this Beltrami stress tensor which, although less general, sometimes will yield a more tractable method of solution for the elastic equations. Beltrami stress functions It can be shown that a complete solution to the equilibrium equations may be written as :\sigma=\nabla \times \Phi \times \nabla Using index notation: :\sigma_=\varepsilon_\varepsilon_\Phi_ : where \Phi_ is an arbitrary second-rank tensor field that is at least twice differentiable, and is known as the ''Beltram ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plate Theory
In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions.Timoshenko, S. and Woinowsky-Krieger, S. "Theory of plates and shells". McGraw–Hill New York, 1959. The typical thickness to width ratio of a plate structure is less than 0.1. A plate theory takes advantage of this disparity in length scale to reduce the full three-dimensional solid mechanics problem to a two-dimensional problem. The aim of plate theory is to calculate the deformation and stresses in a plate subjected to loads. Of the numerous plate theories that have been developed since the late 19th century, two are widely accepted and used in engineering. These are * the Kirchhoff–Love theory of plates (classical plate theory) * The Uflyand-Mindlin theory of plates (first-order shear plate theory) Kirchhoff–Love theory for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variational Calculus
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stress Resultants
Stress resultants are simplified representations of the stress state in structural elements such as beams, plates, or shells. The geometry of typical structural elements allows the internal stress state to be simplified because of the existence of a "thickness'" direction in which the size of the element is much smaller than in other directions. As a consequence the three traction components that vary from point to point in a cross-section can be replaced with a set of resultant forces and resultant moments. These are the stress resultants (also called '' membrane forces'', '' shear forces'', and ''bending moment'') that may be used to determine the detailed stress state in the structural element. A three-dimensional problem can then be reduced to a one-dimensional problem (for beams) or a two-dimensional problem (for plates and shells). Stress resultants are defined as integrals of stress over the thickness of a structural element. The integrals are weighted by integer po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Stress
In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysis is considerably simplified, as the stress state can be represented by a tensor of dimension 2 (representable as a 2×2 matrix rather than 3×3). A related notion, plane strain, is often applicable to very thick members. Plane stress typically occurs in thin flat plates that are acted upon only by load forces that are parallel to them. In certain situations, a gently curved thin plate may also be assumed to have plane stress for the purpose of stress analysis. This is the case, for example, of a thin-walled cylinder filled with a fluid under pressure. In such cases, stress components perpendicular to the plate are negligible compared to those parallel to it. In other situations, however, the bending stress of a thin plate cannot be ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hooke's Law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that distance—that is, where is a constant factor characteristic of the spring (i.e., its stiffness), and is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law since 1660. Hooke's equation holds (to some extent) in many other situations where an elasticity (physics), elastic body is Deformation (physics), deformed, such as wind blowing on a tall building, and a musician plucking a string (music), string of a guitar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Strain Theory
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 conf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Normal
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one (a unit vector) or its length may represent the curvature of the object (a ''curvature vector''); its algebraic sign may indicate sides (interior or exterior). In three dimensions, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the tangent plane of the surface at P. The word "normal" is also used as an adjective: a line ''normal'' to a plane, the ''normal'' component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality (right angles). The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flexural Rigidity
Flexural rigidity is defined as the force couple required to bend a fixed non- rigid structure by one unit of curvature, or as the resistance offered by a structure while undergoing bending. Flexural rigidity of a beam Although the moment M(x) and displacement y may vary along the length of the beam or rod, the flexural rigidity (defined as EI) is a property of the beam itself and is generally constant. The flexural rigidity, moment, and transverse displacement are related by the following equation along the length of the rod, x: :\ EI \ = \int_^ M(x) dx + C_1 where E is the flexural modulus (in Pa), I is the second moment of area (in m4), y is the transverse displacement of the beam at x, and M(x) is the bending moment at ''x''. The flexural rigidity (stiffness) of the beam is therefore related to both E, a material property, and I, the physical geometry of the beam. If the material exhibits Isotropic behavior then the Flexural Modulus is equal to the Modulus of Elasticity (Young ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Bending
Pure bending ( Theory of simple bending) is a condition of stress where a bending moment is applied to a beam without the simultaneous presence of axial, shear, or torsional forces. Pure bending occurs only under a constant bending moment (M) since the shear force (V), which is equal to \frac = V, has to be equal to zero. In reality, a state of pure bending does not practically exist, because such a state needs an absolutely weightless member. The state of pure bending is an approximation made to derive formulas. Kinematics of pure bending #In pure bending the axial lines bend to form circumferential lines and transverse lines remain straight and become radial lines. #Axial lines that do not extend or contract form a neutral surface. Assumptions made in the theory of Pure Bending #The material of the beam is homogeneous1 and isotropic2. #The value of Young's Modulus of Elasticity An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constitutive Relation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or loads to strains or deformations. Some constitutive equations are simply phenomenological; others are derived from first principles. A common approximate constitutive equation frequently is expressed as a simple proportionality using a parameter taken to be a property of the material, such as electrical conductivity or a spring constant. However, it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
