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62 relations: Biharmonic equation, Boundary value problem, Bulk modulus, Cartesian coordinate system, Castigliano's method, Cauchy momentum equation, Cauchy stress tensor, Clapeyron's theorem (elasticity), Constitutive equation, Contact mechanics, Continuum mechanics, Coulomb's law, Deformation (mechanics), Del, Displacement (vector), Displacement field (mechanics), Divergence, Earth, Earthquake, Eigenvalues and eigenvectors, Elastic modulus, Elasticity (physics), Electrostatics, Finite element method, Finite strain theory, Force, GRADELA, Green's function, Homogeneous and heterogeneous mixtures, Hooke's law, Infinitesimal strain theory, Kronecker delta, Lamé parameters, Laplace operator, Linear equation, Mechanical wave, Michell solution, Momentum, Partial differential equation, Plane wave, Plasticity (physics), Poisson's ratio, Saint-Venant's compatibility condition, Seismic wave, Shear modulus, Shear stress, Signorini problem, Solid mechanics, Spring system, Stress (mechanics), ..., Stress functions, Structural analysis, Symmetry of second derivatives, Tensor, Transpose, Transverse isotropy, Viscoelasticity, Voigt notation, Wave equation, William Thomson, 1st Baron Kelvin, Yield (engineering), Young's modulus. Expand index (12 more) »
Biharmonic equation
In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows.
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Boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions.
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Bulk modulus
The bulk modulus (K or B) of a substance is a measure of how resistant to compressibility that substance is.
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Cartesian coordinate system
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.
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Castigliano's method
Castigliano's method, named for Carlo Alberto Castigliano, is a method for determining the displacements of a linear-elastic system based on the partial derivatives of the energy.
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Cauchy momentum equation
The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum.
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Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy.
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Clapeyron's theorem (elasticity)
In the linear theory of elasticity Clapeyron's theorem states that the potential energy of deformation of a body, which is in equilibrium under a given load, is equal to half the work done by the external forces computed assuming these forces had remained constant from the initial state to the final state.
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Constitutive equation
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.
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Contact mechanics
Contact mechanics is the study of the deformation of solids that touch each other at one or more points.
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Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles.
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Coulomb's law
Coulomb's law, or Coulomb's inverse-square law, is a law of physics for quantifying the amount of force with which stationary electrically charged particles repel or attract each other.
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Deformation (mechanics)
Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration.
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Del
Del, or nabla, is an operator used in mathematics, in particular in vector calculus, as a vector differential operator, usually represented by the nabla symbol ∇.
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Displacement (vector)
A displacement is a vector whose length is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a straight line from the initial position to the final position of the point.
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Displacement field (mechanics)
A displacement field is an assignment of displacement vectors for all points in a region or body that is displaced from one state to another.
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Divergence
In vector calculus, divergence is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point.
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Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life.
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Earthquake
An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth, resulting from the sudden release of energy in the Earth's lithosphere that creates seismic waves.
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Eigenvalues and eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
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Elastic modulus
An elastic modulus (also known as modulus of elasticity) is a quantity that measures an object or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it.
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Elasticity (physics)
In physics, elasticity (from Greek ἐλαστός "ductible") 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.
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Electrostatics
Electrostatics is a branch of physics that studies electric charges at rest.
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Finite element method
The finite element method (FEM), is a numerical method for solving problems of engineering and mathematical physics.
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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.
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Force
In physics, a force is any interaction that, when unopposed, will change the motion of an object.
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GRADELA
GRADELA is a simple gradient elasticity model involving one internal length in addition to the two Lamé parameters.
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Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential equation defined on a domain, with specified initial conditions or boundary conditions.
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Homogeneous and heterogeneous mixtures
A homogeneous mixture is a solid, liquid, or gaseous mixture that has the same proportions of its components throughout any given sample.
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Hooke's law
Hooke's law is a principle of physics that states that the force needed to extend or compress a spring by some distance scales linearly with respect to that distance.
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Infinitesimal strain theory
In 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.
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Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers.
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Lamé parameters
In continuum mechanics, the Lamé parameters (also called the Lamé coefficients, Lamé constants or Lamé moduli) are two material-dependent quantities denoted by λ and μ that arise in strain-stress relationships.
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Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.
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Linear equation
In mathematics, a linear equation is an equation that may be put in the form where x_1, \ldots, x_n are the variables or unknowns, and c, a_1, \ldots, a_n are coefficients, which are often real numbers, but may be parameters, or even any expression that does not contain the unknowns.
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Mechanical wave
A mechanical wave is a wave that is an oscillation of matter, and therefore transfers energy through a medium.
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Michell solution
The Michell solution is a general solution to the elasticity equations in polar coordinates (r, \theta \).
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Momentum
In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (pl. momenta) is the product of the mass and velocity of an object.
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Partial differential equation
In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.
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Plane wave
In the physics of wave propagation, a plane wave (also spelled planewave) is a wave whose wavefronts (surfaces of constant phase) are infinite parallel planes.
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Plasticity (physics)
In physics and materials science, plasticity describes the deformation of a (solid) material undergoing non-reversible changes of shape in response to applied forces.
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Poisson's ratio
Poisson's ratio, denoted by the Greek letter 'nu', \nu, and named after Siméon Poisson, is the negative of the ratio of (signed) transverse strain to (signed) axial strain.
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Saint-Venant's compatibility condition
In the mathematical theory of elasticity the strain \varepsilon is related to a displacement field \ u by where 1\le i,j \le 3.
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Seismic wave
Seismic waves are waves of energy that travel through the Earth's layers, and are a result of earthquakes, volcanic eruptions, magma movement, large landslides and large man-made explosions that give out low-frequency acoustic energy.
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Shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of shear stress to the shear strain: where The derived SI unit of shear modulus is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousands of pounds per square inch (ksi).
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Shear stress
A shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section.
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Signorini problem
The Signorini problem is an elastostatics problem in linear elasticity: it consists in finding the elastic equilibrium configuration of an anisotropic non-homogeneous elastic body, resting on a rigid frictionless surface and subject only to its mass forces.
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Solid mechanics
Solid mechanics is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and other external or internal agents.
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Spring system
In engineering and physics, a spring system or spring network is a model of physics described as a graph with a position at each vertex and a spring of given stiffness and length along each edge.
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Stress (mechanics)
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.
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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: where \sigma is the stress tensor, and the Beltrami-Michell compatibility equations: A general solution of these equations may be expressed in terms the Beltrami stress tensor.
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Structural analysis
Structural analysis is the determination of the effects of loads on physical structures and their components.
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Symmetry of second derivatives
In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility under certain conditions (see below) of interchanging the order of taking partial derivatives of a function of n variables.
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Tensor
In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.
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Transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At).
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Transverse isotropy
A transversely isotropic material is one with physical properties which are symmetric about an axis that is normal to a plane of isotropy.
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Viscoelasticity
Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation.
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Voigt notation
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order.
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Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves.
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William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin, (26 June 1824 – 17 December 1907) was a Scots-Irish mathematical physicist and engineer who was born in Belfast in 1824.
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Yield (engineering)
The yield point is the point on a stress–strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior.
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Young's modulus
Young's modulus, also known as the elastic modulus, is a measure of the stiffness of a solid material.
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3-D Elasticity, 3-D elasticity, 3D Elasticity, Beltrami-Michell compatibility equations, Beltrami–Michell compatibility equations, Christoffel equation, Elastic Wave, Elastic wave, Elastic waves, Elastodynamic equation, Elastodynamics, Elastostatic equation, Linear elastic material, Linear material, Navier-Cauchy equations, Stress wave.
References
[1] https://en.wikipedia.org/wiki/Linear_elasticity
