|
Linear Elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics. The fundamental assumptions of linear elasticity are infinitesimal strains — meaning, "small" deformations — and linear relationships between the components of stress and strain — hence the "linear" in its name. Linear elasticity is valid only for stress states that do not produce yielding. Its assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis. Mathematical formulation Equations governing a linear elastic boundary value problem are based on three tensor partial differential equations for the balance of linear momentum and six in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deformation (physics)
In physics and continuum mechanics, deformation is the change in the shape or size of an object. It has dimension of length with SI unit of metre (m). It is quantified as the residual displacement of particles in a non-rigid body, from an configuration to a configuration, excluding the body's average translation and rotation (its rigid transformation). A ''configuration'' is a set containing the positions of all particles of the body. A deformation can occur because of external loads, intrinsic activity (e.g. muscle contraction), body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc. In a continuous body, a ''deformation field'' results from a stress field due to applied forces or because of some changes in the conditions of the body. The relation between stress and strain (relative deformation) is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials. Deform ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Laws Of Motion
Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body remains at rest, or in motion at a constant speed in a straight line, unless it is acted upon by a force. # At any instant of time, the net force on a body is equal to the body's acceleration multiplied by its mass or, equivalently, the rate at which the body's momentum is changing with time. # If two bodies exert forces on each other, these forces have the same magnitude but opposite directions. The three laws of motion were first stated by Isaac Newton in his ''Philosophiæ Naturalis Principia Mathematica'' (''Mathematical Principles of Natural Philosophy''), originally published in 1687. Newton used them to investigate and explain the motion of many physical objects and systems. In the time since Newton, new insights, especially around t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3D Spherical
3D, 3-D, 3d, or Three D may refer to: Science, technology, and mathematics * A three-dimensional space in mathematics Relating to three-dimensionality * 3D computer graphics, computer graphics that use a three-dimensional representation of geometric data * 3D display, a type of information display that conveys depth to the viewer * 3D film, a motion picture that gives the illusion of three-dimensional perception * 3D modeling, developing a representation of any three-dimensional surface or object * 3D printing, making a three-dimensional solid object of a shape from a digital model * 3D television, television that conveys depth perception to the viewer * 3D projection * 3D rendering * 3D scanning, making a digital representation of three-dimensional objects * Video game graphics#3D, 3D video game * Stereoscopy, any technique capable of recording three-dimensional visual information or creating the illusion of depth in an image * Three-dimensional space Other uses in science and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deformation (mechanics)
In physics and continuum mechanics, deformation is the change in the shape (geometry), shape or size of an object. It has dimension (physics), dimension of length with SI unit of metre (m). It is quantified as the residual displacement (geometry), displacement of particles in a non-rigid body, from an configuration to a configuration, excluding the body's average translation (physics), translation and rotation (its rigid transformation). A ''configuration'' is a set containing the position (geometry), positions of all particles of the body. A deformation can occur because of structural load, external loads, intrinsic activity (e.g. muscle contraction), body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc. In a continuous body, a ''deformation field'' results from a Stress (physics), stress field due to applied forces or because of some changes in the conditions of the body. The relation between stre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A ''Solution (mathematics), solution'' to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the Tuple, ordered triple (x,y,z)=(1,-2,-2), since it makes all three equations valid. Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A Nonlinear system, system of non-linear equations can often be Approximation, approximated by a linear system (see linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stress (physics)
In continuum mechanics, stress is a physical quantity that describes Force, forces present during Deformation (physics), deformation. For example, an object being pulled apart, such as a stretched elastic band, is subject to Tension (physics), ''tensile'' stress and may undergo Elongation (materials science), elongation. An object being pushed together, such as a crumpled sponge, is subject to Compression (physics), ''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 (physics), dimension of force per area, with SI Units, SI units of newtons per square meter (N/m2) or Pascal (unit), pascal (Pa). Stress expresses the internal forces that neighbouring particles of a continuous material exert on each other, while Strain (mechanics), ''strain'' is the measure of the relative deformation (mechanics), deformation of the material. For example, when a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinate
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular oriented lines, called '' coordinate lines'', ''coordinate axes'' or just ''axes'' (plural of ''axis'') of the system. The point where the axes meet is called the '' origin'' and has as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame. Similarly, the position of any point in three-dimensional space can be specified by three ''Cartesian coordinates'', which are the signed distances from the point to three mutually perpendicular planes. More generally, Cartesian coordinates specify the point in an -dimensional Euclidean space for any dimension . These coordinates are the signed distances from the point to mutually perpendicular fixed h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Material Derivative
In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation. For example, in fluid dynamics, the velocity field is the flow velocity, and the quantity of interest might be the temperature of the fluid. In this case, the material derivative then describes the temperature change of a certain fluid parcel with time, as it flows along its pathline (trajectory). Other names There are many other names for the material derivative, including: *advective derivative *convective derivative *derivative following the motion *hydrodynamic derivative *Lagrangian derivative *particle derivative *substantial derivative *substantive derivative *Stokes derivative *total derivative, although the material derivati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transpose
In linear algebra, the transpose of a Matrix (mathematics), 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, often denoted by (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. Transpose of a matrix Definition The transpose of a matrix , denoted by , , , A^, , , or , may be constructed by any one of the following methods: #Reflection (mathematics), Reflect over its main diagonal (which runs from top-left to bottom-right) to obtain #Write the rows of as the columns of #Write the columns of as the rows of Formally, the -th row, -th column element of is the -th row, -th column element of : :\left[\mathbf^\operatorname\right]_ = \left[\mathbf\right]_. If is an matrix, then is an matrix. In the case of square matrices, may also denote the th power of the matrix . For avoiding a possibl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nabla Operator
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a ''field'' (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl (rotation) of a vector field. Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be formally defined as a vector operator whose components are the corresponding partial derivative operators. As a ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stiffness Tensor
Stiffness is the extent to which an object resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. Calculations The stiffness, k, of a body is a measure of the resistance offered by an elastic body to deformation. For an elastic body with a single degree of freedom (DOF) (for example, stretching or compression of a rod), the stiffness is defined as k = \frac where, * F is the force on the body * \delta is the displacement produced by the force along the same degree of freedom (for instance, the change in length of a stretched spring) Stiffness is usually defined under quasi-static conditions, but sometimes under dynamic loading. In the International System of Units, stiffness is typically measured in newtons per meter (N/m). In Imperial units, stiffness is typically measured in pounds (lbs) per inch. Generally speaking, deflections (or motions) of an infinitesimal ... [...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] |
