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Lamé Parameters
In continuum mechanics, 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. In general, ''λ'' and ''μ'' are individually referred to as ''Lamé's first parameter'' and ''Lamé's second parameter'', respectively. Other names are sometimes employed for one or both parameters, depending on context. For example, the parameter ''μ'' is referred to in fluid dynamics as the dynamic viscosity of a fluid (not expressed in the same units); whereas in the context of elasticity, ''μ'' is called the shear modulus, and is sometimes denoted by ''G'' instead of ''μ''. Typically the notation ''G'' is seen paired with the use of Young's modulus ''E'', and the notation ''μ'' is paired with the use of ''λ''. In homogeneous and isotropic materials, these define Hooke's law in 3D, \boldsymbol = 2\mu \boldsymbol + \lambda \; \operatorname(\boldsymbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mechanics deals with ''deformable bodies'', as opposed to rigid bodies. A continuum model assumes that the substance of the object completely fills the space it occupies. While ignoring the fact that matter is made of atoms, this provides a sufficiently accurate description of matter on length scales much greater than that of inter-atomic distances. The concept of a continuous medium allows for intuitive analysis of bulk matter by using differential equations that describe the behavior of such matter according to physical laws, such as mass conservation, momentum conservation, and energy conservation. Information about the specific material is expressed in constitutive relationships. Continuum mechanics treats the physical properties of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strain Tensor
In mechanics, strain is defined as relative deformation, compared to a position configuration. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the metric tensor or its dual is considered. Strain has dimension of a length ratio, with SI base units of meter per meter (m/m). Hence strains are dimensionless and are usually expressed as a decimal fraction or a percentage. Parts-per notation is also used, e.g., parts per million or parts per billion (sometimes called "microstrains" and "nanostrains", respectively), corresponding to μm/m and nm/m. Strain can be formulated as the spatial derivative of displacement: \boldsymbol \doteq \cfrac\left(\mathbf - \mathbf\right) = \boldsymbol'- \boldsymbol, where is the identity tensor. The displacement of a body may be expressed in the form , where is the reference position of material po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elasticity Tensor
The 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 \mathbf and \mathbf. The defining equation can be written as : T^ = C^ E_ where T^ and E_ are the components of the Cauchy stress tensor and infinitesimal strain tensor, and C^ are the components of the elasticity tensor. Summation over repeated indices is implied.Here, upper and lower indices denote contravariant and covariant components, respectively, though the distinction can be ignored for Cartesian coordinates. As a result, some references represent components using only lower indices. This relationship can be interpreted as a generalization of Hooke's law to a 3D continuum. A general fourth-rank tensor \mathbf in 3D has 34 = 81 independent components F_, but the elasticity tensor has at most 21 independent components. This fact follows from the symmetry of the stress and st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gabriel Lamé
Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity and finite strain theory elaborate the mathematical abstractions). Biography Lamé was born in Tours, in today's ''département'' of Indre-et-Loire. He became well known for his general theory of curvilinear coordinates and his notation and study of classes of ellipse-like curves, now known as Lamé curves or superellipses, and defined by the equation: : \left, \,\,\^n + \left, \,\,\^n = 1 where ''n'' is any positive real number. He is also known for his running time analysis of the Euclidean algorithm, marking the beginning of computational complexity theory. In 1844, using Fibonacci numbers, he proved that when finding the greatest common divisor of integers ''a'' and ''b'', the algorithm runs in no more than 5''k'' steps, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulk Modulus
The bulk modulus (K or B or k) of a substance is a measure of the resistance of a substance to bulk compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume. Other moduli describe the material's response ( strain) to other kinds of stress: the shear modulus describes the response to shear stress, and Young's modulus describes the response to normal (lengthwise stretching) stress. For a fluid, only the bulk modulus is meaningful. For a complex anisotropic solid such as wood or paper, these three moduli do not contain enough information to describe its behaviour, and one must use the full generalized Hooke's law. The reciprocal of the bulk modulus at fixed temperature is called the isothermal compressibility. Definition The bulk modulus K (which is usually positive) can be formally defined by the equation :K=-V\frac , where P is pressure, V is the initial volume of the substance, and dP/dV deno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elastic Modulus
An elastic modulus (also known as modulus of elasticity (MOE)) is a quantity that describes an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. Definition 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: :\delta \ \stackrel\ \frac 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. Since strain is a dimensionless quantity, the units of \delta will be the same as the units of stress. Elastic constants and moduli Elastic constants are specific parameters that quantify the stiffness of a material in response to applied stresses and are fundamental in defining the elastic pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\text i=j. \end or with use of Iverson brackets: \delta_ = =j, For example, \delta_ = 0 because 1 \ne 2, whereas \delta_ = 1 because 3 = 3. The Kronecker delta appears naturally in many areas of mathematics, physics, engineering and computer science, as a means of compactly expressing its definition above. Generalized versions of the Kronecker delta have found applications in differential geometry and modern tensor calculus, particularly in formulations of gauge theory and topological field models. In linear algebra, the n\times n identity matrix \mathbf has entries equal to the Kronecker delta: I_ = \delta_ where i and j take the values 1,2,\cdots,n, and the inner product of vectors can be written as \mathbf\cdot\mathbf = \sum_^n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (linear Algebra)
In linear algebra, the trace of a square matrix , denoted , is the sum of the elements on its main diagonal, a_ + a_ + \dots + a_. It is only defined for a square matrix (). The trace of a matrix is the sum of its eigenvalues (counted with multiplicities). Also, for any matrices and of the same size. Thus, similar matrices have the same trace. As a consequence, one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the row and column of . The entries of can be real numbers, complex numbers, or more generally elements of a field . The trace is not defined for non-square matrices. Example Let be a matrix, with \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context. I_1 = \begin 1 \end ,\ I_2 = \begin 1 & 0 \\ 0 & 1 \end ,\ I_3 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ,\ \dots ,\ I_n = \begin 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end. The term unit matrix has also been widely used, but the term ''identity matrix'' is now standard. The term ''unit matrix'' is ambiguous, because it is also used for a matrix of on ... [...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] |
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] |
