The Continuum Concept

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The material points forming a closed curve at any instant will always form a closed curve at any subsequent time. In the Lagrangian description, the material derivative of P i j … {\displaystyle P_{ij\ldots }} is simply the partial derivative with respect to time, and the position vector X {\displaystyle \mathbf {X} } is held constant as it does not change with time. Thus, we have The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.

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which is a mapping of the initial configuration κ 0 ( B ) {\displaystyle \kappa _{0}({\mathcal {B}})} onto the current configuration κ t ( B ) {\displaystyle \kappa _{t}({\mathcal {B}})} , giving a geometrical correspondence between them, i.e. giving the position vector x = x i e i {\displaystyle \mathbf {x} =x_{i}\mathbf {e} _{i}} that a particle X {\displaystyle X} , with a position vector X {\displaystyle \mathbf {X} } in the undeformed or reference configuration κ 0 ( B ) {\displaystyle \kappa _{0}({\mathcal {B}})} , will occupy in the current or deformed configuration κ t ( B ) {\displaystyle \kappa _{t}({\mathcal {B}})} at time t {\displaystyle t} . The components x i {\displaystyle x_{i}} are called the spatial coordinates. The material derivative of any property P i j … {\displaystyle P_{ij\ldots }} of a continuum, which may be a scalar, vector, or tensor, is the time rate of change of that property for a specific group of particles of the moving continuum body. The material derivative is also known as the substantial derivative, or comoving derivative, or convective derivative. It can be thought as the rate at which the property changes when measured by an observer traveling with that group of particles.

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Thus, the sum of all applied forces and torques (with respect to the origin of the coordinate system) in the body can be given by v = x ˙ = d x d t = ∂ χ ( X , t ) ∂ t {\displaystyle \mathbf {v} ={\dot {\mathbf {x} }}={\frac {d\mathbf {x} }{dt}}={\frac {\partial \chi (\mathbf {X} ,t)}{\partial t}}}

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A continuum model assumes that the substance of the object completely fills the space it occupies. This ignores the fact that matter is made of atoms, however 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. The continuum concept featured on television in the UK in the 2007 Channel 4 series Bringing Up Baby. It was chosen [ by whom?] as one of three influential parenting "methods" of the 20th century which a number of new parents tested out.

the River Continuum Concept with geostatistical Testing the River Continuum Concept with geostatistical

For the mathematical formulation of the model, κ t ( ⋅ ) {\displaystyle \kappa _{t}(\cdot )} is also assumed to be twice continuously differentiable, so that differential equations describing the motion may be formulated. An additional area of continuum mechanics comprises elastomeric foams, which exhibit a curious hyperbolic stress-strain relationship. The elastomer is a true continuum, but a homogeneous distribution of voids gives it unusual properties. [2] Formulation of models [ edit ] Figure 1. Configuration of a continuum body When analyzing the motion or deformation of solids, or the flow of fluids, it is necessary to describe the sequence or evolution of configurations throughout time. One description for motion is made in terms of the material or referential coordinates, called material description or Lagrangian description.

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Sensing (and fulfilling) elders' expectations that the infants are innately social and cooperative and have strong self-preservation instincts, and that they are welcome and worthy (yet without making them the constant center of attention) Jean Liedloff (November 26, 1926 – March 15, 2011) [1] was an American author, born in New York City, and best known for her 1975 book The Continuum Concept. She is the aunt of writer Janet Hobhouse, [2] and is represented by the character Constance in Hobhouse's book "The Furies." F = F C + F B {\displaystyle {\mathcal {F}}=\mathbf {F} _{C}+\mathbf {F} _{B}} Surface forces [ edit ] A change in the configuration of a continuum body results in a 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 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 κ 0 ( B ) {\displaystyle \kappa _{0}({\mathcal {B}})} to a current or deformed configuration κ t ( B ) {\displaystyle \kappa _{t}({\mathcal {B}})} (Figure 2).