The Continuum Concept

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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 A particular particle within the body in a particular configuration is characterized by a position vector F = ∫ V a d m = ∫ S T d S + ∫ V ρ b d V {\displaystyle {\mathcal {F}}=\int _{V}\mathbf {a} \,dm=\int _{S}\mathbf {T} \,dS+\int _{V}\rho \mathbf {b} \,dV} M = ∫ S r × T d S + ∫ V r × ρ b d V {\displaystyle {\mathcal {M}}=\int _{S}\mathbf {r} \times \mathbf {T} \,dS+\int _{V}\mathbf {r} \times \rho \mathbf {b} \,dV} Kinematics: motion and deformation [ edit ] Figure 2. Motion of a continuum body. 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

Continuum concept - Wikipedia

The motion of a continuum body is a continuous time sequence of displacements. Thus, the material body will occupy different configurations at different times so that a particle occupies a series of points in space which describe a path line. In the Lagrangian description the position and physical properties of the particles are described in terms of the material or referential coordinates and time. In this case the reference configuration is the configuration at t = 0 {\displaystyle t=0} . An observer standing in the frame of reference observes the changes in the position and physical properties as the material body moves in space as time progresses. The results obtained are independent of the choice of initial time and reference configuration, κ 0 ( B ) {\displaystyle \kappa _{0}({\mathcal {B}})} . This description is normally used in solid mechanics. d d t [ P i j … ( X , t ) ] = ∂ ∂ t [ P i j … ( X , t ) ] {\displaystyle {\frac {d}{dt}}[P_{ij\ldots }(\mathbf {X} ,t)]={\frac {\partial }{\partial t}}[P_{ij\ldots }(\mathbf {X} ,t)]} This function needs to have various properties so that the model makes physical sense. κ t ( ⋅ ) {\displaystyle \kappa _{t}(\cdot )} needs to be: 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.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). In continuum mechanics a body is considered stress-free if the only forces present are those inter-atomic forces ( ionic, metallic, and van der Waals forces) required to hold the body together and to keep its shape in the absence of all external influences, including gravitational attraction. [9] [10] Stresses generated during manufacture of the body to a specific configuration are also excluded when considering stresses in a body. Therefore, the stresses considered in continuum mechanics are only those produced by deformation of the body, sc. only relative changes in stress are considered, not the absolute values of stress. The material points forming a closed curve at any instant will always form a closed curve at any subsequent time.

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F = F C + F B {\displaystyle {\mathcal {F}}=\mathbf {F} _{C}+\mathbf {F} _{B}} Surface forces [ edit ] 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.

It is convenient to identify a reference configuration or initial condition which all subsequent configurations are referenced from. The reference configuration need not be one that the body will ever occupy. Often, the configuration at t = 0 {\displaystyle t=0} is considered the reference configuration, κ 0 ( B ) {\displaystyle \kappa _{0}({\mathcal {B}})} . The components X i {\displaystyle X_{i}} of the position vector X {\displaystyle \mathbf {X} } of a particle, taken with respect to the reference configuration, are called the material or reference coordinates. When Good Enough Isn't, Mother Blame in The Continuum Concept, Journal of the Association for Research on Mothering, 6(2) by Chris Bobel (2004)