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. 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 A solid is a deformable body that possesses shear strength, sc. a solid can support shear forces (forces parallel to the material surface on which they act). Fluids, on the other hand, do not sustain shear forces. Born in New York City in 1926, as a teenager she attended the Drew Seminary for Young Women and began studying at Cornell University, but began her expeditions before she could graduate. [1] Following the classical dynamics of Newton and Euler, the motion of a material body is produced by the action of externally applied forces which are assumed to be of two kinds: surface forces F C {\displaystyle \mathbf {F} _{C}} and body forces F B {\displaystyle \mathbf {F} _{B}} . [3] Thus, the total force F {\displaystyle {\mathcal {F}}} applied to a body or to a portion of the body can be expressed as:

The Continuum Concept in Practice - Page 2 - Philosophy Now Forum The Continuum Concept in Practice - Page 2 - Philosophy Now Forum

Physical and kinematic properties P i j … {\displaystyle P_{ij\ldots }} , i.e. thermodynamic properties and flow velocity, which describe or characterize features of the material body, are expressed as continuous functions of position and time, i.e. P i j … = P i j … ( X , t ) {\displaystyle P_{ij\ldots }=P_{ij\ldots }(\mathbf {X} ,t)} . 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 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}}} In the case of gravitational forces, the intensity of the force depends on, or is proportional to, the mass density ρ ( x , t ) {\displaystyle \mathbf {\rho } (\mathbf {x} ,t)\,\!} of the material, and it is specified in terms of force per unit mass ( b i {\displaystyle b_{i}\,\!} ) or per unit volume ( p i {\displaystyle p_{i}\,\!} ). These two specifications are related through the material density by the equation ρ b i = p i {\displaystyle \rho b_{i}=p_{i}\,\!} . Similarly, the intensity of electromagnetic forces depends upon the strength ( electric charge) of the electromagnetic field. 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)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. 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 instantaneous position x {\displaystyle \mathbf {x} } is a property of a particle, and its material derivative is the instantaneous flow velocity v {\displaystyle \mathbf {v} } of the particle. Therefore, the flow velocity field of the continuum is given by 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.

The Continuum Concept in Practice - Page 3 - Philosophy Now Forum The Continuum Concept in Practice - Page 3 - Philosophy Now Forum

Breastfeeding "on cue"—involving infants' bodily signals being immediately answered by their mothers' nursing them; 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. A particular particle within the body in a particular configuration is characterized by a position vector 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).orientation-preserving, as transformations which produce mirror reflections are not possible in nature. The material points forming a closed curve at any instant will always form a closed curve at any subsequent time.