Surface And Body Forces

A “field” associates a physical quantity with a position. It may be time dependent. For example, T(r ⃗,t ) can represent the spatial and time-dependent temperature and ρ(r ⃗,t ) the density. These are examples of scalar valued fields. On the other hand, force or flux are examples of vector valued fields. Furthermore, stress is an example of a rank-two tensor field.
Material Point notion
If we consider a body with volume ”V” which separates parts of a physical space with its surface “A”, the material point “P” is an internal point in this body which has volume ∆V and mass ∆m. Figure 1 depicts part of a body discretized with material points. Figure 1 Discretization in a material point method simulations

Physical properties of materials
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A body is in stress free condition where the only forces which are acting on it are atomic forces which are necessary to maintain the essence of matter. Consequently, we can assume that the stresses are the result of the external forces acting on the body. The force types can be summarized as volumetric forces like gravitational forces that act on all the body and the surface forces that act on a surface which can be internal or external.
Traction vector
A homogenous body where volumetric and surface forces are acting on it is shown in Figure 1. If we cut it through point P into two parts, there would be internal forces acting on the cutting plane “S” due to applied surface or volumetric forces. Assuming the area of the point “P” as ∆A and the amount of force acting on this infinitesimal point as ∆F we can find the traction vector by area to zero as: lim┬(∆S→0)⁡〖∆F/∆A〗=dF/dA=t_i^((n)) where t_i^((n)) is the traction vector acting along the normal vector “n” of cutting plane on the surface “S”. Figure 2 sdfdsf Figure 3
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Lagrangian configurations
When a continuum is in motion, its velocity, stress tensor or other physical or kinematic properties may change with time. In another words, a physical property or a kinematic property of the body can be expressed in terms of a reference coordinate system if we consider that property as a function of its location on the spatial coordinates and time. Meanwhile, where this reference configuration is expressed as a real configuration at time t=0, then it is called the “Lagrangian description”. Following such a description properties like v, T can be expressed as: v= v ̂(X_1,X_2,X_3,t)
T= T ̂(X_1,X_2,X_3,t) where v, T are velocity and stress respectively. Other names for the Lagrangian description are the “material description” and the “reference description”. On the other hand, where the physical properties or the kinematic properties are expressed as the function of a spatial coordinate “x” and time, it is called the spatial description or Eulerian

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