StressTensorNotes

StressTensorNotes - The Stress tensor Introduction In his...

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The Stress tensor Introduction In his Principia , Newton defined the concept of a force in terms of the laws of motion. These laws in effect define the protocol for measuring the force exerted on a particle ( defined as a discrete object) in terms of the rate of change of momentum of that particle. The extension of these ideas to a continuum is attributed to Euler (1775) Euler's First Law can be stated as "The time rate of change of linear momentum of a body relative to the fixed stars (i.e., an inertial frame of refer- ence) is equal to the sum of forces acting on the body." There are essentially two types of forces that can act on a fluid or continuum: body forces and surface forces. Body Forces Body forces are distributed throughout the continuum and are proportional to the mass. They arise as a conse- quence of the continuum being placed in a force field (gravitational, magnetic, electrostatic, or more generally electromagnetic). We will denote these forces with the symbol b, which characterizes the vector field. Body forces can be conservative or nonconservative. A conservative body force can be expressed as a gradient of a scalar potential, i.e., (1) b = -“f where f denotes the scalar potential. Forces that are directed centrally from a source are conservative; examples are gravity, electrostatic, and magnetic. The case of gravity (the body force we will primarily be concerned with in these notes) we can write (2) g = -“f where f = g ÿ x , and » g » ª G is the gravitational constant. Surface Forces Surface forces are short-range forces, molecular in origin, and depend on the interactions of molecules and/or atoms in the body . In a fluid body each molecule interacts with every other molecule of the fluid, but because this interaction (e.g. van der Waals) is short range (the penetration depth of the forces, typically no more than tens of nanometers), molecules only interact strongly with their nearest neighbors. Short-range forces thus decrease rapidly with increase of distance between interacting molecules, and are significant only when that distance is of the order of the molecular separation. Consider now a fluid element that is acted on by short-range forces arising from interactions with another element (either a solid or fluid). Since the short-range forces can act only on a thin layer adjacent to the boundary of the fluid element, the net force acting on the element due to the short-range forces is thus determined by the surface
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StressTensorNotes - The Stress tensor Introduction In his...

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