Download The Non-Linear Field Theories of Mechanics by Professor Clifford Truesdell, Professor Walter Noll (auth.) PDF
By Professor Clifford Truesdell, Professor Walter Noll (auth.)
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Extra info for The Non-Linear Field Theories of Mechanics
Example text
While it is commonly asserted that classical mechanics represents only phenomena reversible in time, this is true only for certain special disciplines such as analytical mechanics. As we shall see below, the mechanical behavior of materials is specified in terms of past experiences only and thus is fully capable of representing irreversibility. It can be shown 2 that an event {x, t} and its image {x*, t*} under a change of frame are related by rigid transformations combined with a time-shift. 2) where c (t) is a point, Q (t) an orthogonal tensor, and- a a real number.
An event is a pair {x, t} consisting of a point x in space and a time t. The totality of all events is called space-time. A change of frame is a one-to-one mapping of space-time onto itself such that distances, time intervals, and temporal order are preserved. Only the concept of a change Of frame, not the concept of frame of reference itself, is regarded as having a mathematical meaning. In particular, frame of reference should not be regarded as a synonym for co-ordinate system. Note that transformations involving a reversal of the sense of time are not considered.
6) below and in which "p is a polynomial invariant. List of non-linear basic products,' ((3) Let D=f(A 1 ,A2 • .... Ar. • t. This theorem is due to SPENCER and RrvLINl. For a polynomial isotropic tensor function D= HA, B) of two variables A, BE 7 we have the explicit representation 2 + D="P01+"PIA+"P2B+ "P3A2+ "P4 B2 + "P5 (AB+ BA) + "Ps (A 2 B+ BA2) + "P7 (AB2+ B2A) + "Ps (A2 B2+ B2A2) } (13·7) where "Po, ... 22). 22) has yet been found. The following results are valid for any dimension n: A vector-valued function v = U(A.