By Wolfgang H. Müller
This booklet introduces box conception as required in sturdy and fluid mechanics in addition to in electromagnetism. It comprises the mandatory utilized mathematical framework of tensor algebra and tensor calculus, utilizing an inductive strategy really suited for newbies. it's aimed toward undergraduate sessions in continuum conception for engineers regularly, and extra in particular to classes in continuum mechanics. scholars will achieve a legitimate uncomplicated realizing of the topic in addition to the power to unravel engineering difficulties by way of making use of the overall legislation of nature by way of the balances for mass, momentum, and effort together with material-specific kin by way of constitutive equations, hence studying the way to use the idea in perform for themselves. this is often facilitated via quite a few examples and difficulties supplied during the text.
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Extra info for An Expedition to Continuum Theory
6. In particular, the second chain of equations shows that the tangent vectors gi do not necessarily need to be normalized. r = const. Fig. 6 Plane polar coordinates and the corresponding (unit) tangent vectors eϑ er e2 ϑ = const. 1: Metric of a plane skew coordinate system Use geometric arguments to show that for the scissor-like coordinates of Fig. 4 the following relations hold: x1 ¼ cos a z1 þ cos b z2 ; x2 ¼ sin a z1 þ sin b z2 ; and the metric tensor reads: gkj ¼ 1 cosða À bÞ cosða À bÞ : 1 ð2:3:9Þ ð2:3:10Þ Discuss and interpret the case a À b ¼ p=2.
1007/978-94-007-7799-6_3, Ó Springer Science+Business Media Dordrecht 2014 47 48 3 Balances (in Particular in Cartesian Systems) Fig. 1 Balance volume with a singular surface ∂A − A v A e V − + A n components of the velocity vector, t ¼ ^tðx; tÞ, and temperature, T ¼ T^ ðx; tÞ, in all material points, x, of a body, V(t), and at all times, t. Initially we will assume that these five fields are described in a Cartesian frame x ¼ xi ei . From an atomistic point of view the three fields can be interpreted as follows: The sum of the mass of all the molecules, dm, divided by the volume of the material point, dV, gives the ^ðx; tÞ.
By renaming the indices m ! k exactly Eq. 9)1. 9)2 can be validated in an analogous manner. It was mentioned in context with Eq. 9) that the metric tensor allows to calculate distances with curvilinear coordinates. We will now show that it can also be used to determine the length of a vector. For this purpose we start from the basic definition for the length of a vector, namely with the scalar product. If the vector is represented by Cartesian components we may write: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A ¼ A Á A ¼ ðAi ei Þ Á ðAj ej Þ ¼ Ai Aj ei Á ej ðxÞ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ Ai Aj dij ¼ Ai Ai ; ð xÞ ð xÞ ðxÞ ð xÞ ð xÞ ð2:4:11Þ ðxÞ ðxÞ where use was made of Eqs.
An Expedition to Continuum Theory by Wolfgang H. Müller