By Frank Rieg, Reinhard Hackenschmidt, Bettina Alber-Laukant
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Extra info for Finite Element Analysis for Engineers
The following equation shows the local vector of a point in a not deformed state (X) and in a deformed state (x). 5-2 shows a point in the part, once in the undeformed and once in the deformed component. The localization of this point in the undeformed part is given by the local vector X, the situation in the deformed part by the local vector x. With the formulation in material coordinates, all of the values deal with the 52 3 Some Elasticity Theory undeformed state, according to the formulation in spatial coordinates, which deal with the deflected state.
With Rn being the remainder term. Applied on our labelling: u (A0 + dx) = u (A0 ) + ∂u (A0 ) ∂2 u (A0 ) dx2 · dx + · + ... + Rn ∂x ∂x2 2 we can neglect the non-linear terms because with dx → 0 they converge faster to 0 than the linear terms do. It remains: u (A0 + dx) = u (A0 ) + ∂u (A0 ) dx = u (B0 ) ∂x With neglecting the non-linear terms we also get: v (B0 ) = v (A0 + dx) = v (A0 ) + = v ( A0 ) + ∂v (A0 ) · dx ∂x ∂v (A0 ) ∂2 v dx2 dx + 2 · + ... = ∂x ∂x 2 For y-axis we get: u (C0 ) = u (A0 + dy) written as a Taylor series: u (A0 + dy) = u (A0 ) + ∂u (A0 ) ∂2 u dy2 dy + 2 · + ...
This law says that physical or technical processes energy cannot get lost, but the manifestation of the energy may change. The thermal transfer or thermal conduction is the transfer of thermal energy on account of a difference in temperature. For a calculation of the thermal transfer in the steadystate case, the general thermal conduction according to Fourier is valid. qx = −λx ∂T ∂x qx stands for the thermal flow density, λx is the heat conductivity and T states the temperature of the body in the position of the coordinate x.