Download e-book for kindle: Advanced mechanics of materials by Boresi A.P., Schmidt R.J.

By Boresi A.P., Schmidt R.J.

ISBN-10: 1601199228

ISBN-13: 9781601199225

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Example text

3 that the solution can be defined as Ur(x, t) on the right-hand side of ;c = x{t) and as U\{x, t) on the left-hand side of X = X~t. The characteristic of the solution in the domain < x < x(t) is the tangent of x = x(t). It is easily seen that the condition {u~ - w^)wo(O) > 0 is necessary for a solution constructed as above. 1) may contain a shock wave (if — u'^)uo{0) < 0) or a left-contact discon­ tinuity (if {u~ - w ^)uq(0) > 0). (iv) Right-contact discontinuity: = o)< X~. This is completely ana­ logous to (hi): {W^ - u ~) uq{0) > 0 ^ shock wave; {u^ - u ~) uq{G) < 0 ^ right-contact discontinuity.

Dr du (2 . 2 . 2) ( m( ± oo), t( ± oo)) = ( m±, X-). 2) satisfies ■ -? P 'W [ du = 0, 1 1 § J 1 d r. 6)2 the Il-centred rarefaction wave or forward centred rarefaction wave, symbolized by R. 10) satisfies the convexity condition p"{x) # 0. 7) which holds for poly tropic gas. 5). 6) on the phase plane (a, x) is called the backward or forward rarefaction wave ^ r v e respectively, symbolized by R or R ^ a in . We denote the curve R passing through a given point (u, t) by ^ ( a , x). We now seek the solution of the Riemann problem.

1. 8) x' = I p (v , Odrj, as so-called Lagrangian coordinates. 1) can be expressed in Lagrangian coordinates with a simple form as (we use x again instead of x') lu, -I- p(r)x = 0 ( 2 . 1. 10) It, - M;, = 0, where r = p “* denotes the specific volume, p = x~^ for polytropic gas. 11) (which holds for a polytropic gas). 5). It is not easy for a gas to undergo an isothermal change as described in mechanics. 10) can be used in the context of one­ dimensional isothermal thermoelasticity where u denotes the velocity, x denotes the deformation gradient and —p denotes the stress.

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Advanced mechanics of materials by Boresi A.P., Schmidt R.J.


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