North American Combustion Handbook: A Basic Reference on the by Richard J. Reed

By Richard J. Reed

North American Combustion guide: A simple Reference at the paintings and technology of commercial Heating with Gaseous and Liquid Fuels, Vol 1.

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Extra info for North American Combustion Handbook: A Basic Reference on the Art and Science of Industrial Heating with Gaseous and Liquid Fuels, Vol. 1

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_ ∂W ∂W 2 ∂W ∂W 2 ∂W þ I1 I1 þ 2I 2 bik bkj bij À δij À 7=3 3J ∂I 1 ∂I 2 ∂I 1 ∂I 2 J ∂I 2 _ þ ∂W δij ∂J ð5:28Þ or " _ _! 2 1 ∂W ∂W þ I1 ti j ¼ bi j À J J 2=3 ∂I 1 ∂I 2 # _ _! _ ∂W ∂W δij 1 ∂W À I1 þ 2I 2 bik bkj ∂I 1 ∂I 2 3 J 4=3 ∂I 2 _ ∂W þ δij ∂J ð5:29Þ With _ ∂W ¼ C10 ; ∂I 1 ð5:30aÞ _ ∂W ¼ C01 ∂I 2 ð5:30bÞ ∂W ¼ 2D1 ðJ À 1Þ ∂J ð5:30cÞ and _ we obtain tij ¼ Á À Áδij 2 À2=3 À J À J À4=3 C01 bik bkj C10 þ I 1 C01 bij À I 1 C10 þ 2I 2 C01 J 3 þ 2D1 ðJ À 1Þδij ! ð5:31Þ The Kirchhoff stress tensor is given by τij ¼ Jtij À Á À Áδij À 2C01 bik bkj ¼ 2 C10 þ I 1 C01 bij À 2 I 1 C10 þ 2I 2 C01 3 þ 2D1 J ðJ À 1Þδij ð5:32Þ 34 5 Stress Measures Then, with G C10 ¼ ; 2 C01 ¼ 0 ð5:33aÞ ð5:33bÞ and D1 ¼ K 2 ð5:33cÞ we get   bkk τi j ¼ G bij À δij þ KJ ðJ À 1Þδij 3 ð5:34Þ Â Ã τij ¼ Gdev bij þ KJ ðJ À 1Þδij ð5:35Þ 1 dev½Š ¼ ðÞ À ½ðÞ : 1Š1 3 ð5:36Þ or where This Eulerian formulation is the one that is employed in the hyperelastic material model in the Abaqus/Standard® finite element code.

Employing the chain rule we can write _ _ _ ∂W ∂W ∂I 1 ∂W ∂I 2 ∂W ∂J ¼ þ þ ∂F jI ∂I 1 ∂F jI ∂I 2 ∂F jI ∂J ∂F jI ð6:58Þ where   ∂I 1 1 ∂I 1 2I 1 ∂J 2 I1 T 2 2 ¼ 2=3 À 5=3 ¼ 2=3 F jI À FJi ¼ 2=3 F jI À I 1 FIjT ð6:59aÞ 3 ∂F jI J ∂F jI 3J ∂F jI J 3 J   ∂I 2 1 ∂I 2 4I 2 ∂J 2 2I 2 T FIj ¼ 4=3 À 7=3 ¼ 4=3 I 1 F jI À b jk FkI À ∂F jI J 3F jI 3J ∂F jI J 3 2 2 4I 2 T ¼ 2=3 I 1 F jI À 4=3 b jk FkI À F 3 Ij J J ð6:59bÞ 6 Tangent Moduli 47 and, from the definition of the derivative of a scalar-valued function of a tensor, ∂J ¼ JFIjT ∂FjI ð6:60Þ where this relationship is derived in Appendix C.

The Biot stress, a Lagrangian-based stress tensor, is also an important stress measure. Only somewhat recently has it been recognized that the Biot stress tensor is helpful in the understanding of certain fundamental problems in elasticity theory. Two detailed numerical examples are presented. There are as many different stress measures as there are strain measures. The second Piola-Kirchhoff stress tensor, also known as the Kirchhoff-Trefftz stress tensor, is the most significant stress measure within the framework of hyperelasticity.

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