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Ch_22

# Ch_22 - Theory of Elasticity Solid Mechanics applications...

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Chapter 22 Page 1 K. Haghighi Theory of Elasticity Solid Mechanics applications Elasticity prob., plates & shells, buckling and stability prob., vibrations, plasticity, elasto- plasticity, viscoelasticity. . . (procedure is the same) Theory, elem. matrices, 2 - D elasticity, axisym. elasticity.

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Chapter 22 Page 2 K. Haghighi Stress, Strain, Hooke’s Law: State of stress at a point - 6 indep. comp. - 9 dep. comp. yy σ yx σ zy σ zx σ zz σ xy σ xz σ xx σ y z x Sign convention: (+): outward normal & comp. are in (+) coord. direction. Similarly for on the negative face. s ' σ σ yz σ
Chapter 22 Page 3 K. Haghighi Stress vector: { } [ ] yz xz xy zz yy xx T τ τ τ σ σ σ = σ ( ) yz xz xy σ σ σ Under Ext. loads. (mechanical/thermal, . . .) body deforms. The resultant displacement of each point of the body has 3 comp., u, v & w parallel to x, y & z axes. Ext. loads displacements strains s ' σ Ext. loads 1 - mechanical elastic strains 2 - thermal thermal strains produce ,,

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Chapter 22 Page 4 K. Haghighi so, { } [ ] yz xz xy zz yy xx T e e e e e e e = total strain vector: { } [ ] yz xz xy zz yy xx T ε ε ε ε ε ε = ε elastic strain vector: and { } [ ] 0 0 0 T T T T T αδ αδ αδ = ε thermal strain vector: temp. change coef. of thermal expansion
Chapter 22 Page 5 K. Haghighi { } { } { } T e ε + ε = Also note that Generalized Hooke’s Law: { } [ ] { } { } [ ] { } ε = σ σ = ε D or C stresses elastic strains

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Chapter 22 Page 6 K. Haghighi [ ] 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 < µ µ µ µ µ µ µ = a a a E C where E = Elastic or Young’s Modulus = Poisson’s ratio a = 2 ( 1 + ) µ µ
Chapter 22 Page 7 K. Haghighi

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