5.2 Elastic Strain Energy

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P Figure 5.2.8: a volume element under stress The strain energy density u is defined as the strain energy per unit volume: Section 5.2 2 σ xx u = (5.2.13) 2E The total strain energy in the bar may now be expressed as this quantity integrated over the whole volume, U = udV (5.2.14) which, for a constant cross-section A and length L reads U = A udx . From Hooke’s 0 law, the strain energy density of Eqn. 5.2.13 can also be expressed as Solid Mechanics Part I 185 Kelly ∫ V ∫ L 1 u = σ xxε xx (5.2.15) 2 As can be seen from Fig. 5.2.9, this is the area under the uniaxial stress-strain curve. u σ xx σ dy dx dz σ xx volume element Figure 5.2.9: stress-strain curve for elastic material Note that the element does deform in the y and z directions but no work is associated with those displacements since there is no force acting in those directions. The strain energy density for an element subjected to a σ yy stress only is, by the same arguments, σ yyε yy / 2 , and that due to a σ zz stress is σ zzε zz / 2 . Consider next a shear stress σ xy acting on the volume element to produce a shear strain ε xy as illustrated in Fig. ε z y P x
Section 5.2 5.2.10. The element deforms with small angles θ and λ as illustrated. Only the stresses on the upper and right-hand surfaces are shown, since the stresses on the other two surfaces do no work. The force acting on the upper surface is σ xydxdz and moves through a displacement λ dy . The force acting on the right-hand surface is σ xydydz and moves through a displacement θ dx . The work done when the element moves through angles d θ and d λ is then, using the definition of shear strain, Eqn. 3.6.1, ( σ dxdz)( dλdy) + ( σ dydz)( dθdx) = ( dxdydz) σ ( 2d ) dW = ε (5.2.16) xy xy and, with shear stress proportional to shear strain as in Fig. 5.2.9, the strain energy density is u = 2 ∫σ xydε xy = σ xyε xy (5.2.17) Figure 5.2.10: a volume element under shear stress The strain energy can be similarly calculated for the other shear stresses and, in summary, the strain energy density for a volume element subjected to arbitrary stresses is ( σ ε + σ ε + σ ε ) + ( σ ε + σ ε + σ ) 1 u = xx xx yy yy zz zz xy xy yz yz zxε 2 Using Hooke’s law, Eqns. 4.2.9, with Eqn. 4.2.5, the strain energy density can also be written in the alternative and useful forms {▲Problem 4} 1 u = 2E μ = 1− νμ = 1− 2ν Solid Mechanics Part I 186 Kelly xy xy zx (5.2.18) 2 2 2 ν 1 2 2 2 ( σ xx + σ yy + σ zz ) − ( σ xxσ yy + σ yyσ zz + σ zzσ xx ) + ( σ xy + σ yz + σ zx ) E 2μ 2 2 2 2 2 2 [ ( 1−ν ) ( ε xx + ε yy + ε zz ) + 2ν ( ε xxε yy + ε yyε zz + ε zzε xx ) ] + 2μ( ε xy + ε yz + ε zx ) 2ν 2 2 2 2 2 2 2 ( ε + ε + ε ) + μ( ε + ε + ε ) + 2μ( ε + ε + ε ) xx yy λdy Strain Energy in a Beam due to Shear Stress zz dy y λ xx σ xy θ dx yy θdx zz σ xy xy x yz zx (5.2.19)
Page 1 and 2: 5.2 Elastic Strain Energy Section 5
Page 3 and 4: ΔW Figure 5.2.4: torque - angle of
Page 5: Figure 5.2.7: a beam subjected to a
Page 9 and 10: Section 5.2 2 Similarly, for torsio
Page 11 and 12: But θ A = 0 and so Eqn. 5.2.33 can
Page 13 and 14: 1 2 EA w ( h + Δ ) = kΔ max , k =

5.2 Elastic Strain Energy

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