Strain Energy Methods

Work and EnergyFFdrFdxConsider a solid object acted upon by force, F, at a point, O, as shown in the figure.Let the deformation at the the point be infinitesimal and be represented by vector dr, as shown. The work done = F drxzyFor the general case:W = Fx dxi.e., only the force in the direction of the deformation does work.

Amount of Work doneConstant Force: If the Force is constant, the work is simply the product of the force and the displacement, W = FxFDisplacementxLinear Force: If the force is proportional to the displacement, the work is FDisplacementxFoxo

Strain Energy Consider a simple spring system, subjected to a Force such that F is proportional to displacement x; F=kx.Now determine the work done when F= Fo, from before:

This energy (work) is stored in the spring and is released when the force is returned to zero

Strain Energy DensityyxaaaConsider a cube of material acted upon by a force, Fx, creating stress sx=Fx/a2causing an elastic displacement, d in the x direction, and strain ex=d/ayxaFxdWhere U is called the Strain Energy, and u is the Strain Energy Density.

Try it:A cube of SAE1045 steel is subjected to a uniform uniaxial stress as shown;Determine the strain energy density in the cube when:sx(a) the stress is 300 MPa; (b) the strain in the x-direction is 0.004(a)yx

(a) For a linear elastic materialu=1/2(300)(0.0015) N.mm/mm3=0.225 N.mm/mm3

(b) Consider elastic-perfectly plasticu=1/2(350)(0.0018) +350(0.0022)=1.085 N.mm/mm3

Shear Strain EnergyyxaaaConsider a cube of material acted upon by a shear stress,txycausing an elastic shear strain gxyyx d = gxya txygxy

Total Strain Energy for a Generalized State of Stress

Strain Energy for axially loaded barF= Axial Force (Newtons, N)A = Cross-Sectional Area Perpendicular to F (mm2)E = Youngs Modulus of Material, MPaL = Original Length of Bar, mm FDAL

Comparison of Energy Stored in Straight and Stepped barsFDaALFDbAL/2nAL/2Note for n=2; case (b) has U= which is 3/4 of case (a)(a)(b)