1

Practice #5

1. Use a uniaxial stress-strain curve to distinguish between elastic strain energy density and the

complementary energy density. What is the relationship between the two forms of energy for a linear elastic material?

2. The material data for a foamed polymer, solid rubber and epoxy resin are given below : E (GPa) ν Foamed polymer 0.01 0 Epoxy resin 0.50 0.3 Solid rubber 0.03 0.495 (i) Each material is loaded elastically to σz = 1 MPa in uniaxial free compression. How much energy is stored in a unit volume of each material? (ii) In a different set of tests, each material is now constrained so that it cannot expand sideways when compressed with a uniaxial strain in the same direction as in the first tests (i.e. all other components of strain except εz are zero). Recalculate the energy absorbed per unit volume when the materials are compressed uniaxially in displacement control until σz = 1 MPa. Comment on the implications for packaging and protective padding. [Note that when a thin sheet of padding is glued to a rigid base it may be constrained in this way, e.g. rubber floor tiles] 3. A kitchen floor is to be covered with rubber tiles. Each tile is 2 mm thick and has a

Young’s modulus E = 0.03 GPa and Poisson’s ratio ν = 0.495. The tiles are glued firmly to the rigid floor, thus preventing lateral expansion of the tiles. Obtain the normal deflection of the tiles and the elastic energy stored by each tile when a uniform normal pressure of P = 10 MPa is applied.

4. Determine the strain energy due to bending for the beam shown in Fig. 1 below, and

hence calculate the displacement at the points of application of the loads. Assume the Young's modulus E = 190 GPa.

Figure 1 (Qu. 4) Figure 2 (Qu. 7)

5. A simply supported beam of length L = 3 m is subjected to a uniformly distributed load w = 10 kN/m. Determine the strain energy stored in the beam due to bending. Assume EI = 15 kNm2.

40 mm

80 mm

30 kN 30 kN

0.3 m 0.6 m 0.3 m3 kN

100 mm

R

2

6. If the stresses and strains at a point in a linear elastic material are given by

σ εij =⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

885 154 0154 1038 615

0 615 577

0 002 0 001 00 001 0 003 0 004

0 0 004 0MPa; and ij

. .

. . ..

respectively, determine (i) the total strain energy density in the material, (ii) the hydrostatic component of the strain energy density, and (iii) the bulk modulus of the material.

7. A curved solid bar with a radius of curvature R = 125 mm is loaded as shown in Figure 2

below. The bar has a circular cross section of diameter 50 mm, and a Young’s modulus E = 80 GPa. Determine (i) the strain energy due to bending stored in the bar and (ii) the displacement in the direction of the load, at the point where the load is applied.

8. A clamp is made of two straight members and a curved member, as shown in Figure 3.

The cross-sectional geometry of the clamp, which is also shown in the figure, has a moment of inertia I = 7 x 10−8 m4, and the material from which the clamp is made has a Young's modulus E = 200 GPa. If the blocks of wood being clamped exert a force P = 40 kN on the straight members of the clamp, as shown in the figure, determine (i) the strain energy due to bending in the clamp and (ii) the movement of the ends where the loads are applied.

Figure 3 (Qu. 8) Answers (1) The two forms of energy are equal for a linear elastic material. (2) (i) Polymer, 50 kJ/m3; Epoxy, 1 kJ/m3; Rubber, 17 kJ/m3. (ii) Polymer, 50 kJ/m3; Epoxy, 0.74 kJ/m3; Rubber, 0.5 kJ/m3. Foamed polymer is the most suitable for protective padding and packaging. (3) 0.02 mm; 49.34 kJ/m3 (4) 99.7 J; 3.3 mm, 3.3 mm ; (5) 6.75 kJ; (6) (i) 5.06 MJ/m3; (ii) 4.2 MJ/m3; (iii) 166.7 GPa (7) (i) 0.28 J; (ii) 0.19 mm. (8) (i) 339 J; (ii) 8.48 mm

P

120 mm 45 mm

B C P 25 mm

40 mm

B C 10 mm