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Subject code: CE41615/ CE60041 Subject name: Theory of Elasticity and Plasticity/ Applied Elasticity and Plasticity Exercise 2 (Material Description)

1. Starting from the constitutive relation for an orthotropic material show that the number of

independent elastic constants for a transversely isotropic material is 5.

2. Show that for an isotropic linear elastic solid the principal axes of the stress and strain

tensors coincide, and develop an expression for the relationship among their principal

values.

3. An orthotropic material has the following properties: Ex = 50MPa, Ey = 18MPa, Gxy = 9MPa

and xy = 0.25. Determine the first principal direction for the stresses and strains at a point

on a free surface where the following strains were measured: xx = 800, yy = 200 and xy=

300. [Ans. Stress: 1 = 4.45o or -175.55o, Strains : 1 = 13.3o or -166.7o]

4. Let the stress and strain tensors be decomposed into their respective spherical and deviator

components as, 휎 = 휎 훿 + 푆 and 휀 = 휀 훿 + 휂 . Show that strain energy density

W may be written as the sum of a dilatation energy density and distortion energy density as,

푊 = 휎 휀 + 푆 휂

5. Assuming a state of uniform compressive stress 휎 = −푝훿 , show that the bulk modulus

(ratio of pressure to volume change) may be given by, 퐾 =( )

= ( ), where 휆 and 휇

are the Lamé constants.

6. If elastic constants E, K and G are required to be positive, show that the Poisson’s ratio must

satisfies the inequality −1 < 휈 < . Where E, k and G are respectively the Young’s

modulus, the bulk modulus and the shear modulus.

7. The stress tensor at P is given with respect to Ox1x2x3 with units of MPa by

흈 =4 푏 푏푏 7 2푏 2 4

where b is unspecified. If 휎 = 3MPa and 휎 = 2휎 . Determine the principal stress direction

of 휎 .

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8. A body under applied loads and body forces produces the following displacement field

푢 = 0 푣 = 퐾푥(푦 − 푧 ) + 퐾푎푥푧 푤 = −2퐾푥푦푧 − 퐾푎푥푦

where, u, v and w are displacements in the x, y and z directions respecitvely. Assume a

linear, elastic, isotropic, homogeneous material with modulus of elasticity E and Poisson

ratio . Determine all the strtess components at x = 2a in terms of K, a, E, , y, and z.

9. Show that the strain energy density may be expressed in terms of strain invariants as,

푈 = 휆 + 휇 퐼 − 2휇퐼퐼

where 휆 and 휇 are are the Lamé constants and 퐼 and 퐼퐼 are respectively the first and second

strain invariants. Explain why U does not depend in the third invariant.

References

Sadd H. M., Elasticity – Theory, Applications and Numerics (Chapter 6)

Mase G. T, Smelser R. E., MAse G. E, Continuum Mechanics for Engineers (Chapter 6.1 – 6.3 )

Reddy J. N, An introduction to continuum mechanics with application (Chapter 6.1 – 6.2.7)