By Prof L S Srinath

This e-book is designed to supply an outstanding origin in Mechanics of Deformable Solids after an introductory path on energy of Materials. This variation has been revised and enlarged to make it a complete resource at the topic. Exhaustive remedy of crucial subject matters like theories of failure, power tools, thermal stresses, tension focus, touch stresses, fracture mechanics make this a whole delivering at the topic.

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While the planes on which the principal normal stresses act are free of shear stresses, the principal shear planes are not free from normal stresses. The normal stresses σ + σ3 σ2 + σ3 associated with the principal shears are respectively 1 , 2 2 σ + σ2 and 1 . These are indicated by points D, E and F in Fig. 16. It will be 2 shown in Sec. 19 that the principal shear planes are at 45° to the principal normal planes. 36) (iv) When s1 = s2 π s3 or s1 π s2 = s3, the three circles reduce to only one 1 circle and the shear stress on any plane will not exceed (s1 – s3) or 2 1 (s1 – s2) according as s1 = s2 or s2 = s3.

20) can then be written as s3 – l1s 2 + l2s – l3 = 0 The quantities l 1, l 2 and l 3 are known as the first, second and third invariants of stress respectively. An invariant is one whose value does not change when the frame of reference is changed. e. and similarly, l1 = l′1 l2 = l2′ and l3 = l3′ The reason for this can be explained as follows. The principal stresses at a point depend only on the state of stress at that point and not on the frame of reference describing the rectangular stress components.

Dividing Eq. 20) by s 2, l l σ − l1 + 2 − 32 = 0 σ σ For appropriate values of s, the quantity on the left-hand side will be equal to zero. 26) − 3 = f (σ ) σ σ2 Since l1, l2 and l3 are finite, f (s ) can be made positive for large positive values of s. Similarly, f (s ) can be made negative for large negative values of s. Hence, if one σ − l1 + 18 Advanced Mechanics of Solids f (s) plots f (s ) for different values of s as shown in Fig. 13, the curve must cut the s axis at least once as shown by the dotted curve and for s this value of s, f (s ) will be equal to A C B zero.

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