By O A Bauchau
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Vanishes. This is a special case of two-dimensional hydrostatic stress. 4. 49) expresses the direct and shear stresses acting on a face oriented at an arbitrary angle θ with respect the axis ¯ı1 , but the presence of trigonometric functions involving the angle 2θ makes it difficult to give a simple, geometric interpretation of these formulæ. A useful geometric interpretation, however, called Mohr’s circle, can be developed. Let the state of stress at a point be defined by its principal stresses, σp1 and σp2 .
32) τns = − σ1 − σ2 σ1 − σ2 τ12 σ1 − σ2 = 0. + τ12 sin 2θp + τ12 cos 2θp = − 2∆ ∆ 2 2 It is also interesting to find the orientation of the faces leading to the maximum value of the shear stress. Indeed, in view of eq. 32), the shear stress is also a 24 1 Basic equations of linear elasticity function of the face orientation angle. 38) where the last equality follows from eq. 34). Here again, this equation presents two solutions, θs and θs + π/2, corresponding to two mutually orthogonal faces. 39) where ∆ is again given by eq.
30 1 Basic equations of linear elasticity Mohr’s circle is a graphical representation of all the possible stress components corresponding to a single state of stress. 7 Lam´e’s ellipse Lam´e’s ellipse provides an elegant geometric interpretation of the state of stress at a point. Consider a material in a plane state of stress and let τ n be the stress vector acting on the face with a unit normal n ¯ at an angle θ with respect to axes ¯ı∗1 , as depicted in fig. 17. As angle θ varies, the tip of the stress vector, τ n , draws an ellipse, called Lam´e’s ellipse, with its center at O and its semi-axes given by the absolute value of the principal stresses, |σp1 | and |σp2 |, respectively.