Shear strain energy density

Equation (14.10) corresponds to the condition that yield occurs when the elastic shear strain energy density in the stressed material reaches a critical value. 

Von MiSOS Yiold Critorion. The Von Mises yield criterion (also known as the maximum distortional energy criterion or the octahedral stress theory) (25) states that yield will occur when the elastic shear-strain energy density reaches a critical value. There are a number of ways of expressing this in terms of the principal stresses, a common one being... 

Boundaries between solids transmit shear stress, particularly if they are coherent or semicoherent. Therefore, the strain energy density near boundaries changes over the course of solid state reactions. Misfit dislocation networks connected with moving boundaries also change with time. They alter the transport properties at and near the interface. Even if we neglect all this, boundaries between heterogeneous phases are sites of a discontinuous structural change, which may occur cooperatively or by individual thermally activated steps. 

Example 2.8 Calculate the kinetic and strain energy densities for the shear wave examined in the Example 2.7. 

Solution Since the shear wave is z-polarized and jr-propagating, the only nonzero derivative in Equation 2.39 is du dx = —jkuj. Thus, the strain energy density is given... 

Neo-Hookean material This is an extension of the linear elastic Hooke s law to include large deformations. The main difference between the classical Hooke s law and the Neo-Hookean law is that in the latter the shear modulus is a function of the deformations. The strain energy density for the compressible Neo-Hookean model is given by... 

The strain energy density released by rock bursts rock in compression-shear, tensile-shear and pure tensile stress state could be written as ... 

The in-plane tensile stress in the direction normal to the hole and the inplane shear stress on the surface of the hole are zero because of the condition of zero applied traction. The strain energy density as a function of position along the surface of the hole is [/(ly) = (t /2E where E = E/ X — v ) is the plane strain elastic modulus. The chemical potential field over the surface of the hole is then... 

Stated for the simple shear case. Again, modulus changes during the test must be monitored independently in order to correct the strain energy density. 

Single-edge crack Pure shear G = 2k aW, k njXt G, = IW, Ac = extension ratio at onset of crack growth Wc = critical stored elastic strain energy density 1 = initial length a,b,c a... 

An argument to resolve the discrepancy between the failure envelopes obtained for different modes of straining is indicated by the work of Blatz, Sharda, and Tschoegl [42]. These authors have proposed a generalized strain energy function as constitutive equation of multiaxial deformation. They incorporated more of the nonlinear behavior in the constitutive relation between the strain energy density and the strain. They were then able to describe simultaneously by four material constants the stress-strain curves of natural rubber and of styrene-butadiene rubber in simple tension, simple compression or equibiaxial tension, pure shear, and simple shear. 

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