Read e-book online Mechanics of incremental deformations; theory of elasticity PDF

By Maurice A Biot

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Extra resources for Mechanics of incremental deformations; theory of elasticity and viscoelasticity of initially stressed solids and fluids, including thermodynamic foundations and applications to finite strain

Example text

These surface distributions of mass appear in boundary conditions for the solutions of equation 7,43b and involve discontinuities of the normal derivatives of the potential. 42. 44) Ik OXj may be referred to as the curvature terms. 45) depends essentially on the body force and the initial stress gradient' and is of a different nature. The significance of the curvature terms was discussed in section 6 for two-dimensional strain. A similar discussion in three dimensions introduces the twist of an element.

9) is satisfied identically. 16) flyy = Y22 flzx = Y31 flzz = Y33 flxy = Y12 Kinematics of Three-Dimensional Strain Sec. 3). 4) thus related represent the same state of strain and can differ only by a rigid body rotation. 3) will be called the local rotation of the material. 3) contains nine independent coefficients, while the state of strain is determined by only six quantities. 3). 13) are linearly related to the change of the square of the length. 10). 10). 16 is obtained immediately as follows.

14). 1) ) - 0 - The components of the body force per unit mass are X, Y, Z, and the mass density at the point x, y, z, is p(x, y, z). Consider a volume V bounded by a surface S before deformation. After deformation this surface becomes S', and it encloses a volume V'. 2) The surface integral is extended to the boundary after deformation, and a~~, a~1/' a~~ are stress components at a point of coordinates g, TJ, ~. We remember that the symmetry of stress components implies a~~ = a~~, a1/~ = a~1/' and a~1/ = a1/~' We shall therefore pay no attention to the order of the indices and choose it for convenience.

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