Limits to elastic response. Yielding and failure

This chapter presents conditions for determining the limits of elastic behaviour for isotropic materials. The stress invariants of equivalent pressure, equivalent shear stress, and equivalent tensile stress are defined. These are then used to define common yield conditions, viz. the pressure-independent Mises and Tresca yield conditions, as well as the pressure-dependent Coulomb-Mohr and the Drucker-Prager yield conditions. Rankine’s failure criterion for brittle materials in tension, that is failure in a brittle material will initiate when the maximum principal stress at a point in the body reaches a critical value, is also discussed.

Linear thermoelasticity

This chapter presents a theory for the coupled thermal and mechanical response of solids under circumstances in which the deformations are small and elastic, and the temperature changes from a reference temperature are small --- a framework known as the theory of linear thermoelasticity. The basic equations of the fully-coupled linear theory of anisotropic thermoelasticity are derived. These equations are then specialized for the case of isotropic materials. Finally, as a further specialization a weakly-coupled theory in which the temperature affects the mechanical response, but the deformation does not affect the thermal response, are discussed; this is a specialization which is of importance for many engineering applications, a few of which are illustrated in the examples.

Elastodynamics. Sinusoidal progressive waves

This chapter briefly considers linear elasticity under circumstances in which inertial effects are accounted for, and states the initial-boundary-value-problem of linear elastodynamics. Sinusoidal progressive waves form an important class of solutions to the equations of linear elastodynamics. Such waves for isotropic media in the absence of a conventional body force are considered and it is shown that for an isotropic medium only two types of sinusoidal progressive waves are possible: longitudinal and transverse.

Principles of minimum potential energy and complementary energy

With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.

Finite elasticity of elastomeric materials

This chapter presents several technologically important constitutive relations for elastomeric materials. In particular, the Neo-Hookean, Mooney-Rivlin, Ogden, Arruda-Boyce, and Gent free energy functions are discussed in the context of incompressible response. Extensions to the slightly compressible case are also detailed, this includes a presentation of a number of possible volumetric response relations and their properties.

Fatigue

This chapter introduces methods for analysing fatigue failure of materials under repeated loads. The notions of defect-free and defect-tolerant failure analysis are discussed. For defect free analysis the notion of S-N curves is introduced and Coffin-Mason low cycle as well as Basquin high cycle relations are discussed. Miner’s rule is also introduced. For a defect-tolerant approach Paris’s law for fatigue crack growth is presented.

Energy-based approach to fracture

This chapter introduces the concept of energy release rates for linear elastic fracture mechanics. The concept of an energy release rate is defined and related to the criteria of Griffith with application in the context of bodies with point loads. Eshelby’s energy momentum tensor is also introduced and Rice’s path independent J-integral is derived, related to energy release rate, and applied to fracture problems.

Rigid-perfectly-plastic materials. Two extremal principles

This chapter introduces the notion of rigid-perfect plasticity, and provides proofs of the extremum principles of applied loads and velocities. The variational principles are applied to generate the upper- and lower-bound theorems for collapse loads in rigid-perfect plasticity models. The method of sliding rigid blocks is presented for the solution of practical problems in forming operations and structural collapse. To facilitate such modelling the hodograph graphical device is also discussed.

Linear viscoelasticity

This chapter introduces the essential elements of linear viscoelastic material behaviour and modeling in one- and three-dimensions. Both relaxation and creep phenomena are introduced and modeled using Boltzmann’s superposition integral. Various common kernel functions are introduced, as is the standard and generalized standard linear model in differential and integral form. The correspondence principle is discussed for the solution of practical problems and to connect relaxation and creep formulations. Storage and loss moduli for oscillatory loadings are discussed, as are loss tangents and dissipation. For the generalized standard linear solid its time integration via the Herrmann-Peterson recursion relation is discussed. Effects of temperature are discussed, and the concept of time-temperature equivalence is introduced.

Small deformation rate-independent plasticity based on a postulate of maximum dissipation

This chapter introduces the concept of maximum dissipation. The elastic set is introduced, and the plastic dissipation is maximized over the elastic set using classical methods from linear programming theory. The plastic flow direction is seen to be generally normal to the yield surface when the plastic dissipation is maximized. The Kuhn-Tucker complementarity conditions are seen in this context to arise from the postulated optimization problem, and the elastic set is seen to be necessarily convex. The concept of maximum dissipation is applied to a Mises material and the models of the earlier chapters are seen to be recovered.