Large amplitude waves in bounded media III. The deformation of an impulsively loaded slab: the second reflexion

Abstract

Representations are derived that describe the interactions of the waves that are reflected from both perfectly rigid and perfectly free interfaces during the arrival of a centred wave with any wave travelling in the opposite direction. These are used to analyse the early stages of the deformation produced when the traction at the loaded boundary continues to vary after changing discontinuously.

The Korteweg capillarity system. Integrable reduction via gauge and reciprocal links

“…Accordingly, the latter is here seen to be gauge equivalent to the canonical integrable cubic NLS equation in . Moreover, if, in addition, q c = 0 then (25) reduces to the Kundu-Eckhaus equation, linked to a linear Schrödinger equation in via gauge transformation (20). Here, we shall be concerned with the general case with q c = 0, q q = 0, s = 0 and alignment of (24) is made with the nonlinear capillarity model (19).…”

A novel class of model constitutive laws in nonlinear elasticity: Construction via Loewner theory

“…Such model laws were extensively used to study uniaxial pulse transmission and reflection in bounded nonlinear elastic and plastic media [2]- [4], [19]. But a limitation of these model laws, as noted by Kazakia and Varley [3], is that they are restricted in their use to approximate the response of nonlinear elastic materials in which the signal speed A is either a monotonically increasing or decreasing function of the strain e. Hence, the models cannot accommodate an internal change in the concavity associated with a change from soft to hard elastic behavior (or vice verse).…”

“…The infinitesimal BTs for this case have compatibility conditions that lead to the model constitutive laws with the signal speed determined by the integrable hyperbolic sinh-Gordon equation. We show that the solution of this equation generated via a single application of a BT to the vacuum solution leads to certain model constitutive laws adopted by Cekirge and Varley [2] and subsequently by Kazakia and Varley [3], [4] in elegant and extensive studies of the uniaxial transmission and reflection of pulses in nonlinear elastic slabs. But these model laws, as noted in [4], cannot describe materials with stress-strain laws with an interior change in the concavity corresponding to a change from soft to hard elastic behavior (or vice versa).…”

“…We show that the solution of this equation generated via a single application of a BT to the vacuum solution leads to certain model constitutive laws adopted by Cekirge and Varley [2] and subsequently by Kazakia and Varley [3], [4] in elegant and extensive studies of the uniaxial transmission and reflection of pulses in nonlinear elastic slabs. But these model laws, as noted in [4], cannot describe materials with stress-strain laws with an interior change in the concavity corresponding to a change from soft to hard elastic behavior (or vice versa). Such an intriguing material response is encountered, for instance, in the compression of polycrystalline materials [4] and in materials like superelastic nickel-titanium (Ni-Ti) as used extensively in shape memory alloys with a wide range of medical device applications [5], [6].…”

“…But these model laws, as noted in [4], cannot describe materials with stress-strain laws with an interior change in the concavity corresponding to a change from soft to hard elastic behavior (or vice versa). Such an intriguing material response is encountered, for instance, in the compression of polycrystalline materials [4] and in materials like superelastic nickel-titanium (Ni-Ti) as used extensively in shape memory alloys with a wide range of medical device applications [5], [6].…”