In the previous chapter, a series of examples was given to illustrate a range of material responses that stem from impact or explosion and result from a transient loading pulse within the material. These drivers propel waves travelling through solids, liquids and gases and place the material they have swept through into a state of compression, tension or shear. This chapter will describe these disturbances in more detail and attempt to give simple mathematical descriptions of the phenomena and the material's response. This basic approach is really a development of solid (a branch of continuum) mechanics to embrace additional features of loading at higher speeds and amplitudes; there are many, more complete texts available on the basics of solid mechanics that the reader may consult. The strategy here is to keep the derivations as simple as possible; again there are texts that derive relations with more generality than here but it is vital that the reader realises the assumptions, and more importantly their limits, in what follows. Particularly, it should be noted that solid mechanics assumes material behaviour based upon observations made in ambient states. Electronic bonding itself changes nature at around 300 GPa, so it is unrealistic to expect theory extended from the elastic state to apply in these regimes. Thus assumptions made and their limitations in the loading states the reader wishes to consider must be fully understood before using the formulae below. The basic laws of conservation of mass, momentum and energy, and classical mechanics will drive the descriptions of the thermodynamic states. To that will be added the concepts of elastic and inelastic (in metals, plastic) deformation bounded by a yield surface. To focus on material response, it is generally the simplest loading that is applied experimentally. Thus these states will be mentioned below to highlight particular relations to which the text will return.

Akrology

Extremes

The dynamic processes operating around us are often treated as transients that are not important when compared with the fixed states they precede. However, an ever-increasing knowledge base has illuminated this view of the operating physics and confirmed that extreme regimes can be accessed for engineering materials and structures. Matter is ever-changing, its form developing in a series of nested processes which complete on the timescales on which mechanisms operate; processes that occur on ever smaller timescales as length scales decrease. This book is concerned with the response that occurs when loads exceed the elastic limit. This affects behaviour in the regime beyond yield which encompasses a range of amplitudes and responses. However, it concerns condensed materials and loading, eventually taking them to a state where they bond in a different manner such that strength is not defined; this limit represents the highest amplitude of loading considered here. Nonetheless the driving forces are vast and awe-inspiring, while the different rates of change observed in operating processes are on scales that span many orders of magnitude. The following pages will highlight prime examples from the physical world and then provide a set of tools that classify mechanisms in order to analyse significant effects of these processes on the materials involved. The wide range of observations and applications create simple but powerful principles that are outlined in what follows.

Materials are central to the technologies required for future needs. Such platforms will place increasing demands on component performance in a range of extremes: stress, strain, temperature, pressure, chemical reactivity, photon or radiation flux, and electric or magnetic fields. For example, future vehicles will demand lighter-weight parts with increased strength and damage tolerance and next-generation fission reactors will require materials capable of withstanding higher temperatures and radiation fluxes. To counter security threats, defence agencies must protect their populations against terrorist attack and design critical facilities and buildings against atmospheric extremes.

The scientific method

The present chapter gives an overview of experimental platforms showing how they may be used to populate models for materials behaviour. The condensed phase defines the pressure and temperature range of interest, which may be approximately fixed at less than 1 TPa and below 10000 K. Indeed pressure has one of the largest ranges of all physical parameters in the universe (the pressure in a neutron star is c. 1033 Pa), so that most of the materials in nature are under conditions very different from those on Earth. The goal of shock experiments is to track response and mechanisms across the realms of stress and volume that are experienced by condensed-phase matter across the universe. At the highest pressures and temperatures, materials move from the solid to the liquid and then to plasma states as new correlations and bonding are formed. These high-density states have been termed warm dense matter (WDM) and lie beyond the finis extremis – outside the regime of extreme behaviour considered here. A summary of the phase space occupied by matter in these regions is shown in Figure 3.1.

The goal of experimental work is to provide adequate knowledge of the response of matter over the operating regimes of the relevant plasticity mechanisms. By this means, analytical descriptions can be constructed to try and capture the fundamental relationships between the independent variables – stress and stress state, strain and strain rate, and temperature – that determine the constitutive, damage and failure behaviour of materials. A shock impulse provides a pump to drive materials deformation and control of that impulse also allows a window into the operative mechanisms that lead to plasticity and damage evolution. This includes determining dynamic strength as a function of pressure as well as determining equation of state over the range of interest for particular applications.

Specific heat anomaly

Bose liquids (or more precisely 4He) show the characteristic λ-point singularity of their specific heat, while superfluid Fermi liquids like the BCS superconductors exhibit a second-order phase transition accompanied by a finite jump in their specific heat [7]. It was established beyond doubt [277, 278, 279, 280, 281] that the anomaly in high-Tc cuprates differs qualitatively from the BSC prediction. As was stressed by Salamon et al. [282] the heat capacity is logarithmic near the transition, and consequently cannot be adequately treated by the mean-field BCS theory even including Gaussian fluctuations. In particular, estimates using the Gaussian fluctuations yield an unusually small coherence volume comparable with the unit cell volume [278]. Comparison of the specific heat of a few high-Tc superconductors with that of 4He found a perfect fit [283], where the λ-like specific heat anomaly points to real-space pairing.

It has also been observed that the resistive transition reveals an upper critical field with increasingly negative gradient on cooling, Section 7.1, in cuprate and some other unconventional superconductors. However, the λ-point of the specific heat scarcely shifts with applied magnetic field [284]. Either phenomenon is highly unusual in itself but also appears to be irreconcilable with the other under the BCS framework. The bipolaron theory reconciled these observations [151, 287].

Calculations of the specific heat of charged bosons in a magnetic field require an analytical DOS, N(E, B), of a particle scattered by other particles and/or by the random potential. One can use the DOS in the magnetic field with impurity scattering, Eq. (7.16), as in Section 7.1.

The 1986 epoch-making discovery [1] by J. Georg Bednorz and K. Alex Müller of superconductivity in cuprates at incredibly high temperatures by the accepted standards is now recognised as one of the greatest scientific revolutions of the twentieth century. This discovery inspired a multitude of researchers worldwide to synthesise materials that are superconducting at temperatures more than six times higher than the earlier ones. High-temperature superconductivity has completely transformed the landscape of solid state science; it has led to the discovery of new classes of materials, of new states of matter, and of new concepts.

Strikingly, after more than 25 years of extensive experimental and theoretical efforts there is still little consensus on the origin of high-temperature superconductivity. The only consensus there is is that charge carriers are bound into pairs with an integer spin. Pairing of two fermionic particles has been evidenced in cuprate superconductors from the quantization of magnetic flux in units of the flux quantum [2]. Soon after that discovery the late Sir Nevill Mott answering his own question: ‘Is there an explanation?’ [Nature 327 (1987) 185] expressed the view that the Bose–Einstein condensation (BEC) of small bipolarons, predicted by us in 1981, could be the one. Several authors then contemplated BEC of real-space tightly-bound electron pairs, but with a purely electronic mechanism of pairing rather than with an electron–phonon interaction (EPI). However a number of other researchers criticised the bipolaron (or any real-space pairing) scenario as incompatible with some observed angle-resolved photoemission spectra (ARPES), with effective masses of carriers, and an unconventional symmetry of the superconducting order parameter in cuprates.

Motivation

With few exceptions [131] it is widely accepted that the conventional Bardeen–Cooper–Schrieffer (BCS) theory [12] and its intermediate-coupling Eliashberg extension [14] do not suffice to explain high-temperature superconductivity. As shown in the preceding chapters the true origin of high-temperature superconductivity could be found in a proper combination of the bare Coulomb repulsion with the bare electron–phonon interaction in the framework of the many-body polaron theory [51] beyond the BCS–Eliashberg approximation.

Nevertheless some researchers have maintained that the repulsive electron–electron interaction alone should provide pairing at high temperatures without phonons despite substantial experimental evidence that the EPI is an important player and needs to be included in the successful theory of high-Tc superconductivity. Following the original proposal by P. W. Anderson, many authors [22, 74] argued that the electron–electron interaction in high-temperature superconductors was strong but repulsive providing high Tc without phonons via superexchange and/or spin fluctuations in the d-wave pairing channel (l = 2). A motivation for this concept can be found in the earlier works by Kohn and Luttinger (KL) [132, 133] who showed that the Cooper pairing of fermions with any weak repulsion was possible since the two-particle interaction induced by many-body effects is attractive for pairs with large orbital momenta, l ≫ 1.