# Negative Stiffness, Incompressibility, and Strain Localisation in Particulate Materials
## 1 Introduction  
Failure of heterogeneous or particulate materials in compression is a phenomenon important in different engineering fields such as mining, petroleum, and construction. Failure in compression is often preceded by strain localisation, whereby the deformation field becomes nonuniform, with the highest values of strain concentrating in a narrow zone inclined to the direction of compression, eventually leading to so-called shear failure. Strain localisation is often considered as a form of material instability. One of the possible mechanisms of strain localisation in the process of displacement-controlled loading is the emergence of areas of apparent negative stiffness caused by detachments and subsequent rotation of the material constituents. Negative stiffness stipulates that the directions of the force and displacement are opposite to each other such that, in order to increase deformation, the force needs to be reduced. While negative stiffness signifies negative elastic energy, which excludes the stability of such structures, in some cases, they can be stabilised by inserting into an encompassing (and sufficiently stiff) system such that the total energy of the compound system is positive (e.g., [1, 2]). Some systems do exhibit negative stiffness. For instance, negative stiffness is detected in hair bundles in the ear and some joints in human body [3, 4, 5]. It is observed in post-buckled columns, shells, and L-frames when the loading frame is sufficiently stiff [6, 7, 8]. Multivalued carbon nanotubes have also been found to exhibit negative stiffness [9, 10, 11]. The negative stiffness effect can be observed in some topological interlocking assemblies [12], as well as in specially engineered structures consisting of elastic elements such as links and levers with springs [13, 14, 15, 16], arches [17], and structureswith magnetic elements [18]. The latter are based on the fact that the magnetic force reduces with the distance, thus producing the effect of negative stiffness. The macroscopic negative stiffness phenomenon can occur in composite materials that contain negative stiffness inclusions [19; 20]. Apparent negative stiffness is also exhibited by various auxetic systems (e.g., [18; 21]). A number of applications of negative stiffness have been envisaged. A negative stiffness-based loading bearing structure was proposed in [22]. Devices of negative stiffness have been developed for the seismic protection of structures, car seat suspensions [4; 23; 24; 25; 26], and vibration isolation [27; 28; 29; 30; 31; 32; 33]. A novel actuator that combines positive and negative stiffness mechanisms was proposed in [32], and a three-dimensional base isolation system that uses a negative stiffness device was discussed in [33]. The dynamics of rotation of non-spherical particles was studied by modelling the system as an inverted pendulum under displacement-controlled loading [34]. The conditions of stability of materials with negative stiffness elements and the methods of stabilisation were discussed in [34; 35; 36; 37]. Negative compressibility in some anisotropic materials [38] leaves the energy positive definite (see also [39]); such situations are not considered here. Incremental negative stiffness (when strain is increasing) is also observed in the loading of many rocks, concrete, and granular materials during the post-peak softening stage (e.g., [39; 40; 41; 42]), i.e., the stage of stress-strain relationship obtained in displacement-controlled loading when stress decreases with strain increase. As the post-peak softening stage is usually preceded by strain localisation, we assume that the negative stiffness can constitute one of the localisation mechanisms. Another possible mechanism of strain localisation is the emergence of a state of incremental incompressibility (incompressibility in terms of stress/strain increments) emerged due to dilatancy (inelastic volume increase due to fracture propagation in the sample in compression).