# Negative Stiffness, Incompressibility, and Strain Localisation in Particulate Materials
## 3 Incrementally Incompressible Material-Fundamentals  
We discussed in [45] that incompressible elastic isotropic materials can allow non-trivial stress distributions in the absence of loading. Here, we further this concept and analyse a possible structure of such a stress distribution and determine if it can cause strain localisation. Consider a sample of particulate material (e.g., rock or concrete) in uniaxial compression under a constant lateral pressure (can be zero), as shown in Figure 3.  
Under compression, the load particulate materials exhibit dilatancy, which is a nonelastic increase of the volume. Incrementally, the effect of dilatancy leads to an apparent increase in Poisson's ratio such that the material can reach (and even exceed) the state of incompressibility (in plane strain, the 2D approximation we use here, the Poisson's ratio is equal to 1). It needs to be emphasised that dilatancy is irreversible; hence, strictly speaking, the incompressibility is achieved only when the load increment is positive (compression is considered to be positive). This property was used in selecting the appropriate nontrivial (nonzero) solution of the incremental equilibrium equations. In 2D, [PERSON]'s law and equilibrium equations for an incompressible elastic isotropic medium in the coordinate frame shown in Figure 3 lead to the following equations:  
\[\varepsilon_{11}=\frac{\sigma_{11}-\sigma_{22}}{E},\ \ \varepsilon_{22}=- \varepsilon_{11},\ \ \ \ \ \varepsilon_{12}=\frac{2\sigma_{12}}{E}, \tag{4}\]  
\[\frac{\partial^{2}u_{1}}{\partial x_{1}{}^{2}}+\frac{\partial^{2}u_{2}}{ \partial x_{1}\partial x_{2}}=0,\ \ \ \frac{\partial^{2}u_{2}}{\partial x_{2}{}^{2}}+\frac{\partial^{2}u_{1}}{ \partial x_{1}\partial x_{2}}=0, \tag{5}\]  
where \(E\) is the incremental [PERSON]'s modulus, and \(u_{i},\ \varepsilon_{ij},\ \sigma_{ij}\) are incremental displacement, strain, and stress, respectively. A nontrivial solution of Equation (5) is sought in the complex form  
\[u_{j}(x_{1},x_{2})=u_{j}^{0}\exp(i\omega(k_{1}x_{1}+k_{2}x_{2})), \tag{6}\]  
where \(u_{j}^{0},\ \omega\) are parameters and \(k_{1}^{2}+k_{2}^{2}=1\). Substituting this form into Equation (5) leads to a singular system for any unit vector **k** = (\(k_{1}\),\(k_{2}\)). Furthermore, it is shown that \(u_{2}^{0}=-u_{1}^{0}k_{2}/k_{1}\); therefore,  
\[u_{2}(x_{1},x_{2})=-u_{1}(x_{1},x_{2})k_{2}/k_{1}. \tag{7}\]  
Substituting Equation (7) into the condition of incompressibility (second equation in Equation (4)) gives \(k_{1}^{2}=k_{2}^{2}\). Since \(k_{1}^{2}+k_{2}^{2}=1\), one can write  
\[k_{1}=1/\sqrt{2},\ k_{2}=\pm 1/\sqrt{2}. \tag{8}\]  
Figure 3: A 2D model of a sample of particulate material under uniaxial compression applied through a stiff loading frame. The obvious boundary conditions written in _stress increments_ at the lateral surface are \(\sigma_{11}|_{\pm H_{1}}=\sigma_{12}|_{\pm H_{1}}=0\). According to Equations (6) and (7), \(\epsilon_{12}(x_{1},x_{2})=0\); hence, \(\sigma_{12}(x_{1},x_{2})=0\). Therefore, the second boundary condition is satisfied automatically. The first boundary condition due to incompressibility leads to \(\sigma_{22}|_{\pm H_{1}}=E\epsilon_{11}|_{\pm H_{1}}\). From Equations (6) and (8), one obtains  
\[\epsilon_{11}(x_{1},x_{2})=\frac{i\omega}{\sqrt{2}}u_{j}^{0}\exp\biggl{(} \frac{i\omega}{\sqrt{2}}(x_{1}\pm x_{2})\biggr{)}. \tag{9}\]  
By introducing  
\[y=\frac{x_{1}\pm x_{2}}{\sqrt{2}},\ |y|\leq\frac{H_{1}+H_{2}}{\sqrt{2}}, \tag{10}\]  
Strain in Equation (9) can be rewritten as  
\[\epsilon_{11}(x_{1},x_{2})=\frac{i\omega}{\sqrt{2}}u_{j}^{0}\exp(i\omega y)=A \cos(\omega y)+B\sin(\omega y). \tag{11}\]  
Since the dilatancy producing incompressibility is irreversible, only positive strains (compression is assumed positive) should be considered.