Nonaffine Residual Displacements

• Nonaffine residual displacements are local, irreversible particle shifts that restore force balance in disordered solids after an imposed affine deformation.
• They are quantified via forward–backward strain protocols and mean-squared measures that capture critical scaling near yielding and jamming transitions.
• Experimental and computational studies use these metrics to predict stress relaxation, plastic flow, and eventual material failure in glasses, colloids, polymers, and crystalline systems.

Nonaffine residual displacements are the particle-level motions in solids and networks that cannot be described by a homogeneous, affine deformation field. In a crystalline or perfectly ordered solid under macroscopic strain, atomic displacements follow the imposed affine map. However, disorder or structural inhomogeneity generate local force imbalances after the affine step; mechanical equilibrium requires additional, non-affine displacements that restore force balance. The residual component refers specifically to the part of these non-affine motions that persists after external loading is reversed and the system is energy-minimized. This concept is central for understanding stress relaxation, plastic flow, mechanical rigidity, and the nature of yielding in glasses, colloids, polymers, and disordered crystalline systems.

1. Formal Definition and Decomposition

Let $\{\mathbf{r}_i^0\}$ be the reference positions of $N$ particles in a solid. Application of an affine shear of magnitude $\gamma$ transforms each particle’s position by a global deformation tensor $F(\gamma)$: $\mathbf{r}_i^A = F(\gamma)\cdot\mathbf{r}_i^0$ Subsequent energy minimization at fixed boundary conditions moves each particle to its true equilibrium position $\mathbf{r}_i^{\text{min}}$, and the nonaffine displacement is: $$\mathbf{u}^{\text{NA}}_i \equiv \mathbf{r}_i^{\text{min}} - \mathbf{r}_i^A$$ To measure nonaffine residual displacement, one performs a forward–backward strain protocol: apply a shear, minimize (to $\mathbf{r}_i^{\text{min}}$), reverse the affine part, and re-minimize to $\mathbf{r}_i^{\text{final}}$. The net residual is: $$\Delta \mathbf{r}_i \equiv \mathbf{r}_i^{\text{final}} - \mathbf{r}_i^0$$ The corresponding mean-squared measures are: $$\Delta^2_{\text{NA}}(\gamma) = \langle \|\mathbf{u}^{\text{NA}}_i\|^2 \rangle = \frac{1}{N} \sum_{i=1}^N \|\mathbf{u}_i^{\text{NA}}\|^2$$

$$\Delta^2_{\text{res}} = \frac{1}{N}\sum_{i=1}^N \|\Delta \mathbf{r}_i\|^2$$

Nonaffine residuals capture local rearrangements induced by disorder and are strictly zero for perfectly elastic or reversible motion; nonzero values indicate irreversible structural reorganization (Saw et al., 2016).

2. Physical Origin: Disorder, Connectivity Loss, and Force Balance

In amorphous solids or disordered networks, geometric frustration and absence of inversion symmetry ensure that affine deformation does not guarantee force balance on each particle. The resulting nonaffine field restores mechanical equilibrium by spatially heterogeneous shifts. In glasses and colloids, the key driver for increasing nonaffinity under shear is the loss of nearest-neighbor connectivity ("cage erosion"): $n_b(\gamma) = n_b^0 e^{-A \gamma}$ with $n_b^0$ the initial coordination and $A$ a fitted decay constant. The affine modulus $G_A$ is proportional to this connectivity, while the nonaffine component $G_{NA}$ introduces a negative (softening) correction: $G(\gamma) = G_A - G_{NA}(\gamma)$ which decreases monotonically with strain (Dang et al., 2015). In central-force models, the nonaffine equilibrium displacement is the unique solution of the force-balance equation: $$\mathbf{u}^{\mathrm{NA}} = -\mathbf{H}^{-1} \, \boldsymbol{\Xi}$$ where $\mathbf{H}$ is the Hessian matrix of the system and $\boldsymbol{\Xi}$ is the affine force imbalance vector (Zaccone et al., 2011, Zaccone et al., 2012).

3. Statistical Properties and Correlation Functions

Nonaffine residual displacement fields exhibit broad, often power-law tailed distributions. In molecular simulations, the probability density $P(D^2)$ for the scalar nonaffine measure $D^2$ follows: $P(D^2) \sim (D^2)^{-\alpha}$ with $\alpha$ dependent on strain and proximity to yielding (e.g., $\alpha \simeq 2.6-2.8$ below yield, increasing to $4.8$ in steady flow) (L et al., 2022, Priezjev, 2016, Priezjev, 2015). Two-point spatial correlation functions

$$C_{D^2}(r) = \frac{\langle D^2(\mathbf{r}_0) D^2(\mathbf{r}_0 + \mathbf{r}) \rangle - \langle D^2 \rangle^2}{\langle D^4 \rangle - \langle D^2 \rangle^2}$$

reveal exponential decay in the elastic regime, indicating finite correlation length $\xi$, and power-law decay ($C_{D^2}(r)\sim r^{-\beta}$) after yielding, reflective of scale-free plastic flow. Residual nonaffine fields thus retain memory of collective, system-spanning plastic rearrangements (L et al., 2022, Priezjev, 2019, Priezjev, 2016).

4. Connection to Mechanical Response and Yielding

Rigidity and stress relaxation in amorphous solids are universally controlled by the magnitude of nonaffine displacements. Saw et al. showed that stress–strain response under large, instantaneous athermal shear collapses onto a universal curve when parameterized by the nonaffine mean-squared displacement, mirroring equilibrium modulus softening by thermal fluctuations. Specifically, for both thermal and mechanical sampling, the normalized response

$$R(\gamma) \equiv \frac{\langle \sigma \rangle}{\gamma G_0}$$

versus $\langle \Delta_{NA}^2 \rangle$ falls on a master curve. Yielding occurs when $\langle \Delta_{NA}^2 \rangle$ exceeds a threshold (e.g., $0.0478$ at $\gamma_y = 0.05$), corresponding to average particle displacements much larger than vibrational amplitudes. Hence, the nonaffine mean-square is a microscopic order parameter for transition from elasticity to plasticity (Saw et al., 2016, Dang et al., 2015).

5. Experimental and Computational Quantification

Both experimental particle-tracking (colloids, polymer gels) and atomistic simulations employ geometric least-squares fitting to determine the best local affine map in a neighborhood $N_i$, with the nonaffine residual defined as

$$\mathbf{u}_i^{NA} = \mathbf{u}_i^{measured} - \mathbf{u}_i^{affine}$$

and mean-square $$\mathcal{A} = \frac{1}{N} \sum_{i=1}^{N} |\mathbf{u}_i|^2$$ (Basu et al., 2010). The correlated statistics, clustering analysis, and spatial profiles of $D^2_{min}$ extract clusters of irreversible events and predict nucleation loci for shear bands and macroscopic failure (Priezjev, 2016, Priezjev, 2019).

In crystalline systems, coarse-grained approaches project relative displacement fields onto affine and non-affine subspaces via minimization and projection operators, yielding a spectrum of non-affine modes (defect precursors) (Popli et al., 2019, Ganguly et al., 2012). Higher-order pseudo-continuum characterizations extend these concepts to experimental DIC or molecular statics, linking nonaffine hierarchy to incompatibility, lattice curvature, and generalized Nye tensors (Khorrami et al., 2021).

6. Scaling Near Marginality and Jamming

In marginally stable networks or systems near jamming, nonaffine residuals exhibit critical scaling. The squared norm of nonaffine displacement diverges as a power law with respect to distance to the jamming point: $$D^2 \propto (\phi_J - \phi)^{-\lambda}, \quad \lambda \simeq 2.7$$ while the participation ratio vanishes as spatial localization increases ($P \sim N^{-\gamma}, \gamma \simeq 0.32$), linked to a fractal dimension $d_f \simeq 2.0$. The tail exponent of the distribution relates directly to the fractal dimension of the nonaffine field [$\alpha \simeq 2.5$, $d_f =(3+\gamma)/(1+\gamma)$], matching predictions for dynamical heterogeneity near jamming and critical glasses (Ikeda et al., 2020).

7. Implications for Structural Relaxation, Plasticity, and Material Design

Nonaffine residual displacements underpin microscopic mechanisms of stress relaxation, plastic yielding, and the loss of rigidity in disordered materials. Their growth with strain marks transitions from elastic to plastic behavior, shear band nucleation, and ultimate material failure. Experiments and simulations conclusively link nonaffine magnitudes to softening of modulus, decline in free energy, and spatial signatures of plastic rearrangement clusters. The concept also provides predictive tools for fatigue life, localization of damage, and tuning mechanical response in polymer networks, colloidal glasses, or designed amorphous assemblies (Dang et al., 2015, Maestro et al., 2017, Jha, 27 Dec 2025).

Nonaffine residual displacements thus serve as a unifying microscopic order parameter across thermal, athermal, mechanical, and structural transitions in disordered solids, directly quantifying the accessible configuration space and the corresponding rigidity of the material.