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We have created a flat piling of disks in a numerical experiment using the Distinct Element Method (DEM) by depositing them under gravity. In the resulting pile, we then measured increments in stress and strain that were associated with a small decrease in gravity. We first describe the stress in terms of the strain using isotropic elasticity theory. Then, from a micro-mechanical view point, we calculate the relation between the stress and strain using the mean strain assumption. We compare the predicted values of Youngs modulus and Poissons ratio with those that were measured in the numerical experiment.
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Molecular dynamic simulation enables one to correlate the evolution of the micro-structure with anisotropic stress when a material is subject to strain. The anisotropic stress due to a constant strain-rate load in a cross-linked polymer is primarily dependent on the mean-square bond length and mean-square bond angle. Excluded volume interactions due to chain stacking and spatial distribution also has a bearing on the stress response. The bond length distribution along the chain is not uniform. Rather, the bond lengths at the end of the chains are larger and uniformly decrease towards the middle of the chain from both ends. The effect is due to the presence of cross-linkers. As with linear polymers, at high density values, changes in mean-square bond length dominates over changes in radius of gyration and end-to-end length. That is, bond deformations dominate over changes in size and shape. A large change in the mean-square bond length reflects in a jump in the stress response. Short-chain polymers more or less behave like rigid molecules. Temperature has a peculiar effect on the response in the sense that even though bond lengths increase with temperature, stress response decreases with increasing temperature. This is due to the dominance of excluded volume effects which result in lower stresses at higher temperatures. At low strain rates, some relaxation in the bond stretch is observed from $epsilon=0.2$ to $epsilon=0.5$. At high strain rates, internal deformation of the chains dominate over their uncoiling leading to a rise in the stress levels.
The mechanical strength and flow of granular materials can depend strongly on the shapes of individual grains. We report quantitative results obtained from photoelasticimetry experiments on locally loaded, quasi-two-dimensional granular packings of either disks or pentagons exhibiting stick-slip dynamics. Packings of pentagons resist the intruder at significantly lower packing fractions than packings of disks, transmitting stresses from the intruder to the boundaries over a smaller spatial extent. Moreover, packings of pentagons feature significantly fewer back-bending force chains than packings of disks. Data obtained on the forward spatial extent of stresses and back-bending force chains collapse when the packing fraction is rescaled according to the packing fraction of steady state open channel formation, though data on intruder forces and dynamics do not collapse. We comment on the influence of system size on these findings and highlight connections with the dynamics of the disks and pentagons during slip events.
We perform computational studies of repulsive, frictionless disks to investigate the development of stress anisotropy in mechanically stable (MS) packings. We focus on two protocols for generating MS packings: 1) isotropic compression and 2) applied simple or pure shear strain $gamma$ at fixed packing fraction $phi$. MS packings of frictionless disks occur as geometric families (i.e. parabolic segments with positive curvature) in the $phi$-$gamma$ plane. MS packings from protocol 1 populate parabolic segments with both signs of the slope, $dphi/dgamma >0$ and $dphi/dgamma <0$. In contrast, MS packings from protocol 2 populate segments with $dphi/dgamma <0$ only. For both simple and pure shear, we derive a relationship between the stress anisotropy and dilatancy $dphi/dgamma$ obeyed by MS packings along geometrical families. We show that for MS packings prepared using isotropic compression, the stress anisotropy distribution is Gaussian centered at zero with a standard deviation that decreases with increasing system size. For shear jammed MS packings, the stress anisotropy distribution is a convolution of Weibull distributions that depend on strain, which has a nonzero average and standard deviation in the large-system limit. We also develop a framework to calculate the stress anisotropy distribution for packings generated via protocol 2 in terms of the stress anisotropy distribution for packings generated via protocol 1. These results emphasize that for repulsive frictionless disks, different packing-generation protocols give rise to different MS packing probabilities, which lead to differences in macroscopic properties of MS packings.
The structural properties of a linear polymer and its evolution in time have a strong bearing on its anisotropic stress response. The mean-square bond length and mean bond angle are the critical parameters that influence the time-varying stress developed in the polymer. The bond length distribution along the chain is uniform without any abrupt changes at the ends. Among the externally set parameters such as density, temperature, strain rate, and chain length, the density as well as the chain length of the polymer have a significant effect on the stress. At high density values, changes in mean-square bond length dominates over changes in radius of gyration and end-to-end length. In other words, bond deformations dominate as opposed to changes in size and shape. Also, there is a large change in the mean-square bond length that is reflected as a jump in the stress. Beyond a particular value of the chain length, $n = 50$, called the entanglement length, stress-response is found to have distinctly different behavior which we attribute to the entanglement effects. Short chain polymers more or less behave like rigid molecules. There is no significant change in their internal structure when loaded. Further, temperature and rate of loading have a very mild effect on the stress. Besides these new results, we can now explain well known polymeric mechanical behavior under dynamic loading from the point of view of the evolution of the molecular dynamics and the derived structural properties. This could possibly lead to polymer synthesis with desired mechanical behavior.
Numerous models have been developed in the literature to simulate the thermomechanical behavior of amorphous polymer at large strain. These models generally show a good agreement with experimental results when the material is submitted to uniaxial loadings (tension or compression) or in case of shear loadings. However, this agreement is highly degraded when they are used in the case of combined load cases. A generalization of these models to more complex loads is scarce. In particular, models that are identified in tension or compression often overestimate the response in shear. One difficulty lies in the fact that 3D models must aggregate different physical modeling, described with different kinematics. This requires the use of transport operators complex to manipulate. In this paper, we propose a mechanical model for large strains, generalized in 3D, and precisely introducing the adequate transport operators in order to obtain an exact kinematic. The stress strain duality is validated in the writing of the power of internal forces. This generalized model is applied in the case of a polycarbonate amorphous polymers. The simulation results in tension/compression and shear are compared with the classical modeling and experimental results from the literature. The results highly improve the numerical predictions of the mechanical response of amorphous polymers submitted to any load case.
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