# Small Displacements

### Editorial Reviews

Significant research has already been conducted on the soil-geosynthetic interaction under ultimate conditions, which is relevant to reinforcement of retaining walls and steep slopes. However, little research has been undertaken to investigate the properties and mechanisms that govern the soil-geosynthetic interaction under small displacements, which is relevant to applications such as the geosynthetic stabilization of pavement layers. While characterization of the maximum geosynthetic strength e.

The objective of this research is to develop a better understanding of the soil-geosynthetic interaction under small displacements using analytical, experimental, and field evaluations. Three studies were conducted on different aspects of soil-geosynthetic interaction under small displacements: Each study provides lessons and conclusions on specific aspects investigated in that study. Collectively, they suggest that the analytical model proposed in this study provides a good basis towards predicting the general performance of geosynthetic-stabilized pavements.

The analytical formulation of the SGC model indicates that soil-geosynthetic interaction under small displacements can be characterized by the stiffness of soil-geosynthetic composite , which is the slope of the linear relationship defined between the unit tension squared T2 versus displacements u in each point along the active length of a geosynthetic.

The linearity and uniqueness of the relationship between the unit tension squared T2 and displacements u throughout the active length of specimens tested in a comparatively large soil-geosynthetic interaction device were experimentally confirmed. Overall, the experimental results from the large-scale soil-geosynthetic interaction tests were found to be in good agreement with the adopted constitutive relationships and with the analytical predictions of the SGC model.

## Assuming Small Displacement: The Large Displacement Flag

Evaluation of experimental results from tests conducted to assess repeatability indicated that the variability of the estimated values for the constitutive parameters and and the stiffness of soil-geosynthetic composite are well within the acceptable ranges when compared to variations of other soil and geosynthetic properties. Suitability of the assumptions and outcomes of the model was also confirmed for a variety of testing conditions and materials.

Evaluation of the experimental data obtained from a subsequent experimental program involving small-scale soil-geosynthetic interaction tests indicates that although the assumptions of the analytical model do not fully conform to the conditions in a small-scale test, experimental results confirm the linearity and uniqueness of the relationship between the unit tension squared T2 and the displacements u throughout the specimen.

Evaluation of the results obtained from small- and large-scale interaction tests on five geosynthetics with a range of properties indicates that both large and small testing scales can be used for comparative evaluation of the stiffness of soil-geosynthetic composite among geosynthetics.

### Result Filters

This linearization implies that the Lagrangian description and the Eulerian description are approximately the same as there is little difference in the material and spatial coordinates of a given material point in the continuum. Therefore, the material displacement gradient components and the spatial displacement gradient components are approximately equal. Also, from the general expression for the Lagrangian and Eulerian finite strain tensors we have. From the geometry of Figure 1 we have. For very small displacement gradients, i. For small rotations, i. From finite strain theory we have.

Therefore, the diagonal elements of the infinitesimal strain tensor are the normal strains in the coordinate directions. In that case the components of the tensor are different, say. Certain operations on the strain tensor give the same result without regard to which orthonormal coordinate system is used to represent the components of strain. The results of these operations are called strain invariants. The most commonly used strain invariants are. Since there are no shear strain components in this coordinate system, the principal strains represent the maximum and minimum stretches of an elemental volume.

If we are given the components of the strain tensor in an arbitrary orthonormal coordinate system, we can find the principal strains using an eigenvalue decomposition determined by solving the system of equations. The dilatation the relative variation of the volume is the trace of the tensor:.

Real variation of volume top and the approximated one bottom: The deviatoric strain tensor can be obtained by subtracting the mean strain tensor from the infinitesimal strain tensor:.

An octahedral plane is one whose normal makes equal angles with the three principal directions. The engineering shear strain on an octahedral plane is called the octahedral shear strain and is given by.

## Assuming Small Displacement: The Large Displacement Flag

The normal strain on an octahedral plane is given by. A scalar quantity called the equivalent strain , or the von Mises equivalent strain, is often used to describe the state of strain in solids.

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Several definitions of equivalent strain can be found in the literature. A definition that is commonly used in the literature on plasticity is. Thus, a solution does not generally exist for an arbitrary choice of strain components. Therefore, some restrictions, named compatibility equations , are imposed upon the strain components. With the addition of the three compatibility equations the number of independent equations are reduced to three, matching the number of unknown displacement components.

These constraints on the strain tensor were discovered by Saint-Venant , and are called the " Saint Venant compatibility equations ". If the elastic medium is visualised as a set of infinitesimal cubes in the unstrained state, after the medium is strained, an arbitrary strain tensor may not yield a situation in which the distorted cubes still fit together without overlapping.

In real engineering components, stress and strain are 3-D tensors but in prismatic structures such as a long metal billet, the length of the structure is much greater than the other two dimensions. The strains associated with length, i. Plane strain is then an acceptable approximation.

The strain tensor for plane strain is written as:. This strain state is called plane strain. The corresponding stress tensor is:. This stress term can be temporarily removed from the analysis to leave only the in-plane terms, effectively reducing the 3-D problem to a much simpler 2-D problem.