Relationship Between Displacement Gradient and Strain

In the study of continuum mechanics, understanding how a body changes shape under external loads is fundamental. This process, known as deformation, is mathematically described through the relationship between the movement of points within a body and the resulting internal distortions. To bridge the gap between the observed movement (kinematics) and the internal forces (mechanics), we must move from the concept of simple displacement to the more sophisticated concept of the displacement gradient and, ultimately, to strain.
Consider a solid body initially in a reference configuration. Any point in this body is identified by its position vector $\mathbf{X}$. After a load is applied, the body deforms and moves to a new, current configuration, where the same material point is now located at position $\mathbf{x}$. The vector representing this change is the displacement field, $\mathbf{u}(\mathbf{X})$, defined as:

$$\mathbf{u}(\mathbf{X}) = \mathbf{x}(\mathbf{X}) - \mathbf{X}$$

While the displacement field tells us where a point has moved, it does not, by itself, describe how the material has been stretched or sheared. To quantify local deformation, we must examine how the displacement varies from one point to another. This spatial rate of change is captured by the displacement gradient tensor, $\mathbf{H}$ (often denoted as $\nabla \mathbf{u}$):

$$H_{ij} = \frac{\partial u_i}{\partial X_j}$$

The displacement gradient is a complete kinematic description of the local motion. However, it is a "mixed" quantity: it contains both the information regarding the pure deformation of the material and the information regarding the rigid-body rotation of the local neighborhood. In structural analysis, we are primarily interested in the former, as rotations do not induce internal stresses, whereas deformations do.

The Small Deformation Regime: Linear Strain Theory

In many engineering applications—such as the analysis of steel structures under service loads—the displacements are extremely small relative to the dimensions of the body. Under this small deformation assumption (where $|\nabla \mathbf{u}| \ll 1$), we can simplify the mathematical treatment by decomposing the displacement gradient tensor into its symmetric and antisymmetric parts:

$$\mathbf{H} = \frac{1}{2}(\mathbf{H} + \mathbf{H}^T) + \frac{1}{2}(\mathbf{H} - \mathbf{H}^T)$$

1. The Small Strain Tensor (Infinitesimal Strain)

The symmetric part, $\boldsymbol{\epsilon} = \frac{1}{2}(\mathbf{H} + \mathbf{H}^T)$, is known as the infinitesimal strain tensor (or Cauchy strain tensor). This tensor isolates the pure deformation, representing the relative change in distance between neighboring points. Its components have specific physical meanings:

  • Normal Strains ($\epsilon_{11}, \epsilon_{22}, \epsilon_{33}$): These diagonal components represent the longitudinal stretching or compression along the coordinate axes.
  • Shear Strains ($\epsilon_{ij}$ where $i \neq j$): These off-diagonal components represent the change in angle between originally perpendicular lines. Note that the engineering shear strain ($\gamma_{ij}$) is exactly twice the tensor shear strain: $\gamma_{ij} = 2\epsilon_{ij}$.

2. The Rotation Tensor

The antisymmetric part, $\boldsymbol{\omega} = \frac{1}{2}(\mathbf{H} - \mathbf{H}^T)$, is the spin tensor (or rotation tensor). It describes the rate at which the local material element rotates without changing its shape or volume.

Example in 2D:
If a 2D displacement field is defined by $u_x = ax + by$ and $u_y = cx + dy$, the displacement gradient is:
$$\mathbf{H} = \begin{bmatrix} a & b \ c & d \end{bmatrix}$$
The resulting linear strain tensor is:
$$\boldsymbol{\epsilon} = \begin{bmatrix} a & \frac{b+c}{2} \ \frac{b+c}{2} & d \end{bmatrix}$$
Here, $a$ and $d$ are the normal strains, while $\frac{b+c}{2}$ is the tensor shear strain component.

Finite Deformation: Non-linear Strain Measures

When dealing with large displacements or large rotations—common in the analysis of polymers, rubber, or biological tissues—the linear approximation fails. In these cases, the geometric non-linearities become significant, and we must use the deformation gradient tensor, $\mathbf{F} = \mathbf{I} + \mathbf{H}$, to derive more robust strain measures.

Depending on whether we wish to describe deformation relative to the original state or the current state, we use different tensors:

  • Green-Lagrange Strain Tensor ($\mathbf{E}$):
    This is a Lagrangian (material) strain measure, meaning it is expressed in terms of the reference configuration. It is defined as:
    $$\mathbf{E} = \frac{1}{2}(\mathbf{F}^T\mathbf{F} - \mathbf{I})$$
    Because it is based on the initial geometry, it is the standard choice for formulating constitutive equations for hyperelastic materials, where the energy density is a function of the initial state.

  • Almansi Strain Tensor ($\mathbf{e}$):
    This is an Eulerian (spatial) strain measure, defined in the current, deformed configuration:
    $$\mathbf{e} = \frac{1}{2}(\mathbf{I} - \mathbf{F}^{-1}\mathbf{F}^{-T})$$
    The Almansi strain is particularly useful in fluid mechanics and certain types of stress analysis where the physical quantities are tracked in the deformed state.

Physical Significance and Engineering Application

At its core, any valid strain measure must quantify two fundamental geometric changes:

  1. Change in length: How much a material fiber has been stretched or compressed.
  2. Change in angle: How much the orthogonality between two material lines has been lost.

In modern computational engineering, specifically within Finite Element Analysis (FEA), the relationship between displacement and strain is the engine of the simulation. The software first calculates the nodal displacements, then uses interpolation functions to determine the displacement gradient at every integration point within an element. From this gradient, the appropriate strain tensor is computed. Finally, by applying a constitutive law (such as Hooke's Law for linear elastic materials, $\boldsymbol{\sigma} = \mathbf{D}:\boldsymbol{\epsilon}$), the software translates these strains into stresses, allowing engineers to predict material failure or structural stability.

Summary

The transition from displacement to strain is a journey from observing motion to quantifying deformation. While the displacement gradient provides a complete picture of local kinematics, it is the strain tensor that filters out rigid-body motions to reveal the true internal state of the material. Choosing between linear strain for small-scale engineering problems and Green-Lagrange or Almansi strain for large-deformation problems is a critical decision that ensures the mathematical model accurately reflects the physical reality of the material being studied.