In the field of continuum mechanics, the fundamental objective is to describe how a body responds to external forces. To achieve this, we must establish a mathematical link between the movement of material points—known as the displacement field—and the resulting internal deformation, known as the strain field. This link is provided by the geometric equations (also referred to as kinematic relations).

In most practical engineering scenarios, such as the analysis of steel bridges, aircraft wings, or building frames, the magnitude of deformation is significantly smaller than the overall dimensions of the structure. This allows us to invoke the small deformation assumption (or infinitesimal strain theory). By linearizing the geometric equations, we drastically reduce the mathematical complexity of the problem, enabling the use of linear elasticity to solve complex structural issues.

Fundamental Assumptions

The validity of the small deformation theory rests on several critical physical and mathematical premises:

• Displacement Magnitude: The displacement vector $\mathbf{u}$ must be negligible compared to the characteristic length $L$ of the body ($|\mathbf{u}| \ll L$).
• Displacement Gradients: The spatial derivatives of the displacement, $\partial u_i / \partial x_j$, must be extremely small ($\ll 1$). This ensures that the change in geometry is minimal.
• Neglect of Higher-Order Terms: In the derivation of strain, products of displacement gradients (e.g., $\frac{\partial u_i}{\partial x_j} \cdot \frac{\partial u_k}{\partial x_l}$) are considered of a higher order and are discarded. This step is what transforms the non-linear relationship into a linearized one.
• Material Continuity: The material is assumed to be a continuum, meaning it is homogeneous and continuous within the domain of interest, without macroscopic cracks or voids that would invalidate the gradient-based approach.

When these conditions are met, we can move from the complex, non-linear finite strain theory to the much more manageable small strain theory.

The Mathematical Framework of Small Strains

1. The Displacement Gradient

Consider a material point moving from its initial (reference) position $\mathbf{X} = (X_1, X_2, X_3)$ to a new (deformed) position $\mathbf{x} = (x_1, x_2, x_3)$. The displacement vector $\mathbf{u}$ is defined as the difference between these two positions:

$$\mathbf{u} = \mathbf{x} - \mathbf{X}, \quad \text{where } u_i = x_i - X_i$$

The displacement gradient tensor describes how the displacement changes from one point to another:

$$\frac{\partial u_i}{\partial X_j}$$

This tensor contains all the information regarding both the deformation (stretching and shearing) and the rigid-body rotation of the material.

2. The Infinitesimal Strain Tensor

Under the small deformation assumption, we are primarily interested in the actual shape change of the material, not its rotation in space. Mathematically, the displacement gradient can be decomposed into a symmetric part and an anti-symmetric part. The symmetric part represents the strain, while the anti-symmetric part represents rigid-body rotation.

The small strain tensor (or linear strain tensor) $\varepsilon_{ij}$ is defined as the symmetric part of the displacement gradient:

$$\varepsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial X_j} + \frac{\partial u_j}{\partial X_i} \right), \quad i,j = 1, 2, 3$$

This equation is the core of the geometric relations. Because we have neglected the higher-order terms, $\varepsilon_{ij}$ is a linear function of the displacement derivatives, which is the cornerstone of linear structural analysis.

3. Engineering Notation (Voigt Notation)

For computational efficiency and ease of use in engineering practice, the strain tensor is often represented in a vectorized form known as Voigt notation. This maps the $3 \times 3$ symmetric tensor into a $6 \times 1$ column vector:

$$
\begin{bmatrix}
\varepsilon_{xx}\ \varepsilon_{yy}\ \varepsilon_{zz}\
\gamma_{xy}\ \gamma_{yz}\ \gamma_{zx}
\end{bmatrix}

\begin{bmatrix}
\displaystyle \frac{\partial u}{\partial x} \[6pt]
\displaystyle \frac{\partial v}{\partial y} \[6pt]
\displaystyle \frac{\partial w}{\partial z} \[6pt]
\displaystyle \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \[6pt]
\displaystyle \frac{\partial v}{\partial z}+\frac{\partial w}{\partial y} \[6pt]
\displaystyle \frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}
\end{bmatrix}
$$

In this notation, $\gamma_{ij} = 2\varepsilon_{ij}$ represents the engineering shear strain, which is more intuitive for describing the change in angle between two originally perpendicular lines.

Compatibility and Volumetric Change

1. The Compatibility Equations

A critical requirement in solid mechanics is that the strain field must correspond to a physically possible, continuous displacement field. If we were to assign arbitrary strain values to different points in a body, the material might "tear" or "overlap," which is physically impossible.

To prevent this, the strain components must satisfy the compatibility equations. In three dimensions, these are expressed as:

$$\frac{\partial^2 \varepsilon_{ij}}{\partial X_k \partial X_l}

• \frac{\partial^2 \varepsilon_{kl}}{\partial X_i \partial X_j}

• \frac{\partial^2 \varepsilon_{ik}}{\partial X_j \partial X_l}
• \frac{\partial^2 \varepsilon_{jl}}{\partial X_i \partial X_k} = 0$$

For simplified cases like plane strain or plane stress, these equations reduce to more manageable forms, such as:

$$\frac{\partial^2 \varepsilon_{xx}}{\partial y^2} + \frac{\partial^2 \varepsilon_{yy}}{\partial x^2} - 2\frac{\partial^2 \varepsilon_{xy}}{\partial x \partial y} = 0$$

These equations serve as a mathematical constraint that ensures the integrity of the displacement field.

2. Volumetric Strain

The change in volume of a material element under small deformation is directly related to the trace of the strain tensor. The volumetric strain (the ratio of the change in volume to the original volume) is given by:

$$\frac{\Delta V}{V} \approx \varepsilon_{xx} + \varepsilon_{yy} + \varepsilon_{zz} = \operatorname{tr}(\boldsymbol{\varepsilon})$$

This relationship is vital when analyzing phenomena such as thermal expansion or the effects of hydrostatic pressure.

Illustrative Examples

Example 1: Uniaxial Tension of a Rod

Consider a uniform rod of length $L$ and cross-sectional area $A$, subjected to an axial pulling force $P$ along the $x$-axis. We assume the displacement $u$ occurs only in the $x$-direction.

• Displacement Gradient: $\frac{du}{dx}$
• Small Strain: $\varepsilon_{xx} = \frac{du}{dx}$

According to Hooke's Law ($\sigma = E\varepsilon$) and the equilibrium condition for a constant force ($\sigma = P/A$), we have:

$$\varepsilon_{xx} = \frac{P}{AE} = \text{constant}$$

Integrating this gives the displacement field:
$$u(x) = \frac{P}{AE}x + C$$

If the end at $x=0$ is fixed ($u(0)=0$), then $C=0$, resulting in the linear displacement $u(x) = \frac{P}{AE}x$. This demonstrates the direct, linear link between force and displacement in the small deformation regime.

Example 2: Pure Shear in a Rectangular Plate

Imagine a rectangular plate of width $b$ and height $h$ undergoing plane strain deformation. If we apply a shear displacement $\pm \delta$ to the top and bottom boundaries, the displacement field can be approximated as:

$$u(x,y) = \gamma y, \quad v(x,y) = 0$$

where $\gamma$ is the shear strain.

• Strain Calculation:
  • $\varepsilon_{xx} = \frac{\partial u}{\partial x} = 0$
  • $\varepsilon_{yy} = \frac{\partial v}{\partial y} = 0$
  • $\gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} = \gamma$

For a linear elastic material with shear modulus $G$, the shear stress is $\tau = G\gamma$. Given the boundary condition $\delta = \gamma (h/2)$, we find:
$$\gamma = \frac{2\delta}{h}, \quad \tau = G\frac{2\delta}{h}$$

This example shows how the geometric equations allow us to solve for the internal state of a body using simple boundary conditions.

Summary

The assumption of small deformations is a cornerstone of modern engineering analysis. By linearizing the geometric equations, we define the small strain tensor $\varepsilon_{ij}$ as the symmetric part of the displacement gradient. This simplification allows for:

• Efficient Computation: Linear equations are significantly easier to solve than non-linear ones.
• Integration with Constitutive Laws: The linear strain tensor pairs perfectly with linear elastic models (like Hooke's Law) to form a complete analytical framework.
• Physical Consistency: Through compatibility equations, we ensure that the mathematical models respect the physical reality of material continuity.

Understanding these fundamental relations is an essential prerequisite for advancing into more complex domains, such as non-linear finite element analysis (FEA) and large-deformation plasticity.