Definition and Physical Significance of the Strain Tensor
In the realm of continuum mechanics, understanding how a body deforms under external loads is a fundamental pursuit. Strain serves as the quantitative measure of this deformation. Because deformation in three-dimensional space is inherently directional, a simple scalar falls short of capturing the complete state of distortion at a given point. Consequently, we must introduce the strain tensor, a second-order tensor that rigorously encapsulates the geometry of deformation.
When a continuous medium is subjected to external forces, the position of any arbitrary point within it shifts. Let the coordinates of a point in the undeformed state (the reference configuration) be denoted by $X_i$, and its coordinates in the deformed state (the current configuration) by $x_i$. The difference between these two positions defines the displacement vector $u_i$:
$$u_i = x_i - X_i \quad (i=1, 2, 3)$$
However, displacement alone does not equate to deformation. A body can undergo rigid body translation or rotation, changing the spatial coordinates of its points without altering the relative distances or angles between them. To isolate pure deformation from rigid body motion, we must examine the spatial rate of change of displacement, formally known as the displacement gradient tensor $\frac{\partial u_i}{\partial X_j}$.
In the vast majority of engineering applications—ranging from civil infrastructure to machine components—deformations are exceedingly small. Under this small deformation assumption, higher-order terms become negligible, allowing us to define the infinitesimal strain tensor $\varepsilon_{ij}$.
The displacement gradient tensor can be mathematically decomposed into a symmetric part and an antisymmetric (skew-symmetric) part:
$$\frac{\partial u_i}{\partial x_j} = \underbrace{\frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)}{\text{Symmetric part (Strain)}} + \underbrace{\frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right)}{\text{Antisymmetric part (Rotation)}}$$
The symmetric part of this decomposition yields the strain tensor $\varepsilon_{ij}$:
$$\varepsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)$$
While a general second-order tensor in 3D space consists of 9 components, the inherent symmetry of the strain tensor ($\varepsilon_{ij} = \varepsilon_{ji}$) reduces the number of independent components to just 6.
Physical Significance of the Strain Components
The individual components of the strain tensor correspond to distinct modes of geometric alteration within the material.
1. Normal Strain
The diagonal elements $\varepsilon_{11}, \varepsilon_{22},$ and $\varepsilon_{33}$ (frequently written as $\varepsilon_x, \varepsilon_y, \varepsilon_z$) are designated as normal strains.
- Physical Meaning: They represent the change in length per unit length (elongation or contraction) along the respective coordinate axes.
- Mathematical Form: For instance, $\varepsilon_{11} = \frac{\partial u_1}{\partial x_1}$.
- Sign Convention: A positive value ($\varepsilon_{ii} > 0$) indicates tensile stretching, whereas a negative value ($\varepsilon_{ii} < 0$) signifies compressive shortening.
2. Shear Strain
The off-diagonal elements $\varepsilon_{12}, \varepsilon_{23},$ and $\varepsilon_{31}$ are known as the shear strain components.
- Physical Meaning: They quantify the change in the angle between two line segments that were initially perpendicular to each other.
- Engineering Shear Strain: In conventional engineering practice, it is customary to utilize the engineering shear strain $\gamma_{ij}$, defined as $\gamma_{ij} = 2\varepsilon_{ij}$.
- Geometric Interpretation: For example, $\varepsilon_{12}$ represents half of the angular deviation from the original right angle ($\pi/2$) between the $x$ and $y$ axes.
Illustrative Examples: Two-Dimensional Deformation
To solidify these concepts, consider a rectangular element in the $xy$-plane subjected to a simple displacement field:
$$u_x = 0.01x, \quad u_y = 0.02y$$
Applying the definition of the strain tensor, we compute the components:
- $\varepsilon_{xx} = \frac{\partial u_x}{\partial x} = 0.01$ (a 1% tensile stretch in the $x$-direction)
- $\varepsilon_{yy} = \frac{\partial u_y}{\partial y} = 0.02$ (a 2% tensile stretch in the $y$-direction)
- $\varepsilon_{xy} = \frac{1}{2} \left( \frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x} \right) = \frac{1}{2}(0 + 0) = 0$ (no angular distortion)
Now, alter the displacement field to $u_x = 0.01y$ and $u_y = 0$. The strain components become:
- $\varepsilon_{xx} = 0, \quad \varepsilon_{yy} = 0$
- $\varepsilon_{xy} = \frac{1}{2} (0.01 + 0) = 0.005$
In this scenario, the rectangular element is distorted into a parallelogram, demonstrating pure shear deformation with no change in length along the coordinate axes.
Essential Properties of the Strain Tensor
The strain tensor possesses several profound mathematical and physical characteristics:
- Coordinate Invariance: As a true second-order tensor, the physical reality of strain remains unaltered regardless of the coordinate system chosen. While the individual components transform with the basis, intrinsic properties such as principal strains are invariant.
- Volumetric Strain and the Trace: The trace of the strain tensor, $\text{tr}(\varepsilon) = \varepsilon_{11} + \varepsilon_{22} + \varepsilon_{33}$, carries a distinct physical interpretation: it represents the volumetric strain, or the relative change in volume per unit volume ($\Delta V/V_0$), under the assumption of small deformations.
- Principal Strains: By performing an eigenvalue decomposition on the strain tensor, one can identify a specific orthogonal coordinate system where all shear strains vanish. The diagonal elements in this principal coordinate system are the principal strains ($\varepsilon_1, \varepsilon_2, \varepsilon_3$). This is crucial for evaluating material failure, as it identifies the orientations experiencing the most severe stretching or compression.
Conclusion
The strain tensor acts as the indispensable bridge connecting geometric deformation to a material's internal mechanical response. By extracting the symmetric part of the displacement gradient, it successfully filters out rigid body rotations and rigorously quantifies states of tension, compression, and shear. A firm grasp of the definition and physical significance of the strain tensor is the cornerstone for delving into constitutive relationships—such as the generalized Hooke's Law ($\sigma = C\varepsilon$)—and for predicting the structural failure of complex engineering systems.