7.2 Normal and shear strain

Normal vs Shear Strain

Definition and Characteristics

Normal strain measures how much a material stretches or compresses along the direction of an applied force. You calculate it by dividing the change in length by the original length. If you pull on a bar and it gets longer, that elongation relative to its starting length is the normal strain.

Shear strain measures angular distortion. When a force acts parallel to a surface rather than perpendicular to it, the material doesn't stretch or compress; it skews. Shear strain is defined as the tangent of the deformation angle, representing how much the shape distorts without changing volume.

Key Differences

The core distinction is the direction of deformation relative to the applied force:

• Normal strain occurs parallel to the force. It changes the length of the material but not its shape. Think of a rubber band stretching when you pull it: it gets longer, but it's still a band-shaped object.
• Shear strain occurs perpendicular to the force. It changes the shape of the material but not its volume. Picture a deck of cards sitting on a table: push the top card sideways, and the whole deck deforms into a parallelogram. No card got longer or shorter, but the overall shape changed.

Strain Calculation

Normal Strain Formula

Normal strain ($\varepsilon$) is the ratio of change in length to original length:

$\varepsilon = \frac{L - L_0}{L_0} = \frac{\delta}{L_0}$

where $L$ is the final length, $L_0$ is the original length, and $\delta = L - L_0$ is the change in length (often called the deformation or elongation).

Example: A steel bar has an initial length of 1 m. Under tension, it elongates to 1.001 m.

1. Find the change in length: $\delta = 1.001 - 1.000 = 0.001 \text{ m}$

2. Divide by the original length: $\varepsilon = \frac{0.001}{1.000} = 0.001$

3. Express as a percentage if needed: $0.001 \times 100 = 0.1\%$

Normal strain is dimensionless (length divided by length), but you'll often see it reported in units like mm/mm, in/in, or as a percentage. A positive value means tension (elongation); a negative value means compression (shortening).

Shear Strain Formula

Shear strain ($\gamma$) is defined as:

$\gamma = \tan(\theta)$

where $\theta$ is the angle of deformation (the angle the deformed edge makes with its original orientation). For small angles, $\tan(\theta) \approx \theta$ (in radians), so you'll often see shear strain written simply as the angle in radians.

An equivalent form that's useful in practice:

$\gamma = \frac{\Delta x}{y}$

where $\Delta x$ is the lateral (transverse) displacement and $y$ is the height (perpendicular distance from the fixed face to the displaced face).

Example: A rectangular block deforms by an angle of 2° under shear stress.

1. Convert to radians if using the small-angle approximation: $\theta = 2° \times \frac{\pi}{180°} = 0.0349 \text{ rad}$
2. Calculate: $\gamma = \tan(2°) = 0.0349$
3. As a percentage: $0.0349 \times 100 \approx 3.5\%$

Notice that for this small angle, $\tan(\theta)$ and $\theta$ in radians are nearly identical. This is why the small-angle approximation works well for most engineering applications where deformations are tiny.

Consistency in Units

Strain is dimensionless, but you still need to be careful: the numerator and denominator must use the same units. If your original length is in meters, your change in length must also be in meters. Mixing mm and m is one of the most common mistakes in strain calculations.

Strain and Deformation

Normal Strain Deformation

Picture a rectangular bar fixed at one end with a tensile force pulling on the other end. The bar elongates along its length, and every cross-section along the bar moves slightly away from the fixed end. Under compression, the opposite happens: the bar shortens. In both cases, the cross-sectional shape stays the same; only the length changes.

A real-world example: a concrete column shortening under the weight of a building above it. The column gets slightly shorter (compressive normal strain), even though the deformation is too small to see with the naked eye.

Shear Strain Deformation

Now picture a rectangular block glued to a table with a horizontal force pushing across its top face. The block skews into a parallelogram. The bottom face stays fixed, the top face slides sideways, and the angle between the vertical edge and its new tilted position is the shear strain angle $\theta$.

A simple analogy: a book on a shelf tilting when you push its top edge sideways. The pages slide relative to each other, and the book's profile goes from a rectangle to a parallelogram.

Combined Deformations

In textbook diagrams, deformations are exaggerated so you can actually see what's happening. Real engineering strains are tiny: a normal strain of 0.001 (0.1%) is common for metals under working loads.

In practice, materials often experience normal and shear strains simultaneously. A beam under bending, for instance, has normal strain (tension on one side, compression on the other) and shear strain (from internal shear forces). Analyzing these combined states is where strain concepts become especially powerful.

Stress and Strain in Materials

Definitions

• Stress ($\sigma$ for normal, $\tau$ for shear) is the internal force per unit area acting within a material.
• Strain ($\varepsilon$ for normal, $\gamma$ for shear) is the resulting deformation.

Stress is the cause; strain is the effect. You apply a load (creating stress), and the material deforms (producing strain).

Hooke's Law

Within the elastic region, stress and strain are directly proportional. This linear relationship is Hooke's Law:

For normal stress and strain:

$\sigma = E \cdot \varepsilon$

where $E$ is the modulus of elasticity (also called Young's modulus). A higher $E$ means the material is stiffer and deforms less for a given stress. Steel, for example, has $E \approx 200 \text{ GPa}$, while aluminum has $E \approx 70 \text{ GPa}$.

For shear stress and strain:

$\tau = G \cdot \gamma$

where $G$ is the shear modulus (also called the modulus of rigidity). For steel, $G \approx 77 \text{ GPa}$.

Both $E$ and $G$ are material properties, meaning they depend on what the material is, not on the geometry of the part.

Elastic Limit and Plastic Deformation

• Elastic behavior: Below the elastic limit, the material returns to its original shape when the load is removed. The stress-strain relationship is linear (Hooke's Law applies).
• Plastic behavior: Beyond the elastic limit, permanent deformation occurs. The stress-strain curve becomes nonlinear, and removing the load leaves the material in a deformed state.

A familiar example: bend a metal paperclip slightly and it springs back (elastic). Bend it far enough and it stays bent (plastic deformation). The transition point between these behaviors is the yield point.

Stress-Strain Curve

The stress-strain curve plots stress (vertical axis) against strain (horizontal axis) for a material tested to failure. For a ductile material like mild steel, the key regions are:

• Elastic region — linear, governed by Hooke's Law
• Yield point — where plastic deformation begins
• Strain hardening (plastic region) — stress continues to increase but the relationship is nonlinear
• Ultimate tensile strength — the maximum stress the material can sustain
• Fracture point — where the material breaks

This curve is one of the most important tools for understanding how a material will behave under load. Different materials produce very different curves: brittle materials like cast iron fracture with little plastic deformation, while ductile materials like steel show significant yielding before failure.