Pure shear

Key takeaways

Pure shear belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Pure shear to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pure shear from memory before moving on to harder problems.

Reference excerpt

In mechanics and geology, pure shear is a three-dimensional homogeneous flattening of a body. It is an example of irrotational strain in which body is elongated in one direction while being shortened perpendicularly. For soft materials, such as rubber, a strain state of pure shear is often used for characterizing hyperelastic and fracture mechanical behaviour. Pure shear is differentiated from simple shear in that pure shear involves no rigid body rotation. The deformation gradient for pure shear is given by:

F = [ 1 γ 0 γ 1 0 0 0 1 ] {\displaystyle F={\begin{bmatrix}1&\gamma &0\\\gamma &1&0\\0&0&1\end{bmatrix}}}

Note that this gives a Green-Lagrange strain of:

E = 1 2 [ γ 2 2 γ 0 2 γ γ 2 0 0 0 0 ] {\displaystyle E={\frac {1}{2}}{\begin{bmatrix}\gamma ^{2}&2\gamma &0\\2\gamma &\gamma ^{2}&0\\0&0&0\end{bmatrix}}} Here there is no rotation occurring, which can be seen from the equal off-diagonal components of the strain tensor. The linear approximation to the Green-Lagrange strain shows that the small strain tensor is:

ϵ = 1 2 [ 0 2 γ 0 2 γ 0 0 0 0 0 ] {\displaystyle \epsilon ={\frac {1}{2}}{\begin{bmatrix}0&2\gamma &0\\2\gamma &0&0\\0&0&0\end{bmatrix}}}

which has only shearing components. For the aforementioned deformation gradient, the eigenvalues of the right Cauchy-Green deformation tensor ( C = F T F = 2 E + I {\displaystyle \mathbf {C} =\mathbf {F} ^{T}\mathbf {F} =2\mathbf {E} +\mathbf {I} } , see Finite strain theory) are 1 , 1 + 2 γ + γ 2 {\displaystyle 1,1+2\gamma +\gamma ^{2}} and 1 − 2 γ + γ 2 {\displaystyle 1-2\gamma +\gamma ^{2}} . The volume change is given as J = det ( F ) = 1 − γ 2 {\displaystyle J=\det(F)=1-\gamma ^{2}} , which is not unity. In literature, a volume preserving formulation for F {\displaystyle F} is used to denote pure shear in large deformation. This is written in the principal coordinate frame as:

F = [ λ 0 0 0 1 0 0 0 1 / λ ] {\displaystyle F={\begin{bmatrix}\lambda &0&0\\0&1&0\\0&0&1/\lambda \end{bmatrix}}}

where λ {\displaystyle \lambda } is the principal stretch.

Worked examples

Example 1 — a first encounter with Pure shear

Start with the simplest possible case. Write down what Pure shear claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pure shear before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pure shear ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pure shear

In research: Pure shear appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Pure shear in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Pure shear is common in secondary-school and first-year university syllabi. It links to neighbouring topics Classical mechanics stubs, Continuum mechanics, Fluid mechanics, so understanding it makes those chapters shorter.
In everyday life: Look for Pure shear outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Pure shear in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pure shear means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Pure shear out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pure shear in simple terms?

In mechanics and geology, pure shear is a three-dimensional homogeneous flattening of a body. It is an example of irrotational strain in which body is elongated in one direction while being shortened perpendicularly.

Why does Pure shear matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pure shear?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pure shear.