mathematics

Kirsch equations

Kirsch equations is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Kirsch equations rather than just read about it. In short: The Kirsch equations describe the elastic stresses around a hole in an infinite plate under one directional tension. They are named after Ernst Gustav Kirsch.

Key takeaways

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

Reference excerpt

The Kirsch equations describe the elastic stresses around a hole in an infinite plate under one directional tension. They are named after Ernst Gustav Kirsch.

Result Loading an infinite plate with a circular hole of radius a with stress σ, the resulting stress field is (the angle is with respect to the direction of application of the stress):

σ r r = σ 2 ( 1 − a 2 r 2 ) + σ 2 ( 1 + 3 a 4 r 4 − 4 a 2 r 2 ) cos ⁡ 2 θ {\displaystyle \sigma _{rr}={\frac {\sigma }{2}}\left(1-{\frac {a^{2}}{r^{2}}}\right)+{\frac {\sigma }{2}}\left(1+3{\frac {a^{4}}{r^{4}}}-4{\frac {a^{2}}{r^{2}}}\right)\cos 2\theta }

σ θ θ = σ 2 ( 1 + a 2 r 2 ) − σ 2 ( 1 + 3 a 4 r 4 ) cos ⁡ 2 θ {\displaystyle \sigma _{\theta \theta }={\frac {\sigma }{2}}\left(1+{\frac {a^{2}}{r^{2}}}\right)-{\frac {\sigma }{2}}\left(1+3{\frac {a^{4}}{r^{4}}}\right)\cos 2\theta }

σ r θ = − σ 2 ( 1 − 3 a 4 r 4 + 2 a 2 r 2 ) sin ⁡ 2 θ {\displaystyle \sigma _{r\theta }=-{\frac {\sigma }{2}}\left(1-3{\frac {a^{4}}{r^{4}}}+2{\frac {a^{2}}{r^{2}}}\right)\sin 2\theta }

References Kirsch, 1898, Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre. Zeitschrift des Vereines deutscher Ingenieure, 42, 797–807.

Worked examples

Example 1 — a first encounter with Kirsch equations

Start with the simplest possible case. Write down what Kirsch equations claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Kirsch equations 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 Kirsch equations 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 Kirsch equations

In research: Kirsch equations appears in mathematics 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 Kirsch equations 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: Kirsch equations is common in secondary-school and first-year university syllabi. It links to neighbouring topics Solid mechanics, so understanding it makes those chapters shorter.
In everyday life: Look for Kirsch equations 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.

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How to study Kirsch equations in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Kirsch equations 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 Kirsch equations out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Kirsch equations in simple terms?

The Kirsch equations describe the elastic stresses around a hole in an infinite plate under one directional tension. They are named after Ernst Gustav Kirsch.

Why does Kirsch equations matter?

Because it connects several mathematics 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 Kirsch equations?

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 Kirsch equations.