physics

Willam–Warnke yield criterion

Willam–Warnke yield criterion is a physics 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 Willam–Warnke yield criterion rather than just read about it. In short: The Willam–Warnke yield criterion is a function that is used to predict when failure will occur in concrete and other cohesive-frictional materials such as rock, soil, and ceramics. This yield criterion has the functional form f ( I 1 , J 2 , J 3 ) = 0 {\displaystyle f(I_{1},J_{2},J_{3})=0\,} where I 1 {\displaystyle I_{1}} is the first invariant of the Cauchy stress tensor, and J 2 , J 3 {\displaystyle J_{2},J_{3}}…

Key takeaways

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

Reference excerpt

The Willam–Warnke yield criterion is a function that is used to predict when failure will occur in concrete and other cohesive-frictional materials such as rock, soil, and ceramics. This yield criterion has the functional form

f ( I 1 , J 2 , J 3 ) = 0 {\displaystyle f(I_{1},J_{2},J_{3})=0\,}

where I 1 {\displaystyle I_{1}} is the first invariant of the Cauchy stress tensor, and J 2 , J 3 {\displaystyle J_{2},J_{3}} are the second and third invariants of the deviatoric part of the Cauchy stress tensor. There are three material parameters ( σ c {\displaystyle \sigma _{c}} - the uniaxial compressive strength, σ t {\displaystyle \sigma _{t}} – the uniaxial tensile strength, σ b {\displaystyle \sigma _{b}} - the equibiaxial compressive strength) that have to be determined before the Willam-Warnke yield criterion may be applied to predict failure. In terms of I 1 , J 2 , J 3 {\displaystyle I_{1},J_{2},J_{3}} , the Willam-Warnke yield criterion can be expressed as

f := J 2 + λ ( J 2 , J 3 ) ( I 1 3 − B ) = 0 {\displaystyle f:={\sqrt {J_{2}}}+\lambda (J_{2},J_{3})~({\tfrac {I_{1}}{3}}-B)=0}

where λ {\displaystyle \lambda } is a function that depends on J 2 , J 3 {\displaystyle J_{2},J_{3}} and the three material parameters and B {\displaystyle B} depends only on the material parameters. The function λ {\displaystyle \lambda } can be interpreted as the friction angle which depends on the Lode angle ( θ {\displaystyle \theta } ). The quantity B {\displaystyle B} is interpreted as a cohesion pressure. The Willam-Warnke yield criterion may therefore be viewed as a combination of the Mohr–Coulomb and the Drucker–Prager yield criteria.

Willam-Warnke yield function

In the original paper, the three-parameter Willam-Warnke yield function was expressed as

f = 1 3 z I 1 σ c + 2 5 1 r ( θ ) J 2 σ c − 1 ≤ 0 {\displaystyle f={\cfrac {1}{3z}}~{\cfrac {I_{1}}{\sigma _{c}}}+{\sqrt {\cfrac {2}{5}}}~{\cfrac {1}{r(\theta )}}{\cfrac {\sqrt {J_{2}}}{\sigma _{c}}}-1\leq 0}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Willam–Warnke yield criterion

Start with the simplest possible case. Write down what Willam–Warnke yield criterion 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 Willam–Warnke yield criterion 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 Willam–Warnke yield criterion 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 Willam–Warnke yield criterion

In research: Willam–Warnke yield criterion 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 Willam–Warnke yield criterion 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: Willam–Warnke yield criterion is common in secondary-school and first-year university syllabi. It links to neighbouring topics Plasticity (physics), Yield criteria, so understanding it makes those chapters shorter.
In everyday life: Look for Willam–Warnke yield criterion 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Willam–Warnke yield criterion” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Willam–Warnke yield criterion in 20 minutes

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

Frequently asked questions

What is Willam–Warnke yield criterion in simple terms?

Why does Willam–Warnke yield criterion 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 Willam–Warnke yield criterion?

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 Willam–Warnke yield criterion.