physics

Piola–Kirchhoff stress tensors

Piola–Kirchhoff stress tensors 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 Piola–Kirchhoff stress tensors rather than just read about it. In short: In the case of finite deformations, the Piola–Kirchhoff stress tensors (named for Gabrio Piola and Gustav Kirchhoff) express the stress relative to the reference configuration. This is in contrast to the Cauchy stress tensor which expresses the stress relative to the present configuration.

Key takeaways

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

Reference excerpt

In the case of finite deformations, the Piola–Kirchhoff stress tensors (named for Gabrio Piola and Gustav Kirchhoff) express the stress relative to the reference configuration. This is in contrast to the Cauchy stress tensor which expresses the stress relative to the present configuration. For infinitesimal deformations and rotations, the Cauchy and Piola–Kirchhoff tensors are identical. Whereas the Cauchy stress tensor σ {\displaystyle {\boldsymbol {\sigma }}} relates stresses in the current configuration, the deformation gradient and strain tensors are described by relating the motion to the reference configuration; thus not all tensors describing the state of the material are in either the reference or current configuration. Describing the stress, strain and deformation either in the reference or current configuration would make it easier to define constitutive models (for example, the Cauchy Stress tensor is variant to a pure rotation, while the deformation strain tensor is invariant; thus creating problems in defining a constitutive model that relates a varying tensor, in terms of an invariant one during pure rotation; as by definition constitutive models have to be invariant to pure rotations). The 1st Piola–Kirchhoff stress tensor, P {\displaystyle {\boldsymbol {P}}} is one possible solution to this problem. It defines a family of tensors, which describe the configuration of the body in either the current or the reference state. The first Piola–Kirchhoff stress tensor, P {\displaystyle {\boldsymbol {P}}} , relates forces in the present ("spatial") configuration with areas in the reference ("material") configuration.

P = J σ F − T {\displaystyle {\boldsymbol {P}}=J~{\boldsymbol {\sigma }}~{\boldsymbol {F}}^{-T}~}

where F {\displaystyle {\boldsymbol {F}}} is the deformation gradient and J = det F {\displaystyle J=\det {\boldsymbol {F}}} is the Jacobian determinant. In terms of components with respect to an orthonormal basis, the first Piola–Kirchhoff stress is given by

P i L = J σ i k F L k − 1 = J σ i k ∂ X L ∂ x k {\displaystyle P_{iL}=J~\sigma _{ik}~F_{Lk}^{-1}=J~\sigma _{ik}~{\cfrac {\partial X_{L}}{\partial x_{k}}}~\,\!}

Because it relates different coordinate systems, the first Piola–Kirchhoff stress is a two-point tensor. In general, it is not symmetric. The first Piola–Kirchhoff stress is the 3D generalization of the 1D concept of engineering stress. If the material rotates without a change in stress state (rigid rotation), the components of the first Piola–Kirchhoff stress tensor will vary with material orientation. The first Piola–Kirchhoff stress is energy conjugate to the deformation gradient. It relates forces in the current configuration to areas in the reference configuration. The second Piola–Kirchhoff stress tensor, S {\displaystyle {\boldsymbol {S}}} , relates forces in the reference configuration to areas in the reference configuration. The force in the reference configuration is obtained via a mapping that preserves the relative relationship between the force direction and the area normal in the reference configuration.

S = J F − 1 ⋅ σ ⋅ F − T . {\displaystyle {\boldsymbol {S}}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}~.}

In index notation with respect to an orthonormal basis,

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Piola–Kirchhoff stress tensors

Start with the simplest possible case. Write down what Piola–Kirchhoff stress tensors 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 Piola–Kirchhoff stress tensors 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 Piola–Kirchhoff stress tensors 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 Piola–Kirchhoff stress tensors

In research: Piola–Kirchhoff stress tensors 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 Piola–Kirchhoff stress tensors 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: Piola–Kirchhoff stress tensors is common in secondary-school and first-year university syllabi. It links to neighbouring topics Tensor physical quantities, so understanding it makes those chapters shorter.
In everyday life: Look for Piola–Kirchhoff stress tensors 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 Piola–Kirchhoff stress tensors in 20 minutes

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

Frequently asked questions

What is Piola–Kirchhoff stress tensors in simple terms?

Why does Piola–Kirchhoff stress tensors 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 Piola–Kirchhoff stress tensors?

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 Piola–Kirchhoff stress tensors.