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Infinitesimal strain theory

Infinitesimal strain theory 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 Infinitesimal strain theory rather than just read about it. In short: In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as density and stiffness) at each point of space…

Infinitesimal strain theory — main illustration
Infinitesimal strain theory — illustration

Key takeaways

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

Reference excerpt

In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as density and stiffness) at each point of space can be assumed to be unchanged by the deformation. With this assumption, the equations of continuum mechanics are considerably simplified. This approach may also be called small deformation theory, small displacement theory, or small displacement-gradient theory. It is contrasted with the finite strain theory where the opposite assumption is made. The infinitesimal strain theory has wide applications in engineering. Stress analysis, for example, tries to predict the behavior of structures built from relatively stiff elastic materials, such as concrete and steel. Such analysis can be used to minimize a structure design's deformation under typical loads. However, this approximation demands caution in the case of thin flexible bodies, such as rods, plates, and shells which are susceptible to significant rotations, thus making the results unreliable.

Infinitesimal strain tensor For infinitesimal deformations of a continuum body, in which the displacement gradient tensor (2nd order tensor) is small compared to unity, i.e. ‖ ∇ u ‖ ≪ 1 {\displaystyle \|\nabla \mathbf {u} \|\ll 1} , it is possible to perform a geometric linearization of any one of the finite strain tensors used in finite strain theory, e.g. the Lagrangian finite strain tensor E {\displaystyle \mathbf {E} } , and the Eulerian finite strain tensor e {\displaystyle \mathbf {e} } . In such a linearization, the non-linear or second-order terms of the finite strain tensor are neglected. Thus we have

E = 1 2 ( ∇ X u + ( ∇ X u ) T + ( ∇ X u ) T ∇ X u ) ≈ 1 2 ( ∇ X u + ( ∇ X u ) T ) {\displaystyle \mathbf {E} ={\frac {1}{2}}\left(\nabla _{\mathbf {X} }\mathbf {u} +(\nabla _{\mathbf {X} }\mathbf {u} )^{T}+(\nabla _{\mathbf {X} }\mathbf {u} )^{T}\nabla _{\mathbf {X} }\mathbf {u} \right)\approx {\frac {1}{2}}\left(\nabla _{\mathbf {X} }\mathbf {u} +(\nabla _{\mathbf {X} }\mathbf {u} )^{T}\right)}

or

… excerpt ends here. Continue reading the full article.

Illustrations

Infinitesimal strain theory illustration
Infinitesimal strain theory: Plane strain state in a continuum.
Plane strain state in a continuum.
Infinitesimal strain theory: Spherical coordinates (r, θ, φ) as commonly used in physics: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.
Spherical coordinates (r, θ, φ) as commonly used in physics: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.

Worked examples

Example 1 — a first encounter with Infinitesimal strain theory

Start with the simplest possible case. Write down what Infinitesimal strain theory 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 Infinitesimal strain theory 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 Infinitesimal strain theory 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 Infinitesimal strain theory

In research
Infinitesimal strain theory 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 Infinitesimal strain theory 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
Infinitesimal strain theory is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elasticity (physics), Materials science, Mechanics, so understanding it makes those chapters shorter.
In everyday life
Look for Infinitesimal strain theory 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 Infinitesimal strain theory in 20 minutes

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

Frequently asked questions

What is Infinitesimal strain theory in simple terms?

In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally smaller) than any relevant dimension of the body; so…

Why does Infinitesimal strain theory 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 Infinitesimal strain theory?

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 Infinitesimal strain theory.

Tags

  • Elasticity (physics)
  • Materials science
  • Mechanics
  • Physical quantities
  • Solid mechanics

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