Hooke's law and elastic constants E, G, nu

Hooke’s law states that within the proportional limit, stress is proportional to strain: and. The elastic constants,,, and are related by (RK Bansal / GATE SOM).

State Hooke’s law for uniaxial loading and name the constant of proportionality.

Within the proportional limit, stress is proportional to strain:. The constant is Young’s modulus (modulus of elasticity).

Write Hooke’s law in shear form.

where is the modulus of rigidity (shear modulus) and is the shear strain (in radians).

Why does Hooke’s law fail beyond the elastic limit?

Dislocations and permanent slip make the – relation nonlinear; unloading does not return to the origin. The material is no longer uniquely described by a single.

How is obtained experimentally from a tension test?

From the initial linear portion of the engineering stress–strain curve: slope (secant/tangent in the proportional range).

Relate,, and Poisson’s ratio.

. Measuring any two of determines the third for isotropic materials.

What is the difference between elastic limit and proportional limit?

Proportional limit: end of linearity. Elastic limit: maximum stress with no permanent set on unloading. They are close for many metals; proportional limit elastic limit.

Express strain energy density for uniaxial Hookean behaviour.

. This is the area under the linear – line.

An isotropic metal has and Poisson’s ratio. Compute the shear modulus and verify that the result is consistent with.

**Given:**,. **Find:** Shear modulus. **Assumptions:** Homogeneous isotropic linear elastic solid; two independent elastic constants. **Solution:** From the isotropic identity Check: — consistent. **Result:**. **Exam trap:** Using or (algebra flipped). Also mixing GPa and MPa in later steps.

A thin plate is under plane stress with,,. Take and. Find,, and.

**Given:**,,,,. **Find:** Normal strains. **Assumptions:** Plane stress; isotropic Hooke’s law; stresses within elastic range. **Solution:** Generalized Hooke: Thickness strain **Result:**,,. **Exam trap:** Treating plane stress as plane strain, or forgetting that even when. Sign error on.

Uniaxial tension produces stress in a bar with. Find the elastic strain energy density and the corresponding axial strain.

**Given:**,. **Find:** Strain energy density and strain. **Assumptions:** Linear elastic uniaxial Hooke behaviour; recoverable elastic energy only. **Solution:** Strain Energy density **Result:**,. **Exam trap:** Using (missing ), or writing without the factor 2.

A prismatic bar of length and area elongates by under axial load. Assuming linear elasticity, determine Young’s modulus.

**Given:**,,,. **Find:** Young’s modulus. **Assumptions:** Uniform axial stress; Hookean material; load along centroidal axis. **Solution:** From **Result:**. **Exam trap:** Inconsistent units (kN with mm without converting to N), or using instead of.

For a material, and. Find and the bulk modulus.

**Given:**,. **Find:** and. **Assumptions:** Isotropic linear elasticity; relations among valid. **Solution:** Bulk modulus **Result:**,. **Exam trap:** Using (wrong Poisson factor), or computing without the 2.

Under pure shear with, find the shear strain and the shear strain energy density.

**Given:**,. **Find:** Shear strain and shear energy density. **Assumptions:** Linear elastic shear Hooke’s law; pure shear state. **Solution:** **Result:**,. **Exam trap:** Reporting in degrees, or using normal-stress energy formula for shear.

Plane stress:,,,. Find the in-plane volumetric strain of the plate thickness-averaged sense and the through-thickness strain.

**Given:**,,,,. **Find:** and. **Assumptions:** Isotropic plane stress; equal biaxial tension. **Solution:** In-plane sum **Result:**,. **Exam trap:** Using 3-D hydrostatic formulae, or setting (that is plane strain, not plane stress).

A tensile coupon of gauge length stretches by at stress. Estimate. If, also find the change in diameter for an original diameter of.

**Given:**,,,,. **Find:** and diameter change. **Assumptions:** Uniaxial tension; isotropic Poisson effect; linear elasticity. **Solution:** Lateral strain (contraction). **Result:**,. **Exam trap:** Sign of (must contract in tension), or using without converting to strain.

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Core concept

Hooke's Law & Elastic Constants

Hooke’s law states that within the proportional limit, stress is proportional to strain: and.

31 min15 Interview8 GATEGATEInterview

Level

Advanced

Topic tags

Hookes law, Young modulus, Poisson ratio

Useful for

GATE favourite, Interview ask

Source

Strength of Materials

Learning outcomes

What you'll learn in this topic

1
Uniaxial Hooke: , .
2
Shear Hooke: ; $G$ = modulus of rigidity (shear modulus).
3
Bulk modulus: (hydrostatic pressure vs volumetric strain).
4
Interrelations: , , .

Key Formulas

Important equations & their meanings

E=2G(1+\nu)

Worked Examples

Step-by-step solved problems

1Find G and K from E and ν
2Elongation using Hooke’s law
3Volumetric strain under hydrostatic pressure

Common Mistakes

Avoid these errors in exams & interviews

Using E=2G(1-ν) instead of E=2G(1+ν).
Forgetting that ν is dimensionless while mixing E in GPa with σ in MPa.
Applying δ=PL/AE beyond the elastic limit.

Practice Questions

Strengthen your concepts with questions

23+Practice Questions

Interview Questions

Most asked interview problems

15Interview Questions

GATE Questions

Previous year GATE questions

8GATE Questions

Applications in real world

BeamsFrames and structural members
ShaftsTorsion and combined loading
Pressure vesselsHoop and longitudinal stress
Machine framesStiffness and strength checks

Visual concept

Problem

Model

Compute

Verify

Model the physics, compute, then verify against limits.

Recommended Book

Strength of MaterialsRK Bansal

Read: Ch. 2–3

View on Amazon

University exams

Important Topic

Industry relevance

High

Concept difficulty

Hard

Average time

31 min

Core assumptions (state these in exams)

1. Linear elastic behaviour — stress ∝ strain (proportional limit not exceeded).
2. Isotropic material — same

E

G

\nu

in all directions.
3. Homogeneous continuum.
4. Small strains — geometry linearised; superposition valid.
5. Isothermal (or constants measured at the working temperature).
6. No time-dependent effects (creep/relaxation neglected).

If the material is anisotropic (composites, wood), a stiffness matrix with more independent constants is required — elementary Hooke formulas do not apply directly.

Generalised Hooke’s law (3-D)

For principal (or Cartesian) stresses on an isotropic body:

\varepsilon_x=\frac{1}{E}\big[\sigma_x-\nu(\sigma_y+\sigma_z)\big]

\varepsilon_y=\frac{1}{E}\big[\sigma_y-\nu(\sigma_z+\sigma_x)\big]

\varepsilon_z=\frac{1}{E}\big[\sigma_z-\nu(\sigma_x+\sigma_y)\big]

Shear strains decouple:

\gamma_{xy}=\frac{\tau_{xy}}{G},\quad\gamma_{yz}=\frac{\tau_{yz}}{G},\quad\gamma_{zx}=\frac{\tau_{zx}}{G}

Volumetric strain:

\varepsilon_v=\varepsilon_x+\varepsilon_y+\varepsilon_z=\frac{1-2\nu}{E}(\sigma_x+\sigma_y+\sigma_z)=\frac{3(1-2\nu)}{E}\sigma_{\mathrm{mean}}

with

\sigma_{\mathrm{mean}}=\frac{\sigma_x+\sigma_y+\sigma_z}{3}

Step-by-step problem approach

1. Confirm loading is within the elastic range (or problem states elastic constants apply).
2. Identify known constants; pick the relation that solves for the unknown directly.
3. For

\delta

: use

\delta=\frac{PL}{AE}

with consistent units.
4. For 3-D stress: write generalised Hooke component-wise; sum for

\varepsilon_v

if needed.
5. Convert GPa ↔ MPa ↔ Pa carefully (

1\,\mathrm{GPa}=10^3\,\mathrm{MPa}=10^9\,\mathrm{Pa}

).
6. Sanity-check:

G\approx \frac{E}{2.6}

for

\nu\approx 0.3

;

K

should exceed

\frac{E}{3}

Advanced problem-solving framework

Use this sequence for long-form mastery and repeatable scoring:
1. Identify objective, system boundary, and required output.
2. Write all givens in SI units and classify each as measured, assumed, or estimated.
3. Choose the governing model and relation (Uniaxial Hooke:

\sigma=E\varepsilon

\delta=\frac{PL}{AE}

.) with one-line justification.
4. Solve symbolically first to catch structural mistakes early.
5. Substitute values with careful unit tracking.
6. Cross-check by sign, order of magnitude, and limiting case.
7. Write a short engineering conclusion tied to safety, performance, reliability, or cost.

Next, solve one "variant version" of the same problem by changing one assumption (loading type, losses, property constancy, boundary condition, or uncertainty level). This builds transfer ability — essential for difficult exams where numbers and wording are changed deliberately.

For viva/interviews, convert your written method into a 45-second explanation format:
"Objective -> model selected -> key assumption -> result -> practical implication."
This makes your answers concise and technically credible.

Exam, interview, and note-making strategy

To make this topic genuinely reusable, maintain notes in four blocks: concept summary, assumptions checklist, solved template, and common error-correction logic. This transforms passive reading into active revision material for class tests, semester exams, GATE-style practice, and interviews.

A practical weekly cycle:
- Day 1: read and annotate the topic.
- Day 3: solve one moderate numerical from memory.
- Day 5: give a 60-second oral explanation.
- Day 7: solve one mixed problem integrating this topic with a prerequisite.
- Day 14: do a timed review to test retention.

For interview readiness, prepare concise answers to:
1. Where is this used in real engineering?
2. Which assumption is most risky if wrong?
3. How do you sanity-check the result quickly?
4. What trade-off does this result influence?

These four questions are asked repeatedly in technical panels, and practicing them creates confidence.

Use this page as a living notebook: append class doubts, lab observations, previous-year tricks, and personal mnemonics. That personalization is what turns a study page into a repeat-visit resource students trust.

Industry scenarios and decision context

Engineering decisions are made under constraints: deadline, budget, material availability, process capability, safety requirements, and maintenance realities. So while solving hooke's law & elastic constants, do not treat the answer as "final truth" without context. The numerical output is a decision input, not the decision itself.

Ask these context questions after every solved example:
- If load uncertainty increases, does design margin remain acceptable?
- If manufacturing tolerance drifts, will performance degrade critically?
- If operating temperature/humidity changes, are properties still valid?
- If maintenance is delayed, what failure mode appears first?

Students who practice contextual questioning develop judgment faster and perform better in internships, design tasks, and technical interviews. This context-first style is a major retention driver because learners see immediate real-world value.

Long-form revision worksheet

Use this worksheet when preparing notes:
A) One-paragraph concept explanation in your own words.
B) Symbol and units table for key variables.
C) Validity limits and assumptions list.
D) One standard solved pattern with all steps.
E) One variant problem where an assumption changes.
F) One industry-use explanation with failure consequence.
G) Three common mistakes and their correction rules.

If you can fill all seven blocks without external help, your topic depth is strong enough for repeat use and long retention. If not, revisit the corresponding section and strengthen the missing block.

This structured worksheet approach is intentionally longer than quick revision notes because it is designed for durable mastery. It supports exactly the product goal you mentioned: students should keep coming back because the page is complete enough to build serious notes.

Definition and physical meaning

Hooke’s law (uniaxial): within the proportional limit, normal stress is linearly proportional to normal strain:

\sigma=E\varepsilon\qquad\Rightarrow\qquad E=\frac{\sigma}{\varepsilon}

E

is Young’s modulus (modulus of elasticity) — slope of the initial straight portion of the

\sigma

–

\varepsilon

curve.

Shear form:

\tau=G\gamma\qquad\Rightarrow\qquad G=\frac{\tau}{\gamma}

G

is the modulus of rigidity (shear modulus).

Bulk form (hydrostatic pressure

p=-\sigma_h

K=-\frac{p}{\Delta V/V}=-\frac{p}{\varepsilon_v}

Physical parameters

Symbol	Meaning	SI unit
$E$	Young’s modulus	$\mathrm{Pa}$ (often GPa)
$G$	Shear modulus	$\mathrm{Pa}$
$K$	Bulk modulus	$\mathrm{Pa}$
$\nu$	Poisson’s ratio	—
$\sigma$ , $\tau$	Normal / shear stress	$\mathrm{Pa}$
$\varepsilon$ , $\gamma$	Normal / shear strain	—
$\varepsilon_v$	Volumetric strain	—

Typical values (steel):

E\approx 200\,\mathrm{GPa}

G\approx 80\,\mathrm{GPa}

\nu\approx 0.3

K\approx 140

–

160\,\mathrm{GPa}

Core assumptions (state these in exams)

1. Linear elastic behaviour — stress ∝ strain (proportional limit not exceeded).
2. Isotropic material — same

E

G

\nu

If the material is anisotropic (composites, wood), a stiffness matrix with more independent constants is required — elementary Hooke formulas do not apply directly.

Derivation summary — elongation of a prismatic bar

Consider a uniform bar of length

L

, area

A

, under axial load

P

\sigma=\frac{P}{A},\qquad\varepsilon=\frac{\delta}{L}

Hooke:

\sigma=E\varepsilon

⇒

\frac{P}{A}=E\frac{\delta}{L}\qquad\Rightarrow\qquad\delta=\frac{PL}{AE}

Strain energy (elastic, gradually applied load):

U=\frac{1}{2}P\delta=\frac{P^2 L}{2AE}=\frac{\sigma^2}{2E}\cdot(AL)

Resilience (strain energy density)

u=\sigma^\frac{2}{2E}

; proof resilience uses

\sigma_y

Relations among E, G, K, and ν

For an isotropic linear elastic solid, only two constants are independent. Standard identities:

Equations

$E = 2G(1+\nu)$
$E = 3K(1-2\nu)$
$E = \frac{9KG}{3K+G}$
$\nu = \frac{3K-2G}{2(3K+G)}$
$G = \frac{E}{2(1+\nu)},\qquad K=\frac{E}{3(1-2\nu)}$

Derivation sketch of

E=2G(1+\nu)

: pure shear is equivalent to equal tension and compression at

45^\circ

. Relating shear strain

\gamma

to principal strains and using

\tau=G\gamma

with

\sigma=E\varepsilon

yields the identity.

Exam tip: Given any two of

\{E,G,K,\nu\}

, compute the rest. Check

\nu<0.5

so that

K>0

Generalised Hooke’s law (3-D)

For principal (or Cartesian) stresses on an isotropic body:

\varepsilon_x=\frac{1}{E}\big[\sigma_x-\nu(\sigma_y+\sigma_z)\big]

\varepsilon_y=\frac{1}{E}\big[\sigma_y-\nu(\sigma_z+\sigma_x)\big]

\varepsilon_z=\frac{1}{E}\big[\sigma_z-\nu(\sigma_x+\sigma_y)\big]

Shear strains decouple:

\gamma_{xy}=\frac{\tau_{xy}}{G},\quad\gamma_{yz}=\frac{\tau_{yz}}{G},\quad\gamma_{zx}=\frac{\tau_{zx}}{G}

Volumetric strain:

\varepsilon_v=\varepsilon_x+\varepsilon_y+\varepsilon_z=\frac{1-2\nu}{E}(\sigma_x+\sigma_y+\sigma_z)=\frac{3(1-2\nu)}{E}\sigma_{\mathrm{mean}}

with

\sigma_{\mathrm{mean}}=\frac{\sigma_x+\sigma_y+\sigma_z}{3}

Fundamentals: complementary shear and the E–G link (Bansal Ch. 2)

Principle of complementary shear: shear stresses on mutually perpendicular planes are equal in magnitude (moment equilibrium of a stress element).

A square element under pure shear

\tau

has principal stresses

\pm\tau

45^\circ

planes. Relating the shear strain

\gamma

to the extension of the diagonal and using

\sigma=E\varepsilon

with

\tau=G\gamma

recovers

E=2G(1+\nu).

This is why isotropic elasticity has only two independent constants — the diagonal-strain argument is the classic SOM derivation.

Volumetric strain of a cylindrical rod under axial stress (with free sides):

\varepsilon_v=\varepsilon(1-2\nu)=\frac{\sigma}{E}(1-2\nu).

Step-by-step problem approach

1. Confirm loading is within the elastic range (or problem states elastic constants apply).
2. Identify known constants; pick the relation that solves for the unknown directly.
3. For

\delta

: use

\delta=\frac{PL}{AE}

with consistent units.
4. For 3-D stress: write generalised Hooke component-wise; sum for

\varepsilon_v

if needed.
5. Convert GPa ↔ MPa ↔ Pa carefully (

1\,\mathrm{GPa}=10^3\,\mathrm{MPa}=10^9\,\mathrm{Pa}

).
6. Sanity-check:

G\approx \frac{E}{2.6}

for

\nu\approx 0.3

;

K

should exceed

\frac{E}{3}

Common mistakes in exams

• Using

E=2G(1-\nu)

instead of

E=2G(1+\nu)

.
• Forgetting that

\nu

is dimensionless while mixing

E

in GPa with

\sigma

in MPa.
• Applying

\delta=\frac{PL}{AE}

beyond the elastic limit.
• Using

K=E/[3(1+\nu)]

(wrong sign/factor) instead of

3(1-2\nu)

.
• Treating

E

and

G

as independent for isotropic materials when a third constant is also freely chosen (over-constrained).
• Confusing modulus of rigidity

G

with bulk modulus

K

Detailed conceptual understanding

Hooke's Law & Elastic Constants should be studied as a complete reasoning chain: definition, governing assumptions, physical interpretation, boundary conditions, and limits of validity. In som, strong students do not stop at "what is the formula"; they explain why the model applies, which simplifications are being used, and what error appears when those simplifications break. This is the key difference between memorized learning and engineering understanding.

A high-quality conceptual pass should answer these questions in writing:
1. Which quantity is being predicted or controlled?
2. Which variables dominate sensitivity and why?
3. Which assumptions are explicit, and which are hidden?
4. What real-world effects are neglected in first-pass analysis?
5. Which engineering decision depends on this output?

When revising, rewrite the concept in your own words and attach one real scenario from lab, workshop, project, internship, or industry case. This habit transforms abstract theory into retrievable memory. If a topic cannot be explained without reading the page, it is not yet mastered.

Use this page as a note source: create a "concept map" with cause-effect arrows and keep updating it whenever you solve new problems. Students who maintain evolving concept maps typically retain topics longer and return less to emergency cramming.

Advanced problem-solving framework

\sigma=E\varepsilon

\delta=\frac{PL}{AE}

Exam, interview, and note-making strategy

These four questions are asked repeatedly in technical panels, and practicing them creates confidence.

Industry scenarios and decision context

Common misconceptions and correction patterns

Most weak performance comes from repeated misconception patterns, not from lack of intelligence. Typical patterns include unit inconsistency, wrong model selection, assumption mismatch, and skipping interpretation after substitution.

Correction pattern to practice:
1. Detect: identify exactly where logic diverged.
2. Diagnose: state why that step is invalid.
3. Repair: rewrite with correct model/assumption.
4. Verify: run a sanity check and compare trends.

Maintain a personal "mistake log" with three columns: mistake, reason, correction rule. Reviewing this log before exams has a larger performance impact than reading new theory repeatedly.

Use the same correction discipline in interviews: acknowledge the slip, state corrected logic, and proceed. This demonstrates professional maturity and keeps the discussion positive even when you initially miss a step.

Ask about this topic

What you'll learn in this topic

Key Formulas

Worked Examples

Common Mistakes

Practice Questions

Interview Questions

GATE Questions

Applications in real world

Visual concept

Recommended Book

What should I learn next?

Core assumptions (state these in exams)

Generalised Hooke’s law (3-D)

Step-by-step problem approach

Advanced problem-solving framework

Exam, interview, and note-making strategy

Industry scenarios and decision context

Long-form revision worksheet

Definition and physical meaning

Core assumptions (state these in exams)

Derivation summary — elongation of a prismatic bar

Relations among E, G, K, and ν

Generalised Hooke’s law (3-D)

Fundamentals: complementary shear and the E–G link (Bansal Ch. 2)

Step-by-step problem approach

Common mistakes in exams

Detailed conceptual understanding

Advanced problem-solving framework

Exam, interview, and note-making strategy

Industry scenarios and decision context

Common misconceptions and correction patterns

Long-form revision worksheet