Lecture 5: Elastic vs Viscous vs Viscoelastic Behavior

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Key Topics

How Materials Deform

Contrasting three fundamental behaviors:

1. Elastic Solids - Store energy, recover shape
2. Viscous Fluids - Dissipate energy, flow continuously
3. Viscoelastic Materials - Combine both behaviors

1. Elastic (Hookean) Behavior

Defining Characteristics

Hooke’s Law:

Where:

• σ = stress (force per unit area)
• E = Young’s modulus (material stiffness)
• ε = strain (fractional deformation)

Key Properties:

• Stress proportional to strain (not rate)
• Deform under load but store energy
• Recover original shape when load removed
• Energy stored elastically (like a spring)

Examples

• Spring
• Rubber band (small extension)
• Steel beam (small deformations)
• Hard plastic below glass transition temperature

Stress-Strain Curves

Linear Elastic Regime:

• Straight line
• Slope = Young’s modulus E
• Higher E → more stress needed for same strain (stiffer)

Shear Modulus:

• For shear deformations: τ = G·γ
• G = shear modulus
• Similar concept, different geometry

Typical Moduli Values

• Steel: E ~ 200 GPa (very stiff)
• Rubber: E ~ 1 MPa (soft)
• Silicone gel: E ~ 50 kPa (very soft)
• Brain tissue: E ~ 1 kPa (extremely soft)

2. Viscous (Newtonian) Behavior

Defining Characteristics

Newton’s Law of Viscosity:

Where:

• τ = shear stress
• η = viscosity (Pa·s)
• = shear rate (strain rate)

Key Properties:

• Stress proportional to strain rate (not strain itself)
• Flows continuously under any applied stress
• Dissipates energy as heat (irreversible)
• No shape recovery when stress removed

Examples

• Water (η ≈ 1 mPa·s)
• Honey (η ≈ 10 Pa·s, ~10,000× water)
• Glycerol (η ≈ 1 Pa·s)
• Motor oil

Dimensional Analogy

Behavior Under Load

• If you stop applying stress to viscous fluid:
  • Deformation continues until stress is zero
  • No recovery
  • Fluid remains in deformed state

3. Viscoelastic Behavior

Combined Characteristics

Definition: Materials exhibiting both elastic and viscous behaviors

• Deform like elastic solid on short timescales
• Flow like viscous fluid over long times
• Behavior depends on observation timescale

Common Viscoelastic Materials

• Polymers (quintessential examples)
• Biological tissues (cartilage, muscle, skin)
• Gels
• Silly putty (classic demo material)
• Many food products (dough, cheese, etc.)

Spring-Dashpot Models

Maxwell Model

Configuration: Spring and dashpot in series

Represents: Fluid with memory

Behavior:

• Can relax stress over time
• If sudden strain applied and held:
  • Initially: High stress (spring compressed)
  • Over time: Stress gradually decays (dashpot flows)
  • Eventually: Stress → 0 (full relaxation)

Equation:

Relaxation Time:

Kelvin-Voigt Model

Configuration: Spring and dashpot in parallel

Represents: Solid that creeps under load

Behavior:

• Under sudden load:
  • No instantaneous elastic jump (dashpot resists)
  • Gradual spring deformation (creep)
  • Approaches equilibrium asymptotically
• When load removed:
  • Slow recovery back to original shape

Equation:

Retardation Time:

Stress Relaxation and Creep

Stress Relaxation

Experiment:

1. Apply sudden fixed strain
2. Hold constant
3. Measure stress vs time

Results:

• Elastic material: Constant stress (no decay)
• Viscoelastic (Maxwell): Stress decays exponentially
• Time constant = relaxation time

Creep

Experiment:

1. Apply constant stress
2. Measure strain vs time

Results:

• Elastic: Immediate strain, then constant
• Viscous: Strain increases linearly forever
• Viscoelastic (Kelvin-Voigt):
  • Immediate dashpot resistance
  • Gradual approach to equilibrium

Relaxation Time Concept

Definition

Relaxation Time (τ): Characteristic time for material to transition from elastic to viscous response

Physical Meaning:

• Maxwell time: τ = η/E
• Time for stress to decay to 1/e of initial value
• Separates elastic (short-term) and viscous (long-term) regimes

Time-Scale Dependence

Fast Observation (t << τ):

• Material behaves elastic (solid-like)
• Dashpot “frozen” (hasn’t had time to flow)
• Spring dominates

Slow Observation (t >> τ):

• Material behaves viscous (liquid-like)
• Spring equilibrated
• Dashpot flows freely

Application to Silly Putty:

• τ ~ few seconds
• Quick poke (t << τ): Bounces like ball (elastic)
• Slow pull (t >> τ): Flows like liquid (viscous)

Examples and Case Studies

Silly Putty Demonstration

Classic Demo:

Fast Impact (High Strain Rate):

1. Roll into ball
2. Drop from height
3. Bounces like elastic ball
4. Behavior: Solid-like (t << τ)

Slow Deformation (Low Strain Rate):

1. Take same putty
2. Slowly pull apart
3. Flows and strands out like liquid
4. Can form thin filament that necks
5. Behavior: Liquid-like (t >> τ)

Very Fast (Brittle Failure):

• Quick yank → snaps (brittle behavior)
• Elastic domination with insufficient time for viscous flow

Metal vs Polymer Comparison

Metal Spring:

• Apply weight briefly
• Springs back immediately
• Pure elastic response
• Negligible viscous component

Viscoelastic Polymer Strip:

• Apply same weight
• Might slowly rebound
• Or not fully recover if short time
• Clear viscoelastic behavior

Pitch Drop Experiment

Famous Long-Term Experiment:

Short Timescale (Impact):

• Block of pitch (tar)
• Hit with hammer → shatters
• Feels solid, elastic

Long Timescale (Years):

• Over ~10 years, drops form and fall out
• Flows like extremely viscous liquid
• Demonstrates extreme relaxation time (years!)

Lesson: Time scale of observation matters!

Quantitative Example

Dynamic Testing:

• Water: Phase angle δ ≈ 90° (all viscous, no elastic storage)
• Rubber: Phase angle δ ≈ 0° (all elastic storage)
• Silly putty: Phase angle δ ≈ 45° at intermediate frequencies (half-half)

Units Reminder:

• Elastic modulus: Pa
• Viscosity: Pa·s
• Example values:
  • Rubber E ~ 1 MPa
  • Honey η ~ 10 Pa·s
  • Silly putty τ ~ seconds

Learning Outcomes

Students will be able to:

1. Differentiate elastic vs viscous response

   • State defining constitutive relations (Hooke’s law vs Newton’s viscosity law)
   • Qualitatively describe behavior under load

2. Give examples of each type

   • Elastic: Steel, hard plastic below , rubber (small strains)
   • Viscous: Water, oil, glycerin
   • Viscoelastic: Silly putty, cartilage, polymer melt
   • Justify classifications

3. Understand viscoelastic models

   • Maxwell model: Initial stress from spring, then decays as dashpot flows
   • Kelvin-Voigt: Sudden stress → dashpot resistance + gradual spring deformation
   • Conceptual understanding (no ODE solving required)

4. Interpret storage vs loss qualitatively

   • Viscoelastic material has:
     • Storage modulus (elastic part, energy stored)
     • Loss modulus (viscous part, energy dissipated)
   • More elastic → bounces back (stores energy)
   • More viscous → damps motion (loses energy as heat)

5. Apply time-scale reasoning

   • Predict material behavior based on deformation rate
   • Quick stretch → stiff (elastic)
   • Slow stretch → soft/flowy (viscous)
   • Articulate relaxation time concept

Discussion Questions

1. Identify Behaviors

Scenarios to classify:

• Bowling ball impacts cement floor and rebounds

  • Mostly elastic (some energy lost)

• Person slowly sinking into memory foam mattress

  • Viscoelastic (slow deformation over time)

• Toothpaste extruded from tube

  • Mostly viscous flow (with yield stress)

• Guitar string vibrating

  • Elastic

Task: Students debate and justify answers

2. Spring vs Dashpot Analogy

Questions:

• If you stretch Maxwell material (spring+dashpot series) and hold, what happens to stress?

  • Initially: High stress from spring
  • Over time: Gradually decays (dashpot relieves it)
  • Stress relaxation

• If suddenly released after long hold, will it return to original length?

  • No – dashpot has flowed (permanent deformation)
  • Only spring part returns
  • Some deformation remains

• How does this differ from Kelvin-Voigt under sudden load?

3. Energy Storage vs Loss

Scenario: Two materials feel similarly stiff when tapped, but one gets warm when bent repeatedly, the other doesn’t.

Question: What does this indicate?

• Warming material: Dissipating energy → higher loss modulus
• Non-warming: Stores-returns energy → higher storage modulus

Examples:

• Rubber bands heating up after stretching
• Car tires heating up (viscoelastic damping)

4. Design Considerations

Shoe Sole for Runners:

• More elastic or more viscous?
• Discussion:
  • Elastic: Returns energy (spring in step)
  • Viscous: Absorbs energy (shock absorption)
  • Likely want mix (viscoelastic)
  • Different answers possible
  • Shows engineering balance

Seismic Dampers:

• Should be highly viscous
• Dissipate earthquake energy
• Not elastic (would spring back and forth)
• Connects material behavior to function

Key Concepts to Remember

Elastic Behavior

• Relation: σ = E·ε
• Energy: Stored (recoverable)
• Response: Immediate
• Examples: Springs, metals, rubber (small strain)

Viscous Behavior

• Relation: τ = η·
• Energy: Dissipated as heat
• Response: Continuous flow
• Examples: Water, honey, oils

Viscoelastic Behavior

• Models: Maxwell (fluid-like) vs Kelvin-Voigt (solid-like)
• Key parameter: Relaxation time τ = η/E
• Time dependence: Solid when fast, liquid when slow
• Examples: Polymers, silly putty, tissues

Storage vs Loss

• Storage modulus (G’): Elastic energy stored
• Loss modulus (G”): Viscous energy dissipated
• Phase angle (δ): tan δ = G”/G’
• Sets up next lecture on rheology!

Preparation for Next Lecture

Next lecture: Rheology – Measuring “Softness”

Consider:

• How do we experimentally measure these properties?
• What is a rheometer?
• How do G’ and G” relate to oscillatory tests?
• What does “soft” really mean quantitatively?

References

1. “What is a Viscoelastic material?” - Biolin Scientific
2. Ferry, J.D. “Viscoelastic Properties of Polymers” (1980)
3. Lakes, R.S. “Viscoelastic Materials” (2009)
4. Newtonian fluid - Wikipedia

Previous Lecture: Lecture 4: Finite-Time Singularities
Next Lecture: Lecture 6: Sheets, Bubbles and Drops
Course Home: Course Overview

Metadata

Author:: Vatsal Sanjay
Date published:: Jul 19, 2026
Date modified:: Jul 19, 2026

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