<!-- PDF-EXPORT-IGNORE-START --> > [!info] 📄 PDF Version > [Download PDF](./5-Viscoelasticity.pdf) <!-- PDF-EXPORT-IGNORE-END --> # Lecture 5: Elastic vs Viscous vs Viscoelastic Behavior ## 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:** $\sigma = E \cdot \varepsilon$ 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:** $\tau = \eta \cdot \dot{\gamma}$ Where: - τ = shear stress - η = viscosity (Pa·s) - $\dot{\gamma}$ = 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 | Property | Elastic | Viscous | |----------|---------|---------| | **Relation** | σ = E·ε | τ = η·$\dot{\gamma}$ | | **Units** | Pa = N/m² | Pa·s | | **Response to** | Strain | Strain rate | | **Energy** | Stored | Dissipated | ### 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:** $\dot{\varepsilon} = \frac{\dot{\sigma}}{E} + \frac{\sigma}{\eta}$ **Relaxation Time:** $\tau_M = \frac{\eta}{E}$ ### 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:** $\sigma = E\varepsilon + \eta\dot{\varepsilon}$ **Retardation Time:** $\tau_K = \frac{\eta}{E}$ ## 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 $\sigma(t) = \sigma_0 e^{-t/\tau_M}$ - **Time constant τ_M** = 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 - $\varepsilon(t) = \frac{\sigma_0}{E}(1 - e^{-t/\tau_K})$ ## 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 T_g, 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·$\dot{\gamma}$ - **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:** [[4-Soft-Matter-Singularities|Lecture 4: Finite-Time Singularities]] **Next Lecture:** [[6-Rheology|Lecture 6: Rheology]] **Course Home:** [[0-README|Course Overview]] > [!significance]- Metadata > Author:: [Vatsal Sanjay](https://vatsalsanjay.com)<br> > Date published:: Jul 19, 2026<br> > Date modified:: Jul 19, 2026 > [!link] Back to main website > [Home](https://comphy-lab.org/), [Team](https://comphy-lab.org/team), [Research](https://comphy-lab.org/research), [Github](https://github.com/comphy-lab) > > 📝 [Edit this page on GitHub](https://github.com/comphy-lab/CoMPhy-Lab-Blogs/blob/main/Lecture-Notes/Intro-Soft-Matter/5-Viscoelasticity.md)