Smart materials: piezoelectric, shape memory alloys

Direct and converse piezoelectric effect, poling, d and g coefficients, piezo sensors and stack actuators, and shape memory alloys (transformation temperatures, one-way effect, superelasticity, wire actuators), with charge, voltage, stroke and force calculations.

Drafted with Aria, reviewed by the AiCanCode.org team. Spotted an error? Use Give Feedback at the bottom of the page.

Why it matters

Smart materials convert one form of energy into another inside the material itself, so a single element can be both structure and sensor or actuator. Piezoelectric ceramics sit in accelerometers, ultrasonic transducers, fuel injectors, nanopositioning stages and vibration energy harvesters; shape memory alloys drive compact valves, locks, grippers and self-expanding medical stents. A mechatronics engineer needs to size these elements with real numbers: charge, voltage, stroke and force.

Key ideas

Piezoelectricity. Some crystals without a centre of symmetry develop an electric polarisation when stressed (direct effect, used in sensors) and strain when an electric field is applied (converse effect, used in actuators). The effect is linear and reversible within limits.

  • Materials: quartz (single crystal, very stable, low coupling, used in oscillators and reference sensors), lead zirconate titanate PZT (polycrystalline ferroelectric ceramic with high coefficients, the workhorse for actuators and transducers), barium titanate, and PVDF polymer film (flexible, low stiffness, good for large-area and wearable sensors).
  • Poling: a ferroelectric ceramic is piezoelectric only after poling, by applying a strong DC field at elevated temperature to align its domains. Heating above the Curie temperature, or applying a large reverse field or excessive stress, depolarises it. Practical operation is kept well below the Curie point.
  • Notation: subscript 3 is the poling direction. d₃₃ relates charge or strain along 3 to stress or field along 3 (thickness mode); d₃₁ relates field along 3 to strain perpendicular to it (lateral mode, used in bimorph benders). d is in C/N, which is the same as m/V.
  • Sensor behaviour: a piezo element is a charge source in parallel with its own capacitance. Under a static force the charge leaks away through the finite insulation and amplifier resistance, so piezo sensors measure dynamic force, pressure and acceleration, not static loads. A charge amplifier is used so cable capacitance does not change the sensitivity.
  • Actuator behaviour: strains are small (about 0.1 %) but forces are large and response is in microseconds. Multilayer stacks multiply stroke; bimorphs and flextensional mechanisms trade force for displacement. Hysteresis (roughly 10–15 % for PZT) and creep need closed-loop control in precision stages.

Shape memory alloys (SMAs). Alloys such as Ni–Ti (nitinol, near-equiatomic) and Cu–Zn–Al, Cu–Al–Ni undergo a reversible, diffusionless transformation between a high-temperature austenite (ordered cubic, stiffer) and a low-temperature martensite (twinned, easily deformed by detwinning). Four temperatures describe it: on cooling, martensite start M_s and finish M_f; on heating, austenite start A_s and finish A_f, with M_f < M_s and A_s < A_f, and a hysteresis between the cooling and heating branches.

  • One-way shape memory effect: deform the alloy as martensite (below M_f); the strain stays after unloading because it is accommodated by detwinning, not by slip. Heat above A_f and the reverse transformation to austenite restores the original shape. Nitinol can recover up to about 8 % strain once; for long cyclic life designers use about 2–4 %.
  • Two-way effect: after special training, the alloy switches between two shapes on heating and cooling, with smaller recoverable strain.
  • Superelasticity (pseudoelasticity): above A_f, stress itself induces martensite at a nearly constant plateau stress, and the martensite reverts on unloading, so strains of several per cent are recovered with a hysteresis loop and no heating. Used in orthodontic arch wires, eyeglass frames, guidewires and self-expanding stents.
  • Actuation: an SMA wire is usually heated by its own electrical resistance (Joule heating) and works against a bias spring that resets it on cooling. Advantages: very high work per unit mass, silent, simple. Limits: low bandwidth (cooling is slow), low energy efficiency (a few per cent), hysteresis, and fatigue at large strains.

Related smart materials. Magnetostrictive alloys (Terfenol-D, Galfenol) strain in a magnetic field; electrorheological and magnetorheological fluids change viscosity in a field and are used in adaptive dampers; electroactive polymers give large strains at low force.

Formulas

D = d·T + ε^T·E (direct effect)

  • D: electric displacement, charge per unit area (C/m²); d: piezoelectric charge coefficient (C/N); T: stress (Pa); ε^T: permittivity at constant stress (F/m), ε^T = ε_r·ε₀ with ε₀ = 8.854 × 10⁻¹² F/m; E: electric field (V/m). With electrodes shorted (E = 0), D = d·T.

S = s^E·T + d·E (converse effect)

  • S: strain (dimensionless); s^E: compliance at constant field (m²/N). Free strain of an unloaded actuator: S = d·E.

Q = d₃₃·F

  • Q: charge generated (C) by a force F (N) along the poling axis of a thickness-mode element. Independent of the element's area.

C = ε^T·A / t, V = Q / C

  • C: element capacitance (F); A: electrode area (m²); t: thickness (m); V: open-circuit voltage (V).

g₃₃ = d₃₃ / ε₃₃^T, V = g₃₃·σ·t

  • g: voltage coefficient (V·m/N); σ: stress (Pa). Equivalent to the two equations above.

Δ = n·d₃₃·V

  • Δ: free stroke of a multilayer stack (m); n: number of layers; V: voltage across each layer (V). Under an external spring load the stroke is reduced.

k² = d² / (s^E·ε^T)

  • k: electromechanical coupling factor (dimensionless), the fraction of input energy converted to the other form.

F = σ_r·π·d_w² / 4, δ = ε_r·L

  • SMA wire actuator: F: recovery force (N); σ_r: allowable recovery stress (Pa); d_w: wire diameter (m); δ: stroke (m); ε_r: design recoverable strain; L: active wire length (m). Take σ_r and ε_r from the alloy supplier's data.

Worked examples

Example 1 (standard): piezo force sensor. A PZT disc has d₃₃ = 400 pC/N, ε_r = 1800, electrode area 1.0 cm² and thickness 1.0 mm. A 50 N force is applied along its axis. Find the charge and the open-circuit voltage.

  1. Charge: Q = d₃₃·F = 400 × 10⁻¹² C/N × 50 N = 2.0 × 10⁻⁸ C = 20 nC.
  2. Capacitance: C = ε_r·ε₀·A / t = 1800 × 8.854 × 10⁻¹² F/m × 1.0 × 10⁻⁴ m² / 1.0 × 10⁻³ m = 1.594 × 10⁻⁹ F.
  3. Voltage: V = Q / C = 2.0 × 10⁻⁸ / 1.594 × 10⁻⁹ = 12.5 V.
  4. Check with g₃₃: g₃₃ = 400 × 10⁻¹² / (1800 × 8.854 × 10⁻¹²) = 0.0251 V·m/N; σ = 50/10⁻⁴ = 5 × 10⁵ Pa; V = 0.0251 × 5 × 10⁵ × 10⁻³ = 12.5 V. Agreement confirms the result.

Example 2 (GATE level): actuator sizing. (a) A multilayer PZT stack has 100 layers with d₃₃ = 500 pm/V and is driven at 150 V. Find its free stroke. (b) A Ni–Ti wire of diameter 0.25 mm and active length 100 mm is designed for 4 % recoverable strain and 200 MPa recovery stress. Find its stroke and force.

  1. Stack: Δ = n·d₃₃·V = 100 × 500 × 10⁻¹² m/V × 150 V = 7.5 × 10⁻⁶ m = 7.5 µm.
  2. Wire stroke: δ = ε_r·L = 0.04 × 100 mm = 4.0 mm.
  3. Wire area: π/4 × (0.25 × 10⁻³ m)² = 4.909 × 10⁻⁸ m².
  4. Wire force: F = 200 × 10⁶ Pa × 4.909 × 10⁻⁸ m² = 9.8 N.

The piezo stack gives micrometre stroke with kilohertz bandwidth; the thin SMA wire gives millimetre stroke but takes a second or more to cool and reset.

Common mistakes

  • Writing the charge in "pC" and calling it a voltage. Charge needs the capacitance to become a voltage.
  • Using ε₀ alone as the permittivity of a piezoceramic. PZT has ε_r of the order of 1000–3000.
  • Expecting a piezo sensor to hold a static force reading; the charge decays.
  • Operating PZT near its Curie temperature or with a large reverse field and losing the poling.
  • Saying an SMA deformed above A_f "behaves like an ordinary metal". Within its limits it is superelastic and recovers on unloading; only beyond the plateau and into slip is the deformation permanent.
  • Designing SMA actuators at the 8 % one-shot limit; cyclic actuators need much lower strain.
  • Confusing the martensite in SMAs (thermoelastic, reversible) with the hard, brittle martensite of quenched steel.

For GATE ME

This topic is not a core GATE ME area, but it supports mechatronics instrumentation questions and university examinations. Expect definitions (direct and converse effect, Curie temperature, poling, the four transformation temperatures, superelasticity) and short numericals on piezoelectric charge, voltage, stack stroke and SMA wire force or stroke. Practise keeping units straight between pC/N, m/V and V·m/N.

Quick check

  1. Why can a piezoelectric sensor not measure a constant force over a long time?
  2. A quartz element has d₃₃ = 2.3 pC/N. What charge does a 100 N force produce?
  3. Above which transformation temperature must a deformed one-way SMA be heated for full shape recovery?
  4. What is superelasticity?
  5. What resets a one-way SMA wire actuator after it contracts?

Answers: 1. The charge leaks away through the finite insulation and amplifier resistance. 2. 230 pC. 3. A_f, the austenite finish temperature. 4. Large strain recovered on unloading above A_f because stress-induced martensite reverts to austenite. 5. A bias spring or load stretches it again as it cools back to martensite.

Interview questions

All Engineering Materials and Manufacturing Processes interview questions

Try answering each one aloud before you open it.

  1. 1.What are piezoelectric materials and how do they work?Concept

    They are materials whose crystal structure lacks a centre of symmetry, so stress displaces positive and negative ions unequally and produces a net polarisation and surface charge (direct effect); conversely an applied electric field produces strain (converse effect). Quartz is naturally piezoelectric, while ferroelectric ceramics such as PZT must first be poled with a strong DC field to align their domains. The charge produced is Q = d33 F for a thickness-mode element, with d33 of a few pC/N for quartz and several hundred pC/N for PZT.

  2. 2.Explain the concept of shape memory alloys (SMAs).Concept

    Shape memory alloys are materials that can return to a pre-defined shape when subjected to the appropriate thermal procedure. This behavior is due to a reversible phase transformation between two solid phases: martensite and austenite. When deformed at a lower temperature, SMAs can recover their original shape upon heating above a certain temperature.

  3. 3.How do piezoelectric materials differ from shape memory alloys?Concept

    Piezoelectric materials generate an electric charge in response to mechanical stress and can change shape when an electric field is applied. In contrast, shape memory alloys change shape in response to temperature changes due to phase transformations. While piezoelectric materials are used for sensing and actuation, SMAs are used for applications requiring shape change and recovery.

  4. 4.Why are piezoelectric materials used in sensors?Application

    They convert force, pressure or acceleration directly into charge with no external excitation, are very stiff (so the sensor barely deflects and has a high natural frequency), and respond over a wide bandwidth, which suits accelerometers, dynamic force and pressure sensors, microphones and ultrasonic probes. The charge is read through a charge amplifier so cable capacitance does not affect sensitivity. Their limitation is that charge leaks away under a constant load, so they measure dynamic, not static, quantities.

  5. 5.What happens if a shape memory alloy is deformed at a temperature above its austenite finish temperature?Application

    It behaves superelastically (pseudoelastically): the applied stress induces martensite at a nearly constant plateau stress, and when the load is removed the martensite reverts to austenite and the strain, up to several per cent for Ni-Ti, is recovered without any heating, tracing a hysteresis loop. The deformation becomes permanent only if the strain exceeds the plateau and slip begins, or if the temperature is so high that slip occurs before martensite can be stress-induced.

  6. 6.Describe an application where shape memory alloys are preferred over traditional materials.Application

    Self-expanding Ni-Ti stents are a classic case: the stent is crimped into a narrow catheter, released in the artery and opens to its set diameter, then applies a gentle, nearly constant outward force because body temperature is above its A_f and it behaves superelastically. Stainless steel could not recover such large strains elastically. In mechatronics, Joule-heated SMA wires replace solenoids or small motors in valves, latches and grippers where silent operation and very high work per unit mass matter more than speed.

  7. 7.A piezoelectric element with d33 = 5 pC/N is loaded with 10 N along its poling axis, and its capacitance (including cable) is 1 nF. Find the charge and the open-circuit voltage.Numerical

    Charge Q = d33 F = 5 pC/N x 10 N = 50 pC = 5 x 10^-11 C. The voltage follows from the capacitance: V = Q/C = 5 x 10^-11 C / 1 x 10^-9 F = 0.05 V. A common mistake is to call the 50 pC a voltage; the d coefficient gives charge, and the voltage depends on the total capacitance, which is why charge amplifiers are used.

  8. 8.What are the limitations of using piezoelectric materials in energy harvesting?Application

    The limitations of using piezoelectric materials in energy harvesting include their relatively low energy conversion efficiency and the need for continuous mechanical input to generate electricity. Additionally, piezoelectric materials can be brittle and may not withstand high mechanical stresses over time, limiting their durability in certain applications.

  9. 9.Explain how temperature affects the performance of shape memory alloys.Concept

    Temperature affects the performance of shape memory alloys by determining the phase of the material. Below the transformation temperature, the material is in the martensitic phase and can be deformed easily. Upon heating above the transformation temperature, the material transforms to the austenitic phase and recovers its original shape. The transformation temperature is critical for the material's shape memory effect and must be considered in applications.

  10. 10.An SMA actuator has an austenite finish temperature of 70°C. What happens if it is used in an environment held at 80°C?Application

    At 80 °C the alloy stays fully austenitic, so it never cools back into martensite and the bias spring cannot reset it; a one-way shape-memory actuator would simply stay in its hot shape and stop cycling. If loaded it would instead show superelastic behaviour, recovering strain on unloading. The alloy must be chosen so that M_f lies comfortably above the highest ambient temperature, otherwise the actuator does not work.

Finished this topic? Mark it so your progress, study plan and readiness keep up.

Stuck on something here?

PreviousHeat treatment and surface hardening of steelsNext Sand casting: patterns, moulds, gating and risers