Piezoelectric, magnetostrictive and coupled effects

One kind of input can produce another response

The direct piezoelectric effect changes electrical polarisation when a suitable material is mechanically loaded; the converse effect produces strain under an electric field. Magnetostriction is a change in dimensions associated with a changed magnetic state. Coefficients depend on direction, material symmetry, state and the relevant constrained quantities.

A pressure sensor and a small positioning element can use the same kind of piezoelectric coupling in different directions of energy transfer. Mechanical loading changes the electrical dipole arrangement; an applied electric field changes the mechanical equilibrium. The output is supplied by the mechanical or electrical input, not created from nothing. A coefficient describes a relation under stated conditions, not a complete device.

In a piezoelectric solid, strain and the relative positions of positive and negative charge can be coupled by the structure’s symmetry. A bulk centrosymmetric crystal cannot have an ordinary linear piezoelectric response: inversion leaves mechanical stress unchanged but reverses polarisation, requiring that linear coefficient to vanish. Appropriate symmetry is necessary; a generic insulator is not automatically piezoelectric.

Poled ferroelectric ceramics are one important family. Poling prepares an oriented polar state, and small later changes can be described around that state. Piezoelectricity does not require every material to have electrically switchable spontaneous polarisation: quartz is also piezoelectric. A linear coefficient for one prepared state does not describe switching, depoling or every field amplitude.

Magnetostriction couples a changed magnetic state to strain. In a magnetic solid, spin–orbit interactions connect magnetic orientation to the electronic environment and lattice; changing that orientation or the distribution of magnetic regions can alter the preferred dimensions. Stress can also change the magnetic response. This reciprocal magnetomechanical coupling differs from piezoelectric coupling and from a coil simply pulling a separate piece by magnetic force.

Declare the directions, signs and constrained quantities

A coefficient with subscripts identifies electrical and mechanical directions. For example, d₃₃ relates a field along axis 3 to a longitudinal strain along 3, and the corresponding direct response to stress along 3. A transverse d₃₁ can instead relate that field to strain along 1. A tensor is the collection of directional relations allowed by symmetry. The following one-dimensional model keeps only a supplied longitudinal mode; it is not a full tensor calculation.

Use a straight homogeneous specimen of reference length L along +z, with full electrodes on its end faces of area A. Its lower end is fixed; lateral faces are mechanically free, and only uniform axial loading is represented. Define S = ΔL/L as axial strain, positive for extension; σ is axial stress, positive in tension. E is the uniform axial electric field, in V/m. The signed voltage is U = φ_lower − φ_upper, so E = U/L when edge fields are neglected.

D is the incremental axial electric displacement, in C/m². For the ideal full electrodes, Q = AD is the incremental free charge on the lower electrode; the upper electrode has −Q. All S, σ, E, D, U and Q below describe changes from the stated prepared reference. Background polarisation and its screening charge are not being silently set to zero.

The supplied reciprocal linear model is S = sᴱσ + dE and D = dσ + ε_σE. Here sᴱ is axial compliance at fixed electric field, in m²/N; ε_σ is axial permittivity at fixed axial stress, in F/m; d is the effective longitudinal coefficient, in C/N or equivalently m/V. The σ subscript on permittivity marks the fixed-stress condition used here, not electrical conductivity. A compliance is inverse stiffness: 1/sᴱ is the relevant Young-type modulus in this scalar mode.

The two terms in each equation have compatible units. sᴱσ and dE are strains; dσ and ε_σE are C/m². The equivalence C/N = m/V follows from 1 J = 1 N·m = 1 C·V. The same d occurs in the two reciprocal small-signal relations under this convention; changing to a stress-form coefficient or another constrained quantity changes the equations.

Use the original fictional values L = 10.0 mm, A = 25.0 mm², sᴱ = 20.0 × 10⁻¹² m²/N, d = +400 × 10⁻¹² m/V and ε_σ = 20.0 × 10⁻⁹ F/m. Assume these constants apply throughout the specific small-strain comparisons below, at a common temperature, with no leakage, switching, inertia or loss. They are not data for a named ceramic or a field/stress rating.

The actuator must satisfy the load as well as the material law

Hold U = +200 V across the single end-electroded specimen. Then E = 200/0.0100 = 2.00 × 10⁴ V/m. With zero axial stress, S_free = dE = 8.00 × 10⁻⁶ and free extension δ₀ = LS_free = dU = 80.0 nm. “Free” here means no resisting axial force, with the lower end serving as the position reference.

Let F_L be the positive force the extending actuator exerts on a resisting load. That load exerts a compressive force −F_L on the actuator, so σ = −F_L/A. Substitution gives δ = δ₀ − (sᴱL/A)F_L = δ₀ − F_L/K_a, where K_a = A/(sᴱL) = 1.25 × 10⁸ N/m = 125 N/µm. This is the fixed-voltage force–displacement line, not a history-dependent magnetic loop.

A perfect axial block imposes δ = 0, requiring F_block = K_aδ₀ = 10.0 N. The material’s axial stress is then −0.400 MPa. Conversely, δ = 80.0 nm requires F_L = 0. The blocked force and free extension are different endpoints; the actuator does not deliver both simultaneously.

For an initially uncompressed linear load spring K_L, connected so positive δ compresses it, equilibrium requires F_L = K_Lδ. Combining the two relations gives δ = δ₀/(1 + K_L/K_a). With K_L = K_a, δ = 40.0 nm and F_L = 5.00 N. A smaller motion under a stiffer restraint need not indicate changed chemistry.

The electrical response also changes with restraint. At the same U, Q = A[dσ + ε_σ(U/L)]. The free, equal-stiffness-load and blocked states give 10.0, 8.00 and 6.00 nC. The table and first figure compare compatible simultaneous states, rather than collecting the largest output from each boundary condition.

The equal-stiffness load stores ½K_Lδ² = 1.00 × 10⁻⁷ J, or 0.100 µJ. A blocked endpoint has force but no incremental displacement there, so force alone is not mechanical work. Multiplying the incompatible free and blocked endpoints, 10 N × 80 nm, is not the work delivered to that spring. The voltage source, stored electrical/mechanical energy and losses must all enter a complete cycle account.

An open electrical circuit is not a mechanically free specimen

Now use the same scalar specimen as a sensor under a supplied tensile force F = +1.00 N, not the actuator’s compressive load convention. The uniform stress is σ = F/A = +4.00 × 10⁴ Pa. Combining the charge relation with E = U/L gives Q = dF + C_σU, where C_σ = ε_σA/L = 50.0 pF is its capacitance at constant force/stress in this model.

For ideal short-circuit sensing, U = 0 and E = 0 while charge can flow between electrodes. The lower-electrode increment is Q_sc = dF = +0.400 nC. The mechanical strain is sᴱσ = 8.00 × 10⁻⁷, giving extension 8.00 nm. Short circuit specifies the electrical boundary, not zero applied force.

For an initially isolated ideal open circuit, no incremental electrode charge is supplied: Q = 0. Thus U_oc = −dF/C_σ = −8.00 V and E = −800 V/m under the declared voltage sign. The induced field opposes the tensile strain. Substituting back gives S = (sᴱ − d²/ε_σ)σ = 4.80 × 10⁻⁷, so extension is 4.80 nm. The corresponding fixed-charge compliance is 12.0 × 10⁻¹² m²/N, smaller than sᴱ.

The same tensile input has therefore produced either a short-circuit charge or an open-circuit voltage, and different strains. They are alternative electrical boundaries, not simultaneous maximum outputs. A real meter has finite resistance and capacitance; leakage and observation time can change a sensor signal. A high resistance alone does not remove the meter’s charge-sharing capacitance.

This is a reversible incremental model. In an actual material, frequency, temperature, dielectric loss, mechanical constraints, field amplitude and poling state affect which measured coefficients apply. A static no-leakage illustration is not a promise of indefinite voltage retention or a full sensor calibration.

Magnetic state changes a preferred dimension

Define longitudinal magnetostrictive strain for a specified reference as λ = [L(H) − L₀]/L₀ at the stated temperature, load, direction and magnetic history. It is dimensionless; 1 ppm means 10⁻⁶ strain. Depending on material and direction, the length change can be positive or negative. Magnetostriction is not the same quantity as magnetisation, and a strain measured in ppm is not a universal linear coefficient per unit field.

A field can reorient magnetisation or alter the distribution of ordered regions. Because electronic magnetic orientation and lattice strain are coupled, the elastic equilibrium can change. Conversely, applied or residual stress can alter magnetic preferences. Clamping therefore can affect the magnetic state as well as preventing a displacement; it is not always valid to subtract a free strain using an unchanged magnetic response.

As a real evidence example, annealed Co–Fe thin-film studies found that structural heterogeneity and processing were associated with a changed effective magnetostriction. The reported microscopic explanation was a proposed heterogeneous mechanism supported by structural and magnetic evidence, not a universal single-phase coefficient. A film constrained by a substrate, a free rod and a rotated crystal need not supply interchangeable numbers. The fictional rods below do not reproduce those films or their reported values.

For a distinct original comparison, take two mechanically free rods A and B of the same declared length L₀ = 50.0 mm. At 298 K, they share composition, geometry and measurement direction but have different prepared processing states. Both start from their stated zero-applied-field reference and follow the same increasing applied-field sequence. The table supplies longitudinal strain; it does not predict strain from chemistry or infer the internal field.

H_app is the imposed field-strength record along the rod, in kA/m, rather than magnetic induction in tesla. The local internal field can differ because of specimen response and geometry. The lines joining supplied points in the second figure are guides between observations, not a claimed constitutive law, saturation curve or complete reversal loop.

Compare the relevant change and field interval

At 20.0 kA/m, A’s 130 ppm gives ΔL_A = (130 × 10⁻⁶)(50.0 mm) = 6.50 µm. At 40.0 kA/m, its 170 ppm gives 8.50 µm from the original reference. The additional movement from 20.0 to 40.0 kA/m is only 2.00 µm. Reference-to-state and state-to-state changes are different quantities.

At 10.0 kA/m, B’s 80 ppm exceeds A’s 50 ppm. At 40.0 kA/m, A’s 170 ppm exceeds B’s 150 ppm. A single ranking therefore does not cover both supplied operating points. For A the interval-average slope is 8.00 ppm/(kA/m) from 10.0 to 20.0 kA/m, but only 2.00 ppm/(kA/m) from 20.0 to 40.0 kA/m. Neither is established as a universal differential coefficient at every field.

A strain alone also does not supply actuator force: a compatible stiffness, mechanical restraint and magnetic-state response are needed. The fixed-voltage piezoelectric force line was derived from its explicit coupled law. Applying that same line to a magnetostrictive specimen merely because both devices move would replace missing evidence with an assumption.

Temperature drift provides another competing length-change route. Compare magnetic observations at a common measured thermal state and load, or include a justified thermal-expansion contribution. Likewise, a measured sensor voltage requires the electrical circuit and electrode convention. A coupled response must be interpreted with the other quantities allowed to change.

A device adds its own limits

A material coefficient does not directly equal sensor voltage or actuator displacement. Geometry, electrodes, load, clamping and drive conditions enter. Some measurements include hysteresis or nonlinear effects that require more than one constant.

Piezoelectric field-induced strain need not imply a freely moving body, and a magnetically induced preferred strain need not appear as the same external displacement when constrained. Record material symmetry, orientation, geometry, electrodes, poling or magnetic preparation, temperature, load and drive waveform. State whether a coefficient was measured at fixed field, fixed stress, fixed strain or fixed charge.

Use Elastic response and elastic modulus for the mechanical quantities and Polarisation, dielectric response and breakdown for electrical response. Magnetic moments and collective order distinguishes magnetic arrangement from the net sum; Domains, hysteresis and magnetic losses develops the history of magnetic response. The present linear examples deliberately omit switching and dissipation, so they do not supply those neighbouring models.

To test an application, first translate its required force, motion, charge or voltage into the matching boundary conditions. Then compare representative evidence within the supported field, stress, frequency and temperature range. A high d, a large free λ or one bright response plot does not establish a component’s output, efficiency, lifetime or qualification.

Worked examples

Try the reasoning

1. Can direction matter in piezoelectric response?

   Show the answer and reasoning

   Yes.

   Different loading and response directions require different coefficients; a tensor is the collection describing those directional relations, constrained by material symmetry.

2. Does a large coefficient guarantee unlimited actuator work?

   Show the answer and reasoning

   No.

   Field, stress, geometry and losses constrain operation.

3. A separate freely deforming directional record supplies d₃₁ = −160 × 10⁻¹² m/V and d₃₃ = +400 × 10⁻¹² m/V. Apply E₃ = +2.00 × 10⁴ V/m with no applied stress. For a 5.00 mm transverse width and 10.0 mm axial length, find their changes. May d₃₃ be substituted for d₃₁?

   Show the answer and reasoning

   Transverse strain is −3.20 × 10⁻⁶ and width change −16.0 nm; axial strain is +8.00 × 10⁻⁶ and length change +80.0 nm. The two directional coefficients cannot be interchanged.

   Use the supplied free-strain relation in each direction and its matching reference dimension. The sign indicates contraction or extension under the declared field; it is not a negative efficiency.

4. Retain the single-layer +200 V specimen but use K_L = 3K_a. Find extension, force and lower-electrode charge. Did the smaller extension establish a smaller d?

   Show the answer and reasoning

   δ = 20.0 nm, F_L = 7.50 N and Q = +7.00 nC. The changed restraint does not establish a changed d.

   δ = 80.0/(1 + 3) nm. The compressive stress is −0.300 MPa; D = (400 × 10⁻¹²)(−3.00 × 10⁵) + (20.0 × 10⁻⁹)(2.00 × 10⁴) = 2.80 × 10⁻⁴ C/m². Multiply by 25.0 × 10⁻⁶ m² for charge.

5. An ideal comparison divides the same 10.0 mm total length into 20 equal active layers of unchanged area, wired electrically in parallel and poled so their free elongations add. Each layer receives 200 V. Assume the same linear coefficients remain valid at the new layer field, and ignore inactive layers, edges and losses. Find the free total extension and fixed-stress capacitance. Is this the same field as the single-layer case?

   Show the answer and reasoning

   Free extension is 1.60 µm and capacitance is 20.0 nF. Each layer field is 4.00 × 10⁵ V/m, twenty times the single-layer field.

   Each 0.500 mm layer extends dU = 80.0 nm. Twenty mechanical-series elongations add. Each layer has 20 times the single-layer capacitance and twenty parallel layers give 400 × 50.0 pF = 20.0 nF. The assumed coefficient range and field constraints need evidence; a shared terminal voltage is not a shared internal field or electrical burden.

6. For the +1.00 N tensile sensor, connect an initially uncharged ideal 50.0 pF meter capacitance in parallel with the specimen, with no external charge supplied to the combined pair. Keep the constant-force model and neglect leakage. Use Q_sample = dF + C_σU and Q_sample + C_meterU = 0. Find U and specimen extension.

   Show the answer and reasoning

   U = −4.00 V and extension is 6.40 nm, differing from the ideal isolated sample’s −8.00 V and 4.80 nm.

   U = −dF/(C_σ + C_meter) = −0.400 nC/100 pF. E = −400 V/m, so S = 8.00 × 10⁻⁷ − 1.60 × 10⁻⁷ = 6.40 × 10⁻⁷. Charge can redistribute between the specimen and meter even though the combined isolated charge increment is zero.

7. A separate 50.0 mm rod gives an observed 130 ppm length increase during a field change, but temperature also rises by 2.00 K. In a stipulated additive comparison, thermal expansion is α = 12.0 × 10⁻⁶ K⁻¹ and all remaining strain is assigned to the magnetic route. Estimate that remaining strain and displacement. Is 130 ppm alone a magnetostriction value for a constant-temperature comparison?

   Show the answer and reasoning

   The model assigns 24.0 ppm to thermal expansion and 106 ppm, or 5.30 µm, to the remaining route. The uncorrected 130 ppm does not establish the same constant-temperature magnetic response.

   αΔT = 24.0 × 10⁻⁶; subtract it before multiplying by L₀. Additivity and the supplied α are assumptions here. In a real specimen, changing temperature can also change the magnetic state, so merely subtracting one term does not prove the full mechanism.

8. A report supplies rod A’s free 170 ppm strain at 40.0 kA/m and claims this alone gives its blocked actuator force. What is missing? Can the piezoelectric specimen’s 10.0 N blocked force be copied?

   Show the answer and reasoning

   A compatible mechanical stiffness/restraint and magnetic-state response are missing. The unrelated piezoelectric blocked force cannot be copied.

   Force follows from a constrained coupled model or suitable measurement, not strain alone. Stress can change magnetisation and magnetostrictive response, while length, area and preparation can differ. The fixed-voltage piezoelectric force line depended on its own complete stated law.

What to carry forward

• State coupled variables and restraints.
• Translate coefficients into device response with a model.
• Piezoelectric charge and voltage, or free motion and blocked force, belong to different boundaries; compare simultaneous compatible states.
• The same reciprocal scalar law can make electrical restraint change mechanical compliance.
• Direction, poling or magnetic preparation, temperature and observation conditions are part of a coupling comparison.
• Magnetostrictive strain is dimensionless and state/reference dependent; an interval slope is not a universal coefficient.
• Energy output needs an input and a full account; a large coefficient alone does not establish device work, efficiency or lifetime.