1. Nonlinear Magnetostrictive Transducer: Actuator and Sensor Characterization
© 2013 COMSOL. All rights reserved.

2. Magnetostriction
Joule effect – Magnetostrictive materials exhibit free strain (λ) when exposed to magnetic field (H)
Villari effect – The magnetization (M) in this type of materials can be changed by applying mechanical stress (σ) in the presence of a bias magnetic field
Applications
Acoustic transducer: SONAR, Hydrophone, Ultrasonic cleaning, Ultrasonic friction welding,
Actuator: Pump, Active valve, Rotary motor
Sensor: Vibration sensor, Position sensor, Torque sensor

3. Magnetostrictive Transducer
Piston
Steel housing
Drive coil
Magnetostrictive rod

4. Description of transducer
The transducer has a magnetostrictive core
Magnetic field is applied to the magnetostrictive core by a current-carrying drive coil around it
The transducer has a steel housing which encloses the drive coil and the core
The steel housing, piston and core acts as a closed magnetic flux path

5. Modeling objectives
In this tutorial we will look at how the non-linear bi-directional magneto-mechanical coupling involved in magnetostrictive transduction can be modeled in COMSOL
Case studies:
Actuator characterization: Transducer displacement as a function of coil current for different preloads on the piston
Evaluation of the blocked force: Clamp the piston and evaluate the force experienced by it as a function of coil current
Sensor characterization: Change in magnetic flux density in the magnetostrictive material as a function of axial load on the piston for different DC bias coil current

6. Model features
2D axial symmetry geometry used to reduce computation time and memory
Drive coil is modeled as a homogenized current carrying domain where the individual wires are not resolved
The magnetic and structural simulations do not include time-dependent effects (inertial force, eddy current, damping, etc.)
A parametric sweep of current in the drive coil and total force applied on the piston is used to represent quasi-static operation of the transducer
Effects of both positive (tension) and negative (compression) loads are studied
Effect of only positive coil current is studied as magnetostrictive response is symmetric with respect to the direction of magnetic field
Non-linear constitutive relation is implemented by using interpolated data of B(σ,H) and λ(σ,H)

7. Coupled constitutive relations
The magneto-mechanical coupling takes place due to quantum mechanical interaction but can be modeled at macro-level using magnetomechanically coupled constitutive relations
Structural constitutive equation
Magnetic constitutive equation
ε: Total strain
σ: Mechanical stress
E: Young’s modulus at magnetic saturation
λ: Magnetoelastic strain
H: Magnetic field
B: Magnetic flux density
μo: Magnetic permeability of free space
M: Magnetization

8. References
The modeling approach shown here is based on the following references
C. Mudivarthi, S. Datta, J. Atulasimha and A. B. Flatau, “A bidirectionally coupled magneto-elastic model and its validation using a Galfenol unimorph sensor,” Smart Materials and Structures, Vol. 17(3) , 2008.
F. C. Graham, C. Mudivarthi, S. Datta and A. B. Flatau, “Development and validation of a bidirectionally coupled magnetoelastic actuator model,” Smart Materials and Structures, Vol. 18(10), , 2009.
The material data was obtained from the following reference
S. Datta, J. Atulasimha and A. B. Flatau, “Figures of merit of magnetostrictive single crystal iron-gallium alloys for actuator and sensor applications,” Journal of Magnetism and Magnetic Materials, Vol. 321, No. 24, pp , 2009.

9. Modeling steps The following slides show the important modeling steps
More details can be seen in the accompanying model file:
nonlinear_magnetostriction_transducer_43b.mph

10. First few steps Physics interfaces: AC/DC > Magnetic Fields (mf)
Structural Mechanics > Solid Mechanics (solid)
Domain ODEs and DAEs (dode)
Change the Field name and Dependent variables name to Lambda

11. Global Definitions > Parameters
We will later use the parameters I0 and Fz in a Parametric Sweep to vary the current in the drive coil and the load on the piston

12. Implementing the constitutive relations
The non-linear constitutive relations require us to provide information on M(σ,H) and λ(σ,H) as a model input
M(σ,H) and λ(σ,H) can be obtained from experimental data or from nonlinear magnetostriction material models for a set of discrete values of (σ,H)
We would need to reformat the data to get H(B, σ) and λ(B, σ) to use it in COMSOL
This is better suited to work with how the constitutive equations are implemented in COMSOL

13. How should we arrange the data?
Grid-data format in COMSOL
Our data file should be arranged as follow

14. Global Definitions > Functions > Interpolation
Import the data file that contain H(B,σ) in a grid format

15. Global Definitions > Functions > Interpolation
Import the data file that contain λ(B,σ) in a grid format

16. Create the geometry
See model file for dimensional details

17. 2D axisymmetric representation of 3D
Piston
Steel housing
Drive coil
Magnetostrictive rod
Air domain to model magnetic flux leakage

18. Magnetic Fields – solve on all domains

19. Domain ODEs and DAEs – solve on Domain 3
Check the option to see the Discretization

20. Evaluate Lambda
Here the variable Lambda at each point in space is evaluated from the interpolation function L_B_Stress using the arguments mf.normB (magnitude of the B-field) and solid.sz (z-component of stress)

21. Solid Mechanics – solve on Domains 2-4
Domain 3 only
Lambda is obtained from the Domain ODEs and DAEs interface
Magnetostriction can be assumed to be a volume-preserving inelastic strain, hence its values along the x and y-directions are –Lambda/2

22. Why can’t we directly use the interpolation function to specify λ(B,σ) as an initial strain?
COMSOL computes the stress (σ) from the following equation
Here the initial strain
If the initial strain (εi) is directly specified as a function of stress then this creates a circular dependency preventing COMSOL from evaluating both stress and initial strain
The variable Lambda in the Domain ODE and DAE interface is introduced to circumvent this problem by creating extra degrees of freedom which are obtained as a function of stress and B-field
In the Solid Mechanics interface the initial strain becomes an “indirect” function of stress
The problem expressed in this form can now be handled by COMSOL’s nonlinear solver

23. Solid Mechanics – boundary conditions
Boundary Load is applied on the top surface of the piston (Boundary 10)
The same boundary is Fixed for evaluating the transducer blocked force

24. Materials Air Soft Iron (without losses) Magnetostrictive
Select from Material Browser > Built-in > Air
Soft Iron (without losses)
Select from Material Browser > ACDC > Soft Iron (without losses)
Assign it to Domains 2 and 4 (transducer housing and piston)
Magnetostrictive
Right-click Materials and select Material
Rename it to Magnetostrictive
Assign it to Domain 3 (Galfenol rod)

25. Magnetostrictive material properties
These are typical properties of Galfenol (Fe81Ga19)
Note how the function H_B_Stress is called with arguments normB and solid.sz

26. Mesh
Domains 2, 3 and 4 have a significantly finer mesh to help resolve the material nonlinearities

27. Study 1 – Actuator characterization
Turn-off parametric solver to improve convergence as Fz is varied from negative to positive values
Disable Fixed Constraint 2 in Study 1
Right-click Study 1 and select Show Default Solver prior to computing
Increase the number of Newton iterations so that the Nonlinear solver does not stop prematurely

28. Results
Choose the value of Load and Drive current to see the results for different operating conditions

29. Structural displacement and Magnetic flux density

30. Magnetic flux density and Stress

31. Magnetic flux density (mf. normB) vs
Magnetic flux density (mf.normB) vs. Drive current (I0) at different Preloads (Fz) evaluated at the center of the rod (Point 4)
Saturation effect
Stress-induced anisotropy

32. Magnetostriction (Lambda) vs
Magnetostriction (Lambda) vs. Drive current (I0) at different Preloads (Fz) evaluated at the center of the rod (Point 4)
Stress-induced magnetoelastic strain (λs)
Magnetic moments are randomly oriented in the absence of pre-load
Pre-load aligns the magnetic moments along an easy axis in a plane perpendicular to the direction of magnetic field thereby yielding the saturation magnetostriction of (3/2)λs
(3/2)λs

33. Study 2 – Transducer blocked force
Disable Boundary Load 1 in Study 2
All other setting are similar to Study 1

34. Evaluating the Blocked force
Find the total reaction force on the top surface of the piston that has been fixed (clamped)
Create a plot of the data is this Table

35. Results – Blocked force

36. Study 3 – Sensor characterization
Setting are similar to Study 1

37. Magnetic flux density (mf. normB) vs
Magnetic flux density (mf.normB) vs. Load (Fz) at different bias current (I0) evaluated at the center of the rod (Point 4)
I0 = 0: No sensing
I0 > 0: Higher load is required to overcome the magnetic field-induced anisotropy
Bias current can be used to control sensitivity and operating range

38. Strain (solid.eZ) vs. Load (Fz) at different bias current (I0) evaluated at the center of the rod (Point 4)
Delta E-effect:
Apparent change in elastic modulus due to rotation of magnetic moments
Stiffness can be tuned with bias current and load
Useful for tunable acoustic waveguide

39. Magnetic field (mf. normH) vs
Magnetic field (mf.normH) vs. Load (Fz) at different bias current (I0) evaluated at the center of the rod (Point 4)
Contrary to popular belief, a constant current in the coil does not create constant field inside the magnetostrictive material!

40. Summary
This tutorial showed how to simulate a complete quasi-static transducer characterization of a magnetostrictive material
The model incorporates full nonlinear material properties and bidirectional physics coupling between structural and magnetic inputs and response
Key points:
How to implement the nonlinear constitutive relations
How to use nonlinear material data as a function stress and B-field
How to create the appropriate mesh and solver settings
How to create the important transducer characterization plots