2-D Nickel Titanium Truss FEA

MATLAB simulation (top) and Solidworks simulation (bottom)

For a graduate course on smart and intelligent materials I designed a deployable 2-D nickel titanium truss cell. Nickel titanium alloys, commonly referred to as Nitinol, can undergo incredibly high reversible strains. This is achieved via stress- or temperature-induced phase transformations. The goal of this project is to find a unit cell composed of truss members such that the total strain of the structure is maximized in a specified direction. The advantage of using truss members is that their deformation can be modeled using 1-D laws, which simplifies the analysis of the overall structure.

The only temperature in consideration is at a temperature less than so-called martensitic start, M_s, and below this temperature the phase of the Nitinol is guaranteed to be 100% martensite. In order to be useful, the geometry must be set at a temperature higher than so-called austenitic finished, A_f, where the material is purely in the austenite phase, or the geometry must be cut from a plate initially at that temperature and cooled below M_s. In any case, the truss is designed so that it is deformed below M_s then deployed by heating to some temperature above austenitic finish, A_f. The shape-memory properties of the Nitinol allow these deformation steps to occur. At temperatures below M_s, the material can be seemingly permanently deformed, but after heating to temperatures above A_f, the deformation is fully reversed.

First the nonlinear stress-strain curve is determined from material parameters and the strain is converted from Green's strain to true strain (conversion from the 2nd Piola-Kirchhoff stress isn't necessary for 1-D modeling). In this case, what appears to be the ''yield point" is simply the critical stress at which phase transformations from martensite to austenite begin. In MATLAB, the direct stiffness method (DSM) is used to solve for unknown displacements of the truss nodes as the truss is loaded. In order to handle nonlinear changes in stiffness, which result when one or members begin deforming nonlinearly, a Newton-Raphson method is employed to iteratively close so a solution where the difference between applied nodal loads and internal nodal forces is sufficiently small. This method is used to test different truss geometries before expending too much time analyzing the design in Solidworks.

A comparison between the MATLAB simulation and the Solidworks realization is shown in the first GIF. Note that the color schemes representing the stresses are different. Specifically, in the MATLAB simulation, cyan corresponds to zero stress, magenta to the critical stress, and red the maximum reversible stress. The difference in timing is due to the fact that Solidworks uses adaptive time-stepping. When I tried using the true stress-strain in Solidworks, the solver chose to unload in reverse along the stress-strain curve instead of linearly according to the material modulus. Because of this issue, a bilinear model was used to approximate the nonlinear behavior. As can be seen in the GIF, the design of the truss is realized using flexures for pinned joints. The displacement of the top node for each simulation is shown in the following figure.

Displacement of top node

Note that the Solidworks realization using a bilinear stress-strain curve provides results that closely match what was expected from MATLAB.

Total vertical engineering strain of the structure.

The total vertical strain is shown and the reason for the difference is simply that the initial vertical dimension is necessarily larger in the Solidworks realization. The final strains are -1.34 in the MATLAB simulation and -1.10 in the Solidworks simulation. The magnitude of the Solidworks simulation is more than 16 times that of the material itself.

Thanks to Prof. Carman for his direction. The 1-D constitutive model is based on Brinson's 1993 paper.