Finite Element Simulation for In-plane Tensile Behavior of ShapeMemory Alloy Honeycombs
Jie HE 1 and Yutaka TOI 111

Institute of Industrial Science, University of Tokyo, Japan

Introduction

Shape memory alloy (SMA) is considered as a promising
function material. By proper external stimulus, SMA could
recover its original shape after large inelastic deformation.
Honeycomb structure as one type of cellular structures, has ahigh stiffness-to-weight ratio on certain direction. Common typehoneycomb structures in the form of honeycomb Sandwich
Panel (HSP) could be found as structural components in manyfields. Another type of honeycomb, the low shear stiffness typehoneycomb structure (Fig. 1) is considered as a good candidate
for smart structures. Early researches include SMA honeycombcore actuator [1], in-plane tensile testing [2] and in-plane
compressive simulation [3].
To fully discover its potential, this research focuses on lowshear stiffness honeycomb's tensile behaviour. Simulations inthis research adopt an improved version of SMA computationalmodel. This model considers different material properties oftwinned martensite phase and crystallographic re-orientation
process between variants of martensite phases. The research canbe considered as essential part of fundamental research about
low shear stiffness honeycomb.

properties between twinned martensite and detwinned
martensite, as well as the irreversible phase transformation
kinetics (or crystallographic re-orientation) from twinned
martensite phase to detwinned martensite phase. FEM
techniques are specialized for in-plane frame which could
provide both axial and radical distribution of internal variables.
3

Numerical studies

3.1

Material level validation
Since conventional models do not differentiate stiffness oftwinned martensite and detwinned martensite, a much betterstress strain fitting could be obtained by using the new model. In
Fig. 2, new model reproduced the different stiffness of twovariants. This fitting could be considered as material levelvalidation of the new model.

Fig. 2.Experiment [2] and fitting result of stress strain relation

Fig. 1. Different kinds of honeycomb structures2

Numerical studies include two kind of honeycomb structure:
OX type honeycomb structure with positive Poisson's ratio, andauxetic type honeycomb structure with negative Poisson's ratio.
Their original shapes and boundary conditions are in Fig. 3.

SMA numerical modelThis research adopts an improved version of SMA based onphenomenon model proposed by Brinson [4] and Toi [5]. The
incremental form of stress strain relation is as follows:(1)   D   S  T  

 is the transformation coefficient vector; isstiffness;   is the thermal elastic coefficient vector;   is thewherestress vector;   is the strain vector; T is the temperature;  S isthe stress-induced martensite volume fraction.
As improvements to conventional model, new elements
were embedded into model, which are:
1. Consider tensile and compressive asymmetry by using
Drucker-Prager equivalent stress
2. Distinguish twinned martensite phase and detwinned
martensite phase
3. FEM techniques such as Euler-Bernoulli element andlayered beam element
Extensive discussions related to improvement No.1 could befound in Toi [5], which are supported by experiment results in
Aurrichio [6]. Improvement No.2 considers different material

Fig. 3. Original shape and boundary conditions of auxetic type (left) andOX type (right) honeycomb structure

3.2

Auxetic type honeycomb structure
Auxetic type honeycomb structure simulation consists of
two full cycle simulations.
One is in relatively low temperature when only variants ofmartensite phase exist. In this situation, temperature is lower
than reverse phase transformation start temperature. Irreversible
phase transformation from relatively symmetry twinned tosingle oriented detwinned martensite phase occurs in thissituation.
Average stress and strain relation is plotted in Fig. 4.
Loading part starts when honeycomb structure in elastic status,
the average stress and strain relation is approximately a straightline. Then crystallographic re-orientation starts along withstiffness weakening. This behaviour is identical to experimental
result in Hassan's research [2]. New model is validated instructural level in this case. During unloading, honeycomb
structure is in elastic status. Therefore stress strain curve is astraight line in Fig. 4.

Simulations were conducted by fitting SMA experimental
stress strain curve for material level validation, and honeycomb
structure simple tensile loading for structure level validation.
OX type's positive Poisson's ratio and auxetic type's negative
Poisson's ratio were reproduced.
Finally full cycle tensile simulations were conducted for OX
and auxetic type honeycomb structure. Bifurcation behaviour
was discovered in OX type simulation. This behaviour is due tostress concentration and the resulting SMA phase transformation.
This discovery demonstrated better structural stability of auxetic
type honeycomb structure.
References[1]Fig. 4. Full cycle tensile loading of auxetic type honeycomb structure in
low temperature (left) and high temperature (right)

Another is simulation in temperature higher than reverse
phase transformation finish temperature. Average stress strain
curve could be found in Fig. 4. In this case, martensite phasetransformation and reverse phase transformation could occur. As
a result, two stiffness weakening region in stress strain curveand full recovery could be observed. Deformation shape andmartensite phase fraction under maximum loading could befound in Fig. 5. Martensite phase was induced in elements neareach joint. Maximum fraction is around 50%. In this state,expansion in both vertical and horizontal directions could beobserved, which means auxetic honeycomb structure has anegative Poisson's ratio.

[2]

[3]

[4]

[5]

[6]

Okabe, Y.; Sugiyama, H. & Inayoshi, T. Lightweight ActuatorStructure With SMA Honeycomb Core and CFRP Skins Journal of
Mechanical Design, 133, 011006, 2011
Hassan, M.; Scarpa, F. & Mohamed, N. In-plane Tensile Behaviorof Shape Memory Alloy Honeycombs with Positive and Negative
Poisson's Ratio Journal of Intelligent Material Systems andStructures, 20, 897-905, 2009
Michailidis, P.; Triantafyllidis, N.; Shaw, J. & Grummon, D.
Superelasticity and stability of a shape memory alloy hexagonal
honeycomb under in-plane compression International Journal of
Solids and Structures, 46, 2724-2738, 2009
Brinson, L. & Lammering, R. Finite element analysis of thebehavior of shape memory alloys and their applications
International journal of Solids and Structures, 30, 3261-3280, 1993
Toi, Y.; Lee, J. & Taya, M. Finite element analysis of superelastic,large deformation behavior of shape memory alloy helical springs
Computers & structures, 82, 1685-1693, 2004
Auricchio, F.; Taylor, R. & Lubliner, J. Shape-memory alloys:
macromodelling and numerical simulations of the superelastic
behavior Computer methods in applied mechanics and engineering,
146, 281-312, 1997

Fig. 5. Deformation and martensite phase fraction distribution undermaximum loading. Honeycomb is in high temperature

3.3

OX type honeycomb structure
For OX type honeycomb structure, both high temperatureand low temperature simulations are conducted. The average
stress strain curves of whole structure in two situations are very
similar. Fig. 6 is the loading histories in both temperatures. Its
loading part could be validated with experiment result in
Hassan's [2]. Similar behaviour in two results proved anotherqualitative validation in structural level for improved model.
Different from auxetic type honeycomb structure,bifurcation behaviour was observed in deformation plots of Fig.
7 and 8. Stress concentration in "X" shape region in OX typehoneycomb
inducedphasetransformation.
Phasetransformations in these regions weaken stiffness, whichbecomes the main cause of localized deformation. No matter inhigh temperature or in low temperature, or in other words, nomatter the phase transformation is reversible or irreversible,
similar bifurcation behaviour could be observed. Convex cells in"X" regions become concave during unloading process.
Further studies supported discussion above. When loading
force is small, or the material is elastic, no bifurcation happens.
4Fig. 6. Full cycle tensile loading of OX type honeycomb structure in low
temperature (left) and high temperature (right)

Fig. 7. Deformation and martensite phase fraction distribution undermaximum loading. Honeycomb is in high temperature

Conclusion

An improved computational model for SMA is developed,validated and finally applied in SMA honeycomb structure
simulation. The model takes account of twinned martensite
phase's different material property and its phase transformation
process to detwinned martensite phase. The improvement could
provide better accuracy in low temperature condition.

Fig. 8. Deformation and martensite phase fraction distribution afterunloading. Honeycomb is in high temperature