Available online at www.sciencedirect.com -

ScienceDirect Physics

Procedía

Physics Procedía 10 (2010) 33-38 :

www.elsevier.com/locate/procedia

3rd International Symposium on Shape Memory Materials for Smart Systems

Modeling of SMA superelastic behavior with nonlocal approach

Arnaud Duval, Mohamed Haboussi, Tarak Ben Zineb*

LEMTA, Nancy University, CNRS, 2 rue Jean Lamour, 54500, Vandoeuvre-lès-Nancy, France.

Abstract

Due to their thermomechanical properties (high mechanical work/volume ratio), shape memory alloys (SMA) are particularly interesting to be adopted in the design of micro-sensors and micro-actuators. Thus, various constitutive models have been developed and implemented in finite element codes in order to design such applications. If these "local" models are well adapted to describe the behavior of the bulk material, they fail to satisfactorily describe phenomena such as transformation localization or size effects observed in small samples.

A gradient constitutive model is presented in order to describe the localization of phase transformation in SMA structures. To achieve this development restricted to superelasticity, a non local variable (martensite volume fraction) is defined at a material point as the weighted average over the entire material domain of the local transformation variable. Using Taylor expansions, the non local definition is substituted by a gradient based equation, introducing a material length parameter which controls the size of the localization zone.

The gradient and mechanical equilibrium constitutive equations have been numerically integrated by using the finite element method. Two kinds of finite elements have been developed (1D truss and 2D quadrilateral) and implemented in Abaqus® via a UEL subroutine. Several simulations have been performed which exhibit the localization phenomena of phase transformation in structures undergoing mechanical loading.

© 2010 Published by Elsevier Ltd

Keywords: Shape Memory Alloy Finite Element Modeling Superelasticity Softening Instability Nonlocal gradient models

1. Introduction

Shape memory alloys are particularly well adapted for the design of microcomponents [1] [2]. This is due to their specific behavior and their high mechanical work / volume ratio. If the behavior of the bulk material is well described by macroscopic models developed in a local context [3] [4] [5] [6] [7], a softening behavior is observed in small sized samples as wires or thin films and can not be taken into account by these models. This softening gives rise to a localized deformation followed by propagation of transformation fronts.

The localization phenomenon takes place on a finite domain around the considered material point. Thus, it can not be described by a classical local approach with no reference to an internal length parameter. Indeed, the presence of softening leads to a loss of ellipticity of the governing equations and a mesh dependent solution in the case of a finite element model.

* Corresponding author. Tel.: +33 383685090; fax: +33 383685001. E-mail address: tarak.ben-zineb@esstin.uhp-nancy.fr

1875-3892 © 2010 Published by Elsevier Ltd doi:10.1016/j.phpro.2010.11.071

To solve this problem, a nonlocal approach is presented in this paper. A nonlocal counterpart of an internal variable of interest (here, the martensite volume fraction) is introduced. A gradient approach developed by Peerlings et al. [8] is adopted to preserve a strong nonlocality. An SMA macroscopic phenomenological model [7] which proves its efficiency in the description of the bulk material is modified by introducing a nonlocal variable of phase transformation.

The numerical exploitation of such a modelling requires the development of specific finite element in Abaqus® software via the user subroutine UEL where the nonlocal variable is treated as an additional degree of freedom.

2. Definition of a nonlocal variable

The presented approach has for aims to describe the behavior of superelastic films underlying instable phase transformation. As this behavior can be described by a unique internal variable, the martensite volume fraction f, a nonlocal martensite volume fraction f can be introduced:

f(-) = £ oft, ~x)f (-?) dfl(?) (1)

where O (-y,~£j = exp p) is a kernel function with - an internal length parameter and p = |~x A Taylor expansion of f (-y) around the point"x gives:

d f 1 d2 f f V) =f P) + dXi (yi - xi) + 2! dXidXj (yi - xi) - j

+ 3! dXdTdr(yi - xi) - (yk - xk) + •••

3! dxidxjdxk

Replacing this expression of f ("y) in equation (1) gives:

f (~x) = f p) + c V2 f - + d (0 v4 f - + ••• (3)

where V2 is the laplacian operator V2 = £i JX?. This kind of gradient definition is an explicit definition as f is directly computed from the local quantity f, which leads to a weak nonlocality according to [8]. This definition can be improved by combining equation 3 and its laplacian to get:

f - - c (-0 V2f (?) = f("X) (4)

which is an implicit definition of the nonlocal quantity. This partial derivative equation (PDE) is completed by adding a Neumann boundary condition Vf = 0 on the border r of the considered domain D. This ensures the conservation of the total quantity of martensite inside the considered domain: fa f dD = fa f dD.

3. Constitutive equation of an instable superelastic behavior

The local superelastic behavior is described by a simplified expression of the Gibbs free energy proposed by [7]. Here, the effect of martensite twinning and reorientation are avoided. Thus, the model exhibits only one internal variable : the martensite volume fraction f. The associated driving force is written:

T f - 1 f

Ff = sTSAT^eq - B (T - To) - Hff - aof - a, (5)

f 1 - f

where £eq is the Von Mises equivalent stress, B the stress-temperature slope of transformation limits, T the current temperature, T0, the equilibrium temperature of martensitic phase transformation, Hf, an interaction term taking into account the incompatibilities between martensite variants and eSAT, the saturation value of the transformation

strain. The two last terms correspond to nonlinear penalties to ensure the condition 0 < f < 1. The dissipative process associated to the phase transformation is modeled by a dry friction relation, introducing a critical value Ff"': |F-1 < Fj". A local instable behavior, as developed by Churchill [9], can be described by introducing the following expression of the critical force, which decreases from an initial value, as the transformation progresses:

Ff = Ffl exp (-H-j) (6)

The imposed total strain tensor is decomposed in a transformation part and an elastic one as follows:

Eij = S ijki^ki + ET (7)

were E is the total strain, S the fourth order isotropic elasticity tensor and ET = 2 Y^feSAT the transformation strain, with the deviatoric stress tensor. From the previous equations, the incremental constitutive equations are derived

as follows: _ _ _

SZij = Hj6Eu + HUjf Sf and Sf = H^SEu + Hff sf (8)

4. Variational form of the governing equations

The developed constitutive equations (4), (6) and (7) are used with the mechanical equilibrium equation:

(-) = 0 (9)

and the appropriated boundary conditions in order to describe the phase transformation in SMA superelastic films subjected to quasi-static loading.

The variational form of the governing equations is obtained by multiplying the field equations (4) and (9) by appropriate test functions ~wu and wj and integrating the result on the whole domain D:

• VSdD = 0 VWU € Wu (10)

£ ff - ¿2V2f) dD = J WffdD Vw-j e Wj (11)

The test functions belong to the functional spaces:

Wu = j"wu\"wu e [C0] ,~wu = 0 where~u is prescribedj (12)

Wf = [wJ\wj e [C], wj = 0 where f is prescribed} (13)

After partial integration of (10) and (11), the equations are expressed under their linearized form around the equilibrium state at the iteration (i - 1):

£ ["VW^ : (Huu : SE) dD + £ ["V-W^ : HufSf) dD = £ "Wu •"? (i)dr - £ [Vw^ : 2(i-1)dD

- £ w-H- : SEdD - £ w- - H— Sf + i2~V w- • V SfdD

D J ' (14)

= - I wJ-(i-1) + *2Vwj •"Vf(i-1)dD + £ w-ff (i-1)dD

Figure 1: Transition from a structural quadrangular finite element to a reference element by transformation F.

5. Finite element spatial discretization and implementation in Abaqus®

5.1. Spatial discretization

For 2D problems, a solid isoparametric finite element has been developed. It exhibits, in addition to classical displacement degree of freedom, additional ones which represent the nonlocal martensite volume fraction at each node. Bilinear shape functions are adopted to interpolate the degrees of freedom inside the element:

where n) is a given point in the reference element (see Fig. 1).

Interpolation matrices are then introduced in the weak equations of equilibrium (14) and (15). The increments of nodal displacements and nonlocal martensite volume fraction can now be linked to internal and external nodal residual forces by introducing a tangent matrix:

It should be noticed that no external force is considered for the equation of nonlocal martensite volume fraction because the only boundary condition is Neumann type and is intrinsically taken into account in the equilibrium weak formulation.

The residual forces and tangent submatrices are expressed as integrals forms on the element domain ile. For example, term [Kuu] reads:

A Gauss full integration method is adopted to evaluate these integrals. Thus, 4 Gauss points are considered in the reference element at coordinates ,

5.2. Implementation in Abaqus®

The developed finite element has been implemented in Abaqus1®, using the user subroutine UEL(User Element). The UELsubroutine is called for each element of the structure mesh. Initial value and increment of degree of freedom are given as input data and the routine must return the residual subvector and tangent submatrix. Abaqus®® is then in charge to assemble the elementary residuals and matrices to solve the problem on the whole structure.

6. Results and discussion

A holed sheet under tension is studied (see Fig. 2). Each of its extremities is loaded by an imposed displacement to reach a 7 % nominal strain. Due to symmetry properties, only 1/4 of the sheet is modelled. Several values of the internal length parameter £ have been tested (5 mm, 10 mm and 50 mm). Modelling parameters are recalled in Tab. 1. The Fig. 2 shows the spatial distribution of the nonlocal martensite volume fraction during loading.

Thickness £ B T

0.1 mm 5, 10 and 50 mm 5 MPa/°C 25 °C -40 °C 150 MPa IMPa 4

70 000 MPa 0.05 0.1 0.1

Table 1: Modelling parameters for the simulation of a tensile test performed on a holed sheet.

Figure 2: Tensile test on a holed sheet. The hole generate a stress concentration that induce a localization of the phase transformation. The value of the internal length parameter £ allows the control of the spatial range of the localization.

It can be observed that when the internal length parameter £ is lower than the size of the hole, the structure exhibits a localization of the phase transformation. This localization totally disappears for high value of £ (phase transformation is homogeneous). This kind of approach can be useful to study the size effect of a default in a given structure. A comparison with experimental field measurement should be useful to identify the value of the internal length parameter for a given material.

7. Conclusion

A finite element based on a nonlocal description of the martensitic phase transformation has been developed and implemented in the commercial finite element code Abaqus®. It allows the description of the localization of phase transformation in thin films by introducing an internal length parameter which should be linked to the microstructure of the material.

The present work is focused on the description of the softening effect on the superelastic behavior of SMAs. It will be extended to the full thermomechanical behavior in order to take into account the shape memory effect.

Another point of improvement will be the development of a nonlocal approach that permits the description of the size effect without introducing a softening behavior as it has been made for more standard materials [10], [11] [12], [13].

The purpose of these improvements is to develop accurate numerical tools for the design of smart applications made of SMA thin films.

References

[1] Y. Bellouard, Shape memory alloys for microsystems: A review from a material research perspective, Materials Science and Engineering: A 481-482 (2008) 582-589.

[2] M. Tabib-Azar, B. Sutapun, M. Huff, Applications of TiNi thin films shape memory alloys in micro-opto-electro-mechanical systems, Sensors and Actuators 77 (1999) 34-38.

[3] K. Tanaka, A thermomechanical sketch of shape memory effect: One dimensional tensile behavior, Res Mechanica (18) (1986) 251-263.

[4] B. Raniecki, C. Lexcellent, Thermodynamics of isotropic pseudoelasticity in shape memory alloys, European Journal of Mechanics - A/Solids 17 (2) (1998) 185-205.

[5] M. Panico, L. C. Brinson, A three-dimensional phenomenological model for martensite reorientation in shape memory alloys, Journal of the Mechanics and Physics of Solids 55 (11) (2007) 2491-2511.

[6] P. Popov, D. C. Lagoudas, A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite, International Journal of Plasticity 23 (10-11) (2007) 1679-1720.

[7] Y. Chemisky, A. Duval, E. Patoor, T. Ben Zineb, Modeling behavior of shape memory alloys, including phase transformation and twinning inside martensite effects, submitted in Mechanics of Materials.

[8] R. Peerlings, M. Geers, R. D. Borst, W. Brekelmans, A critical comparison of nonlocal and gradient-enhanced softening continua, International Journal of Solids and Structures 38 (2001) 7723-7746.

[9] C. Churchill, J. Shaw, M. Iadicola, Tips and tricks for characterizing shape memory alloy wire: Part2 - Fundamental isothermal responses, Experimental Techniques 33 (2009) 51-62.

[10] N. A. Fleck, J. W. Hutchinson, A phenomenological theory for strain gradient next term effects in plasticity, Journal of the Mechanics and Physics of Solids 41 (12) (1993) 1825-1857.

[11] N. A. Fleck, J. W. Hutchinson, Strain Gradient Plasticity, Advances in Applied Mechanics 33.

[12] H. Gao, Y. Huang, W. D. Nix, J. W. Hutchinson, Mechanism-based strain gradient plasticity -1. Theory, Journal of the Mechanics and Physics of Solids 47 (1999) 1239-1263.

[13] H. Gao, Y. Huang, Taylor-based nonlocal theory of plasticity, International Journal of Solids and Structures 38 (2001) 2615-2637.