Jarno Lukasz CL

JUNIORSTAV 2011 2.1 Concrete and masonry structures

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CONCRETE PRESTRESSED STRUCTURES USING SHAPE MEMORY ALLOYS.

CONSTITUTIVE MODELLING.

Łukasz Jarno1

Abstract

This paper is an attempt to answer for a question if its possible to apply a new kind, in civil engineering, materials

which are known as shape memory alloys to effective prestessing for new building concrete structures and also for

strengthening existing structures.

Keywords

Shape memory alloys, smart materials, concrete, prestressed concrete, modelling.

1 GENERAL INFORMATION

Shape memory alloys (SMA) are kind of the smart materials which has found increasing application in many areas.

These are materials which have some specific properties, useful to applying it in very different systems of structures

which can be able to automatically adjust their characteristics according to external condition, such a unexpected

quantity of loading, and their allow to build slender, wide spanned structures with very high ability to changes in use.

There are also proposition to use shape memory alloy for sustainment of existing structures. They have been used as

actuators in many mechanical devices and for some seismic suppressors, because of its damping capacity. From

macroscopic point of view there are two major properties responsible for these abilities, shape memory effect and

pseudoelasticity. Until now there are about 30 well - known alloys which are reported as shape memory alloys, but not

all of them can be used in civil engineering structures. [5], [8].

2 SHAPE MEMORY ALLOYS DESCRIPTION

2.1 Shape memory effect and pseudoelasticity

SMA are materials which can be able to return to their previously remembered shape after deformation during to

operating with temperature and this phenomenon is called one way shape memory effect. Like many other materials

(metals or alloys), SMA’s have more than one crystal structures, so it is called as polymorphism. Certain crystal

structures depend highly on both temperatures and external stress. There are two stable phases corresponding with these

crystal nets, stronger high temperature phase called austenite and weaker low temperature phase called martensite and

there is a possibility to reversible switching from one to the other. Each of these phases has completely different

mechanical, physical, thermal and electrical characteristics. The austenite has a body – centered cubic crystal structures

and the martensite has a parallelogram. When SMA in martensite phase is under external stress, they deform through

a detwining mechanism and are not a just elastic strains, this is a reorientation of atoms in crystal structures which can

handle with maximum elongation. Martensite phase is very weak and soft, can be deformed very easily due to its

parallelogram structure of crystals. In the other hand the austenite phase has only one possible crystal orientation and it

has strong resistance for external stresses.

1 Lukasz Jarno, MSc C.Eng., Cracow University of Technology, Civil Engineering Faculty, Institute for Building Materials and Structures, Warszawska street, 31-155 Cracow, mobile +48 604575600, e-mail: [email protected]


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Fig. 1 Shape memory effect.

Another effect for shape memory alloys is pseudoelasticity. This effect has place in higher range of temperature then

shape memory effect. We are talking about pseudoelastic behavior of shape memory alloys when we apply to the

material external mechanical loading, the border of some critical stress is crossed through, and the corresponding strain

is about several percent, and it disappear after unloading. Cycle of the loading – unloading is not runs on the same path

as in nonlinear elastic materials but shows hysteresis loop. [1], [5], [8], [9].

2.2 Reversible diffusionless transformation

The internal crystal structure depends on energy level. For certain temperature, crystal net needs to gain the

minimum energy state. Only by operating with external force the phase transformation can has place: martensite

transformation and its reverse transformation. This external driving force for thermodynamic transformation is the

difference between the Gibbs free energy of the two stable phases, martensite and austenite. This transformation can be

launched by either temperature gradient or mechanical loading. When consider thermomechanical laws, the role of

temperature and external stress is equivalent in case of transformation. As a result of this, there are two different ways

to go through martensite transformation, caused by temperature what leads to shape memory effect and caused by

external stress what has a result as pseudoelasticity.

Fig. 2 Stress free martensitic phase transformation

Figure 2 shows a general stress-free martensitic transformation and its invert transformation launched by activity of

temperature. There are four characteristics temperatures in those transformations. Ms is a temperature when

transformation from austenite to martensite begins, and at Mf, the transformation is finished. On the opposite direction

As is a temperature when the invert transformation begins (from martensite to austenite) and Af is temperature of the

end of transition. The important thing is that temperature – deformation loops is hysteretic because of internal phase

friction. These four characteristics temperatures are highly dependent on kind of the alloy, its microscopic composition,

treatment during processing and on operating by external stresses from mechanical loading. Typical stress – strain graph

for shape memory alloy in situation when there are 100 % of martensite fraction and temperature T < Mf. An example

o SMA is stretched, so the elastic deformation is followed by great strain corresponding with stress. This kind of

reaction has place because of the hysteresis – mobility of the twined variation and some defects inside the martensite

phase. The deformation is not a deformation within the meaning of mechanical theory, but it is an atom reorientation in

crystal structures inside the martensite. When unloading, only the elastic part of strain can be recovered, the “strain”

from this reorientation inside martensite can be recovered only in reverse transformation from martensite to austenite


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during to heating. If the deformation is too high for the martensite phase capacity (in the meaning of reorientation

mechanism) the permanent plastic deformation takes place. When designing structure it is necessary to remember that

applied stress should not exceed the maximum value corresponding with plastic deformation. [9].

2.3 Other kinds of shape memory alloys

There are also some other kinds of shape memory alloys. Fe-Mn-Si-X alloys is the represent of so called iron

– based shape memory alloys, which are also refers as shape memory steels. The main advantage of these kinds of

alloys is the 60 – 65 % ratio of iron fraction and this is the reason of low cost compares to the Ni – Ti alloys with

remaining high strength and a high Young’s modulus. Nowadays it is not possible to have two – way memory effect in

these kind of iron – based alloys. Corrosion resistance in outdoor environment is not good until addition of about 10%

chromium and nickel, and then the resistance is comparative with stainless steel. There are also other non iron – based

shape memory alloys founded. [5].

3 CONSTITUTIVE MODEL FOR CONCRETE SECTION

3.1 Boltzmann principle for viscoelastic behavior of concrete under continuous loading.

The rules of viscoelastic theory describe well behavior of concrete under long – term loading. However, classical

theory works only with constant loading (applied stress or strain), it was necessary to find a solution in case of

continuous and changeable process of loading. Based on fundamental laws of viscoelastisity is difficult to find this

solution, because of number of variables of the loading history. In case of this finite time period problem, a new

incremental form of constitutive law for material was built. Theory based on Boltzmann principle and linearity, propose

to divide finite period of time on to infinitely small parts, and write is as integral form of superposition principle.

(1)

With as a characteristic function representing material properties. There is more general form of this equation

with including value of the age at loading time t0 :

(2)

Those “action” and “response” express each of them either local strain or local stress, and it is depend on

formulation of equation. A characteristic function in a kernel of integral describes relation between stress and strain and

can be used either as a creep function in creep equation or as relaxation function in relaxation equation.

Creep equation:

(3)

Relaxation equation:

(4)

There are also another formulation (just mathematical rearrangement) of these equations and are called as Volterra

equations but for a purpose of this project are useless so will be ignored. [DES-99], [GWO-99].

Solution of Boltzmann equation for a finite time period is considered:

(5)

where:

is stress corresponding to free strain (thermal, shrinkage).


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Fictitious value of elasticity modulus of material:

(6)

And history term:

(7)

Formula (5) shows a linear relation between increase of local strain and stress over a finite time period in uni – axial

state of stress. [2], [4].

4 CONSTITUTIVE MODEL FOR SHAPE MEMORY ALLOYS

4.1 General macroscopic description of the thermo – mechanical behavior of SMA (Lexcellent

model)

Basic assumptions:

Total martensite fraction z was split into two parts:

zT - obtained under pure thermal action, which was self-accommodating, only reorientation of crystals not macroscopic

strain.

zσ – obtained by an external mechanical stress, corresponding to macroscopic strain.

Total strain was divided into two parts, elastic (e) and transformation (tr) strain:

(8)

Form of the Helmholtz free energy was chosen like:

(9)

is the free energy of phase:

1 – parent phase (austenite)

2 – self-accommodating martensite

3 – oriented marteniste

Expression of this energy is:

(10)

Where and are the specific energy and entropy of the α phase.

The constitutive laws resulting from this model are for the forward phase transformation (austenite to martensite):

(11)

(12)

For reverse transformation (martensite to austenite):

(13)


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(14)

There are thirteen materials parameters different kinds of alloys determine by some anisothermal and mechanical tests.

5 INTRODUCING OF MODELED OBJECT

The purpose of this thesis is to try preparing a mathematical model for a pre – stressed plain concrete beam

introduced below, using Shape Memory Alloys tendons. The model is build using different models for these two

materials, and connecting them together. For behavior of Shape Memory Alloy wire acting as a tendon in pre – stressed

concrete structure, the Thermo – Mechanical model based on Helmholtz free energy is used. For concrete behavior

under this pre – stressing, model was build in to steps, the first is pure elastic model based on a beam theory, and then

visco – elastic model based on Boltzmann equation was developed.

5.1 General information. Geometry and material properties

The first step of preparation was to calculate the dimensions of three beams. Then to specify the dimensions and

numbers of tendons with Shape Memory Alloys. The best settings were determined by simulations. All data concerning

the wire of SMA’s are obtained from preliminary tests. Each of beams were pre - stressed by 4, 8, and 12 wires.

Geometry and material characteristics (figure 3) in pre-dimensioning procedure were:

- Preliminary dimensions of beam is 48 x 60 x 520 mm, distance between supports l0 = 450mm.

- The class of concrete used to build – up the beam was C20/25, which corresponds to an average strength

and modulus of elasticity 19.9 GPa.

- The wire of SMA’s has a diameter equal 1 mm, and their yield strength was . The modulus

of elasticity for martensitic phase was , and for austenite phase it was

.

- The SMA’s wires were placed in 4 mm from underside surface of the concrete beam.

- Special anchorage was developed because of very small diameter of SMA’s wires, and not allowing them to

slip.

Fig.3 Geometry and cross-section of beam

5.2 Shape memory alloys behavior

The first step was to find a model for Shape memory alloys behavior. The one chosen was thermo – mechanical

model developed by S. Leclercq and C. Lexcellent and programmed by Hanh Tran. This is a termo – dynamic

constitutive model, basing on thermo – dynamic laws. The model describes changes of Helmholtz free energy to

determine participation of each phase during transformation. Basic assumptions and equations for this model are

explained in chapter 4.1. In our case it was not necessary to programming the whole cycle of transformation. The SMA

model was divided onto two parts. First part was build to simulate increase of temperature, producing austenite fraction

and second part for decrease of temperature. Input data for model itself is initial strain which was applied to wires in

martensite phase at ambient temperature. Type of alloy used allows to keep either martensite or austenite phase at room

temperature. It was 49.2 % Titanium and 50.8% Nickel alloy.

Programming cycle is started at room temperature (20 deg. C) with 100 % martensite wires anchored and stretched

with certain strains (eps_mart) which are presented in table 1. Knowing these values of initial strains and uniaxial

pseudoelastic strain equals 0.061 fraction of stress induced martensite was calculated as (values are shown in table 2)


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(15)

Then temperature was increased to 60 deg. Celsius to be sure that no martensite fraction remained and there was 100%

austenite. The speed of increasing was 1.6 deg. C per minute. The whole time of increasing was 25 minutes.

eps_mart * (10^-3)

Beam 1 Beam 2 Beam 3

series 4 wires 8 wires 12 wires

SMA1 1,583 1,639 1,69

SMA2 2,729 2,379 2,469

SMA3 3,916 4,123 4,17

SMA4 6,009 6,032 6,152

SMA5 8,184 8,195 8,173

SMA6 10,027 10,265 10,408

Tab. 1 Initial values of strain applied tot he SMA wires.

z_sig

Beam 1 Beam 2 Beam 3

series 4 wires 8 wires 12 wires

SMA1 0,0260 0,0269 0,0277

SMA2 0,0447 0,0390 0,0405

SMA3 0,0642 0,0676 0,0684

SMA4 0,0985 0,0989 0,1009

SMA5 0,1342 0,1343 0,1340

SMA6 0,1644 0,1683 0,1706

Tab. 2 Quantity of stressed – induced martensite fraction

When maximum temperature was reached, 50 minutes process of decreasing began from 60 deg. C to 20 deg. C with

speed of decreasing 0.8 deg. C per minute. Both increasing and decreasing paths are shown on figure 5.1.

Fig.3 Shape memory alloys temperature acting

The results that we was looking for was stress in SMA’s wire changing due to increasing and decreasing temperature.

Those stresses were inputs to concrete part of modeling to show changing of flexural curvature of the pre - stressing

beam. Next step was to find last value of stress (sig_crit) and put it inside another part of concrete behavior where only

pure creep was considered. The whole cycle of stress is shown on fig 4. We start from room temperature (20 deg. C)

and when increasing temperature the As border was reached and the martensite – austenite transformation had place

finished at Af. Depending on initial values of martensite strain, maximum level of stress can be reached. There is

a possibility during decreasing temperature from maximum to ambient, to reach the border (Ms) and then some fraction


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of martensite can be produced, and level of stress decrease along to this border. Model of SMA’s has taken into account

this kind of possibility and thermal expansion of wires due to these processes.

Fig.4 Stress depending on temperature

6 ADAPTATION OF THE THEORETICAL APPROACH TO THE TEST RESULTS

Homegeneous visco – elastic beam (generalized variables).

Constitutive law of homogeneus viscoelastic beam cross – section in terms of the generalized variables of the classic

beam theory.

(16)

Generalized variables:

Generalized stress (increment over the given time interval

(17)

Generalized strains (increment over the given time interval

(18)

are increments of the normal force N and the bending moment M related to a given reference axis in the cross

– section. The axis is chosen arbitrary.

are increments of the longitudinal strain and the flexural curvature of the bent cross section (calculated at the

given reference axis chosen previously).

Fictitious stiffness matrix calculated with the fictitious modulus of elasticity:

(19)

Where K is the geometry matrix given by :

(20)


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A, B, I : are beam cross section area, first moment of inertia, second moment of inertia, calculated according to the

given reference axis.

Free – strain (shrinkage, thermal expansion):

where

is increment of the longitudinal free strain at the given reference axis

Term of history:

(21)

Hence:

(22)

(23)

Where:

(24)

(25)

Initial values of stresses from normal force and bending moment were calculated with classic beam theory and linear

elasticity and they are given by:

(26)

(27)

Where:

is equal to which is the force calculated from shape memory behavior part, is the output from that model, and

input for concrete part at the same time.

is equal to multiply by eccentricity, distance from central point of SMA’s wire to geometric center of

concrete part.

This model was validate by comparing creep function calculated with one from uni – axial visco – elastic formulation

7 CALCULATION RESULTS PRESENTATION AND DISCUSSION

In this part, results and conclusion of calculation are presented. Calculation was divided onto two parts, first part is

cooperation between, shape memory alloys model and visco – elastic modeling of concrete. Firstly, forces in SMA’s

wires are presented on charts and curvature of concrete beam pre – stressed. Second part is a presentation of long – time

concrete behavior.

All of curvature calculation was done for three beams, pre – stressed with 4, 8 and 12 shape memory alloys wires. For

each beam, different initial martensite strain of wire was applied. Calculation of force in wires, was done only for only

one beam with pre – stressing by 4 wires.

All of the initial values of martensite pre – strain and quantity of initial fraction are presented in tab. 1 and tab. 2.


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Fig.5 Forces in Shape memory alloys wires

Figure 5 presents changes in forces, during increasing temperature from room (20 deg.C) to maximum temperature (60

deg.C), and then decreasing to room temperature again. Each force, starting with Force 1, correspond to SMA series

with specific initial martensite strain. First increasing part of diagram is due to martensite to austenite transformation

which is done at the maximum point. Then, next is a decreasing part, correspond with thermal expansion due to

increasing temperature from austenite transformation finish temperature to the maximum. This maximum was chosen as

60 deg. C to be sure that all transformation was done. Next part is decreasing temperature and increasing force because

of thermal contraction of the wires. As we can see, on the diagram, when consider Force 4, which corresponding to

eps_mart equal about 6 promiles, the “new” part of diagram is shown about 26 deg.C the force is decreasing to a value

of 780,67 N. Afterwards, forces 5 and 6 are below force 4 but ending about the same value, 779,28 N and 740,95 N.

The decreasing slop at the and of force 4 is because reaching a border for martensite start transformation and some of

martensite fraction was produced in our case this phenomena is consider as instant pre – stressing loss, which has to be

taking into account during calculation and designing process. It was unknown why force 5 and 6 are lower than force 4.

It is explained using figure 6.

Fig.6 Forces in SMA’s wires with maximum temperature 100 deg.C

As we can see on figure 6, maximum temperature was change to 100 deg. C. Then forces 4, 5 and 6 at the end point

have almost the same value. It is causing by some production of martensite which has result of decreasing slope at the

end. When compare Fig. 5 and Fig 6 we can see that forces 5 and 6 now are above force 4. This is a proof that

previously with maximum temperature equals to 60 deg.C in case of martensite pre – strain about 6 promiles and above

transformation wasn’t completed. Temperature didn’t crossed a finish martensite to austenite finish border.

Up to SMA 3 research serie, results from experiments and from the model are comparable, especially for 4 wires case.

For SMA 3 and next series results are good for 4 wires, but for 8 and 12 are overestimated. Curvature didn’t increasing

during increasing of martensite pre – strain, the are still the same two reasons of that behavior, production of martensite

because of crossing the border of martensite transformation start and not completed austenite transformation.


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Literature

[1] Z. Deng, Q. Li, H. Sun - Behavior of concrete beam with embedded shape memory alloy wires. Engineering

Structures, October 2006, 28(12), 1691-1697

[2] J.-F. Destrebecq, P. Gwoździewicz, B. Jurkiewicz – Time analysis of prestressed concrete members subjected

to axial plus shear loads. Wroclaw, Poland 1999.

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deposited shape memory alloys Ti – Ni and Ti – Ni – Cut thin films. Thin Solid Films, May 2000.

[4] P. Gwoździewicz – Time analysis of prestressed concrete structures with accounting for Bendig and shear

effects. Clermont – Ferrand 1999.

[5] L. Janke, C. Czaderski, M. Motavalli, J. Ruth - Applications of shape memory alloys in civil engineering

structures: Overviews, limits and new ideas. Materials and Structures, June 2005, 38(5), 578-592

[6] B. Jurkiewicz, J-F. Destrebecq, A. Vergne – Incremental analysis of time – dependent effects in composite

structures. Computers & Structures 73 (1999).

[7] C. Lexcellent, S. Leclercq – A general macroscopic description of the thermomechanical behavior of shape

memory alloys. November 1995

[8] G. Song, N. Ma, H.N. Li - Applications of shape memory alloys in civil structures. Engineering Structures,

July 2006, 28(9), 1266-1274

[9] A. Ziółkowski – Pseudosprężystość stopów z pamięcią kształtu. Badania doświadczalne i opis teoretyczny.

Prace IPPT. Czerwiec 2006.

Reviewer

Piotr Gwoździewicz, PhD. C. Eng., Cracow University of Technology, Civil Engineering Faculty, Institute for

Building, Materials and Structures, Warszawska street 24, 31-155 Cracow.

Jarno Lukasz CL

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