THESIS TITLE Characterization and Peridynamic Modelling of Shape Memory Alloy based

Self-Healing Composite in Aerospace

UNIVERSITY UNIVERSITY KEBANGSAAN MALAYSIA(UKM)

DEPARTMENT Built Environment & Engineering

MAIN SUPERVISOR

Prof. Dr. –Ing. Nik Abdullah Nik Mohamed

CO SUPERVISOR Dr. Salahuddin bin Mohamed Harris

STUDENT Shakib Sharifian

REPORT TITLE Numerical solution of ordinary nonlinear equations of motion for

Peridynamics model

DATE 9 May. 2012

Abstract

Until now I used Peridynamics rules to model a fully elastic material under shear loading and solve

the extracted equation of motion numerically by forth order Runge-Kutta (Appendix A).

To observe the deformation of a model or finding final states of particles, I need to have static or

steady state solution. Actually, the solution for ordinary nonlinear equation of motion for

Peridynamics includes dynamic terms that need to be eliminated. Although, searching for a method

to converge dynamic solution to static is the main effort for this report. When I draw the result of

simulation during time, I tried to catch a general solution that just observe the states and estimate

the position of desired static solution.

The proposed method1 was successful to estimate the steady state solution of equation over 1 and 2

dimension systems like strings and thin plates, but I understand that there are some other

conventional methods to get the final quasi-static deformation, dynamic relaxation methods, that I’m

working on this topic at present, and I will mention my results about this method on next reports.

In continue I’m going to describe the method that was used to match dynamic and static solution.

Estimation of Steady state solution by observing the general numerical solution

Assume that we have a simple spring system (Table1) which is under a constant load (Fig.1); static

solution of final displacement is like (1)

∑

(1)

Table 1 . Mechanical properties of a simple spring system

Amount Property

2 External load [N]

1 Spring constant [N/m]

1 Mass [Kg]

On the other hand, from the equation of motion (2), we can calculate the position of mass during

time (Fig.2)

∑ (2)

Figure 1 Simple mass spring system

1 . This method is described over presented report

Figure 2. Dynamic solution of simple mass spring solved by our code in matlab

But the question is how could we matching dynamic and static solutions; the profit of the finding a

way to this matching will lead to investigation about dynamical events on particles of material from

start of loading until reaching final deformation, very similar to what do happen on a specimen in an

mechanical experimental test.

Matching dynamic solution with static for Peridynamics

Actually in Peridynamics, the main equation is equation of motion (2), so the solution of particle’s

position will be in frequency domain and the answers are changing during time, but for our purpose

which is simulation of deformation by Peridynamics method under shear loads, the initial and final

configuration of particles are needed, in other words, we are looking for the static solution that on

this position the total force on nodes are zero (3).

∑ ⇒

(3)

As it can be seen from (Fig. 2) during time of simulation the position of mass is passing through static

stability point frequently and in this point the amount of is in maximum, so one method for finding

stability point can be investigation about zero slope of , but there are two main problem in this type

of solver, until when I have to continue for solving the equation to catch this maximum amount,

estimation of local extremum, and which step time is appropriate for our simulation.

Thus, the system should be capable to estimate this extremum point however we passed through it

during time. In first step I replace the axis of time in the diagram and used Lyapanov diagram instead

(Fig.3) which is applicable for investigation of stability point.

Figure 3. Lyapanov diagram for a simple spring system

As it can bee seen from (Fig.3) the system has a neutral stability because of the pure elastic and

without disspative propertiy of theconsidered system,and static stability happened on x=2 which is

the answer of static solution for our system which is described in (Table.1).

Fast estimation of stability point for neutral stability

Now consider that we just run our dynamical simulation just for very small steps from initial position,

so the position and velocity diagram and Lyapanov diagram are like (Fig. 4) and (Fig.5), respectively.

Although we know that we are faced with a neutral system, but by analyzing (Fig.4), it is not possible

to catch stability point for this system, because its information for velocity and the energy of system

is not sufficient, on the other hand, from (Fig.5) for a neutral system it is obvious to find the center of

the mentioned circle and find the stability point by the method that will be described in continue.

Figure 4. Dynamical numerical solution of simple spring system during small time after initial position

Figure 5. Lyapanov diagram for simple spring system during small time after initial position

Figure 6. Attributed circle by 3 points

Equation of a circle from 3 points

Consider that we have 3 point that cannot be connected by a straight line, so we can pass a circle

through them. The technique just need to consider these points pairwise, for instance consider that

we have 3 points, P1,P2 and P3 (Fig.6), then we can pass through P1 and P2 line a, and line b

contains P2 and P3, the equation of these lines are,

{ ( ) ( )

(4)

Where and are the slopes of the lines; the center of this circle comes from the intersection of

perpendicular lines on line a and line b which can be described by (5)

{

( ) ( ) ( )

( )

(

)

(5)

Extension the model

Until now, I just consider a single mass-spring system, but what will happen to more complex

systems that are modeled by numerous mass-springs, actually the Lyapanov diagram for these

systems doesn’t have a circle shape, but elliptical (Fig.7), so we need to use another method to

estimate the stability point, using Taubin’s Method [2] to attribute an ellipse, and using ellipse

properties to calculate its center. (Appendix B)

Figure 7. Lyapanov diagram for 3 mass-spring system

Testing the capability of method

Till now the system should have this ability to estimate the equilibrium during observing the motion

of particles, so we described three scenarios to challenge the method.

Table 2. Scenarios for testing the extracted method

Scenario 1

(1 Dimension system)

* +

* +

* +

*

+

Scenario 2 (2 Dimension system)

K is 1 but with Peridynamics horizon equal to 2.5

Fext=.05N shear loading on the upper surface

Scenario 3 (2 Dimension system)

K is 1 but with Peridynamics horizon equal to 2.5

Fext=1N tensile loading on the upper surface

The displacement behavior of the scenario 1 that is solved by Runge Kutta (Fig. 8) for mass number 1

is not smooth and over the simulated time, it is not periodic yet, because of its periodic time

constant. The amplitude is same as the amount that gains from static solution, and the steady state

solution is in the average between minimum and maximum point for displacement. In (Fig. 9) the

centers of fitted ellipses for scattered data are presented that estimate the steady state solution of

system and in (Fig. 10) and (Fig. 11) the convergence of proposed method is proved for estimation of

steady state solution during increasing simulation time. The final deformation of system in the

scenario 1 is like (Fig. 12) that also plots the trajectory of dynamic solution.

Figure 8. displacement of mentioned springs in table.2

Figure 9. Lyapanov diagram of mentioned springs in table.2

Figure 10. Estimated Steady stae solutions of mentioned springs in table.2 during simulation time

Figure 11. Variation of Estimated steady state solution during simulation time

Figure 12. Estimated Static solution during simulation and trajectory of dynamic solution

Figure 13. Trajectory of particles under shear loading (scenario 2)

Figure 14. Estimated static solution (scenario 2), crosses: initial position, circles: equilibrium posision after loading

As it can be seen from (fig.13), the simulation for a thin plate under shear loading is rational, the first

layer deformed more than other layers and there is a proportional relation between displacement of

each layer, so the method is working well in estimating the static solution and neglecting transient

responds of equation of motion by analyzing the trajectory of particles (fig.14).

Figure 15. Dynamic solution of particles under tension loading (scenario 3)

Figure 16. Estimated static solution (scenario 3) , crosses: initial position, circles: equilibrium posision after loading

In (fig. 15) at time the particles are meeting different positions where are different by final

equilibrium position (fig. 16), because particles meet their equilibrium answers in various times.

Next Task in simulation

As I searched more I faced with Adaptive Dynamic Relaxation methods (ADR) that also is useful for

extracting steady state solution from dynamic or pseudo-transient response [1]. Moreover, I want to

add the critical stretch value for connectors in simulator and define local damage parameter that can

introduce subject of growth of the crack on a model.

Appendix A

Two codes have been developed in MATLAB to create desired models for simulation and define loads

on this model and then it was solved numerically by simple Euler's method, but during simulation

because of lack of accuracy in mentioned method, simulator didn't present true results, although the

step size of time was decreased, so I need to change the solver part from Euler's method to another

solver that have been used for solving ODE systems.

The most popular method for solving ODE systems, as it could be seen in other solvers like ODE45 in

MATLAB, is Runge Kutta method, therefore as the first step of improving the simulator I decided to

implement Runge Kutta also. Usually to solve an ODE system by numerical methods, at first the

system must be written in state matrix form. In below it could be seen that how I rearrange the

simulator states by considering of their implemented dimensions in my simulator.

[

]

( )

[

]

[

]

[

]

⇒

}⇒ {

Because of the accepted dimension for ODE45 solver, I have to change some of upper mentioned

parameters like below,

[

]

*

+

( )

( )

[

]

*

+

* ( )

( )+ * ( )

( )+

[

]

[

]

(| | | |

| |)

[

]

( )

[

]

| |

[ | | |

|

| | |

|

| | | |

| | | |]

(

| |)

*

+

[

]

[ | |

| |

| |

| |

| |

| |

| |

| |]

[ ]

( )

[

]

[ ∑

∑

∑

∑ ]

[

]

[

]

[

]

{

}

,

-

[

]

⁄

{

}

,

-

{

[

]

⁄

}

Calculate internal force among pointed node

(i) and family nodes (j)

Loop

All nodes are

considered

Point next node

𝑭𝑻𝒐𝒕𝒂𝒍𝒊 𝑭𝑬𝒙𝒕𝒆𝒓𝒏𝒂𝒍𝒊 𝑭𝑰𝒏𝒕𝒆𝒓𝒏𝒂𝒍𝒊

{

��𝟏 𝑿𝟐

��𝟐 𝑿𝟑 𝑭𝑻𝒐𝒕𝒂𝒍𝝆

}

,��𝟏

��𝟐

-𝑶𝑫𝑬𝟒𝟓 {

𝑿𝟏(𝒕)

𝑿𝟐(𝒕)}

Go to

next

time

step

No

Ye

s

𝑿𝟏 : States of Positions

Figure A1. Simulator in Schematic

Appendix B

Ellipse curvature properties

Through the general curvature equation of an ellipse (Fig.1b), it is possible to find the centers of

ellipse, F1 and F2, and then the average of them will be like calculated as below

And we have below relation among parameters

And through these parameters we can reach center of C in (Fig.1.b)

{

Figure1B. An ellipse with its properties

Reference(s)

[1].”Adaptive Dynamic Relaxation Algorithm for non-linear hyper elastic structures, Part I

formulation”, D. R. Oakley, N. F. Knight, Jr. , Computer Methods in Applied Mechanics and

engineering, Elsevier, 1993

[2]. G. Taubin, "Estimation Of Planar Curves, Surfaces And Nonplanar Space Curves Defined By Implicit Equations, With Applications To Edge And Range Image Segmentation", IEEE Trans. PAMI, Vol. 13, pages 1115-1138, (1991)