This document describes a peridynamic model for simulating shape memory alloy composites. It presents the key equations of peridynamic theory, including the balance law accounting for internal forces between particles within a horizon. A simple 2D simulation in MATLAB is outlined using linear springs to model internal forces between particles within a defined horizon based on their displacement. The simulation calculates net internal and external forces on particles over time to model deformation, with the goal of upgrading the model in the future to include failure mechanisms.

1. Date: 7/10/2011
Report Number: 4
Title: Characterization and Peridynamic Modeling of Shape Memory Alloy based Self‐Healing
Composite Aerospace
Keywords: Gravitational force, Elasticity force, Balance law, Simulation, 2 Dimensions
Gravitational and elasticity forces
For interaction energy between particles with conservative and dissipative forces we have below
equation:
ቄௗటೣೣᇲሺ௥ೣೣᇲሻ
ௗ௥ೣೣᇲ
௖ ሺݎ௫௫ᇱሻቅ ݏ௫௫ᇱ ൅ ݂௫௫ᇱ
൅ ݂௫௫ᇱ
ௗ ሺݎ௫௫ᇱ, ݏ௫௫ᇱሻݏ௫௫ᇱ ൑ 0 (1)
So in situation that only conservative events are considered, we will have:
௖ ൌ െ ௗటೣೣᇲሺ௥ೣೣᇲሻ
݂௫௫ᇱ
ௗ௥ೣೣᇲ
(2)
The classical law of universal gravitation considers the interaction energy and force which are
exerted from particle x to x' by respective mass ݉௫ and ݉௫ᇱ as below forms, and G>0 is the
universal gravitational constant.
߰௫௫ᇱ ൌ ܩ ௠ೣ௠ೣᇲ
௥ೣೣᇲ
(3)
݂௫௫ᇱ ൌ െܩ ௠ೣ௠ೣᇲ
మ ݊௫௫ᇱ (4)
௥ೣೣᇲ
Moreover the net of interaction energy in an area like P will be calculated from below summation
that ܲሼ݌௫ሽ means all of particles in area except ܲ௫.
߰ሺܲሻ ൌ ଵ
ଶ
Σ Σ ܩ ௠ೣ௠ೣᇲ
௣ೣ ఢ ௉ ௣ೣᇲ ఢ ௉ሼ௣ೣሽ (5)
௥ೣೣᇲ
Similarly, for two particles x, x' connected by linear spring with constant ܭ௫௫ᇱ ൐ 0 and reference
length of ݈௫௫ᇱ ൐ 0, the interaction energy and force are like below
߰௫௫ᇱ ൌ ଵ
ଶ ܭ௫௫ᇱሺݎ௫௫ᇱ െ ݈௫௫ᇱሻଶ (6)
݂௫௫ᇱ ൌ െܭ௫௫ᇱሺݎ௫௫ᇱ െ ݈௫௫ᇱሻ݊௫௫ᇱ (7)

2. Figure 1. Relation between bond force density and bond strain in a composite
Master balance law in Peridynamic
As we can see from (Fig.1) bond forces are considered as a linear spring (elastic forces) based on
bond displacement, so these parts will be mentioned on the first part in right hand of Peridynamic
equation and "Van der Waals" forces which act as gravitational forces are mentioned in body force
part of Peridynamic equation (second part in right hand), moreover from (Fig.1) we can see that in a
composite if the forces act on fibers of composite, maximum sustainable amount of bond force will
be higher than available amount for matrix.
ௗ
ௗ௧ ׬ ߩݕሶሺݔ, ݐሻܸ݀ ௉ ൌ ׬ ׬ ݂ሺݔᇱ ௉ ஻௉ , ݔ, ݐሻܸ݀௫ᇱܸ݀௫ ൅ ׬௉ ܾሺݔ, ݐሻܸ݀௫ (8)
There is a master balances law in Peridynamic (eq.9) which define rate of a property like E that
depends on dimension of region B has scalar or vector form and D is known as dual interaction
density part and the first part shows all interaction between sub‐region P and its exterior and final
part source rate s.
ܧሶሺܲሻ ൌ ׬ ׬ ܦܸ݀Ԣܸ݀ ௉ ஻௉ ൅ ׬ ݏܸ݀ ௉ (9)
If we consider that dual interaction density remain same in whole of region B and we have two
different sub‐regions P1 and P2 in B like below
ܲଵ ת ܲଶ ൌ ׎ (10)
In continue we can write (eq.9) for P1 and P2 and also other different sub‐regions
ቂܧሶሺܲ௡ሻ ൌ ׬ ׬஻௉ ܦܸ݀௫ᇱܸ݀௫ ௉೙ ೙
൅ ׬௉ ݏܸ݀௫ ೙
ቃ
௡ୀଵ,ଶ,ڮ
(11)
And if we plus right hand of equation for two different sub‐regions we will reach additive property
for E.

3. (12)
න න ܦܸ݀௫ᇱܸ݀௫
௉భ ஻௉భ
൅ න ݏܸ݀௫
௉భ
൅ න න ܦܸ݀௫ᇱܸ݀௫
௉మ ஻௉మ
൅ න ݏܸ݀௫
௉మ
ൌන න ܦܸ݀௫ᇱܸ݀௫
௉భת௉మ ஻௉భת௉మ
൅ න ݏܸ݀௫
௉భת௉మ
ൌ ܧሶሺܲଵ ת ܲଶሻ
If we have an anti‐symmetric dual interaction density then we can write below equalities:
׬ ׬ ܦܸ݀Ԣܸ݀ ௉௉ ൌ 0 ௉ (13)
ܧሶሺܲሻ ൅ න න ܦܸ݀Ԣܸ݀
௉ ௉௉
ൌ න න ܦܸ݀Ԣܸ݀
௉ ஻௉
൅ න න ܦܸ݀Ԣܸ݀
௉ ௉௉
൅ න ݏܸ݀
௉
ൌ න න ܦܸ݀Ԣܸ݀
௉ ஻
൅ න ݏܸ݀
௉
Now, if we consider a density function like e with below definition then we can extract a local
balance for each point in region B which has same form of balance law and parameters for linear
momentum, angular momentum and energy can be select from (Table 1).
ܧሺܲሻ ൌ ׬ ܸ݁݀ ௉ (14)
݁ሶൌ ׬ ܦܸ݀Ԣ ஻ ൅ ݏ (15)
Table 1. Definitions of parameters in master law balance for linear, angular and energy equation
Balance (Eq.) e D s
Linear mom. ߩݕሶ ݐ െ ݐԢ ܾ
Angular mom. ݕ ൈ ߩݕሶ ݕᇱ ൈ ݐ െ ݕ ൈ ݐԢ ݕ ൈ ܾ
Energy ߝ ൅
ߩݕሶ· ݕሶ
2
ݍ ൅ ݐ · ݕሶԢ െ ݐԢ · ݕሶ ݎ ൅ ܾ · ݕሶ
In [1] Silling suppose below equation for dual density interaction that K has positive quantity and
usually ݈ ൏ 3.
|ܦሺݔ, ݔԢሻ| ൑ ܭ|ݔ െ ݔԢ|ି௟ ݔ ߳ ܤ, ݔԢ߳Թଷܤ (16)
In above table ࢿ is a kind of scalar quantity with name internal energy density which can be
searched in thermodynamic laws and ࢚ is known as bond force density and can be produced by
constitutive model in the Peridynamic theory and its dimension is same as Pairwise force (f) which
was force per unit volume squared.
First simulation of relative movement of particles in MATLAB
As the first model in moment, we want to make a simple model of some particles that behave by
Peridynamic rules together. Our software for programming is MATLAB that usually write in m‐ files,

4. and for the first step we decide to consider horizon as an effective parameter in calculation of
internal forces [2] and using simple linear spring equation for simulation of internal force vectors as
the only effective internal force on our model. The model includes separate particles with unit
density and external forces will occur on each particle as initial condition.
After we defined the position of m particles in 2D system by a [2Xm] matrix and the horizon and
initial velocities and acceleration of each particle, we define a timer by constant Δt sequences that
by their summation we will have time of simulation which can be changed to variable domains and
calculate from methods that make a relation between Δt and rate of parameters in integral of time.
Figure 2. Applied algorithm for Peridynamic simulation in the first modeling
In the first step after defining positions of particles (Fig.2) the amount of external forces will be
applied on particles with unit density amount to calculate the deformation of particles in respect to
others, and then by created displacement and by using a method to define the inner forces which in
the first simulation we used Hook's law for internal forces, finally we can reach the amount of net
internal forces on a particle, we must consider this assumption that only the masses in the horizon
will connected to each other and effect on final internal force of central particle.
In the first code we tried to make a simulator which can receive any kinds of forces and any
arrangement of particles and also have the capability of upgrading in the future. The main program
is "model_v104.m " which work with three defined functions namely, SPD_LS.m, Force_ext.m,
length_2D.m that are used respectively for calculation of internal force and updating new positions
and for definition of external force and the last one is for calculation of length of a vector in two
dimension system.
Figure 3. Deformed and Bond definition

5. Figure 4. Deformed spring model
As we mentioned above, in first level we decided to use simple spring forces (Hook's law) to
modeling the internal forces, so deformed bond and bond are needed to measure in each seconds
(Fig.3,4), and the amount of internal force will derive from below equation:
ܨ௜௡௧ ൌ ܥሺܺ′ െ ௑
஼௢௦ఏ ൌ ܺᇱ െ ௑
೉ᇲ.೉
ห೉ᇲห|೉|
ሻ ൌ ܥሺܺᇱ െ ܺ ห௑ᇲห|௑|
௑ᇲ.௑ ሻ (17)
Moreover, there is no limitation for final amount of allowable amount of internal force in this first
simulation, so the breaks cannot be simulated in this model yet.

6. [1]. "Peridynamic Theory of Solid Mechanics", S. A. Silling, R. B. Lehoucq, Sandia National Laboratory,
Apr. 28, 2010.
[2]."Peridynamic Simulation", Report 3, Sh. Sharifian, UKM, Faculty of Engineering & Build
Environment, 07/09/2011