1. Date: 15/08/2011
Report Number: 1
Title: self‐healing structures
Keywords: Continuum Mechanics, Integral Equation, Peridynamic, Equation of Motion, Horizon
For the first report I decided to introduce with some initial fundamental definitions of solid
mechanics parameters and after describe Peridynamic equation for equation of motion in
differential form, we continue the subject by the reasons for using integral equation instead of
differential form in Peridynamic equation of motion.
In this project our final purpose is to cast a composite surface that has the ability of self‐healing, to
reach this goal at first the composite must capable to sense crack in initial moments and then the
related controller must force actuators to rejoin the created gap. We can separate our project to 2
phase, modeling phase and experimental phase, modeling phase begins with theoretical and
mathematical parts which in this section we try to model composite properties and then implement
some external loads on this model, moreover the simulation must have this ability to behave as a
real composite for instance in deformation or growth of crack after insert loads on model, in
continue as the next step shape memory alloy wire will be used as a grid on our base composite to
avoid extension of crack and vanish it, so in modeling phase a proper controller will be designed and
regulated. Finally, in experimental phase a prototype will be produced and theoretical verdicts will
be compared to realistic model.
2 Initial Definitions
If we consider a surface with dimension of ΔS on a body and calculate the amount of ΔF which exerts
from this part to other parts, stress vector will represent the force per unit area acting on this
surface (Figure 1), and if we assume the unit area tends to zero, the stress vector will be like (eq. 1).
Figure 1. Stress principles
2. Figure 2. Notation of stress components
ܶ௩ ൌ ௗி
The superscript of v in (eq.1) shows the direction of stress vector that has same direction with
normal vector of ΔS. In the next step, we consider that the interior of body is filled by cubic elements
that each side of imaginary element has a normal vector along the direction of the coordinate axes
(Figure 2), so we will have ܶ for each Δܵ and each ܶ includes 3 components like,ܶଵ
therefore, we can arrange these components in a matrix (table 1) and use a notification rule (eq.2)
for explicit writing.
ߪ ൌ ܶ
߬ ൌ ܶ
Table 1. Stress tensor
Components of stresses
1 2 3
Surface normal to ߪଵ ߬ଵଶ ߬ଵଷ
Surface normal to ߬ଶଵ ߪଶ ߬ଶଷ
Surface normal to ߬ଷଵ ߬ଷଶ ߪଷ
Strain is a description of deformation in a relative displacement of particles in a body in order to
external loads, temperature changes within the body and body forces like gravity or electromagnetic
Peridynamic equation (eq. 3) is same as equation of motion in classical theory and stress tensor (υ)
in this equation is similar to Piola stress tensor in classical form.
ߩሺݔሻݑሷሺݔ, ݐሻ ൌ . ݒሺݔ, ݐሻ ܾሺݔ, ݐሻ (3)
Above equation includes some differential parts so it can be solved only in differentiable situations,
but actually in real engineering problems we face with non‐ideal environment which includes some
non‐differentiable points on whole domination of function, in addition, in (eq.3) only the first order
relations were considered and higher order relations in stress tensor (υ) are neglected, in our
3. purpose of using this equation for simulation of particles in a body, it means that only the nearest
particles that have stronger effects on our Origin particle will be remained and other side effects of
neighbors are vanished (Figure 3)and this neglect will increase error among realistic model and
Figure 3. In above spring model all of masses are connected together but in below the effect of far masses are vanished
because of smaller amount of effect on origin particle respect to near particles effect
Stewart in  considered a Horizon H around an Origin particle in the mass and wrote (eq.4) and it
involves all of internal forces (Figure 4) when an deformation is occurred (݂ሺݍ, ݔ, ݐሻ) between other
particles in this Horizon and Origin particle by an integral form and also there are body forces which
in equation can be viewed by ܾሺݔ, ݐሻ, the result of these forces as Newton law will occur to
acceleration of ݑሷ to particle in Origin (x) with density of ߩ.
ߩሺݔሻݑሷሺݔ, ݐሻ ൌ ݂ሺݍ, ݔ, ݐሻ ு ܸ݀ ܾሺݔ, ݐሻ (4)
Figure 4. Origin particle in center and its horizon and relative distance
4. In a simpler model for an elastic mass we can consider ݂ሺݍ, ݔ, ݐሻ as a spring force which works by
Hook law and describe the force between two particles proportion to their relative distance (eq. 5).
݂ሺݍ, ݔ, ݐሻ ൌ ܥሺݔ, ݍሻሺݑሺݍ, ݐሻ െ ݑሺݔ, ݐሻሻ (5)
Thus, we will have an integral equation for equation of motion (eq.6) instead of differential equation
ߩሺݔሻݑሷሺݔ, ݐሻ ൌ ܥሺݔ, ݍሻሺݑሺݍ, ݐሻ െ ݑሺݔ, ݐሻሻ ு ܸ݀ ܾሺݔ, ݐሻ (6)
Finally, if we want to simulate this equation by calculator machines we can change integral
formulation by series (eq. 7), which discrete Horizon to n different or same (depend on method of
calculus) elements (Δܸ).
ܥሺݔ, ݍሻሺݑሺݍ, ݐሻ െ ݑሺݔு , ݐሻሻ ܸ݀ ൌ Σୀଵ,ଶ,…, ܥሺݔ, ݍሻ൫ݑ െ ݑ௫൯Δܸ (7)
For the next report we will try to describe about Integral Equations and their solving method and
also we are going to solve Peridynamic equation of motion by extracted method and schedule to
constitute all desired equations for modeling a composite and simulate the growth of crack on it.
. "Length Scales and Time Scales in Peridynamics", Stewart Silling, SIAM Conference on
Mathematical Aspects of Materials Science Philadelphia, May 2010.