In traditional engineering treatments, shot peening by using steel balls to bombard onto metal surfaces has been adopted to leave compressive residual strains within the affected region in promoting the fatigue properties [29, 30]. The balls have typical diameters of 0.1 ∼ 2 mm and gain their speed by compressed air. Normally, these balls bombard the metal surfaces in the frequency range of 20∼ 100 Hz and speed range of 50 ∼ 100 m/s.
A new physical treatment named ultrasonic surface mechanical attrition treatment (SMAT) was firstly introduced in 1999 [31, 32]. The SMAT balls are accelerated and bombarded by the ultrasonic motor on the chamber bottom, as shown in Fig. 1.1 (a).
The diameter and speed of flying balls and the bombarding frequency are in the range, 1 ∼ 10 mm, 1 ∼ 20 m/s, and 10 ∼ 100 kHz [33], respectively. The most important feature is that the incident direction onto the metal surface can be designed to vary with time lapse in making smaller grain size of metallic materials. This will lead to promising properties of metals such as grain refinement and gradient structure. And researchers have made extensive uses of this SMAT treatment on various metallic materials including pure iron [34], stainless steel [35], and pure copper [36] in making gradient and nano-crystalline structure [37–40].
Although SMAT has gradually developed into a matured engineering surface treat-ment, never before has SMAT been treated with rigorous analytical modelling. Therefore, a systematic SMAT model is actually needed. Here, we consider the interaction between flying balls and chamber imagined as a canonical ensemble, where chamber is reservoir giving balls the kinetic, internal, and heat energy. The chamber volume is much greater than balls volume, so collision frequency between balls is small compared to that between balls and chamber and balls interaction can be neglected. The motion of motor top is characterized by longitudinal harmonic motion,
v = 2Aπνsin(2πνt), (1.19)
where vm is the velocity of motor top, A is the amplitude, and ν is the angular frequency.
To construct the relation of energy conversion between motor top and balls, the ball-motor collision can be counted as elastic collision. Thus, the induced velocity of ball, vb, is described by
vb = 2mmvm
mb + mm ≈ 2vm, mm >> mb, (1.20) where mb and mm are the mass of each ball and motor. On the other hand, the ball-sample collision is assumed to be inelastic collision with restitution constant e obeying conservation of momentum,
e = vs′ − vb′
vb− vs
, msvs+ mbvb = msv′s+ mbvb′, (1.21)
where msand vsare sample mass and velocity, respectively.
The kinetic energy of flying balls is not conserved due to the inelastic ball-sample collision. The kinetic energy loss for flying balls and sample in the SMAT chamber can be mainly converted into three parts as indicated in Fig. 1.1 (b). Firstly, it is the strain energy of sample due to the formation of dislocations and vacancies [41–43]. Secondly, it is the heat energy of sample, where the heat flow and its temperature distribution are both important factors for the resulting metal micro-structure [44]. Recrystallization might be taken place in the sample while the experienced temperature reaches some critical values.
Finally, it would be the sonic energy and heat energy in the chamber originated from the inelastic collisions between flying balls.
In SMAT topics, we have established connections among the parameters of flying balls, the ball size, flying speed, the bombarding frequency and amplitude of motor mo-tion, the height of chamber, and the energy and power of sample. During the SMAT processing time, we can find the input energy and power of sample through these con-nections. The condition for the frequency of flying ball reaching a stead speed can also be obtained in this approach. For the heat energy of sample, we have introduced the one-dimensional heat equation with the uniformly-distributed heat source to estimate the heat flow and temperature distribution of sample which are hard to be measured in the SMAT
experiment. With the temperature distribution of sample, we can make connection among the strain rate, hardness, and grain size of sample. With these connections and modelling, one can find an optimized approach to the mechanical performance of metal surface via the SMAT experiment.
Chapter 2
Basic theory for CME, Bio-evolution, and SMAT
2.1 Hamilton-Jacobi Equation
In Classical Mechanics (CM) [45], theoretical physicists use independent variable of position x and momentum p for each particle in constructing Hamiltonian to characterize the particle dynamics, where the correspond equation of motion constructed by Hamilto-nian for x and p is the Hamilton equation. To simplify the equation of motion, we prefer the physical system in which all of the generalized coordinates or all of the canonical mo-mentums are cyclic, namely, Hamiltonian with respect to x and p is a constant. To achieve the goal, a canonical transformation (CT) under which the equation of motion is invariant should be found, and any CT corresponds to a generating function consisted of half new and half old generalized coordinates. Here comes the Hamilton-Jacobi equation (HJE) which generating function satisfies.
In stochastic dynamics, a fictitious Hamiltonian which is similar to that of CM as a tool to formulate the Lagrangian and the action functional by using the WKB expansion, P (xt, t) = e−1ϵuϵ(xt,t)with small ϵ, where ϵ results from diffusion. And such WKB expan-sion we use in bio-evolution is P (x, t) = eN u(x,t)with large N , where the N is genome length or population size. It’s obviously that both expansions are mathematically
equiva-lent for small ϵ or large N . In our work for CME or bio-evolution, we put such expansion into the equation of motion (KFE or evolution model) with condition ϵ→ 0+or N >> 1 to get the HJE for u. As ϵ→ 0+or N >> 1, limϵ→0+uϵ(xt, t) is called the large deviation rate function in probability theory [46] or the principal function in CM [45]. The large deviation rate function asymptotes the behaviour of P (xt, t) as ϵ → 0+ or N → ∞, and the principal function furnishes the entire family of orbits corresponding to a Hamiltonian system in phase space.
In our research, HJE in chemical reaction derives the path probability of each reaction while HJE in bio-evolution derives a series of equation for finite correction of O(N1). Here we give some derivation and introduction to HJE in CM.
2.1.1 Canonical Transformation
In CM, the form of Hamilton’s equations are invariant under canonical transformation (CT) and Hamilton’s principle states that the most possible track of classical system makes the corresponding action functional minimized. Thus, we have the variation of action functional is 0 over the most possible track:
δ
∫ t2
t1
L(q, ˙q, t)dt = 0, δ
∫ t2
t1
L′(q, ˙q, t)dt = 0, δ
∫ t2
t1
[piq˙i − H(q, p, t)]dt = 0, δ
∫ t2
t1
[PiQ˙i− K(q, p, t)]dt = 0,
which implies:
λ[piq˙i− H(q, p, t)] = PiQ˙i− K(q, p, t) +dF dt ,
where q and p are old generalized coordinates, Q and P are new generalized coordinates, L, L′, H, and K are the corresponding Lagrangian respectively, λ is a scale constant, and F is corresponding generating function in terms of half new and half old coordinates. For
the λ = 1 case, it’s the case called CT in CM. With λ = 1, the equation becomes:
piq˙i− H(q, p, t) = PiQ˙i− K(q, p, t) +dF
dt , (2.1)
where F called the generating function has four basic forms by [45],
We start the derivation of Hamilton-Jacobi equation (HJE) from Eq. (2.1) and put the generating function F2 in Eq. (2.2) to arrive:
piq˙i− H(q, p, t) = PiQ˙i− K(Q, P, t) + dF
Then we rearrange it and have the equation:
[H(q, p, t) +∂F2
∂t − K(Q, P, t)] + (∂F2
∂qi − pi) ˙qi+ (∂F2
∂Pi − Qi) ˙Pi = 0, (2.3)
where ˙qi and ˙Pi are separated independently. Since the three terms in Eq. (2.3) are inde-pendent of each other, the three terms must be 0 to hold the equality. Therefore,
∂F2
∂qi = pi,∂F2
∂Pi = Qi, H(q, p, t) +∂F2
∂t = K(Q, P, t). (2.4)
To make all generalized coordinates cyclic, we can set K(Q, P, t) = 0. And the corre-sponding Hamilton’s equations are:
Q˙i = ∂K
∂Pi = 0, P˙i =−∂K
∂Qi = 0,
which means that all generalized coordinates are constants of motion. On the other hand, making K(Q, P, t) = 0 gives the corresponding equation for F2:
H(q, p, t) +∂F2
∂t = 0,
→ H(⃗q,∂F2
∂⃗q , t) + ∂F2
∂t = 0, (2.5)
where Eq. (2.5) is called Hamilton-Jacobi equation and F2 is the Hamilton’s principal function in CM which is the counterpart of u(x, t) in stochastic dynamics. To investigate the physical meaning of F2, we can take its total differential with respect to t:
dF2
dt = ∂F2
∂qi q˙i+∂F2
∂Pi
P˙i+∂F2
∂t
= piq˙i− H(q, p, t) = L(q, ˙q, t),
and then we integrate dFdt2 back with respect to t:
F2 =
∫ t
t0
L(q, ˙q, t′)dt′,
which states that the Hamilton’s principal function F2 is equivalent to action functional.
Thus, in physics, solving the HJE is equivalent to solve the variation equation of action functional, Euler-Lagrange equation.
2.1.3 HJE Application in CME
The HJE method in CM has been well developed for hundreds years, and it is a power and analytical tool to investigate the characteristic of physical system in macro. Thus, we
want to develop the HJE method in CME to help us realize the mechanism of chemical sys-tem. As stated in previous sections, the principal function u(x, t) in stochastic dynamics is similar to the role of generating function in CM. Here we introduce a fictitious Hamil-tonian corresponding to HJE in CME. In the introduction of CME, we have the following CME for general chemical system with n molecules:
∂P (x, t) rate of generation and degradation. By Taylor expansion with largeness of N at x and P (x, t) = exp [N u(x, t)], we have:
Put these expansions above into Eq. (2.6) to get the equation of zero order in N1:
∂u
Equation (2.8) has the exact form of HJE, where u corresponds to generating function in CT and H corresponds to Hamilton in CM. Thus, the fictitious Hamiltonian and equation
of motion are shown as:
H(x, u′) = [R+(x)− R−(x)]u′, x(t) =˙ ∂H(x, u′)
∂u′ = R+(x)− R−(x) = b(x),
where b(x) = R+(x)− R−(x). It is reasonable for chemical reaction of zero order whose concentration obeys the ordinary differential equation. And the variance of P (x, t) has been derived in [15]:
b2(x)
∫ x
x0
c(y)
b3(y)dy, (2.9)
where c(y) = R+(y) + R−(y) and x0 is the reference point. Therefore, each chemical system actually corresponds to a fictitious Hamiltonian system derived from CME of each chemical system.