Elastodynamical disturbances due to laser irradiation in a microstretch thermoelastic medium with microtemperatures

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Abstract

This research is concerned with the study of mechanical disturbances due to presence of ultra-short laser pulse as input heat source in a microstretch thermoelastic medium with microtemperatures. The medium is subjected to normal and tangential forces. The solution of the problems is developed in terms of normal modes. Mathematical expressions have been obtained for normal stress, tangential stress, microstress and temperature change. The numerically computed results are shown graphically. A mathematical model has been developed and various stress quantities have been analyzed. Some particular cases are also derived from the present investigation.

1 Introduction

Eringen [1] developed the theory and basic equations of microstretch thermoelastic solids. Microstretch continuum is a model for Bravais lattice having basis on the atomic level and two-phase dipolar substance having a core on macroscopic level. Examples of microstretch thermoelastic materials are composite materials filled with chopped elastic fibers, porous elastic fluids whose pores have gases or inviscid liquids, or other elastic inclusions and liquid–solid crystal. Marin [2, 3] established the solution of equations in microstretch thermoelasticity and in elasticity of dipolar bodies with voids.

Laser technologies have a lot of utilities in medical science, industries, metallurgies, and nondestructive testing and evolution. High-rated thermal processes are interesting in the development of theories of thermoelasticity, due to thermal-mechanical coupling. The thermal shock creates very fast movements in the internal molecular particles, which causes an increase in very significant inertial forces and vibrations. The ultra-short lasers have pulse durations ranging from nanoseconds to femto seconds. In irradiation of ultra-short pulsed laser, the high-intensity energy flux and ultrashort duration lead to a very large thermal gradient. So, in these cases, Fourier law of heating is no longer valid. Rose [4] developed an analytical mathematical basis for point laser source. Scruby et al. [5] studied the point source ultrasonic generation by lasers. A new laser generation model was presented by Spicer [6] and McDonald [7]. Al-Huniti and Al-Nimr [8] studied a problem related to laser ultrasound in thermoelastic materials. Thermoelastic behavior in metal plates due to laser interactions using fractional theory of thermoelasticity was studied by Ezzat et al. [9]. A comparison in context of four theories of thermoelasticity was presented by Youssef et al. [10]. A generalized thermoelastic diffusion problem for a thick plate irradiated by thermal laser was discussed by Elhagary [11]. Kumar, Kumar and Singh [12] recently studied the thermomechanical interactions of an ultra-laser pulse with microstretch thermoelastic medium.

Grot [13] developed a theory of thermoelasticity of elastic solids with microelements possessing microtemperatures. The Clausius–Duhemin relation is modified to include microtemperatures, and the first-order moment of the energy relations are included. Riha [14] discussed heat conduction in solids with microtemperatures. The linear theory of thermo elasticity with microtemperatures for elastic materials was derived by Iesan and Quintanilla [15]. Iesan and Quintanilla [16] proposed the theory of micro-morphic elastic solids with microtemperatures. Different type of problems in thermoelasticity with microtemperatures were discussed by Iesan [17]. Chirita et al. [18] discussed some important aspects in the theory of thermoelasticity with microtemperatures. Iesan [19] extended thermoelasticity of bodies with microstructures and microtemperatures.

In this research, a model has been developed for microstretch thermoelastic solid with microtemperature in the presence of input ultra-short laser pulse. The stress components and temperature distribution have been computed numerically. The resulting expressions are then applied to the problem of a microstretch thermoelastic medium with microtemperatures whose boundary is subjected to two types of loads, i.e., normal force and tangential load. The resulting quantities are shown graphically to show the effect of microtemperature and input laser heat source.

2 Basic equations

Following Eringen [1], Iesan [16], and Al Qahtani and Dutta [20], the basic equations and constitutive equations for microstretch thermoelastic medium with microtemperatures are

(λ+μ)(.u)+(μ+K)2u+K×ϕ+λ0ϕνT=ρu¨,

(γ22K)ϕ+(α+β)(.ϕ)+K×uμ1×w=ρjϕ¨,

(α02λ1)ϕλ0.u+ν1Tμ2.w=ρj02ϕ¨,

K2T=ρcTt+υT0(.uQ)+ν1T0ϕtK1(.w),

K62w+(K4+K5)(.w)+μ1t(×ϕ)μ2t(ϕ)bwtK2wK3T=0,

tij=(λ0ϕ+λur,r)δij+μ(ui,j+uj,i)+K(uj,iεijkϕk)νδijT,

mij=αϕk,kδij+βϕi,j+γϕj,i+b0εmjiϕ,m,

λi=α0ϕ,i+b0εijmϕj,m,

qi,j=K4wk.kδijK5wi,jK6wj,i,i,j,m=1,2,3

Figure 1
Figure 1

Temporal profile off (t)

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0015

Figure 2
Figure 2

Profile of g(x1)

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0015

Figure 3
Figure 3

Profile of h (x3)

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0015

Figure 4
Figure 4

Geometry of the problem

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0015

The surface of the medium is irradiated by laser heat input (following Al Qahtani and Dutta [20]):

Q=I0f(t)g(x1)h(x3),

f(t)=tt02e(tt0),

g(x1)=12πr2e(x12r2),

h(x3)=γeγx3,

Here I0 is the energy absorbed, t0 is the pulse rise time, and r is the beam radius.

Here λ, μ, α, β, γ, K, λ0,λ1, α0, b0 are constants with values depending on the nature of material, ρ is density of the medium, u = (u1, u2, u3), w = (u1, u2, u3) and ϕ = (ϕ1, ϕ2, ϕ3) are respectively the displacement vector, microtemperature vector the microrotation vector, T represents temperature, ϕ* is the scalar microstretch, T0 is the reference temperature of the medium, c* is the specific heat at constant strain, tij are components of stress, K* is the coefficient of the thermal conductivity, λi is the microstress tensor, j is the microinertia, j0 is the microinertia for the microelements, mij are components of couple stress, eij is the strain tensor, is the dilatation, and δij is Kroneker delta function.

3 Formulation of the problem

A microstretch thermoelastic medium with microtemperatures irradiated by ultra-short laser pulse as input heat source is considered. The origin of the Cartesian coordinate system Ox1 x2x3 is taken on a point of the x1 x2-plane and x3-axis points vertically downwards into the medium.

For two-dimensional problem, the displacement vector u, microtemperature vector w, microrotation vector ϕ can be written mathematically as:

u=(u1,0,u3),ϕ=(0,ϕ2,0)w=(w1,0,w3),

We introduce the following non-dimensional quantities:

ui=uiL,xi=xiL,t=ωt,ϕ=j02L2ϕ,T=TT0,tij=1υT0tij,ϕi=j2L2ϕi,c12=λ+2μ+kρ,mij=1Lν1T0mij,qij=1Lc1νT0qij,L=Kρcc1,wi=Lwi,t=c1Lt,λi=1LνT0λi,Q=QωT0c

Also, it is appropriate to introduce the scalar potentials ϕ, ϕ1 and vector potential ψ, ψ1 through the Helmholtz representation of vector fields u and w as:

u1=ϕx1ψx3,u3=ϕx3+ψx1and w1=ϕ1x1ψ1x3,w3=ϕ1x3+ψ1x1

Making use of Equations (14) and (15) and then using the potential functions defined in (16), in relations (1)(5), yield:

[(a1+1)2a52t2]ϕ+a3ϕa4T=0,

(2a52t2)ψ+a2ϕ2=0,

(22a6a82t2)ϕ2a62ψ+a72ψ1=0,

(2a10a132t2)ϕa112ϕa122ϕ1+a9T=0,

(2a14t)Ta15ϕta162ϕ+a172ϕ1=Q0f(x1,t)eγx3,

[2(1+a18)a21a23t]ϕ1a20ϕta22T=0,

[2(1+a18)a21a23t]ψ1+a19ϕ2t=0

Here, 2=2x12+2x32 is the Laplacian operator and

f(x1,t)=[t+ετ0(1rr0)]ex12r2+tt0,Q0=a14I0γ2πr2t02

Here, ai are defined in Appendix A.

4 Solution of the problem

The solution of the considered physical variables can be decomposed in terms of the normal modes as in the following form:

{ϕ,ϕ1,ϕ,T,ψ,ψ1,ϕ2}(x1,x3,t)={ϕ¯,ϕ¯1,ϕ¯,T¯,ψ¯,ψ¯1,ϕ¯2}(x3)ei(kx1ωt),

Here ω is the angular frequency and k is the wave number.

Making use of (24) in Eqs. (17), (20), (21), and (22) and eliminating (ϕ1, ϕ*, T), (ϕ, ϕ*, T), (ϕ, ϕ1, T) and (ϕ, ϕ1, ϕ*) respectively we obtain the following equations:

[D1D8+D2D6+D3D4+D4D2+D5]ϕ¯=f1(γ,x1,t)eγx3,

[D1D8+D2D6+D3D4+D4D2+D5]ϕ¯1=f2(γ,x1,t)eγx3

[D1D8+D2D6+D3D4+D4D2+D5]ϕ¯=f3(γ,x1,t)eγx3,

[D1D8+D2D6+D3D4+D4D2+D5]T¯=f4(γ,x1,t)eγx3

Also using (24) in relations (18), (19), and (23) and simplifying the resulting equations we obtain:

[D6D6+D7D4+D8D2+D9]ψ¯=0,

Here, g1 = a1 + 1, g2 = a18 + 1, k1 = a5ω2k2g1, k2 = a13 ω2a10k2, k3 = iωa14k2, k4 = a23k2g2a21, k5 = k2a5ω2, k6 = a8ω2k2 − 2a6,

Here, D=ddx3 in Eqs. (25)(29), and

D1=g1g2,D2=g1(iωa20a12+k10)+k1g2+g2(a11a3a4a16),D3=k1(iωa20a12+k10)+g1(k8a12+k11+k13)+k15a3+k18a3+k19a4k23a4,D4=g1(k12+k14k9a12)+k1(k11+k13+k8a12)+k16a3+k20a4+2k2(k23a4k18a3),D5=k1(k12+k14k9a12)k4(k23a4k18a3)k17a3+k21a4,D6=g2,D7=k24,D8=k25,D9=k26,

ki (i = 1, …, 26) are defined in Appendix B.

It is desired to satisfy the radiation conditions, i.e., (ϕ, ϕ1, ψ, ψ1, T, ϕ2, ϕ*) → 0as x3 → ∞, the solutions of Eqs. (25)(29) can be considered in the following form:

ϕ¯=i=14ciemix3+f1f5eγx3,

ϕ1¯=i=14α1iciemix3+f2f5eγx3,

ϕ¯=i=14α2iciemix3+f3f5eγx3,

T¯=i=14α3iciemix3+f4f5eγx3,

(ψ¯,ψ1¯,ϕ2¯)=i=57(1,α4i,α5i)Ciemix3,

Ci (i = 1, 2, …, 7) are arbitrary constants.

mi2(i = 1, 2, 3, 4) are the roots of the characteristic equation (25) and mi2(i = 5, 6, 7) are the roots of characteristic equation (29).

α1i=D1iD0i,α2i=D2iD0i,α3i=D3iD0ii=1,2,3,4&α4i=D4iΔ0i,α5i=D5iΔ0ii=5,6,7

Djj, Δ0i and D0i j = 1, 2, …, 5, are defined in Appendix C.

Substituting the values of (ϕ, ϕ1, ψ, ψ1, T, ϕ2, ϕ*) from the Eqs. (30)(34) in (6)(9), and using (14)(16), (24) and solving the resulting equations, we obtain:

t¯33=i=17G1iemix3+M1eγx3,

t¯31=i=17G2iemix3+M2eγx3,

m¯32=i=17G3iemix3+M3eγx3,

λ3¯=i=17G4iemix3+M4eγx3,

T¯=i=17G5iemix3+M5eγx3,

q33¯=i=17G6iemix3+M6eγx3,

q31¯=i=17G6iemix3+M6eγx3,

Here, Gmi = gmiCi, i, m = 1, 2, …, 7

Grs, (r, s =1, 2, …, 7) and Mr, (r =1, 2,3, …, 7) are described in Appendix D.

5 Boundary conditions

We consider normal and tangential force acting on the surface x3 = 0 along with vanishing of couple stress, microstress, and temperature gradient with insulated and impermeable boundary at x3 = 0 and I0 = 0. Mathematically this can be written as:

t33=F1ei(kx1ωt),t31=F2ei(kx1ωt),m32=0,λ3,Tx3=0,q33=q31=0,

Substituting the expression of the variables considered into these boundary conditions, we can obtain the following equations:

i=17G1ici=F1,

i=17G2ici=F2,

i=17G3ici=0,

i=17G4ici=0,

i=17miG5ici=0,

i=17G6ici=0,

i=17G7ici=0,

Eqs. (43)(49) can be written in matrix form in the following manner:

g11g12g13g14g15g16g17g21g22g23g24g25g26g27g31g32g33g34g35g36g37g41g42g43g44g45g46g47g51g52g53g54g55g56g57g61g62g63g64g65g66g67g71g72g73g74g75g76g77c1c2c3c4c5c6c7=F1F200000,

Equation (50) is solved by using the matrix method as follows:

c1c2c3c4c5c6c7=g11g12g13g14g15g16g17g21g22g23g24g25g26g27g31g32g33g34g35g36g37g41g42g43g44g45g46g47g51g52g53g54g55g56g57g61g62g63g64g65g66g67g71g72g73g74g75g76g771F1F200000

5.1 Particular case

5.1.1 Microstretch thermoelastic medium

In absence of microtemperature effect, i.e.,

μ1=μ2=K1=K2=K3=K4=K5=K6=0

in Eqs. (43)(49), we obtain the dispersion equation for microstretch thermoelastic medium.

5.1.2 Generalized thermoelastic medium

In the absence of microtemperature and microstretch effect in Eqs. (43)-(49), we obtain the dispersion equation for microstretch thermoelastic medium.

6 Numerical results and discussions

The following values of relevant parameters are taken for numerical computations. Following Eringen [22], the values of micropolar constants are:

λ = 9.4 × 1010N.m−2, μ = 4.0 × 1010N.m−2, K = i.0 × 1010N.m−2, ρ = 1.74 × 103Kg.m−3, j = 0.2 × 10−19m2, γ = 0.779 × 10−9N

Thermal parameters are give n by (following Dhaliwal [23]):

c* = 1.04 × 103J.Kg−1.K−1, K* = 1.7 × 106J.m−1. s−1 .K−1, a = 2.9 × 104m2.s−2.K−1, T0 = 298K, τ1 = 0.613 × 103s

Microstretch and microtemperature parameters are taken as (following Kumar and Kaur [24]):

j0 = 0.000019 × 10−13m2, α0 = 0.8 × 10−9N, λ0 = 2.1 × 1010N.m−2, λ1 = 0.7 × 1010N.m−2, b = 1.5 × 10−9Kg−1.m5.s−2, b0 = .5 × 10−9Kg−1.m5.s−2, K1 = .0035Ns−1, K2 = .045Ns−1, K3 = 0.055NK−1s−1, K4 = 0.065Ns−1m2, K5 = 0.076Ns−1m2, K6 = 0.096Ns−1m2, μ1 = 0.0085N, μ2 = 0.0095N

A comparison of the dimensionless form of the field variables for the cases of microstretch thermoelastic medium with microtemperature and ultra-short laser pulse as input heat source (MTPL), microstretch thermoelastic medium with microtemperature but without laser pulse (MTP) and microstretch thermoelastic medium, i.e., without microtemperature effect and without laser pulse (MSTH) subjected to normal force is presented in Figures 511. The values of all physical quantities for all cases are shown in the range 0 ≤ x3 ≤ 20.

Figure 5
Figure 5

Variation of normal stress

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0015

Figure 6
Figure 6

Variation of tangential stress

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0015

Figure 7
Figure 7

Variation of coupled tangential stress

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0015

Figure 8
Figure 8

Variation of micro-stress

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0015

Figure 9
Figure 9

Variation of temperature distribution

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0015

Figure 10
Figure 10

Variation of q33

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0015

Figure 11
Figure 11

Variation of q33

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0015

Solid lines, small dash lines, and large dash lines including a dot corresponds to microstretch thermoelastic solid with microtemperature and laser pulse (MTPL), microstretch thermoelastic solid with microtemperature (MTP), and microstretch thermoelastic solid without microtemperature and laser pulse (MSTH) respectively.

The computations were carried out in the absence and presence of laser pulse (I0 = 105,0) and on the surface of plane x1 = 1, t = .1

Figure 5 presents the variation of normal stress t33 with the distance x3. It is noticed that for MTPL, MTP and MSTH, the normal stress t33 shows a similar behavior. The value of normal stress for MT PL and MSTH monotonically decreases as x3 increases. MTP shows a similar behavior followed by some oscillatory trend. The values of normal stress approaches the boundary surface at large distance from the point of application of source.

Figure 6 shows the change in tangential stress t31 with respect to distance x3. It is noticed that initially the trend of t31 for the two cases, i.e. MTPL and MSTH is monotonically increasing but in case of MTP it is decreasing. After this, i.e., x1 ≥ 4i.e. far away from the point of application of ultra-laser heat source and normal force the values of t33 for all the cases tends to approach value zero.

Figure 7 exhibits the trend of couple stress m32 w.r.t. the displacement x3. For MTPL and MTP, the initial behavior of couple stress is monotonically increasing but behavior of MSTH is opposite to it. For higher values of x1 the value of m32 is approaching to boundary surface for all the three cases.

Figure 8 shows the trend of micro stress λ3 with distance x3. The trend and variation of λ3 is oscillatory for MTP indicating that the micro-stress is much affected by including the microtemperature effect. In case of MTPL and MSTH the micro-stress approaches to the boundary surface uniformly.

Figure 9 shows the behavior of temperature distribution T with distance x3. The values of temperature for MTPL, MTP, and MTH decreases monotonically and approaches to the boundary surface away from the point of application of source.

Figure 10 shows the behavior of q33 with distance x3. The values of q33 for MTPL and MTP decreases monotonically and approaches to the boundary surface away from the point of application of source. Here we observe that the magnitude of q33 in case of MTP is larger than the magnitude of q33 in MTPL.

Figure 11 shows the behavior of q31 with distance x3. The values of q31 for MTPL, MTP increases monotonically and approaches to the boundary surface away from the point of application of source.

Variation of temperature with respect to time

Figure 12 shows the variation of temperature distribution with respect to time. It is clear from the figure that initial trend of variation is monotonically increasing until the temperature reaches a maximum value. After that the trend of variation of temperature change is monotonically decreasing and approaches boundary surface away from the point of laser heat irradiation.

Figure 12
Figure 12

Variation of temperature with respect to time

Citation: Mechanics and Mechanical Engineering 23, 1; 10.2478/mme-2019-0015

7 Conclusions

In this problem, we have investigated the displacement components, stress components, and temperature change in a microstretch thermoelastic medium with microtemperature. The solution of the physical variables has been obtained in terms of normal modes. Theoretically computed variables are also discussed graphically.

This analysis of the results obtained give the following conclusions:

  1. It can be concluded from Figures 511 that all the physical variables have nonzero values only in the bounded region. This indicates that all the results obtained here are in agreement with the generalized theory of thermoelasticity.
  2. It is clear from the results that the input laser heat source (value of I0) has a significant role in the variation of all field quantities.
  3. If the microtemperature parameters are absent, then the results are obtained for generalized thermoelastic problem, which are in agreement with Kumar, Kumar and Singh [12].
  4. The variation of various stress components differs significantly due to the presence of normal force / thermal source.
  5. The temperature change is also affected due to input laser heat source as well as load/source applied.
  6. The microtemperature effect has also significant effect on the physical quantities.

The new model is employed in a microstretch thermoelastic medium with microtemperatures as a new concept in the field of thermoelasticity. The subject becomes more interesting due to presence of an ultra-short input laser heat source. The method of solution in this research can be applied to a large number of problems in engineering and science. It is hoped that this model will serve as more realistic model and will motivate the other authors to solve problems in microtemperature thermoelasticity. Solutions to such problems also have utilities in geophysical mechanics.

Funding: No fund received for this research from any agency.

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Appendix A

a1=λ+μμ+K,a2=Kμ+K,a3=λ0μ+K,a4=νT0μ+K,a5=ρc12μ+K,a6=KL2γ,a7=μ1γ,a8=ρjc12γ,a9=υ1T0L2α0,a10=λ0L2α0,a11=λ1L2α0,a12=μ2α0,a13=ρj0c122α0,a14=ρcc1LK,a15=υ1c1LK,a16=νc1LK,a17=k1KT0,a18=k4+k5k6,a19=μ1c1Lk6,a20=μ2c1Lk6,a21=k2L2k6,a22=k3T0L2k6,a23=bc1L2k6

Appendix B

k1=a5ω2k2g1,k2=a13ω2a10k2,k2=iωa14k2,k4=iωa23a21k2g2,k5=k2a5ω2,k6=a8ω2k22a6,k6=iωa23a21k2g2,k8=(iωa5a22+iωa20k3iωa20k2),k9=k2(iωa5a22+iωa20k3),k10=(g2k3+k4+a17a22+g2k2),k11=(k3k4a17a22k2+k2k3g2+k2k4+k2a17a22),k12=k2(k3k4a17a22k2),k13=iωa9(a17a20a5g2),k14=iωa9(a17a20k2+a5k4)k15=g2(a11(a11k3k2)a9a16)+a11k4,k18=a22(a11a17+a12a16),k16=g2(a9a16k2k2k3)+k3(a16(a16a16k2)a9a16),k17=k4k2(a9a16k3),k19=g2(iωa5a11a16(k2k2))a16k4,k23=iωa20(a17a11+a12a16)k20=g2(k2k2a16iωa5a11k2)+k4(iωa5a11a16(k2k2)),k21=k4(k2a16k2iωa5a11k2)k24=iωa7a19(k7+)2k6)+g2k5a2a6g2,k25=iωa7a19(k5+k2)k6k7+k5(k7+g2k6)a2a6(k7g2k2),k26=iωa7a19k5k2+k5k6k7+a2a6k7k2

Appendix C

For i = 1, 2, 3, 4

D0i=a12(D2k2)D2+k2a9a17(D2k2)iωa5D2+k3g2D2+k4iωa20a22,D1i=a11(D2k2)D2+k2a9a16(D2k2)iωa5D2+k30iωa20a22D2i=a11(D2k2)a12(D2k2)a9a16(D2k2)a17(D2k2)D2+k30g2D2+k4a22,D3i=a11(D2k2)a12(D2k2)D2+k2a16(D2k2)a17(D2k2)iωa50g2D2+k4iωa20

For i = 5, 6, 7

Δ0i=a7(D2k2)D2+k60iωa19,D4i=a6(D2k2)D2+k6g2D2+k7iωa19,D5i=a6(D2k2)a7(D2k2)g2D2+k70

Appendix D

d1=ρc12νT0,d2=λνT0,d3=L2λ0j02νT0,d4=μνT0,d5=L2j2KνT0,d6=L2j2γL2νT0,d7=b0j02νT0,d8=α0j02νT0,d9=L2j2b0L2νT0,d10=K4L3c1νT0,d11=(K5+K6)L3c1νT0,d12=K5L3c1νT0,d13=K6L3c1νT0,d17=μ1L2νT0,d19=μ2L2νT0g1i=[b2(mi2k2)b3mi2]+b1α2iα3i,g2i=ιkb3mi,g3i=ιkb8α2i,g4i=mib9α2i,g5i=miα3i,g6i=k4(mi2k2)(k5+k6)mi2,g7i=ιkmi(k5+k6)α1i, for i=1,2,3,4

and

g1i=ιkb3mi,g2i=(b4ml2+b5k2)b6α5i,g3i=b7α5imi,g4i=ikb10α5i,g5i=0,g6i=ikmi(k5+k6)α4i,g7i=ιkmi(k5k2+k6ml2)α4i for i=5,6,7M1=[b2(γ2k2)b3γ2]f1+b1f3f4f5,M2=ιkb3γf1f5,M3=ιkb8f3f5,M4=γb9f3f5,M5=γf4f5,M6=(k4(γ2k2)(k5+k6)γ2)f2f5,M7=(ιkγ(k5+k6)f2)f5.

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