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Some Aspects of Superstring Theory
Published in Harish Parthasarathy, Supersymmetry and Superstring Theory with Engineering Applications, 2023
is the antisymmetric gauge field tensor and λ is its superpartner, ie, the gaugino field. D is an auxiliary field that is eliminated by the equations of motion. So effectively our gauge-gaugino action is S[Vμ,λ]=∫[ƒuvƒuv+c0λTγ5ϵγμλ,μ]d4x
Variational Methods and Functional Legendre Transforms
Published in A.N. Vasiliev, Patricia A. Millard, Functional Methods in Quantum Field Theory and Statistical Physics, 2019
Let us derive several expressions which will be useful in making the substitution α → β. We introduce the auxiliary field variable φ = φ(x) and the functionals () α(φ)=∑n=0∞αnφn,β(φ)=∑n=1∞1n!βnφn.
Soliton signals propagating in fiber waveguides and slow light generation
Published in Iraj Sadegh Amiri, Abdolkarim Afroozeh, Harith Ahmad, Integrated Micro-Ring Photonics, 2016
Iraj Sadegh Amiri, Abdolkarim Afroozeh, Harith Ahmad
In 1966, Basov et al. showed the propagation of a pulse through a laser amplifier in which the intensity of the pulse was high enough to make a nonlinear optical response (Schweinsberg et al., 2006). The nonlinear optical saturation of the amplifier gave rise to fast light, an unexpected result since the linear dispersion is normal at the center of an amplifying resonance so that vg < c is expected for low intensity pulses. The pulse development can be attributed to a nonlinear pulse reshaping effect where the front edge of the pulse depletes the atomic inversion density so that the trailing edge propagates with much lower amplification. In addition, it was found that the effects of dispersion give a negligible contribution to the pulse propagation velocity in comparison to the nonlinear optical saturation effects. Such pulse advancement due to amplifier saturation is referred to as super luminous propagation. The propagation of pulses is sufficiently weak so that the linear optical properties of the medium take into consideration. These properties can be modified in a nonlinear fashion by applying an intense auxiliary field.
Calculations of crack stress intensity factors based on FEM and XFEM models
Published in Australian Journal of Mechanical Engineering, 2023
Yashi Liao, Xuhui Zhang, Bisheng Wang, Miaolei He
Interactive integral method was compiled based on Fortran programming language. The specific process is shown in Figure 5: 1) The seamless combination of urdfil and XFEM was used to read the unit number and node information after the operation (including the current increment step, node number, coordinates and displacement, etc.). 2) The coordinates of crack tip were obtained by interpolations of horizontal set values of cell nodes. 3) Its integral radius was determined. It was used to judge whether the cell was in the integral region and to assign q-functions to cell nodes in the integral region. 4) The auxiliary field was established to obtain the stress and strain of auxiliary field. 5) The stress and strain of real field and auxiliary field were substituted into calculate J1, J2, KI and KII. The incremental step was updated to determine whether it was the last incremental step.
Life enhancement of cracked structure by piezoelectric patching underneath thermo-mechanical loading environment
Published in Mechanics of Advanced Materials and Structures, 2022
Ritesh Kumar, Akhilendra Singh, Mayank Tiwari
Here, denotes displacement, strain, stress of the auxiliary field, respectively, denotes displacement, strain and stress field obtained numerically. The interaction integral is correlated with the SIF as: where, and denotes SIF of mode I and II, and denotes auxiliary SIF of mode I and II, for plane strain and for plane stress, Y = elastic modulus, = shear modulus, = Poisson’s ratio.
Interaction between interfacial damage and crack propagation in quasi-brittle materials
Published in Mechanics of Advanced Materials and Structures, 2021
Pranavi Dhaladhuli, Rajagopal Amirtham, Junuthula N. Reddy
Phase-field method (PFM) [23] has emerged as one of the best methods for solving problems with interfaces. The total potential energy of cracking solids [24] is minimized to obtain both displacement field and crack set simultaneously. A variational framework for rate-independent and viscous over-force models are proposed in [25] and [26]. The phase-field approach combined with natural neighbor Galerkin method is used to model fracture in quasi-brittle materials [27] and for evolution of composition [28]. The PFM is also used for predicting damage in transient dynamic analysis (see [29] and [30]). Ductile fracture is modeled using PFM in a 2D framework in [31]. Fracture in anisotropic solids can also be modeled [32]. There has been recent works on modeling cohesive fracture using PFM [33]. The displacement field, crack phase field and an auxiliary field representing displacement jump is considered for modeling cohesive fracture. This model requires an additional constraint on the auxiliary field that needs to be satisfied. Crack infringing on an interface is studied in detail [34] and the results obtained are in well accordance with LEFM.