Energy Variational Method Research Articles

An innovative deformation coordination method for analyzing distortion effects on box girders

A deformation coordination method is proposed in this study to account for the distortion effects on a box girder. The differential equation for distortion in vertical web box girders is derived based on the deformation coordination condition of the distortion angle, considering both external loads and internal forces. Subsequently, a comparative analysis is conducted to explore the similarities and differences between the differential equations derived from the proposed deformation coordination method, the plate element analysis method and the total potential energy variation method. The accuracy of the proposed approach is verified through bench-scale tests and numerical simulations. The findings indicate that the derived governing distortion differential equation and distortion attenuation coefficients in the proposed method align with those obtained from the plate element analysis method and the total potential energy variational method, which enhances the applicability to allow for the distortion equations to be obtained simply by calculating the distortion displacements. The analytical findings regarding the distortion warping normal stresses on the cross-sections of the box girders demonstrate favorable correspondence with the experimental results, displaying an acceptable error ranging from − 0.3% to 5.4%. Moreover, the peak of distortion warping normal stresses on the mid-span cross-section increases with higher span-to-depth ratios and height-to-thickness ratios of the web. Consequently, augmenting the thickness of the box wall proves to be an effective means of reducing the distortion effect in box girders.

Modeling of contact resistivity and simplification of 3D homogenization strategy for the H formulation

The finite element method (FEM) provides a powerful support for the calculations of superconducting electromagnetic responses. It enables the analysis of large-scale high-temperature superconducting (HTS) systems by the popular H formulation. Nonetheless, modeling of contact resistivity in three-dimensional (3D) FEM is still a matter of interest. The difficulty stems from the large aspect ratio of the contact layer in numerical modeling. Nowadays, an available solution is to model the contact layer with zero thickness but requires the discontinuity conditions of the magnetic field. In this paper, the energy variational method is utilized to incorporate the contribution of contact resistivity into the H formulation. From the perspective of energy transfer, the contact resistivity is related to the energy dissipation of the radial current flowing through the contact interface. In terms of applications, this method can be employed to calculate the charging delay of no-insulation coils and the current sharing behaviors of CORC cables. One advantage of this model is that the magnetic field is continuous and hence can be easily implemented in FEM. Additionally, it requires fewer degrees of freedom and hence presents advantages in computational efficiency. Moreover, this method can be employed to simplify the 3D H homogeneous model for insulated coils. The above discussions demonstrate that the proposed model is a promising tool for the modeling of contact resistivity.

Experimental and theoretical investigation of shear lag effect in twin box‐shaped composite girders of long‐span cable‐stayed bridges

AbstractSteel‐concrete composite box girders offer notable advantages such as high stiffness, excellent torsional performance, and light self‐weight, making them prevalent in large‐span bridges, particularly in cable‐stayed configurations. However, a persistent challenge arises from the shear lag effect in the concrete slab within the flanges induced by shearing interactions between the webs and flanges. Limited attention has been devoted to investigating the experimental and theoretical aspects of shear lag in long‐span cable‐stayed bridges with twin box‐shaped composite girders. This research addresses this gap by designing a segmental scale twin box‐shaped composite model specimen to simulate the cable spacing of the main girder in a cable‐stayed bridge, specifically targeting the shear‐lag effect. The study successfully captures the non‐uniform distribution of cross‐sectional strain in the top concrete slab. Subsequently, analytical strain solutions for the composite girder under mid‐span concentrated load and axial load are derived using the energy variation method. The theoretical model's applicability is then calibrated against experimental results. The investigation extends to the in‐situ testing of the Chibi Yangtze River Bridge, the world's largest steel‐concrete composite girder cable‐stayed bridge, boasting a main span of up to 720 m. An established finite element (FE) analysis model for the Chibi Yangtze River Bridge simulates the transverse stress distribution under typical construction conditions of CC6. Comparative analyses between the simulated results, calculated values from elementary beam theory, and the theoretical model against field‐measured values reveal that the FE model provides a more accurate estimation of the shear lag effect in the concrete slab. Notably, the FE model effectively reflects the bridge's characteristics and serves as a baseline for subsequent studies. This comprehensive investigation contributes significantly to the understanding and analysis of shear lag phenomena in twin box composite girders of cable‐stayed bridges. The integration of experimental, analytical, and construction site validations establishes robust tools for engineering design and the construction of safe and serviceable cable‐stayed bridges. The research augments knowledge regarding shear lag behavior and underscores the necessity to appropriately consider its effects in bridge analysis and design.

Ispitivanje poprečne unutarnje sile sandučastog nosača primjenom principa varijacije energije

This study investigated the transverse internal force of a box girder with a trapezoidal cross-section under eccentric loading. The effects of the box girder distortion and frame shear difference on the transverse internal force were considered. An energy variation method based on the minimum potential energy principle was adopted for transverse internal force analysis of the frames. The variable shear difference of the top slab was considered an unknown quantity, and a fourth-order governing differential equation was established. The transverse bending moment of the frame was obtained by solving for the shear difference. Two examples were used to verify the proposed method and analyse the differences between the transverse internal force results calculated using different methods. The effect of the distribution of the box-girder distorted transverse bending moment on each slab was investigated for various stiffness ratios. The results showed that the transverse internal force of the box girder calculated using the proposed method agreed well with the finite element results, with a maximum error of less than 9.68 %. When the stiffness ratio of the box girder increased, the distribution of the distorted transverse bending moment on the top slab increased, whereas that on the bottom slab decreased.

Nonlinear analysis of unimorph piezoelectric nanobeam with a variable cross-section and flexoelectric effect

An analytical method is presented to investigate the nonlinear electromechanical coupling response of an unimorph piezoelectric nanobeam with a variable cross-section, taking into account the flexoelectric effect. Considering the geometric nonlinearity, strongly nonlinear governing equation with variable coefficients is derived using the energy variation method and solved by the differential quadrature method. The nonlinear effects of geometric nonlinearity, electromechanical loadings, flexoelectric effect and width ratio on the electromechanical coupling response are analyzed in detail. The results show that under different loadings, the flexoelectric parameter and width ratio have distinct effects on the electromechanical response of the piezoelectric variable cross-section nanobeam. When the mechanical loads are applied on the nanobeam, a larger flexoelelctric parameter results in a greater bending deflection and higher output of electric field and polarization. The width ratio of variable cross-section can also influence the bending deflection, but its impact on the output of electric field and polarization is limited. When the applied voltage loads are applied to the nanobeam, the bending direction can be controlled by the applied voltage, flexoelectric parameter or width ratio. Due to the strongly nonlinear governing equation, these results in this study differ significantly from those of piezoelectric nanobeam with uniform cross-section. This study can contribute to the understanding of electromechanical coupling response in piezoelectric variable cross-section nanobeam with flexoelectric effect. Additionally, it can be helpful in guiding the structure optimization of piezoelectric nanodevices.

Strain-induced solid-state coating of TWIP steel sheets with zinc

We propose the strain-induced solid-state coating process of twinning-induced plasticity (TWIP) steel with zinc, achieved through concurrent rolling of TWIP steel and zinc sheets. Our study explores the effects of strain, strain rate, and rolling temperature on the morphology and intermixing at the Zn-TWIP steel interface. We found that two simultaneous mechanisms are responsible for the formation of a joint interface between the sheets in the rolling temperature range of 100–380 °C, namely mechanical intermixing at the interface and accelerated diffusion along moving dislocations and other defects introduced during rolling. Since it is impossible to deconvolute these mechanisms, we introduced the concept of an ‘apparent diffusion coefficient’ to represent the diffusion-like mass transfer resulting from mechanical intermixing and diffusion. These diffusion coefficients were determined by primary electron energy variation method in scanning electron microscope and by energy-dispersive X-ray spectroscopy in transmission electron microscope. Notably, these diffusion coefficients were found to be higher than the grain boundary diffusion coefficients of Zn in γ-Fe, primarily due to the dynamic nature of interface microstructure formation during rolling.

Ruled Shells of Conical Type on Elliptical Base

The information about main results on geometry of developable surfaces with an edge of regression which have a directrix ellipse in the base is gathered. These surfaces constitute a group called “Ruled surfaces of conical type on elliptical base”. This group includes elliptical cones, torses with two ellipses defined in the parallel planes, equal slope surfaces, and ruled surfaces with the main frame of three superellipses that are ellipses in one coordinate plane and broken straight lines in the other two coordinate planes. The paper presents a method for developing torses onto a plane, approximation of torses by folded surfaces, and parabolic ending of a thin sheet from elastic material into a torse shell. A brief review of the methods of stress-strain and buckling analysis of the considered ruled shells is given, including the displacement-based finite element method and variational energy method. It is shown that analytical methods can be used only in the case of applying the momentless shell theory for ruled thin shells of conical type. The analytical formulae for determining the normal and tangent internal forces in any momentless conic shell with a superellipse in the base are derived. References to forty four scientific articles of other authors, working or having worked on the subject of the paper are given. These references confirm the conclusions of the author and the perspectives of investigations of the considered ruled surfaces and shells.

Study on the effect of tunneling and secondary grouting on the deformation of existing pipelines in bedrock raised strata

Shield crossing the bedrock raised strata easily leads to large deformation of the soil, resulting in deformation and damage of the upper pipelines. To investigate the law of pipeline deformation caused by shield underpassing in bedrock raised strata, the changes in the action range of the additional thrust of the cutter head (p1), the friction resistance of the shield shell (p2), and the additional grouting force (p3) were analysed. A convergence model of the excavation face suitable for bedrock raised strata was proposed. The calculation formula of soil displacement at the axis of the pipeline was derived by using Mindlin’s solution and stochastic medium theory. The solutions of the vertical displacement, bending moment and strain of the pipeline were obtained by the energy variational method and the principle of minimum potential energy. Pipeline deformation analysis and reliability verification were carried out by an engineering case in Hangzhou, China. The influence of bedrock distribution and secondary grouting behind the pipe wall on pipeline deformation was studied. The results show that the calculated results are similar to the measured data with a high degree of coincidence, and the degree of agreement is higher compared to the existing method. When the shield tunneling direction is well controlled, the settlement and bending moment of the pipeline caused by the shield crossing the raised bedrock stratum will be reduced, and the reduction increases with increasing bedrock length and intrusion depth. The secondary grouting behind the pipe wall can reduce the settlement of the pipe line, but it is necessary to reasonably control the three parameters of grouting amount per unit length (Vinj), grouting ring length (Lh), and grouting ring angle (δ). Lh should not be too long and can be controlled below 30 m, δ should be controlled within 60°, and Vinj can be adjusted according to the deformation size of the pipeline to prevent excessive or insufficient correction.

Analytical Method of the Shear Lag Effect in Thin-Walled Box Girders Based on the Shear Flow Distribution Law

This paper presents an analytical method based on the shear flow distribution law to study the shear lag effect of thin-walled single- and double-cell box girders. The first step in this method is to determine the box girder’s shear flow distribution. Subsequently, a series of novel improved longitudinal displacement functions mathematically expressed as cubic parabolas are established. The parabolic origin of these functions is located at the zero point of the shear flow corresponding to each plate; the unknown parameters used to describe the function form can be determined according to the shear flow distribution, the continuity conditions, and the axial force balance condition. Then, the variational energy method is adopted to derive the governing differential equations. The shear lag effect in thin-walled single- and double-cell box girders under several boundary conditions and load cases is studied and analytical expressions for the shear lag coefficient are derived. Finally, results obtained using the proposed method are validated via comparison with numerical results. The results show that the proposed method can provide reasonable predictions for the shear lag effect of single- and double-cell box girders, and that this method is more straightforward and practical. In addition, the shear lag coefficients at different webs are not entirely equal, which is related to the distance from the web to the zero point of the shear flow.

Coupled chemical reactions: Effects of electric field, diffusion, and boundary control.

Chemical reactions involve the movement of charges, and this paper presents a mathematical model for describing chemical reactions in electrolytes. The model is developed using an energy variational method that aligns with classical thermodynamics principles. It encompasses both electrostatics and chemical reactions within consistently defined energetic and dissipative functionals. Furthermore, the energy variation method is extended to account for open systems that involve the input and output of charge and mass. Such open systems have the capability to convert one form of input energy into another form of output energy. In particular, a two-domain model is developed to study a reaction system with self-regulation and internal switching, which plays a vital role in the electron transport chain of mitochondria responsible for ATP generation-a crucial process for sustaining life. Simulations are conducted to explore the influence of electric potential on reaction rates and switching dynamics within the two-domain system. It shows that the electric potential inhibits the oxidation reaction while accelerating the reduction reaction.

Mathematical Model for Chemical Reactions in Electrolytes Applied to Cytochrome c Oxidase: An Electro-Osmotic Approach

This study introduces a mathematical model for electrolytic chemical reactions, employing an energy variation approach grounded in classical thermodynamics. Our model combines electrostatics and chemical reactions within well-defined energetic and dissipative functionals. Extending the energy variation method to open systems consisting of charge, mass, and energy inputs, this model explores energy transformation from one form to another. Electronic devices and biological channels and transporters are open systems. By applying this generalized approach, we investigate the conversion of an electrical current to a proton flow by cytochrome c oxidase, a vital mitochondrial enzyme contributing to ATP production, the ‘energetic currency of life’. This model shows how the enzyme’s structure directs currents and mass flows governed by energetic and dissipative functionals. The interplay between electron and proton flows, guided by Kirchhoff’s current law within the mitochondrial membrane and the mitochondria itself, determines the function of the systems, where electron flows are converted into proton flows and gradients. This important biological system serves as a practical example of the use of energy variation methods to deal with electrochemical reactions in open systems. We combine chemical reactions and Kirchhoff’s law in a model that is much simpler to implement than a full accounting of all the charges in a chemical system.

Theoretical analysis and a parametric study on forced vibration of large statically deformed curved beam with rigid links and moving support

Forced vibration characteristics around large statically deformed configuration of curved beam system with rigid links and moving support are analyzed. In addition, the paper also investigates various parametric effects of height to span ratio and roller radius to height ratio coupled with support connection mechanism and profile shape, on dynamic characteristics of the curved beam system. The subject problem includes two interrelated problems: determining deformed configuration under static load and dynamic response of the loaded beam under harmonic excitation. The static problem is analyzed incrementally through the variational principle-based energy method in body-embedded curvilinear frame considering geometric nonlinearities due to combined bending-stretching and non-uniform initial curvature. In each incremental step, the nonlinear governing equation is solved iteratively, and kinematic constraints due to interactive deformation of the flexible curved beam with rigid links are imposed to obtain the global solution. Concerning the statically deformed configuration, governing equation for forced vibration is derived through Hamilton’s principle in curvilinear frame. Nonlinear effects caused by combined bending-stretching modes of vibration, non-uniform curvature, and lumped mass of the dead static load are considered. The set of inhomogeneous governing equations is solved through Broyden’s method. A direct substitution technique through successive relaxation is also employed to obtain an initial guess solution. Besides the forced vibration, free vibration analysis at static deformed configuration is also carried out for the completeness of the study. The theoretical model is validated through comparisons of loaded natural frequency results with finite element package. After the validation study, the effects of the mentioned geometric parameters on the dynamic response of the curved beam system are investigated and presented suitably. The physically insightful framework along with the parametric study may facilitate simulation and design of several practical structures involving moving support and rigid links subjected to continuous excitation around large statically deformed configurations.

Applications of common energy variability methods for establishing the equilibrium equation in mechanics

The principles of the energy variation method are commonly utilized in mechanics. Energy is a scalar variable, so these are more convenient and simple to establish the equilibrium equations compared to vector-based approaches (i.e. using forces and displacements). The present article applied the theorem of the energy variation method in order to set the equilibrium equations for various complicated problems. Four examples of applying the energy variation method include the differential equation of the Euler – Bernoulli beam based on the energy method, the system of Equilibrium Equations of the Euler–Bernoulli beam with the theorem of Least Work, the principle of maximum work to establish the equation of motions for the Euler–Bernoulli beam and the equation of motion for the Euler–Bernoulli beam by the virtual work theorem, have been implemented. The results obtained from this study open up further research directions on the application of the energy variation method in mechanics as well as in the analysis theory of beam bridges.

Study on the Mechanical Properties of Continuous Composite Beams under Coupled Slip and Creep

Steel–concrete continuous composite beams are widely used in buildings and bridges and have many economic benefits. Slip has always existed in composite beams and will reduce the stiffness of composite beams. The effect of creep under a long-term load will also be harmful. Many scholars ignore the combined effects of slip and creep. In order to more accurately study the mechanical properties of steel–concrete continuous composite beams under long-term loads, this paper will consider the combined actions of slip and creep. By combining the elastic theory and the age-adjusted effective modulus method, the differential equation of the composite beam is derived via the energy variational method. The analytical solutions of axial force, deflection and slip under a uniform load are obtained by substituting the relevant boundary conditions. The creep equation is used to simulate the behavior of concrete with time in ANSYS. The analytical solution is verified by establishing a finite element model of continuous composite beams considering slip and creep. The results suggest the following: the analytical solution is consistent with the finite element simulation results, which verifies the correctness of the analytical solution. Considering the slip and creep effects will increase the deflection of the composite beam and the bending moment of the steel beam, reduce the bending moment of the concrete slab and have a significant impact on the structural performance of the continuous composite beam. The research results considering the coupling effect of slip and creep on continuous composite beams can provide a theoretical basis for related problems.

Study on Mechanical Properties of Simply-Supported Composite Beams Considering Creep and Slip

In the current industry, steel–concrete composite beams are used in large-span bridges and super-high-rise building structures due to their excellent overall performance. Concrete’s creep and slip effects in the combined structure can adversely affect the structure, thus affecting the safe use of bridges and buildings. It is necessary to study the mechanical properties of the combined structure considering creep and slip. In order to further study the mechanical properties of steel–concrete composite beams under creep–sliding coupling, in this study, based on the energy variational method principle, the energy equation of a composite beam considering creep and slip coupling is established. The second-order differential equation of the axial force of steel–concrete composite beams is derived by introducing basic assumptions. The calculation formulas for the axial force, deflection, and slip of simply-supported composite beams under different loads are obtained using different boundary conditions. Then, the creep effect of composite beams is simulated using the creep criterion in the ANSYS finite element software when the concrete material parameters change with time. The results show that a simply-supported composite beam considering both slip and creep will have a significant effect on the structure; the more strongly the studs constrain the concrete slab, the greater the adverse effect of concrete creep on the combined beam. The formula derived in this paper is consistent with the numerical simulation solution and is suitable for different creep and slip conditions. The research results can provide a theoretical basis for the calculation of the axial force, deflection, and slip of combined beams under uniform and concentrated loads in practical engineering considering slip and creep.

Analysis of the protective effect of setting isolation piles outside the foundation pit on the underpass tunnel side

Excavating and unloading a foundation pit cause deformation or even destruction of any existing adjacent tunnel. In construction projects, isolation piles are often set between the foundation pit and tunnel to protect the existing tunnel. The aim of the present study was to investigate the horizontal displacement law of an existing tunnel outside the isolation piles as a result of foundation pit excavation, so as to analyze the protective effect of isolation piles on the existing tunnel. The unloading effect of the sidewalls, the unloading effect of the base of the foundation pit, and the deflection deformation of the isolation piles were considered. A three-dimensional mechanical calculation model for foundation pits, isolation piles, and existing tunnels was established. Based on the Mindlin’s solution, the calculation formula of the additional stress caused by the unloading of the foundation pit was deduced. Based on the energy variation method, the calculation formula for the deflection deformation of the isolation piles was derived. Using the mirror image method, the calculation formula for the horizontal displacement of the existing tunnel caused by the deformation of the isolation pile was derived. The influence of different spacing distances between isolation piles on the horizontal displacement of an existing tunnel was studied. The research results showed that the calculation results in this study were consistent with the actual engineering data available and numerical simulation results. It was concluded that the installation of isolation piles could effectively reduce the horizontal displacement of tunnels and fulfil a beneficial protective role, and that the protective effect gradually increases as the distance between isolation piles is reduced.

A phase field model for mass transport with semi-permeable interfaces

In this paper, a thermaldynamical consistent model for mass transfer across permeable moving interfaces is proposed by using the energy variation method. We consider a restricted diffusion problem where the flux across the interface depends on its conductance and the difference of the concentration on each side. The diffusive interface phase-field framework used here has several advantages over the sharp interface method. First of all, explicit tracking of the interface is no longer necessary. Secondly, interfacial conditions can be incorporated with a variable diffusion coefficient. Finally, topological changes of interfaces can be handed easily. A detailed asymptotic analysis confirms the diffusive interface model converges to the existing sharp interface model as the interface thickness goes to zero. An energy stable numerical scheme is developed to solve this highly nonlinear coupled system.Numerical simulations first illustrate the consistency of theoretical results on the sharp interface limit. Then a convergence study and energy decay test are conducted to ensure the efficiency and stability of the numerical scheme. To illustrate the effectiveness of our phase-field approach, several examples are provided, including a study of a two-phase mass transfer problem where droplets with deformable interfaces are suspended in a moving fluid.

Stability analysis of three-dimensional thick rectangular plate using direct variational energy method

This study investigated the elastic static stability analysis of homogeneous and isotropic thick rectangular plates with twelve boundary conditions and carrying uniformly distributed uniaxial compressive load using the direct variational method. In the analysis, a thick plate energy expression was developed from the three-dimensional (3-D) constitutive relations and kinematic deformation; thereafter the compatibility equations used to resolve the rotations and deflection relationship were obtained. Likewise, the governing equations were derived by minimizing the equation for the potential energy with respect to deflection. The governing equation is solved to obtain an exact deflection function which is produced by the trigonometric and polynomial displacement shape function. The degree of rotation was obtained from the equation of compatibility which when equated to the deflection function and put into the potential energy equation formulas for the analysis were obtained after differentiating the outcome with respect to the deflection coefficients. The result obtained shows that the non-dimensional values of critical buckling load decrease as the length-width ratio increases (square plate being the highest value), this continues until failure occurs. This implies that an increase in plate width increases the probability of failure in a plate. Hence, it can be deduced that as the in-plane load on the plate increase and approaches the critical buckling, the failure in a plate structure is abound to occur. Meanwhile, the values of critical buckling load increase as the span-thickness ratio increases for all aspect ratios. This means that, as the span-thickness ratio increases an increase in the thickness increases the safety in the plate. It also indicates that the capacity of the plate to resist buckling decreases as the span-depth ratio increases. To establish the credibility of the present study, classical plate theory (CPT), refined plate theory (RPT) and exact solution models from different studies were employed to validate the results. The present works critical buckling load varied with those of CPT and RPT with 7.70% signifying the coarseness of the classical and refined plate theories. This amount of difference cannot be overlooked. The average total percentage differences between the exact 3-D study (Moslemi et al., 2016), and the present model using polynomial and trigonometric displacement functions is less than 1.0%. These differences being so small and negligible indicates that the present model using trigonometric and polynomial produces an exact solution. Thus, confirming the efficacy and reliability of the model for the 3-D stability analysis of rectangular plates.

Fully-decoupled, energy stable second-order time-accurate and finite element numerical scheme of the binary immiscible Nematic-Newtonian model

We consider the numerical approximation of a two-phase complex fluid flow model that couples the immiscible liquid crystal fluid component immersed in the ambient free flow field, where the free interface motion between the two fluids is simulated by using the phase-field approach via the energy variational method. The model is a highly nonlinear coupling system, which is composed of the Navier–Stokes equations for the flow field, the Cahn–Hilliard equations for the free moving interface, and the constitutive equation for the nematic liquid crystal. This paper proposes an effective full-discrete finite element numerical scheme to solve the model and makes the scheme not only have the characteristics of linearity, second-order time accuracy, unconditional energy stability, decoupling structure but also only need to solve a few constant-coefficient elliptic equations at each time step. The unconditional energy stability of the scheme and the detailed implementation process are further given. Various numerical experiments including the well-known “beads-on-string” phenomena are also simulated to illustrate the effectiveness of the scheme.

A second-order time accurate and fully-decoupled numerical scheme of the Darcy-Newtonian-Nematic model for two-phase complex fluids confined in the Hele-Shaw cell

We consider the numerical approximation of the binary immiscible mixture of nematic liquid crystals and viscous Newtonian fluids confined in a Hele-Shaw cell, where the free interface motion is simulated by using the phase-field approach via the energy variational method. The governing system is highly complicated nonlinear and coupled, consisting of the Darcy equations for the flow field, the Cahn-Hilliard equations for the free moving interface, and the constitutive equation for the nematic liquid crystal. The numerical scheme developed in this paper is the first “ideal” scheme, namely, it not only has the following characteristics: linearity, second-order time accuracy, unconditional energy stability, and decoupling structure, but also at each time step, only needs to solve a few elliptic equations with constant coefficients. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical experiments are further carried out to prove the effectiveness of the scheme, in which the influence of the initial orientations and anchoring elastic energy of the liquid crystal on the Saffman-Taylor fingering instability are studied.