Strain Energy Formula Research Articles

A characterization method for equivalent elastic modulus of rock based on elastic strain energy

Energy is an internal variable during rock deformation and failure, and its dissipation and conversion law can reflect the rock’s internal damage and deterioration state. Analysis of rock deformation and failure process from the perspective of energy is helpful to deeply understand the mechanism of rock damage, fracture and instability failure, and has important theoretical and practical significance for the stability evaluation and support control of surrounding rock. In this study, through single cyclic loading and unloading (SCLU) experiments, cyclic triaxial loading and unloading (CTLU) experiments and conventional triaxial compression (CTC) experiments, the equivalent elastic modulus method based on elastic strain energy is proposed to analyze the energy conversion of rock. The results show that the error of the elastic strain energy calculated by the strain energy formula method is generally higher than 10% with the secant and tangent modulus of the loading and unloading curve as input parameters. Taking the equivalent elastic modulus proposed in this study as an input parameter, more accurate elastic strain energy can be obtained by the strain energy formula. During the rock failure process, the equivalent elastic modulus shows a three-stage characteristic of increase, steady and decrease. The equivalent elastic modulus can be estimated by the quadratic function between the equivalent elastic modulus and confining pressure and axial strain. Under the same deformation and deviatoric stress, the elastic strain energy stored in rock increases with increasing confining pressure. The local maximum energy dissipation rate corresponds to stress drop, and the peak energy dissipation rate appears near the peak strength. High energy dissipation mainly occurs in a short time after peak strength, and energy release and dissipation are more sudden and severe under high confining pressure.

Crack Detection in Plate‐Like Structures Using Modal Strain Energy Method considering Various Boundary Conditions

Among vibration‐based damage detection methods, one of the effective approaches for damage localization is the modal strain energy method. In this paper, the modal strain energy method is developed for damage detection in plate‐like structures with various boundary conditions. Firstly, the theory of the modal strain energy method is briefly outlined. In order to overcome the limitation of measuring points, a central difference method is newly employed to compute the partial differential terms in the modal strain energy formula. Finite element analysis is conducted on an aluminum thin plate to obtain the mode shapes before and after the occurrence of damage. Feasibility of the proposed method is verified by investigating plates with different types of boundary conditions. A damage index is presented to identify the location and extent of crack in the plate‐like structures. The analytical results show that the proposed method accurately identifies the crack in the plate structure with various types of boundary conditions by using appropriate mode shapes and damage threshold.

Effects of nonlinear parameter of fluid-solid coupling on acoustic field in porous media.

The purpose of this Letter to the Editor is to demonstrate the effects of the nonlinear parameter of fluid-solid coupling on the acoustic field in porous media. The nonlinear acoustic field excited by a one-dimensional finite-amplitude shear wave propagating in the uniform and infinite saturated porous media is studied. Researchers use the complete strain energy formula with third-order terms of strain in saturated porous media. The formula for calculating the fluid-solid coupling nonlinear parameter γ from the dynamic compatibility condition is given. To illustrate the effect of the fluid-solid coupling nonlinear term on the nonlinear acoustic field excited by the longitudinal and transverse waves, this study compares nonlinear acoustic fields excited by a finite amplitude wave with and without nonlinear parameter γ. The results show that the nonlinear parameter γ has an influence on the nonlinear acoustic field excited by the one-dimensional shear wave, but it has no influence on the nonlinear acoustic field excited by the one-dimensional longitudinal wave.

Formation energy and some mechanical properties of hydrogenated hexagonal monolayer of GeC

Mechanical properties of pristine and fully hydrogenated 2D hexagonal GeC monolayer were investigated by density functional theory calculations. Fully hydrogenated means that there is a hydrogen atom connected to each Ge and C atoms. Formation energy of fully hydrogenated GeC was negative which indicates that hydrogenated GeC is energetically stable. Uniaxial and biaxial strains were applied to the pristine and hydrogenated GeC in harmonic elastic deformation range and in-plane stiffness and Poisson's ratio were calculated by fitting the resulted strained energy to the strain-energy formula. Calculated in-plane stiffness and Poisson's ratio of hydrogenated GeC were respectively about 50% and 30% lower than those of pristine GeC. Pristine and hydrogenated GeCs are semiconductor and the band gap of hydrogenated GeC is larger than pristine. By Mulliken charge analysis we found that, in hydrogenated GeC, charge transfers from H atoms to GeC sheet.

Study and Application of Properties Equivalent Algorithm for Composite Wind Turbine Blade Based on Strain Energy Method

ABSTRACT Zhang, L., 2015. Study and application of properties equivalent algorithm for composite wind turbine blade based on strain energy method. To simplify the modeling and improve calculating efficiency of wind turbine blade, properties equivalent method of composite laminate based on strain energy method is put forward. To take representative volume element of composite as research object, basing on the two-dimension strain energy method, the equations set, which shows the relationship between equivalent stiffness matrix coefficient of composite properties and strain energy of RVE, is deduced on the three-dimension strain energy method through utilizing strain energy conservation theory and transforming of different strain energy formula. And the equation set is in accordance with the stress and strain field of RVE is the same as that of homogeneous equivalent unites under certain conditions. Then energy expression of properties equivalent stiffness matrix for composite is given. Turning the stiffnes...

Elastic Energy of Surfaces and Residually Stressed Solids: An Energy Approach for the Mechanics of Nanostructures

The surface energy plays a significant role in solids and structures at the small scales, and an explicit expression for surface energy is prerequisite for studying the nanostructures via energy methods. In this study, a general formula for surface energy at finite deformation is constructed, which has simple forms and clearly physical meanings. Next, the strain energy formulas both for isotropic and anisotropic surfaces under small deformation are derived. It is demonstrated that the surface elastic energy is also dependent on the nonlinear Green strain due to the impact of residual surface stress. Then, the strain energy formula for residually stressed elastic solids is given. These results are instrumental to the energy approach for nanomechanics. Finally, the proposed results are applied to investigate the elastic stability and natural frequency of nanowires. A deep analysis of these two examples reveals two length scales characterizing the significance of surface energy. One is the critical length of nanostructures for self-buckling; the other reflects the competition between residual surface stress and surface elasticity, indicating that the surface effect does not always strengthen the stiffness of nanostructures. These results are conducive to shed light on the importance of the residual surface stress and the initial stress in the bulk solids.

Detection of matrix cracks in composite laminates by using the modal strain energy method

As the foremost mechanism of damage development, matrix cracking is the critical damage found in the early stage of structural failure of composites. This study aims to nondestructively detect matrix cracks in composite laminates by using an experimental modal analysis (EMA). An AS4/PEEK composite was used to fabricate cross-ply [02/9012/02] and quasi-isotropic [(±45/0/90)4] s laminates. The damage in the form of a matrix crack in the laminates was created by using a tensile load. The EMA was conducted on the laminates to obtain the modal displacements before and after damage. The displacements were then employed to compute the modal strain energy and to define the damage index, which is used for detecting matrix cracks. Limited by the mesh points of measurements, we used the differential quadrature method to calculate the partial differentials in the strain energy formula. The results obtained were validated by using the X-ray radiography method and three-point bending tests. The experimental results showed that the damage index well identified the location of breadthwise matrix cracks inside the laminates. However, the resolution of the damage index became poor if the spans of matrix cracks were short or the matrix cracks were scattered over the laminates.

Molecular structures and electronic properties of helical thiophene carbon–sulfur oligomers, H2(C2S) n C2H2 (n = 1–20)

Structure optimizations of the thiophene carbon–sulfur H2(C2S) n C2H2 (n = 1–20) were carried out using density functional theory calculations at the B3LYP/6-31G(d) level. The B3LYP/6-31G(d) geometrical data for heptamer H2(C2S)7C2H2 and undecamer H2(C2S)11C2H2 are in good agreement with the X-ray crystallographic data for the helical (C2S)n β-heptathiophene and β-undecathiophene, respectively. Structural and electronic properties of helical oligothiophenes obtained at the B3LYP/6-311++G(d,p)//B3LYP/6-31G(d) level are reported. The strain energy formula of n oligothiophenes as a linear function of their molecular length was obtained.

Damage detection of a woven fabric composite laminate using a modal strain energy method

This paper presents the detection of surface cracks in a woven fabric composite laminate using a modal strain energy method. A carbon fabric F3T-282/epoxy (DICY) is used to fabricate a plain woven laminate. In the first place, the unknown material properties are computed by utilizing an inverse method through finite element analysis and experimental modal analysis. Three equivalent models, i.e. cross-ply [ 0 / 90 ] n s , orthotropic and representative cell, are established to simulate the woven laminate. Tensile tests are also performed to measure the mechanical properties for comparison. A surface crack is created to represent nominal damage which is comparatively small and does not significantly affect the global stiffness of the woven laminate. Experimental modal analysis is conducted on the woven laminate to obtain the modal displacements before and after damage. The modal displacements are used to compute the modal strain energies through the three equivalent models. A damage index is defined by employing the fractional modal strain energy of the woven laminate before and after damage, and then used to identify the location of the surface crack. Limited by grid points in measurement, a differential quadrature method is utilized to compute the partial differential terms in strain energy formula. Consequently, the damage indices obtained from global and local measurements successfully locate the surface crack in woven laminate. Only a few measured mode shapes are required in this method, which has a relatively low cost and flexibility in measurement, nondestructive evaluation, and feasibility of real-time detection in woven laminates.

Nondestructive Damage Detection of Two Dimensional Plate Structures Using Modal Strain Energy Method

AbstractA nondestructive detection method of surface cracks in two dimensional plate structures using modal strain energy method is investigated in this paper. Experimental modal analysis (EMA) is conducted on an aluminum alloy 6061 thin plate to obtain the mode shapes before and after damage under a completely free boundary condition. The measured mode shapes are used to compute the strain energy of the plate. Limited by the measured points, a differential quadrature method is employed to compute the partial differential terms in strain energy formula. A damage index is then defined based on strain energy ratio of the plate before and after damage. This damage index is employed to identify the location of surface crack in plate structure. A finite element analysis (FEA) is also performed to access this approach and demonstrate a feasible process for the experimental work. Good correlation between FEA and EMA results is obtained. The damage index obtained from both FEA and EMA successfully identify the location of surface crack in the aluminum plate. Only few measured mode shapes of the plate are required in this method, which provides a quick, flexible, inexpensive and nondestructive technique to identify the damagein local and global 2D plate structures.

Development of scanning damage index for the damage detection of plate structures using modal strain energy method

This work presents a novel approach of nondestructive detection of damage in plate structures by using experimental modal analysis (EMA) and modal strain energy method (MSEM). An aluminum alloy 6061 thin plate with a surface crack is investigated in this study. EMA is conducted on the plate to obtain the mode shapes before and after damage. The modal displacements of each mode shape are then used to compute the modal strain energy. For all measured mode shapes, a damage index is defined by using the ratio of modal strain energies of the plate before and after damage. In fact, small damage causes very little change in system response, but it is an essential early warning of structure damage. As the second-order derivatives, modal strain energy is much more sensitive to the small change of structural response than frequencies and mode shapes. It is therefore feasible to approach the small damage by using a damage index defined by fractional MSE of the structure before and after damage. In this study, a scanning damage index (SDI) is developed by moving damage indices obtained from the local area throughout the structure as if a scanning sensor is used to inspect the structure. The damage indices in overlap areas are added up and the summation may intensify the signals of damage in the plate. Limited by the numbers of measured point, a differential quadrature method is employed to calculate the partial differential terms in strain energy formula. Experimental results show that SDI well identifies a surface crack location by using only few measured mode shapes of the aluminum plate. This novel approach provides a flexible, cost-effective, and nondestructive damage evaluation in either local or global structure. Its applicability to different types of structures and different sizes of damage is to be experimentally validated in the future work.

A simple approach to temperature dependence of strain energy: Application to GaN-based semiconductors

We propose a simple approach to temperature dependence of the strain energy in epitaxial thin films coherently grown on the substrate and apply it to the stability for GaN-based semiconductors. The strain energy is simply described as a function of the bulk modulus and lattice parameters using the continuum elasticity theory. Based on this formula, the temperature dependence of the strain energy is estimated using Mie–Grüneisen equation of state that gives temperature dependence of the bulk modulus and lattice parameter. The versatility of the strain energy formula is confirmed by comparing the calculated strain energy for GaN with that obtained by our empirical potential calculations at 0 K. We also compare the temperature dependence of the bulk modulus and lattice parameter for GaN with experimental results at high temperatures. The calculated results imply that the elastic constants decrease while the lattice parameter increases with increase of temperature. They are consistent with experimental results. Using these calculated results, it is clarified that the strain energy accumulated in zinc blende structured (ZB) GaN coherently grown on ZB–SiC(0 0 1) at 1000 K becomes ∼10% greater than that at 0 K. This gives smaller critical film thickness of misfit dislocation generation at high temperatures.

Damage Detection of Surface Crack in Composite Quasi-Isotropic Laminate Using Modal Analysis and Strain Energy Method

This paper presents a damage detection of surface crack in composite laminate. Carbon/epoxy composite AS4/PEEK was used to fabricate a quasi-isotropic laminate [0/90/±45]2s. Surface crack was created by using laser cutting machine. Modal analysis was performed to obtain the mode shapes of the laminate before and after damage. The mode shapes were then adopted to compute the strain energy, which was used to define a damage index. Consequently, the damage index successfully predicted the location of surface crack in the laminate. Differential quadrature method (DQM) was introduced to calculate the partial differential terms in strain energy formula.

Analysis of a Novel 3D Hybrid Woven/Knitted Fabric Structure

In Part I of this series, a geometric model was developed to predict loop length, yarn consumption, and thickness of a novel 3D hybrid woven/knitted fabric structure. In this part, a mechanical model is generated for estimating the Young's modulus and extension of the 3D hybrid woven/knitted fabric structure. Castigliano's theorem is used to derive the relations. The compression, extension, and bending energies of yarns are considered for the total strain-energy formula of the fabric. The mechanical model is verified for the fabric Young's modulus. There is reasonable agreement between theory and experiment considering the assumptions made and the complexity of the 3D fabric structure. The deviation is less with high performance multifilament fibers in the warp and bias yarns.

The exact thick arch finite element

The exact stiffness matrix is derived for a curved beam element with constant curvature. The plane two-node six degree-of-freedom element is considered in which effects of flexural, axial and shear deformations are taken into account. The analytical shape functions describing radial and tangential displacements as well as cross-section rotations are found in the algebraic–trigonometric form. They contain the coupled influences of shear and membrane effects. Based on these shape functions, using the strain energy formula, the stiffness matrix for shear flexible and compressible arch element is formulated. Obviously, this element is completely free of shear and membrane locking effects. The advantage of the elaborated element is its applicability to any combination of geometrical properties of the arch structure, e.g. the depth–length ratio of element. In presented numerical examples the shear and membrane influences on the displacements for various cases of boundary conditions and loading are investigated. The results coincide exactly with the analytical ones obtained for continuous arches.

AN EFFICIENT CURVED BEAM FINITE ELEMENT

The plane two-node curved beam finite element with six degrees of freedom is considered. Knowing the set of 18 exact shape functions their approximation is derived using the expansion of the trigonometric functions in the power series. Unlike the ones commonly used in the FEM analysis the functions suggested by the authors have the coefficients dependent on the geometrical and physical properties of the element. From the strain energy formula the stiffness matrix of the element is determined. It is very simple and can be split into components responsible for bending, shear and axial forces influences on the displacements. The proposed element is totally free of the shear and membrane locking effects. It can be referred to the shear-flexible (parameter d) and compressible (parameter e) systems. Neglecting d or e yields the finite elements in all necessary combinations, i.e. curved Euler–Bernoulli beam or curved Timoshenko beam with or without the membrane effect. Applying the elaborated element in the calculations a very good convergence to the analytical results can be obtained even with a very coarse mesh without the commonly adopted corrections as reduced or selective integration or introduction of the stabilization matrices, additional constraints, etc., for the small depth–length ratio. © 1997 John Wiley & Sons, Ltd.

Euler's buckling formula and Wirtinger's inequality

This article contains a discussion of Euler's buckling formula for a compressed elastic column. The most commonly used classroom derivations of this formula follow roughly the original arguments of Euler. For this reason the history of this problem is briefly reviewed. Alternative methods of derivation use energy arguments, including Rayleigh's principle and Kirchhof's energy inequality. Here we show that the quadratic strain energy formula can be used directly, in conjunction with the so called Wirtinger‐Poincare‐Almansi inequality to offer an extremely simple proof of the Euler's buckling load formula: Pcritilal = π2.EI/l2. To the best of our knowledge this derivation is new.

Box beams subject to torsion

Stress analysis is carried out for the case where a cantilever box beam is subjected to distributed torque and concentrated torque at the tip. Direct stress σ x acting in a cover plate is assumed to be in the form of ∑ n σ n(x)y n where n is an odd integer and the fundamental equation governing a box beam is introduced, using variational method. The accuracy can be investigated by comparing the values of strain energy by each obtained solution. Since this calculation is very complicated, strain energy formula is simplified by the new device. From the results of theoretical analysis, it became evident that, for the case where approximation improves, there is no necessity for the empirical formula proposed by Kuhn to be used, and that the range affected by the clamped end is about within the breadth of a box beam from the clamped end.

On Simplification of the Strain-Energy Formula

On Simplification of the Strain-Energy Formula

歪みエネルギー式の簡単化とその応用について

This paper presents the method of simplification of strain energy formula by using the fundamental equations and restrained conditions, both of which are derived by variational calculus of strain energy formula. By this method, total strain energy can easily be calculated and arbitrary constant contained in solution is determined without troublesome work.