Conventional Strain Energy Research Articles

Theoretical verification of the rationality of strain energy storage index as rockburst criterion based on linear energy storage law

The rationality of using strain energy storage index ( W et ) for evaluating rockburst proneness was theoretically verified based on linear energy storage (LES) law in this study. The LES law is defined as the linear relationship between the elastic strain energy stored inside the solid material and the input strain energy during loading. It is used to determine the elastic strain energy and dissipated strain energy of rock specimens at various loading/unloading stress levels. The results showed that the W et value obtained from experiments was close to the corresponding theoretical one from the LES law. Furthermore, with an increase in the loading/unloading stress level, the ratio of elastic strain energy to dissipated strain energy converged to the peak-strength strain energy storage index ( W et p ). This index is stable and can better reflect the relative magnitudes of the stored energy and the dissipated energy of rocks at the whole pre-peak stage than the strain energy storage index. The peak-strength strain energy storage index can replace the conventional strain energy storage index as a new index for evaluating rockburst proneness.

Experimental and theoretical assessment of crack kinking in marble stones in the presence of a V-notch under mode I loading

Marble has been used extensively in the historical monuments as a building material. Notches and sharp corners in these buildings are potential locations for crack formation, crack extension and in some cases their eventual failure. In this study, some fracture tests are accomplished on a number of V-notched marble specimens under mode I loading condition. The specimens are in the configurations of double cantilever beam (DCB) and compact tension (CT) each embedding two various notch opening angles. It is substantiated that although both the loading conditions and geometry were symmetric, the crack initiation angle is not necessarily along the notch bisector line. It is also shown that the classical fracture criteria are not able to predict crack kinking in the tested marble specimens. Thus, the conventional strain energy density and maximum tangential stress criteria used for V-notches, V-SED and V-MTS respectively, are generalized by introducing the first non-singular term of notch stress field into the calculations. Finally, it is demonstrated that the generalized criteria, V-GSED and V-GMTS, can predict the fracture strength and the crack initiation angle of the tested marble specimens well.

Crack bifurcation in sharp V-notches

In this paper, the crack initiation angle is studied experimentally and theoretically in sharp V-notched specimens subjected to mode I loading. Some experiments are performed on the V-notched compact tension (CT) and double cantilever beam (DCB) specimens made of polymethylmetacrylate (PMMA) with two different notch opening angles. It is demonstrated that although symmetry exists in both geometry and loading of the specimens, the initiated crack would not be necessarily along the notch bisector line. Hence, a proper theory is necessary for predicting the crack initiation angle. An attempt is then made to generalize the conventional strain energy density (SED) criterion by considering the first non-singular term in the fracture model. It is shown that the first non-singular stress term has a significant role in predicting both the crack initiation angle and the notch fracture resistance. Moreover, the results of theoretical predictions when the first non-singular term is taken into account have reasonable consistency with the experimental data.

The s‐homodesmotic method for the computation of conventional strain energies of bicyclic systems and individual rings within these systems

AbstractThe s‐homodesmotic method for computing conventional strain energies (CSE) has been extended for the first time to bicyclic systems and to individual rings within these systems. Unique isodesmic, homodesmotic, and hyperhomodesmotic reactions originate from the s‐homodesmotic method. These are used to investigate 12 bicyclic systems comprising cyclopropane and cyclobutane and how the CSE of each system compares to the sum of the individual rings within each. Equilibrium geometries, harmonic vibrational frequencies, and the corresponding electronic energies and zero point vibrational energy corrections are computed for all relevant molecules using second‐order perturbation theory and density functional theory (B3LYP) with the correlation consistent basis sets cc‐pVDZ and cc‐pVTZ. Single‐point CCSD(T) energies are computed at the MP2/cc‐pVTZ optimized geometries to ascertain the importance of higher order correlation effects. Results indicate that CSEs are additive when the two rings are separated by one or two bonds and somewhat additive in other cases.

Conventional strain energies of 1,2-dihydroazete, 2,3-dihydroazete, 1,2-dihydrophosphete, and 2,3-dihydrophosphete

The conventional strain energies of 1,2-dihydroazete, 2,3-dihydroazete, 1,2-dihydrophosphete, and 2,3-dihydrophosphete are determined within the isodesmic, homodesmotic, and hyperhomodesmotic models. Optimum equilibrium geometries, harmonic vibrational frequencies, and corresponding electronic energies and zero-point vibrational energies are computed for all pertinent molecular systems using SCF theory, second-order perturbation theory, and density functional theory and employing the correlation consistent basis sets cc-pVDZ, cc-pVTZ, and cc-pVQZ. Single-point fourth-order perturbation theory, CCSD, and CCSD(T) calculations employing the cc-pVTZ and the cc-pVQZ basis sets are computed using the MP2/cc-pVTZ and MP2/cc-pVQZ optimized geometries, respectfully, to ascertain the contribution of higher order correlation. Three DFT functionals, B3LYP, wB97XD, and M06-2X, are employed to determine whether they can yield results similar to those obtained at the CCSD(T) level.

Conventional strain energies of azetidine and phosphetane: Can density functional theory yield reliable results?

The conventional strain energies for azetidine and phosphetane are determined within the isodesmic, homodesmotic, and hyperhomodesmotic models. Optimum equilibrium geometries, harmonic vibrational frequencies, and corresponding electronic energies and zero-point vibrational energies are computed for all pertinent molecular systems using self-consistent field theory, second-order perturbation theory, and density functional theory and using the correlation consistent basis sets cc-pVDZ, cc-pVTZ, and cc-pVQZ. Single point fourth-order perturbation theory, CCSD, and CCSD(T) calculations using the cc-pVTZ and the cc-pVQZ basis sets are computed using the MP2/cc-pVTZ and MP2/cc-pVQZ optimized geometries, respectively, to ascertain the contribution of higher order correlation effects and to determine if the quadruple-zeta valence basis set is needed when higher order correlation is included. In the density functional theory study, eight different functionals are used including B3LYP, wB97XD, and M06-2X to determine if any functional can yield results similar to those obtained at the CCSD(T) level.

Energetic Differences between the Five- and Six-Membered Ring Hydrocarbons: Strain Energies in the Parent and Radical Molecules

The C-H bond dissociation enthalpies (BDEs) for the five- and six-membered ring alkanes, alkenes, and dienes were investigated and discussed in terms of conventional strain energies (SEs). New determinations are reported for cyclopentane and cyclohexane by time-resolved photoacoustic calorimetry and quantum chemistry methods. The C-H BDEs for the alkenes yielding the alkyl radicals cyclopenten-4-yl and cyclohexen-4-yl and the alpha-C-H BDE in cyclopentene were also calculated. The s-homodesmotic model was used to determine SEs for both the parent molecules and the radicals. When the appropriate s-homodesmotic model is chosen, the obtained SEs are in good agreement with the ones derived from group additivity schemes. The different BDEs in the title molecules are explained by the calculated SEs in the parent molecules and their radicals: (1) BDEs leading to alkyl radicals are ca. 10 kJ mol (-1) lower in cyclopentane and cyclopentene than in cyclohexane and cyclohexene, due to a smaller eclipsing strain in the five-membered radicals relative to the parent molecules (six-membered hydrocarbons and their radicals are essentially strain free). (2) C-H BDEs in cyclopentene and cyclohexene leading to the allyl radicals are similar because cyclopenten-3-yl has almost as much strain as its parent molecule, due to a synperiplanar configuration. (3) The C-H BDE in 1,3-cyclopentadiene is 27 kJ mol (-1) higher than in 1,4-cyclohexadiene due to the stabilizing effect of the conjugated double bond in 1,3-cyclopentadiene and not to a destabilization of the cyclopentadienyl radical. The chemical insight afforded by group additivity methods in choosing the correct model for SE estimation is highlighted.

Conventional Strain Energy in Dimethyl-Substituted Cyclobutane and the gem-Dimethyl Effect

The gem-dimethyl effect is the acceleration of cyclization by substituents in the chain and is often used in organic synthesis as a ring-closing effect. Calculations on cyclobutane, methylcyclobutane, and 1,1-dimethylcyclobutane are performed. 1,1-Dimethylcyclobutane is a four-membered carbon ring with gem-dimethyl substituents. Optimum equilibrium geometries, harmonic vibrational frequencies, and corresponding electronic energies are computed for all pertinent molecular systems using SCF theory, density functional theory (DFT), and second-order perturbation theory (MP2) with two triple-zeta quality basis sets, 6-311G(d,p) and 6-311G+(2df,2pd). Additional single-point calculations are performed using the optimized MP2/6-311G+(2df,2pd) geometries and coupled-cluster theory including single and double excitations and noniterative, linear triple excitations (CCSD(T)). Calculations indicate that 1,1-dimethylcyclobutane is more than 8 kcal mol-1 less strained than cyclobutane, that is, there is at least some thermodynamic component to the gem-dimethyl effect.

Conventional strain energy in the oxadiazetidines

AbstractThe conventional strain energies for all four isomers of oxadiazetidine are determined within the isodesmic, homodesmotic, and hyperhomodesmotic models. Optimum equilibrium geometries, harmonic vibrational frequencies, and corresponding electronic energies are computed for all pertinent molecular systems using self‐consistent field theory, second‐order perturbation theory, and density functional theory (DFT) and employing two basis sets of triple‐zeta valence quality: 6‐311G(d,p) and 6‐311+G(2df,2pd). The DFT functional employed is Becke's three‐parameter hybrid functional using the Lee, Yang, and Paar correlation functional. Single‐point fourth‐order perturbation theory and coupled‐cluster theory restricted to single and double excitations [CCSD(T)] calculations employing the larger basis set also are computed, at both the second‐order Møller–Plesset (MP2)/6‐311G(d,p) and the MP2/6‐311+G(2df,2pd) optimized geometries, to determine the effect of higher‐order correlation effects on strain energy computation and to gauge the effect of geometry on these effects. Using the same models and methods, the conventional strain energies for both isomers of oxazetidine also are computed, to determine the effect on the strain energy of replacing a nitrogen with a carbon in the ring. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Estimating stresses in a partially inflated high altitude balloon using a relaxed energy

During ascent and at other times during flight, the lifting gas of a high altitude balloon is compressed and only able to partially inflate the balloon. In this condition the surface of the balloon will sag to form folds and wrinkles which are difficult to analyze. Previous numerical work to analyze these types of balloons was based on minimizing extrema of potential energy of balloon shapes that included an explicit representation of excess material as folds. These models used the conventional strain energy for linear isotropic membranes and permitted compressive states to enter the solutions. This paper explores the application of the energy relaxation method to the earlier models to produce solutions free of compressive states. Numerical results computed using the relaxed energy are presented and compared with results computed using the standard strain energy for a membrane.

Computation of the conventional strain energy in oxaziridine

The conventional strain energy for oxaziridine is determined within the isodesmic, homodesmotic, and hyperhomodesmotic models. Using these models, the conventional strain energy for cyclopropane, cyclobutane, cyclopentane, and cyclohexane are also computed for comparison purposes. Optimum equilibrium geometries, harmonic vibrational frequencies, and corresponding electronic energies are computed for all pertinent molecular systems using SCF theory, second-order perturbation theory, and density functional theory and employing two basis sets of triple-zeta valence quality: 6-311G(d,p) and 6-311+G(2df,2pd). Single-point fourth-order perturbation theory and CCSD(T) calculations employing the larger basis set are also computed at the MP2/6-311+G(2df,2pd) optimized geometries to determine the effect of higher-order correlation effects on ring strain computation. Finally, the ring strain of oxaziridine is compared to other small ring systems.

Stacking Fault Tetrahedra Formation During Growth of Si1-xGex Strained Layers on 〈111〉 Oriented Si Substrates: Tem Observations and Defect Modeling

AbstractWe report the observation of stacking fault tetrahedra (SFT) in strained Si1-xGex layers grown via rapid thermal CVD on (111) Si substrates. It is shown that these defects provide a mechanism for strain relief in films strained in compression due the presence of bounding edge-type stair rod partials whose Burgers vectors lie parallel to the strained layer interface. Cross section and plan view TEM were used to characterize this defect structure in epilayers (30 to 650nm thick) of Si1-xGex (0 < xGe < 0.27) grown on 〈111〉 oriented Si wafers. Stacking fault tetrahedra were observed only in alloys in the compositional range xGe ≥ 0.13 and only when growth proceeded on the 〈111〉 surface. A critical strain energy model that identifies conditions for the stable growth of stacking fault tetrahedra in a strained layer is presented. The model was based on conventional strain energy considerations where the energy of the stacking fault area plus the bounding dislocation network (including dislocation interactions but neglecting the free surface) was balanced against the strain energy released by the introduction of the defect. In addition, a formation mechanism consistent with these observations is described that involves the dissociation of Frank partial dislocation loops bounding stacking faults lying in the growth plane.

Enthalpies of combustion of γ-butyrolactone, γ-valerolactone, and δ-valerolactone

The standard ( p o = 0.1 MPa) molar enthalpies of combustion in oxygen at 298.15 K were measured by static-bomb calorimetry and the standard molar enthalpies of vaporization at 298.15 K were measured by microcalorimetry for three lactones. −Δ c H m o/(kJ·mol −1 Δ l g H m o/(kJ·mol −1 ) γ-butyrolactone 2010.6±0.5 54.4±0.4 γ-valerolactone 2649.6±0.8 54.8±0.4 δ-valerolactone 2673.3±0.5 58.0±0.4 The derived standard molar enthalpies of formation of the gaseous compounds were compared with values expected for strain-free structures. The conventional strain-energy in the six-membered ring of δ-valerolactone is greater than that in the five-membered-ring lactones, and this is in reverse order of strain-energies generally observed in alicyclic compounds.

Enthalpies of combustion of succinic anhydride, glutaric anhydride, and glutarimide

The standard ( p o = 0.1 MPa) molar enthalpies of combustion in oxygen at 298.15 K were measured by static-bomb calorimetry for succinic anhydride, glutaric anhydride, and glutarimide. The standard molar enthalpies of sublimation of these compounds were derived from vapour pressures as functions of temperature determined using Knudsen cells in a microcalorimeter. −Δ c H m o(cr)/(kJ·mol −1) Δ cr g H m o/(kJ·mol −1) Succinic anhydride 1537.1±0.4 80.7±1.6 Glutaric anhydride 2206.5±0.4 86.1±1.6 Succinimide 83.6±1.5 Glutarimide 2480.3±0.5 94.1±1.6 From the standard molar enthalpies of formation in the gaseous state it was deduced that the conventional strain-energies in the six-membered-ring compounds, glutaric anhydride and glutarimide, are greater than in the corresponding five-membered-ring compounds, succinic anhydride and succinimide.