Modeling of three-dimensional beam nonlinear vibrations generalizing Hencky’s ideas

In this contribution, a novel nonlinear micropolar beam model suitable for metamaterials design in a dynamics framework is presented and discussed. The beam model is formulated following a completely discrete approach and it is fully defined by its Lagrangian, i.e., by the kinetic energy and by the potential of conservative forces. Differently from Hencky’s seminal work, which considers only flexibility to compute the buckling load for rectilinear and planar Euler–Bernoulli beams, the proposed model is fully three-dimensional and considers both the extensional and shear deformability contributions to the strain energy and translational and rotational kinetic energy terms. After having introduced the model formulation, some simulations obtained with a numerical integration scheme are presented to show the capabilities of the proposed beam model.

Effect of Heat Treatment on the Cyclic Deforming Behavior of As-Extruded ZA81M Magnesium Alloy

In the present work, the effect of heat treatment on the cyclic deformation behavior of as-extruded ZA81M magnesium alloy was investigated. Two heat treatment conditions were applied to the as-extruded ZA81M alloy: a solution treatment (T4, 653 K for 40 h and quenched with 298 K water) and a solution treatment plus artificial aging (T6, 348 K for 32 h (pre-aging at low temperature) and 453 K for 8 h (the second aging) and quenched with 353 K water). The results showed that the fine second phase precipitated after the aging treatment, the tensile yield strength of the T6-treated specimens increased, and the stress amplitude of T6-treated specimens was always higher than that of T4-treated specimens. The T6-treated specimens had a higher total strain energy density and a shorter fatigue life at a strain amplitude of 0.4%, and a lower total strain energy density and a longer fatigue life at a strain amplitude of 0.8%, compared to the T4-treated specimens. All fatigue cracks of the T4 and T6 ZA81M alloy were initiated at the second phase or along the grain boundary and propagated perpendicular to the loading direction.

Assessment of four strain energy decomposition methods for phase field fracture models using quasi-static and dynamic benchmark cases

Strain energy decomposition methods in phase field fracture models separate strain energy that contributes to fracture from that which does not. However, various decomposition methods have been proposed in the literature, and it can be difficult to determine an appropriate method for a given problem. The goal of this work is to facilitate the choice of strain decomposition method by assessing the performance of three existing methods (spectral decomposition of the stress or the strain and deviatoric decomposition of the strain) and one new method (deviatoric decomposition of the stress) with several benchmark problems. In each benchmark problem, we compare the performance of the four methods using both qualitative and quantitative metrics. In the first benchmark, we compare the predicted mechanical behavior of cracked material. We then use four quasi-static benchmark cases: a single edge notched tension test, a single edge notched shear test, a three-point bending test, and a L-shaped panel test. Finally, we use two dynamic benchmark cases: a dynamic tensile fracture test and a dynamic shear fracture test. All four methods perform well in tension, the two spectral methods perform better in compression and with mixed mode (though the stress spectral method performs the best), and all the methods show minor issues in at least one of the shear cases. In general, whether the strain or the stress is decomposed does not have a significant impact on the predicted behavior.

Analysis on the Difference of Energy Evolution in the Process of Energy Storage Failure of Strong Coal Rush Rock Samples under Different Adaptive Modification Regulation Measures

Current energy to “release” after accumulating + first for the mechanism of rock bursts occurred in analysis of the strategy is accepted by many scholars, based on the existing means of prevention and control of percussive ground pressure, from the angle of the prevention and control of design of the mechanism of impact ground pressure energy regulation, namely, “weakened after the first release +” softened water injection measures and “lead after the first release +” drilling pressure relief measures, for the study of mining under the action of a strong shock tendentiousness rock energy regulatory mechanism; based on rock mechanics experiment, the analysis under different modification measures should be variant energy storage mechanisms of induced damage evolution of rock energy. The mechanism of energy evolution in the modification of strong bursting liability roof rock is revealed. The results show that different modification regulation measures can effectively change the physical and mechanical parameters of target rock samples and realize “hard rock softening or soft rock hardening.” Samples under different modification measures are classified as initial consolidation stage, elastic stage, stage of plastic deformation, yield failure stage, and late stage, the energy evolution is roughly the same as the sample complete natural condition, but the yield failure stage and the destruction of late stage have an obvious difference, which provides favorable conditions for impact ground pressure to prevent. With the help of three characteristic energy indexes of total strain, elastic strain energy, and dissipative strain energy of rock samples, the evolution law of energy indexes under different modification control measures is analyzed. The index of elastic energy consumption ratio is introduced as a precursor feature of rock instability and failure, which indicates the rock impact tendency to a certain extent. The energy regulation mechanisms of “first release+then weakening” water injection softening measures and “first release+then guidance” drilling pressure relief measures are explained theoretically, respectively. However, we should focus on the defects of the corresponding control measures and finally try to make a reasonable combination of different modification measures. Finally, the gradient pressure relief scheme should be considered in order to avoid large stress drop caused by large-scale pressure relief in the region and aggravate the instability of rock mass. The instability of rock mass is further aggravated.

A generalized strain approach to anisotropic elasticity

This work proposes a generalized Lagrangian strain function $$f_\alpha$$ f α (that depends on modified stretches) and a volumetric strain function $$g_\alpha$$ g α (that depends on the determinant of the deformation tensor) to characterize isotropic/anisotropic strain energy functions. With the aid of a spectral approach, the single-variable strain functions enable the development of strain energy functions that are consistent with their infinitesimal counterparts, including the development of a strain energy function for the general anisotropic material that contains the general 4th order classical stiffness tensor. The generality of the single-variable strain functions sets a platform for future development of adequate specific forms of the isotropic/anisotropic strain energy function; future modellers only require to construct specific forms of the functions $$f_\alpha$$ f α and $$g_\alpha$$ g α to model their strain energy functions. The spectral invariants used in the constitutive equation have a clear physical interpretation, which is attractive, in aiding experiment design and the construction of specific forms of the strain energy. Some previous strain energy functions that appeared in the literature can be considered as special cases of the proposed generalized strain energy function. The resulting constitutive equations can be easily converted, to allow the mechanical influence of compressed fibres to be excluded or partial excluded and to model fibre dispersion in collagenous soft tissues. Implementation of the constitutive equations in Finite Element software is discussed. The suggested crude specific strain function forms are able to fit the theory well with experimental data and managed to predict several sets of experimental data.

Strain and interface energy of ellipsoidal inclusions subjected to volumetric eigenstrains: shape factors

Thermodynamic modeling of the development of non-spherical inclusions as precipitates in alloys is an important topic in computational materials science. The precipitates may have markedly different properties compared to the matrix. Both the elastic contrast and the misfit eigenstrain may yield a remarkable generation of elastic strain energy which immediately influences the kinetics of the developing precipitates. The relevant thermodynamic framework has been mostly based on spherical precipitates. However, the shapes of actual particles are often not spherical. The energetics of such precipitates can be met by adapting the spherical energy terms with shape factors. The well-established Eshelby framework is used to evaluate the elastic strain energy of inclusions with ellipsoidal shapes (described by the axes a, b, and c) that are subjected to a volumetric transformation strain. The outcome of the study is two shape factors, one for the elastic strain energy and the other for the interface energy. Both quantities are provided in the form of easy-to-use diagrams. Furthermore, threshold elastic contrasts yielding strain energy shape factors with the value 1.0 for any ellipsoidal shape are studied.

Biomechanical feasibility of semi-rigid stabilization and semi-rigid lumbar interbody fusion: a finite element study

Background Semi-rigid lumbar fusion offers a compromise between pedicle screw-based rigid fixation and non-instrumented lumbar fusion. However, the use of semi-rigid interspinous stabilization (SIS) with interspinous spacer and ligamentoplasty and semi-rigid posterior instrumentation (SPI) to assist interbody cage as fusion constructs remained controversial. The purpose of this study is to investigate the biomechanical properties of semi-rigidly stabilized lumbar fusion using SIS or SPI and their effect on adjacent levels using finite element (FE) method. Method Eight FE models were constructed to simulate the lumbosacral spine. In the non-fusion constructs, semi-rigid stabilization with (i) semi-rigid interspinous spacer and artificial ligaments (PD-SIS), and (ii) PI with semi-rigid rods were simulated (PD + SPI). For fusion constructs, the spinal models were implanted with (iii) PEEK cage only (Cage), (iv) PEEK cage and SIS (Cage+SIS), (v) PEEK cage and SPI (Cage+SPI), (vi) PEEK cage and rigid PI (Cage+PI). Result The comparison of flexion-extension range of motion (ROM) in the operated level showed the difference between Cage+SIS, Cage+SPI, and Cage+PI was less than 0.05 degree. In axial rotation, ROM of Cage+SIS were greater than Cage+PI by 0.81 degree. In the infrajacent level, while Cage+PI increased the ROM by 24.1, 27,7, 25.9, and 10.3% and Cage+SPI increased the ROM by 26.1, 30.0, 27.1, and 10.8% in flexion, extension, lateral bending and axial rotation respectively, Cage+SIS only increased the ROM by 3.6, 2.8, and 11.2% in flexion, extension, and lateral bending and reduced the ROM by 1.5% in axial rotation. The comparison of the von Mises stress showed that SIS reduced the adjacent IVD stress by 9.0%. The simulation of the strain energy showed a difference between constructs less than 7.9%, but all constructs increased the strain energy in the infradjacent level. Conclusion FE simulation showed semi-rigid fusion constructs including Cage+SIS and Cage+SPI can provide sufficient stabilization and flexion-extension ROM reduction at the fusion level. In addition, SIS-assisted fusion resulted in less hypermobility and less von Mises stress in the adjacent levels. However, SIS-assisted fusion had a disadvantage of less ROM reduction in lateral bending and axial rotation. Further clinical studies are warranted to investigate the clinical efficacy and safety of semi-rigid fusions.