The modal **strain** **energy** variation of a cantilever sandwich beam (consisting of top and bottom face plates with honeycomb core in between) with multiple delaminations or debonds embedded between the face-layer laminates and the honeycomb core, from that of a sandwich beam without any delamination is studied herein using free vibration analysis. The influence of size and location of delaminations on the natural frequency is investigated for sandwich beams with multiple delaminations. It is observed that modal **strain** **energy** plot can give the location and extent of damage or debond more accurately than a mode shape plot. It is also seen that the delaminations reduce the natural frequency and modal **strain** **energy** and change the modeshape. The effect of second debond is not significant if it is located near to the free end and far away from the first debond.

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best accuracy particularly when the results for the proposed method, Bridgeman and Bridgeman-Leory are very close. If we take the optimization method as a benchmark, it can be observed from Table 6 that best agreement is obtained for **energy** method and Bridgeman approach. We compared the differences of the quantities with respect to their corresponding numerical values given in the 3 rd row of Table 6. The differences are given in parenthesis in the same table. The methods can now be compared more clearly with each other. The first interesting point is that the differences(except one case for Siebel) are well below 10%. As the results indicate and as far as A and B are concerned, Siebel gives the worst results and the proposed model and Birdgeman provide the best results, although differentiation between the proposed model and Bridgeman is difficult. We think that the dominant parameters in defining the corrected stress-**strain** curve are A and B and fracture **strain** is less important. The reason is that fracture **strain** is obtained from the fractured specimen. When the two parts of the broken specimens are put together to measure the neck section, the two parts usually don’t match exactly and therefore, the fractured neck diameter is always accompanied by some errors. As a result, we can conclude that the performances of the proposed model and Bridgeman model are close. It is a fact that all researchers have made some simplifications in their correction models. However, we may argue that optimization aided numerical simulations provide the most accurate prediction, because it provides the best agreement with the experiment for the neck profile. Having accepted this as the benchmark for assessing the accuracy of the models, we can see that the good agreement is obtained for the **energy** method discussed in this work.

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In this study, in order to clarify the cyclic nature of the superelastic behavior of the coil springs of the Ti-Ni shape memory alloy, the loading-unloading cycling tests were carried out under isothermal temperatures. The eﬀects of the magnitude of the repeated shear **strain**, the Ni content, and the temperature of heat treatment for the shape memory on the critical shear stress for inducing martensites and the dissipated **strain** **energy** are discussed in the superelastic regime.

This need is still largely unfulfilled in both transversely- and torsionally loaded elements and structures. Mass or volumetric efficiency measures are used in this paper to evaluate the solutions now in use to withstand transverse and torsional loads. Maximum **strain** **energy** storable in a given amount of material is taken as the reference measure. With this reference measure, common transversely loaded and torsionally loaded designs and components are evaluated to assess relative efficiency in the use of materials and **energy**. It is shown that most common designs offer significant opportunities for improvements.

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This study provides damage indices for reinforced concrete shear walls on the basis of the cumulative plastic **strain** **energy** (consumed **energy**). First, nonlinear finite element analyses of reinforced concrete shear wall tests were performed, and load–displacement relationships were compared to validate the analysis method. Second, the distributions of consumed **energy** of rebar and concrete were calculated on the basis of the numerical results. The relationships between the consumed **energy** and damages are discussed. Finally, the damage assessment index is presented as the amount of **energy** consumed by concrete elements per unit volume.

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Many constitutive choices in the literature of the form W := W (F, τ ) do not satisfy the ISRI restrictions (2.12) and (4.18) presented in this article. In Section 2.1, we gave an example of how these constitutive choices may lead to unphysical behaviour even for simple deformations such as uniaxial extension. This is also true of more complex deformations. Taking an example from biomechanics, where residual stresses play a crucial role, suppose we wish to model the mechanics of an arterial wall that supports an internal pressure. Let us choose two different reference configurations: first, the unloaded configuration where the fluid in the artery has been removed, and second, the opening angle configuration (60, 30) where the fluid has been removed and the artery has been cut along its axis. Both these configurations are subject to no external loads, but there will be less (and differently distributed) internal stress in the opening angle configuration. If we use a **strain** **energy** function W (F, τ ) that does not satisfy ISRI, then each of the two reference configurations will lead to a different stress distribution in the intact, inflated configuration of the arterial wall. We therefore cannot believe the preditions from either reference configuration since a physically correct model should not give different results due to an arbitrary choice of reference configuration.

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It is objective to experimentally verify the inventive approach to extract the dissipation **energy** only by building damping proposed in Part 1, and to compare it with the actual input **energy** to the building evaluated from the earthquake observation records. The building damping dissipation **energy** with **strain** **energy** proportional damping is analytically obtained by the previous simulation model by Hiraki et al. (2007) and the extraction method mentioned above. The earthquake observation records of the Tsuruga-wan Nanpooki earthquake and the Shizuoka-ken Chubu earthquake are selected in this analysis for the following reasons. The former earthquake occurred on October 11th, 1997. It had a JMA magnitude of Mj5.1 and an epicentral distance from the observation site of 24.5km. It had a relatively large input **energy** to the building evaluated from the observed acceleration record on top of a mat-slab foundation. The latter earthquake, occurred on October 5th, 1996. It had a JMA magnitude of Mj4.6 and an epicentral distance from the observation site of 42.3km. It had a relatively small input **energy** to the building. The velocity-equivalent **energy** spectra of the observed acceleration records on top of a mat-slab foundation are shown in Fig.3. The analytical simulation model is the same lattice model as that by Hiraki et al. (2007). The input motion to the lattice model is evaluated from the free-field observation record. The simulation analysis procedure is shown in Fig.4.

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among those electronic centers in addition to bond charges act to increase the value of external bond angles in order to reduce the repulsion potential. In the calculating the **energy** functions of the studied molecules, the useful information about stability can be derived from the value of **strain** **energy** calculated according to the isodesmic reaction method, where the stability of the compound increased with raising nitrogen content in the compounds from tetrahydrane to triazetetrahedrane, this can be attributed to contribution of excitation in reducing the high electron density within the tetrahedral frame [15], in addition, the presence of the nitrogen atom with its loan pair causes a highly degree of polarization represented by increasing the value of the dipole moment as shown in Table (2). By moving from diazetetrahedrane to triazatetrahedrane we note that the change in the value of the dipole moment was not significant, but rather a slight decrease in value.

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Confined concrete ductility and **strain** **energy** are very well improved, proportionally with the diameter of the confining bars, thing that can be noticed due to the regular/ similar shape of the stress - **strain** curves right after the ascending linear - elastic branch. Many of these comments are true for the other case study. Clearly, confinement of the concrete is improved if transverse reinforcement layers are placed relatively close together along the longitudinal axis - there is a positive change of confinement effectiveness coefficient (Eq. 11), and lateral confining stresses (Eq. 13, Eq. 14). However, there will be some critical spacing above which the section midway between the transverse sets will be ineffectively confined, and the assumption of uniform lateral stresses exerted on the concrete core would be inappropriate [1] (meaning that the stress - **strain** model accepted would be not valid anymore, but more important the confinement mechanism would be quite inexistent). On the other hand, aiming a very close space, especially when overlapping hoops are used, would be inappropriate from a practical point of view. In absolute terms, by comparing both of the cases results, it can be concluded that the variation of the confining bars diameter, influences the most the behavior of confined concrete, in terms of strength, ductility, and **strain** **energy**. This conclusion, is somehow specific and should be discussed for other reference conditions in order to take a more representative form, why not adding the economical - technical component also.

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volume fraction of slip-deformed martensite, where the value is normalized against the recovery **strain** **energy** at the first cycle and shown as a function of copper content. The de- graded recovery **strain** **energy** shows the strong dependence on the volume fraction of slip-deformed martensite and in- creases almost linearly with increasing the volume fraction of slip-deformed martensite. The result shows that the volume fraction of slip-deformed martensite is capable of represent-

By increasing the radius size of the -SiC nanowire from 0.432 to 0.864 nm, the curve of **strain** **energy** vs. **strain** is obtained in Fig. 4 at 100, 300 and 500 K. Compared with a previous simulation on the nanowire with a 0.432 nm radius, it shows more complicated and non-homogeneous behavior. Repeated yielding and elongation must have occurred frequently. The Young’s modulus obtained from eq. (4) is shown in Table 2. It decreased with temperatures due to increased softness of the nanowire material but increased slightly with the system size. Remembering that the Young’s modulus in [001] obtained by Lambrecht et al. 14) is 362 GPa

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The outline of a fracture-mechanics model for debonding of external fibre reinforced polymer plates on reinforced concrete beams has been presented. The paper has shown that a modified version of Branson’s effective stiffness model can be used to determine the elastic **strain** **energy** stored in the beam. The **energy** released from the system with the extension of an existing interface-flaw can be calculated and compared with the **energy** required to create new fracture surfaces. This will decide whether the flaw will propagate. The **energy** required to create the associated new surfaces depends on the Mode I concrete fracture **energy**, which can be determined from standard concrete fracture tests.

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E. Ability to detect damage on structures without known response. Damage index method and frequency decomposition MSE method allow detection of damage in structures with unknown damage state. For the MSE method, it can be show that the Fourier series of **strain** **energy** mode shapes of an undamaged pinned-pinned beam consists only of two members for any mode, while the Fourier series of a damaged beam contains significantly higher number of harmonics. Damage location in the beam can be established by removing the two harmonics related to the frequency spectrum of an undamaged beam and performing the inverse of the Fourier series. The same technique can be applied to other symmetrical boundary conditions and beams of more complex shape.

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The deformation, equivalent elastic **strain**, equivalent stress, **strain** **energy** and shear stress are very important for poppet valve. To meet these requirements to perform structural analysis on stainless steel and ceramic composite materials of poppet valves. The finite element analysis was carried out by using Ansys software. This analysis was performed based on the following assumptions.

This study may be considered as an extension of Mitchell’s pioneering study that employs the **Strain** **Energy** Density (SED) function suggested by Silling et al. (2007) for the “ linear PD solid ”. The main contribution concerns the use Prony series for modeling the behavior of viscous material for three- and two-dimensional PD thermoviscoelastic analyses. Also, it employs the PD form of the SED function suggested by Madenci and Oterkus (2014) for a linear material response identical to that of classical continuum mechanics. It is expressed in terms of three PD material parameters specifically for small deformation and small rotations. These PD parameters for both three- and two-dimensional analyses are determined by calibration against the classical SED by considering the two simple loading conditions of isotropic expansion and simple shear.

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According to the author of the present paper, in case of determining **strain** **energy** during rolling of a man’s body as compared with a car wheel, the formula 5 will not apply in full. It is obvious, that **strain** **energy** per a volume unit will be smaller, if the body surface contacting the ground is big. Therefore, it is better to roll over with the biggest area of muscles involved. It is obvious that heavier cars have wheels with both bigger diameter and wider, this ensures a greater con- tact of the tyre surface with the ground. It also hap- pens, that on a single axis there are more wheels than two in order to enlarge the contact surface. Book ref- erences state [43] that, within certain speed limits, when this speed is increasing without sudden accel- erations, deformation of the car wheel tyre decreases. It seems obvious, even when a person is watching first an immobile car and then when it is starting to move. Tyre deformation is reduced when contact- ing the ground along with a slow speed increase. This means, that if we shorten the time of a contact of a tyre with the ground, **strain** **energy** per surface area decreases. The formula 5 will not correctly represent this phenomenon for time variable. This is because it was formulated for a body hitting a ground in a pro- gressive movement, for example, similarly to a car hit- ting an obstacle. It is obvious for this case, that the longer the time of the car speed change, the smaller the force of inertia affecting the passengers. The above considerations show, that when a human body is rolling, some increase in the rolling velocity without big accelerations can reduce decrease of the **strain** the human body is affected by at the moment it contacts the ground. This is a result of the fact, that reducing the time of the body contact with the ground can decrease the **energy** of the body deformation dur- ing a fall, provided the body is rolling in the same way as a car wheel. An experimental analysis of such cases can serve in order to justify the above theory. An interesting form of performing falls forward is pre- sented by the National Geographic by Ryan Doyle and Daniel Illabace exercising on a show jumping course [44]. The film shows a dummy falling from a certain height, and then a fall of a person practic- ing on the course. It was found, that when during a fall a subject assumed a circular technique of the body motion, that is the considered above technique

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In a planar lattice-mismatch system, as the epitaxial film thickness increases, the **strain** **energy** increases and dislocations will appear. The critical thickness can be de- fined as the film thickness at which the film is no longer coherent with the substrate due to **strain** relaxation via the formation of dislocations [22]. In contrast to planar films, for the NW core-shell heterostructure, due to the comparable volumes of the core and shell regions, both the core and shell should be considered. For a certain core radius, there exists a critical shell thickness and vice versa. The critical dimensions of a NW core-shell system are combinations of core and shell dimensions that will lead to coherently strained structures. The dependence of the critical shell thickness as a function of the core radius for different lattice mismatch is shown in Fig. 3. Under a certain mismatch, there is a critical core radius, below which the NW heterostructure will be coherent regardless of shell thickness. For NW core-shell hetero- structure with a core radius larger than the critical core radius, there is a critical shell thickness, below which no dislocations will occur. For a GaAs/In 0.2 Ga 0.8 As NW

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We propose that the major interaction comes from elastic strain, with smaller electrostatic terms: the configuration of the planes is one which minimises the elastic strain energy..[r]

The larger the size of the nanoparticle, the higher the magnetic anisotropic **energy**, which therefore increases with increasing particle size, until reaching the maximum magnetic anisotropic **energy**: D = 6.527 meV in the 6.0(3) nm Au nanoparticles. The results are in good agreement with the molecular field theories, which predict linear or exponential variations for large and small anisotropic energies, depending on whether a classical or quantized system is used for the magnetic moment [29]. In general, magnetic anisotropy means the dependence of the internal **energy** of a system on the direction of the spontaneous magnetization. Most kinds of magnetic anisotropy are related to the deviations in the lattice constant of the **strain**, known as magnetocrystalline anisotropy [30]. Figure 5 shows the **strain** as a function of mean diameter \ d [ . Shown in Fig. 5b (left panel), the relative **strain** can be estimated from the change in the a-axis lattice constant of Au nanoparticles

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reference, the static **strain** could be converted to dynamic **strain** and the coefficient is about 0.4. The increase of total **strain** in the interrupted dynamic tests is probably due to the interruption of the heat accumulation within a specimen during dynamic loading. Dynamic stored **energy** is defined as the **energy** stored in the material by dynamic loading and could be calculated by integrating the stress- **strain** curve.The contribution of dynamic stored **energy** has identical trend with the dynamic **strain** in a failure process, thus it is not appropriate to use dynamic stored **energy** as a criterion for ASB formation for UFG Ti. The overall trend is similar for UFG AZ31 with UFG Ti, but the contribution of static loading to final failure is larger and comparable with that of dynamic loading. In that case, the **strain**-based criterion appears better than the criterion base on dyamic stored **energy** for UFG AZ31, but still the **strain**-based criterion could not describe the fractrue point accurately.

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