Equivalent Strain Energy Research Articles

The effect of GFRP on reinforcing the buried and internally pressurized steel pipes against terrorist attacks

In recent years, the occurrence of numerous intentional and accidental explosions in buried pipelines, especially in oil and gas pipelines, has been turned to a proper subject of research. The present research, for the first time, studies the utilization of GFRP for reducing the deformation of buried and internally pressurized steel pipes against explosions through the non-linear three-dimensional finite element method. The Combined Eulerian-Lagrangian (CEL) algorithm is used for modeling simulation. It means that the pipe and GFRP are modeled using the Lagrangian algorithm and air, soil and explosives are modeled using the Eulerian algorithm. The studied pipe is API-X65 with an outside diameter of 914.4 mm and the burial depth of 2.5 m. The explosive material is TNT with 10 kg of weight and in the 1 m height below the ground. The results of the model were compared to the results of field tests and the experimental equations presented by previous researchers, and good compatibility was observed. Utilization of GFRP decreases maximum equivalent strain, transverse deformation, longitudinal strain and explosion energy imposed on the pressurized pipeline for about 35.3, 29.6, 13 and 63%; respectively. The internally pressurized pipeline of API-X65 with the wall thickness of 15.88 mm and with the GFRP reinforcement with 7 mm of thickness has acceptable resistance against explosion, in a way that transverse deformation of it is in the acceptable limit and its longitudinal strain is lower than the yield strain. Using the results of this research, we can protect the under-construction or operation buried pipelines against the sub-surface explosions.

Analysis of the run-out processes of the Xinlu Village landslide using the generalized interpolation material point method

The prediction of the landslide kinetic features is of great importance in minimizing the potential hazardous impacts and in applying the appropriate stabilization techniques. The present study used the generalized interpolation material point (GIMP) method to analyze the run-out processes of the Xinlu Village landslide that has taken place in Xinlu Village, Chongqing, China, in 2016. The evolutions of equivalent plastic strain, displacement, landslide velocity, and kinematic energy were investigated during the landslide motion. The simulation results indicated that the initial stage of the landslide started with slippage of the mid-front soil part with a maximum velocity of 1.02 m/s (at t = 9 s). The rear rock came to failure at t = 69 s as the tensile crack extended from the landslide surface to the deep weak interlayer. Thereafter, the rear rock further accelerated and pushed the mid-front sliding soil, which formed an overall movement. The kinetic energy of the studied landslide concentrated in the acceleration phases of the soil and rock masses. The predicted landslide geometry and run-out distance had slight differences from the actual ones. Based on the landslide run-out analysis, the studied landslide can be classified as a landslide that simultaneously comprises retrogressive and advancing features. A potential secondary failure of this landslide could happen under specific extreme circumstances.

Reduced cortical bone thickness increases stress and strain in the female femoral diaphysis analyzed by a CT-based finite element method: Implications for the anatomical background of fatigue fracture of the femur

The incidence of hip fractures is increasing in Japan and is high among women older than 70 years. While osteoporosis has been identified as one of the causative factors of fracture, atypical femoral fracture has emerged as a potential complication of bisphosphonate therapy. Atypical femoral fracture is prevalent among Asian women and has been attributed to morphological parameters. Age-related decreases in the morphological parameters of the femoral diaphysis, such as cortical bone thickness, cortical cross-sectional area, and the cortical index, were reported in Japanese women prior to bisphosphonate drugs being approved for treatment. Thus, in the present study, the relationships between biomechanical and morphological parameters were analyzed using a CT-based finite element method.Finite element models were constructed from 44 femurs of Japanese women aged 31–87 years using CT data. Loading conditions were set as the single-leg configuration and biomechanical parameters, maximum and minimum principal stresses, Drucker-Prager equivalent stress, maximum and minimum strains, and strain energy density were calculated in 7 zones from the subtrochanteric region to distal diaphysis. Pearson's correlation coefficient test was performed to investigate relationships with morphological parameters.While absolute stresses gradually decreased from the subtrochanteric region to distal diaphysis, absolute strains markedly declined in the proximal diaphysis and were maintained at the same levels as those in the distal regions. All types of stresses and minimum principal strain in the femoral diaphysis scored higher absolute values in the high-risk group (≥70 years, n = 28) than in the low-risk group (<70 years, n = 16) (p < 0.05). The distribution patterns of equivalent stress and strain energy density were similar to that of Young's modulus, except for the region of the linea aspera. All biomechanical parameters correlated with morphological parameters and correlation efficiencies, with the reciprocal of cortical bone thickness showing the strongest correlation.The present results demonstrated that biomechanical parameters may be predicted by calculating the cortical bone thickness of femurs not treated with bisphosphonates. Furthermore, strain appeared to be repressed at a low level despite differences in stress intensities among the regions by bone remodeling. This remodeling is considered to be regulated by Wolff's law driven by equivalent stress and strain energy densities from the proximal to distal femur. The present results will promote further investigations on the contribution of morphological parameters in the femoral diaphysis to the onset of atypical femoral fracture.

Equivalent Dynamic Model for Triangular Prism Mast with the Tape-Spring Hinges

The development of space mission requires the lightweight, large deploying-to-folding ratios and high-precision deployable structures. The tape-spring hinge can achieve a deflection of 180 deg within a 5% elastic deformation range. For one triangular prism mast (TPM) with the tape-spring hinges, the equivalent beam model for the TPM is derived using an energy equivalence method. The node deflections’, geometrical parameters’, and material properties’ relationship between the TPM and equivalent model are built based on the structural features. The strain and kinematic energy equations of the key points are derived by the equivalent model. The stiffness and mass matrices of the equivalent model are obtained. In addition, the modal of the TPM in free vibration modes is analyzed based on the Timoshenko continuum beam theory, and natural frequencies are solved. Two TPM units are developed to validate the precision of the continuum beam equivalent model. A model test is conducted at the cantilever position, and the finite element (FE) model of the two TPM units is modified. Subsequently, the modeling approach of the two TPM units is applied to establish the FE model of the 10 TPM units. Moreover, the accuracy of the continuum beam equivalent theoretical model is verified.

Clustering Elements of Truss Structures for Damage Identification by CBO

The number of structural elements plays a significant role in detecting damage location and severity; such methods have sometimes failed to provide correct solutions due to the entrapment of damage detection algorithms in the local optimum. To resolve this problem, this study proposed the simultaneous use of mathematical and statistical methods to narrow down the search space. To this end, a two-step damage detection method was proposed. In the first step, the structural elements were initially divided into different clusters using the k-means method. Subsequently, the possibly damaged elements of each cluster were identified. In the second step, the elements selected in the first step were placed in a new set, and a process was applied to identify their respective damage location and severity. Thus, the proposed method reduced the search space as well as the possibility of entrapment in the local optimum. Other advantages of the proposed method include the use of fewer dynamic properties. Accordingly, by narrowing down the search space and the dimensions of the system for governing equations, the proposed method could significantly increase the chance of obtaining favorable results in structures with many elements and those with few vibration modes. A meta-heuristic method, called the colliding bodies optimization (CBO), was used in the proposed damage detection optimization algorithm. The optimization problem was based on the modal strain energy equations. According to the results, the proposed method was able to detect the location and severity of damage, even at its slightest percentage.

Fatigue tests on notched specimens of G20Mn5QT cast steel and life prediction by a new strain-based method

Fatigue performance of notched specimens of G20Mn5QT cast steel was investigated experimentally and analytically. Fatigue tests on a total number of 22 semi-circular notched specimens were conducted with the load ratios of − 1 and 0.1. Nominal stress-fatigue life relationship was obtained, and mean stress correction rules were verified for the notched specimens based on the test results. A new strain-based approach, strain field intensity (SNFI) method, was proposed, in which fatigue life of materials is predicted based on the weighted average strain in the fatigue damage region. The fatigue lives of the test specimens were predicted using three strain-based approaches, modified Neuber’s rule, equivalent strain energy density (ESED) method and the proposed SNFI method. The fatigue life prediction by the SNFI method was in very good agreement with the test results, which verified the rationality and applicability of the proposed SNFI method and also the adopted fatigue properties of G20Mn5QT cast steel. Both the modified Neuber’s rule and the ESED method gave conservative prediction of the fatigue life of the notched specimens. The necessity of the 3-dimensional fatigue damage region was also discussed for the proposed SNFI method.

Static analysis of different spoke structure of airless and conventional tyre

An airless tyre is known as non-pneumatic tyres which are not advocated by air pressure. The major issues related to pneumatic tyre are puncture and flatness due to wear and tear, which lead to the development of airless tyres. The aim of this work is to conduct static analysis of airless tyre, considering various 3D printing materials with different spoke structures. The spoke structures like honeycomb, triangular and diamond and the 3D printing materials such as Acrylonitrile Butadiene Styrene + Polycarbonate (ABS+PC), Polyethylene Terephthalate (PET) and High Impact Polystyrene (HIPS) are considered for modelling of airless tyre. For conventional tyre, Neoprene rubber is considered. The static analysis is conducted using ANSYS workbench 19.2, and the total deformation, equivalent elastic strain, equivalent von-mises stress and strain energy are compared and evaluated for conventional and airless tyre.

Soft composite based hyperelastic model for anisotropic tissue characterization

Soft tissues are complex anisotropic composite systems comprising of multiple differently oriented layers of fiber embedded within a soft matrix. To date, soft tissues have been mainly characterized using simplified linear elastic material models, isotropic viscoelastic and hyperelastic models, and transversely isotropic models. In such models, the effect of fiber volume fraction (FVF), fiber orientation, and fiber-matrix interactions are missing, inhibiting accurate characterization of anisotropic tissue properties. The current work addresses this literature gap with the development of a novel soft composite based material framework to model tissue anisotropy. In this model, the fiber and matrix are considered as separate hyperelastic materials, and fiber-matrix interaction is modeled using multiplicative decomposition of the deformation gradient. The effect of the individual contribution of the fibers and matrix are introduced into the numerical framework for a single soft composite layer, and fiber orientation effects are incorporated into the strain energy functions. Also, strain energy formulations are developed for multiple soft composite layers with varying fiber orientations and contributions, describing the biomechanical behavior of an entire anisotropic tissue block. Stress-strain relationships were derived from the strain energy equations for a uniaxial mechanical test condition. To validate the model parameters, experimental models of soft composites tested under uniaxial tension were characterized using the novel anisotropic hyperelastic model (R2 = 0.983). To date, such a robust anisotropic hyperelastic composite framework has not been developed, which would be indispensable for experimental characterization of tissues and for improving the fidelity of computational biological models in future.

Estimation of elastic‐plastic strain response at two‐dimensional notches

AbstractIt is widely recognized that the accuracy of notch fatigue calculations can be improved significantly when those calculations are based on the elastic‐plastic response strain at the notch root, as opposed to the remotely applied loads or stresses. Two of the most widely used approximations for this response are Neuber's rule and Glinka's equivalent strain energy density method. In the present work, a survey of some of the many published evaluations of these methods was first conducted, and then, additional detailed comparisons with elastic‐plastic finite element analyses for a series of semicircular and V‐shaped notch configurations were performed. Based on the observed limitations of both the Neuber and Glinka approaches, and with the guidance of the elastic‐plastic finite element results, a new (and more robust) approach for the estimation of notch response strains is proposed. This approach calls for the definition of a generalized notch response curve (GNRC), which is dependent on both the material stress–strain curve and the notch geometry. Once defined, the GNRC allows the determination of the response strain for any applied stress.

Design and analysis of a grid patch multi-stable composite

In this paper, the effect of stacking sequences on the number of stable configurations for multi-patch composite panel is studied. The model is designed by connecting a number of bi-stable composite patches in a grid. The total strain energy equation with a fourth order shape function including variations of the curvatures is applied in conjunction with a Rayleigh-Ritz technique for fast predication of stable shapes. A square panel model consisting of four connected patches of asymmetrical laminate is designed with 0° and 90° ply's angle, and then the model is extended by a mix of symmetric and asymmetric laminates. The stable shapes of model are validated using both FE predictions and experimental tests, and the results show quite good agreements in the stable configurations. Each case of panels exhibits two or more distinct stable configurations, which are mainly dependent on distribution of stacking sequences on surface. These stable configurations are interesting shapes that may be used to create a large multi-stable surface as useful device in a morphing skin application. This work could be used as a guideline for design of a large multi-patch morphing skin for highly degree of stability.

Mode shape transformation for model error localization with modal strain energy

A modeling error location method based on modal strain energy is presented in this paper. Errors in the design model with shell elements are located by an error indicator which is based on changes between the equivalent modal strain energy and the modal strain energy of the design model. The equivalent modal strain energy is defined as a quadratic form using the stiffness matrix of the design model and the mode shape of the reference coming from the sophisticated and high fidelity finite-element model, called the supermodel, or the full-field measurement. The major obstacle to obtain the equivalent modal strain energy is how to match the mode shapes of a solid element and those of a shell element since each node of the solid element contains only three translation degrees of freedom (dofs) while each node of the shell element has six dofs, including three translation and three rotation components. In order to solve this problem, a mode shape transformation method from the solid element to the shell element is proposed using the shape functions or linear approximation. Using this approach, the errors in the design model can be determined and the updating parameters can be selected so that the updated model has physical meaning and can represent the dynamic characteristics of the real structure. The simulation of a simple plate is used initially to illustrate the effectiveness of the proposed method. Then, a rotor test rig casing is taken as an example for further investigation. A comparison of the updating parameters selected by the proposed method and the traditional sensitivity analysis technique is then undertaken. It is verified that the updating parameters selected based on error location have physical sense and represent the true errors in the design model through the updating results. The advantage of this technique is that only detailed mode shapes from the reference is required. The approach shows potential for further industrial engineering applications.

An approach for the application of energy-based liquefaction procedure using field case history data

This paper presents an overview to the applicability of the “energy-based liquefaction approach” with regards to the new developments in the subject. The method involves comparing the strain energy for the soil liquefaction (capacity) with the strain energy imparted to the soil layer during an earthquake (demand). The performance of the method was evaluated by using a large database of SPT-based liquefaction case history. The energy-based method and the more commonly used stressbased method were compared in their capability to assess liquefaction potential under the same damaging historic earthquakes and geotechnical site conditions. In the procedure, the predictive strain energy equations were used to estimate the capacity energy values. These empirical equations have been developed based on the initial effective soil parameters. As for the energy of any given strong ground motion, it was computed from a velocity-time history of the ground motion and the unit mass of soil through utilization of kinetic energy concepts. The proposed energy-based method has effective way in evaluating the liquefaction potential based on the seismological parameters, contrary to the stress-based approach, where only peak ground acceleration (PGA) is considered.

Multiaxial fatigue life assessment in notched components based on the effective strain energy density

This paper presents a methodology to predict the fatigue lifetime in notched geometries subjected to multiaxial loading based on the effective strain energy density concept. The modus operandi consists of defining a fatigue master curve that relates the strain energy density with the number of cycles to failure from standard cylindrical specimens tested under low-cycle fatigue conditions. After that, the multiaxial loading history at the geometric discontinuity is reduced to an equivalent uniaxial loading scenario via the calculation of an averaged value of the strain energy, which is done by combining the equivalent strain energy density concept along with the theory of critical distances. Then, this energy is inserted into the fatigue master curve to estimate the fatigue lifetime. The method is tested in solid round bars with lateral notches subjected to in-phase bending-torsion loading. Overall, the comparison between the experimental and predicted fatigue lives shows a very good agreement. Additionally, the proposed approach enables the determination of the most likely initiation sites as well as the crack angles at the early stage of crack growth.

Local strain energy density approach to assess the fatigue strength of sharp and blunt V-notches in cast steel

In this work, the strain energy density approach (SED) is utilized to assess the fatigue strength of circumferential V-notched specimens covering axial loading and rotating bending. Subsequent calculations of the linear-elastic SED lead to a statistical proven accordance among about one-hundred experimental tests, thereby introducing extended H-function values for increased Rc to ρ values as well. Furthermore, low-cycle fatigue tests are conducted and utilized in an extensive elasto-plastic SED assessment. Thus, the scatter index of the evaluated equivalent strain energy density is reduced towards a ratio of about two. Concluding, the finite-volume-based energy approach provides an engineering-applicable fatigue assessment tool.

Stress-strain calculation and fatigue life assessment of V-shaped notches of turbine disk alloys

For local stress-strain behavior modeling at the notch root, both the Neuber's rule and equivalent strain energy density (ESED) method are commonly utilized in engineering practice, in which the former overestimates notch elasto-plastic responses while the latter gives conservative predictions. In this work, a correction coefficient related to the effective Poisson's ratio veff is proposed to quantify the weight of dissipated heat energy term with the modified ESED model. Notch-root strains predicted by the proposed model are compared with nonlinear FE calculations of GH4169 and TC4 V-shaped notched-bars, in which the proposed solution provides good agreement with FE results for both cases. Then, consulting the idea of local stress-strain approaches, a fatigue lifing model is derived assuming that the notched specimens fail once reach the critical total strain energy as same as that of smooth ones. Results indicated that prediction accuracy of local equivalent strains at notch roots is improved, and yields better life predictions than the Ellyin's model for the two alloys.

Discrete element modelling of flexible membrane boundaries for triaxial tests

The discrete element modelling of triaxial tests plays a critical role in unveiling fundamental properties of particulate materials, but the numerical implementation of a flexible membrane boundary for the testing still imposes problems. In this study, a robust algorithm was proposed to reproduce a flexible membrane boundary in triaxial testing. The equivalence of strain energy enables the particle-scale parameters representing the flexible membrane to be directly determined from the real geometric and material parameters of the membrane. Then the proposed flexible membrane boundary was implemented in the context of discrete element simulation of triaxial testing and was validated with laboratory experiments. Furthermore, comparisons of triaxial tests with flexible and rigid boundaries were performed from macro-scale to meso-scale. The results show that the boundary condition has limited influences on the stress-strain behaviour but a relatively large impact on the volumetric change, the failure mode, the distribution of contact forces, and the fabric evolution of particles in the specimen during triaxial testing.

An approximate method to predict the mechanical properties of small volume fraction particle-reinforced composites with large deformation matrix

The aim of this paper is to propose an approximate homogenization method for deriving the effective strain energy density function of small volume fraction particle-reinforced hyperelastic matrix composites. This method is inspired by the finite element (FE) simulation of a single-particle representative volume element (RVE) model, in which an incompressible Yeoh constitutive model is used to describe the mechanical behavior of the matrix and the particle is regarded as a rigid material. According to the FE results, the RVE can be partitioned into different regions with uniform strain distributions, and then, the equivalent strain energy density function of the whole composite can be calculated based on a volume averaging treatment. With the new method, the stress–strain response of a mica particle-filled rubber matrix composite under uniaxial tension is theoretically predicted, and it shows a good agreement with both experimental measurements and numerical calculations, especially when the particle volume fraction is relatively small. Due to the simple derivation and concise formulation of the strain energy density function, the present method should be of practical value for engineering use in predicting the large deformation behavior of hyperelastic composites.

New strain energy-based critical plane approach for multiaxial fatigue life prediction

Based on critical plane approach, this article develops a new damage parameter through combing the equivalent strain energy aspect for multiaxial fatigue analysis, which includes no additional fitted parameters and overcomes the deficiency of using only equivalent stress/strain criterion separately under multiaxial loadings. Then, experimental data of GH4169, TC4, Al 7050-T7451 alloys under different loading conditions are applied for model validation and comparison with other four models. Results indicate that the proposed damage parameter yields better multiaxial fatigue life predictions than others.

Homogenized elastic response of random fiber networks based on strain gradient continuum models

The purpose of this work is to develop anisotropic strain gradient linear elastic continuum models for two-dimensional random fiber networks. The constitutive moduli of the strain gradient equivalent continuum are assessed based on the response of the explicit network representation in so-called windows of analysis, in which each fiber is modeled as a beam and the fibers are connected at crossing points with welded joints. The principle of strain energy equivalence based on the extension to the strain gradient of the Hill–Mandel macro homogeneity condition is employed to identify the classical and strain gradient moduli, based on the application of a sequential set of polynomial displacements on windows of analysis of different sizes. The scaling of the first- and second-order moduli with network parameters, such as network density and the ratio of fiber bending to axial stiffness, is determined. We observe a similar dependency of classical and strain gradient moduli on the same network parameters. The internal length scales associated with the gradient coefficients of the constitutive equation are also defined in terms of the network parameters. The strain gradient moduli prove to be size-independent in the affine regime, and they converge toward a size-independent value in the non-affine deformation regime after a rescaling of physical dimensions by the window size. The obtained results show that the strain gradient moduli scale uniformly with the square of the magnitude of the strain gradients applied to the window of analysis.

Multi-scale damage framework for textile composites: Application to plain woven composite

A damage mechanics-based progressive damage analysis (PDA) of plain woven textile composite (PWTC) is performed in this work. The primary objective is to identify the micro-scale and meso-scale failure modes, and compute the ultimate strength of PWTC under in-plane loadings. Multi-scale PDA is performed on an equivalent cross-ply laminate (ECPL) model by maximum stress theory to predict the meso-scale failure modes of PWTC. The damaged stiffness is computed by 3 different ways: 1) hypothesis of strain energy equivalence by proposed 4th order anisotropic damage tensor, 2) applying isotropic damage law, and 3) eliminating the stiffness terms. Novel multi-scale PDA approaches, applying Mori-Tanaka (MT) theory on ECPL model and on the quarter portion of representative volume element of PWTC, are performed to predict micro-scale failure modes. The macro-scale constitutive behaviour of PWTC is theoretically predicted and the corresponding strengths are computed. A user material (UMAT) subroutine is developed to validate the proposed constitutive behaviour of PWTC solving macro-scale boundary value problems in ABAQUS®. The PDA simulation results of 3D PWTC bar model are found to be reasonably matching with the in-house experiments.