Equivalent Strain Energy Research Articles

A novel stiffness prediction method with constructed microscopic displacement field for periodic composite plates

A novel method for predicting effective stiffnesses of composite plates with in-plane periodicity is proposed according to two-scale strain energy equivalence. The macroscale energy is reached with the Reissner-Mindlin plate theory, while the microscale energy is obtained through the newly constructed displacement functions of a unit cell within the framework of the three-dimensional elasticity theory. The new displacement functions consist of homogenized displacements, two decoupled expansion terms, and a higher-order perturbed displacement term. The elaborate finite element formulations are presented for realizing the present method in Comsol Multiphysics. Several numerical experiments demonstrate the effectiveness of the new stiffness prediction method.

Thermal Damage Constitutive Model and Brittleness Index Based on Energy Dissipation for Deep Rock

In deep underground engineering, rock usually encounters a high temperature problem. The stress–strain relationship of rock under high temperatures is the basis of engineering excavation. Based on the Lemaitre’s equivalent strain hypothesis and energy dissipation theory, the thermal damage constitutive model of rock is established. The results show that the peak stress, ultimate elastic energy and dissipated energy at the peak all decrease with the increase of temperature in a logistic function, which indicates that the increase of temperature aggravates the deterioration of rock’s mechanical properties. Compared with rock’s constitutive model that is established on strength criterion, the thermal damage constitutive model based on energy dissipation better reflects the phenomenon of stress drop after the peak and well describes the whole stress-strain curve of rock failure, which verifies the rationality of the model. The damage model further improves the theoretical system of the rock damage constitutive model and makes up for the defect in that the traditional damage model cannot reasonably explain the nature of rock failure. The brittleness index, defined based on the energy drop coefficient, shows a logistic function with the increase of temperature, which has good physical meaning. Analyzing the phenomenon of the rock stress drop from the perspective of energy is of great significance for deeply understanding the brittle fracture mechanism of rock.

A new anisotropic elasto-plastic-damage model for quasi-brittle materials using strain energy equivalence

An anisotropic elasto-plastic-damage model is developed for quasi-brittle materials within the thermodynamics framework. The anisotropic damage is characterized by second-order damage tensors that are different under tensile and compressive loadings. The hypothesis of strain energy equivalence is adopted with two components, including the hypothesis of elastic strain energy equivalence and the hypothesis of plastic strain energy equivalence. The hypothesis of plastic strain energy equivalence is extended from metals to quasi-brittle materials so as to derive the relations of hardening stresses, which are previously assumed between the undamaged and damaged configurations for quasi-brittle materials. A plastic yield criterion is adopted simultaneously with a damage criterion to account for the coupled elasto-plastic-damage behavior under different loadings. The damage criterion can capture larger axial damage in uniaxial tension and smaller axial damage in uniaxial compression as compared with the lateral damage. The Helmholtz free energy functions are derived for three components: elastic, plastic, and damage. The constitutive laws are derived for three scenarios: the elastoplastic behavior without further damage evolution, the damage behavior without further plasticity evolution, and the coupled elasto-plastic-damage behavior. The anisotropic elasto-plastic-damage model is implemented in ABAQUS as a user material (UMAT) subroutine. The coupled constitutive laws in the continuum domain are implemented in an uncoupled way in the discrete domain with three steps, elastic predictor, plastic corrector, and damage corrector. The UMAT is validated in different conditions, including uniaxial, biaxial, and three-point bending tests. The advantage of considering the plastic free energy in anisotropic damage is illustrated through the strength envelope in biaxial loadings. The numerical results generally agree with experimental results, including the stress-strain curves in uniaxial tests, strength envelope in biaxial tests, and load-deflection curves in three-point bending tests.

Folding stress analysis of a sandwich plate based on transversely isotropic and rate-dependent constitutive relationship of adhesive

ABSTRACT Flexible and foldable electronic products often contain adhesive layers, and failure generally initiates at the adhesive interfaces. To analyze the interfacial stress accurately, correct constitutive relationship of adhesive layers is necessary. However, soft adhesives used in foldable sandwich plates can show rate-dependent and anisotropic mechanical behaviors, which can hardly be implemented by traditional linear elastic and hyperelastic constitutive models in the folding process simulation. Through experiments in material’s principal axes and the strain energy equivalence principle, a transversely isotropic and rate-dependent constitutive relationship of adhesive has been developed in a three-dimensional form. By commercial finite element (FE) software ABAQUS and its user subroutine UMAT, the folding stress of a sandwich plate has been analyzed. The results of FE simulation show that folding time would affect stress distributions significantly due to the anisotropic and rate-dependent properties of adhesive.

Analytical and Experimental Investigation of Elastic–Plastic Strain Distributions at 2-D Notches

Abstract Both elastic and elastic–plastic stress–strain response were studied for a series of part-circular and V-shaped notches in flat specimens. Both the peak strain at the notch root and the strain distribution along the notch bisector were studied. These quantities were determined analytically using detailed finite element analysis (FEA), and it was shown that significant differences can arise between detailed FEA results and two of the commonly used approximations for the peak notch strain response: Neuber’s rule and Glinka’s equivalent strain energy density theorem. These differences can become significant for notches with low acuity. The strain response for these configurations was also studied experimentally using two measurement techniques: optical fibers and surface differential displacement mapping. Agreement was shown between computed response strains and measured strains for some of the eight notch configurations and two aerospace alloys studied, while for others, experimental difficulties prevented agreement. These difficulties are described in detail.

A Study on Low Cycle Fatigue Life Assessment of Notched Specimens Made of 316 LN Austenitic Stainless Steel

Abstract The local strains obtained from the best-known analytical approximations, namely, Neuber's rule, equivalent strain energy density method, and linear rule, were compared with those resulting from finite element analysis. It was found that apart from Neuber's rule with the elastic stress concentration factor Kt, all the aforementioned analytical methods underestimate the local strains for all notch root radius, strain amplitude levels, at room temperature and 550 °C. Neuber's rule with Kt slightly overestimates the maximum strains for lower notch root radius, namely, 1.25 mm, at high temperature. Based on the analytically and numerically obtained notch root strains, the fatigue lives were estimated using the Coffin–Manson–Basquin equation. Besides, a numerical assessment of fatigue lives was made based on Brown–Miller and maximum shear strain multi-axial fatigue life criteria. It was found that all these methods provide inaccurate fatigue life results for all notch root radius, strain amplitude level, and under both temperatures conditions. Therefore, a new method was suggested, for which only the applied strain amplitude is needed to calculate the fatigue life of notched components. It was revealed that the suggested method provides a good fatigue life prediction at a higher temperature loading state.

Analytical examination of mesh-dependency issue for uniaxial RC elements and new fracture energy-based regularization technique

This paper presents a comprehensive analysis of the mesh-dependency issue for both plain concrete and reinforced concrete (RC) members under uniaxial loading. The detailed mechanisms for each case are firstly derived, and the analytical and numerical strain energies for concrete in different cases are compared to explain the phenomena of mesh-dependency. It is found that the mesh-dependency will be relieved or even eliminated with the increasing of the reinforcing ratio. Meanwhile, a concept of the critical reinforcing ratio is proposed to identify the corresponding boundary of mesh-dependency of RC members. In order to verify the above findings, several illustrative examples are performed and discussed. Finally, to overcome the mesh-dependency issue for RC members with lower reinforcing ratios, we propose a unified regularization method that modifies both stress-strain relations of steel and concrete based on the strain energy equivalence. The method is also applied to the illustrative examples for validation, and the numerical results indicate that the developed method can obtain objective results for cases with different meshes and reinforcing ratios.

Optimization and Re-Design of Integrated Thermal Protection Systems Considering Thermo-Mechanical Performance

Integrated thermal protection system (ITPS) is regarded as one of the most promising thermal protection concepts with both thermal insulation and load-bearing capacities. However, the traditional layout of webs could inevitably lead to thermal short effects and high risk of buckling failure of the ITPS. A topological optimization method for the unit cell of the ITPS was established to minimize the equivalent thermal conductivity and elastic strain energy with the constraint of maintaining structural efficiency. The ITPS was re-designed consulting the optimized cell configuration. In order to control the buckling-mode shape and the associated buckling load of the ITPS, the new design was further optimized, subjected to the total weight of the initial design. Detailed finite element models were established to validate the structural responses. By contrast, the optimized design presents lower bottom surface temperature and better thermal buckling characteristics, performing a better balance between thermal insulation and load-bearing constraints.

Solid Truss to Shell Numerical Homogenization of Prefabricated Composite Slabs.

The need for quick and easy deflection calculations of various prefabricated slabs causes simplified procedures and numerical tools to be used more often. Modelling of full 3D finite element (FE) geometry of such plates is not only uneconomical but often requires the use of complex software and advanced numerical knowledge. Therefore, numerical homogenization is an excellent tool, which can be easily employed to simplify a model, especially when accurate modelling is not necessary. Homogenization allows for simplifying a computational model and replacing a complicated composite structure with a homogeneous plate. Here, a numerical homogenization method based on strain energy equivalence is derived. Based on the method proposed, the structure of the prefabricated concrete slabs reinforced with steel spatial trusses is homogenized to a single plate element with an effective stiffness. There is a complete equivalence between the full 3D FE model built with solid elements combined with truss structural elements and the simplified homogenized plate FE model. The method allows for the correct homogenization of any complex composite structures made of both solid and structural elements, without the need to perform advanced numerical analyses. The only requirement is a correctly formulated stiffness matrix of a representative volume element (RVE) and appropriate formulation of the transformation between kinematic constrains on the RVE boundary and generalized strains.

Comparison of different one-parameter damage laws and local stress-strain approaches in multiaxial fatigue life assessment of notched components

This paper aims to compare the predictive capabilities of different one-parameter damage laws and local stress-strain approaches to assess the fatigue lifetime in notched components subjected to proportional bending-torsion loading. The tested fatigue damage parameters are defined using well-known stress-based, strain-based, SWT-based and energy-based relationships. Multiaxial cyclic plasticity at the notch-controlled process zone is accounted for within a 3D-FE linear-elastic framework using three local stress-strain approaches, namely Neuber’s rule, equivalent strain energy density rule (ESED) and the modified ESED rule. Regarding the local stress-strain approaches, irrespective of the fatigue damage parameter, Neuber’s rule always led to more conservative results, and the modified ESED rule resulted in slightly better fatigue life predictions when compared to the original ESED rule. As far as the fatigue damage parameters are concerned, energy-based models were more accurate, irrespective of the local stress-strain approach.

Analysis of Nanoplate with a Central Crack Under Distributed Transverse Load Based on Modified Nonlocal Elasticity Theory

In this paper, using the complete modified nonlocal elasticity theory, the deflection and strain energy equations of rectangular nanoplates, with a central crack, under distributed transverse load were overwritten. First, the deflection of nanoplate was obtained using Levy's solution and consuming it; strain energy of nanoplate was found. As regards nonlocal elasticity theory wasn’t qualified for predicting the static behavior of nanoplates under distributed transverse load, using modified nonlocal elasticity theory, the deflection of nanoplate with a central crack for different values of the small-scale effect parameter was achieved. It was gained with the convergence condition for the complete modified nonlocal elasticity theory. To verify the result, the results for the state of the small-scale effect parameter were placed equal to zero (plate with macro-scale) and then were compared with the numerical results as well as the classical analytical solution results available in the valid references. It was shown that the complete modified nonlocal elasticity theory does not show any singularity at the crack-tip unlike the classical theory; therefore, the method presented is a suitable method for analysis of the nanoplates with a central crack.

Effects of Shear Deformation of Struts in Hexagonal Lattices on their Effective In-Plane Material Properties

Two-dimensional lattices are widely used in many engineering applications. If 2D lattices have large numbers of unit cells, they can be accurately modeled as 2D homogeneous solids having effective material properties. When the slenderness ratios of struts in these 2D lattices are low, the effects of shear deformation on the values of the effective material properties can be significant. This study aims to investigate the effects of shear deformation on the effective material properties of 2D lattices with hexagonal unit cells, by using the homogenization method based on equivalent strain energy. Several topologies of hexagonal unit cells and several slenderness ratios of struts are considered. The effects of struts’ shear deformation on the effective material properties are examined by comparing the results of the present study, in which shear deformation is neglected, with those from the literature, in which shear deformation is included.

Vibration Analysis of Rectangular Kirchhoff Nano-Plate using Modified Couple Stress Theory and Navier Solution Method

In this study, the characteristics of rectangular Kirchhoff nano-plate vibrations are investigated using a modified couple stress theory. To consider the effects of small-scale, the modified couple stress theory proposed by Young (2002) is used as it has only one length scale parameter. In modified couple stress theory, the strain energy density is a function of the components of the strain tensor, curvature tensor, stress tensor, and symmetric part of the couple stress tensor. After obtaining the strain energy, external work, and kinetic energy equation and inserting them in the Hamilton principle, the main and auxiliary equations of nano-plate are obtained. Then, by applying the boundary and force conditions in the governing equations, the vibrations of the rectangular Kirschhof nano-plate with the thickness are investigated with simple support around. The solution method used in this study is the Navier method and the effects of material length scale, length and thickness of the nanoplate on the vibration are investigated and the results are presented and discussed in details.

Numerical study on bending response of auxetic 2D-lattice plates

In this study, the bending response of 2D-lattice plates having auxetic unit cells is determined by using the homogenization method based on equivalent strain energy. The auxetic unit cells in this study are different topologies of re-entrant hexagonal unit cells. In the homogenization method, the effective out-of-plane elastic properties of an auxetic 2D-lattice plate are obtained from the strain energy values of its unit cell under different curvature modes, determined by the finite element method. In the analysis, the auxetic unit cells are considered as frames whose struts are modeled as Euler beams. The effective elastic properties, i.e., the effective bending moduli, Poisson’s ratios, and shear modulus, are described as the bending response of a 2D-lattice plate subjected to out-of-plane bending. In the validation, the effective elastic properties of some auxetic 2D-lattice plates obtained from their unit cells by the homogenization method are numerically compared with those obtained from direct structural analysis of the plates. Besides, the obtained results show how the bending response of the auxetic 2D-lattice plates can be adjusted by varying their unit-cell geometries, especially the internal cell angle.

Isotropized Voigt-Reuss model for prediction of elastic properties of particulate composites

Voigt and Reuss models do not make any assumption regarding composite microstructure, but they are only able to estimate bounds rather than values of composite properties. To enable the models to predict elastic properties of particulate composites, we isotropized Voigt and Reuss models based on the equivalence of normal and shear strain energy respectively in a representative volume element, and derived a set of simple and uncoupled equations for the prediction of composite Young’s modulus and shear modulus. Composite bulk modulus and Poisson’s ratio can then be determined either by the elasticity relationships or by generalization of the isotropized Voigt-Reuss equations. Validation with reported experimental data show that, for composites with phase properties of either small or large contrast, isotropized Voigt-Reuss model has similar accuracy as the widely used micromechanical models, for the prediction of Young’s modulus and shear modulus.

Determination of Transverse Shear Stiffness of Sandwich Panels with a Corrugated Core by Numerical Homogenization.

Knowing the material properties of individual layers of the corrugated plate structures and the geometry of its cross-section, the effective material parameters of the equivalent plate can be calculated. This can be problematic, especially if the transverse shear stiffness is also necessary for the correct description of the equivalent plate performance. In this work, the method proposed by Biancolini is extended to include the possibility of determining, apart from the tensile and flexural stiffnesses, also the transverse shear stiffness of the homogenized corrugated board. The method is based on the strain energy equivalence between the full numerical 3D model of the corrugated board and its Reissner-Mindlin flat plate representation. Shell finite elements were used in this study to accurately reflect the geometry of the corrugated board. In the method presented here, the finite element method is only used to compose the initial global stiffness matrix, which is then condensed and directly used in the homogenization procedure. The stability of the proposed method was tested for different variants of the selected representative volume elements. The obtained results are consistent with other technique already presented in the literature.

Estimation of Mode I Fracture of U-Notched Polycarbonate Specimens Using the Equivalent Material Concept and Strain Energy Density

Polycarbonate (PC) has a wide range of applications in the electronic, transportation, and biomedical industries. In addition, investigation on the applicability to use PC in superstrate photovoltaic modules is ongoing research. In this paper, PC is envisioned to be used as a material for structural components in renewable energy systems. Usually, structural components have geometrical irregularities, i.e., notches, and are subjected to severe mechanical loading. Therefore, the structural integrity of these components shall consider fracture analysis on notched specimens. In this paper, rectangular PC specimens were machined with straight U-notches having different radii and depths. Eight different notch radii with a depth of 6.0 mm were tested. In addition, three notch depths with a radius of 3.5 mm were considered. Quasi-static fracture tests were performed under displacement-controlled loading with a speed of 5 mm/min. Digital image correlation technique was used to capture the strain fields for un-notched and notched specimens. It was assumed that fracture occurs at the onset of necking. The equivalent material concept (EMC) along with the strain energy density criterion (SED) were employed to estimate the fracture load. The EMC-SED combination is shown to be an effective and practical tool for estimating the fracture load of U-notched PC specimens.

Nonlocal phase field approach for modeling damage in brittle materials

In this work, we present a nonlocal phase field model for damage in brittle materials. We define a non-conservative order parameter for representing the damage. The Helmholtz free energy functional of the Ginzburg-Landau type that incorporates a new degradation function for the elastic strain energy is considered. The hypothesis of strain equivalence and strain energy equivalence from continuum damage mechanics are adopted to predict the evolution of damage. A conjugate force to damage is derived for both these hypothesis to arrive at a damage criterion. The strong interrelation between phase field models and gradient damage models has been brought out. The solution of the nonlinear system of coupled equation is achieved by a change in the modified Arc-length method and considering a staggered approach. In this approach a linearization of the Crisfield Arc-length quadratic equation for calculation of load increment factor results in an unique root. Numerical examples are presented to demonstrate the usefulness of the proposed damage model.

Multiaxial fatigue behaviour of maraging steel produced by selective laser melting

This paper studies the multiaxial fatigue behaviour of maraging steel samples produced by selective laser melting. Hollow cylindrical specimens with transverse circular holes are subjected to different in-phase bending-torsion loading scenarios. Fatigue crack initiation sites and fatigue crack angles are predicted from the first principal stress field. Fatigue lifetime is computed using a straightforward approach, based on a one-parameter damage law, developed via uniaxial low-cycle fatigue tests. The cyclic plasticity at the notch-controlled process zone is accounted for by combining the equivalent strain energy density concept and the theory of critical distances within a linear-elastic framework. Regardless of the multiaxial loading scenario, experimental observations and predicted lives are very well correlated.

Topology optimization of compliant mechanisms and structures subjected to design-dependent pressure loadings

This article presents an extended algorithm for topology optimization of compliant mechanisms and structures with design-dependent pressure loadings using the moving iso-surface threshold (MIST) method. In this algorithm, the fluid-structure interface is modeled using the finite element method via considering equivalent virtual strain energy and work and is tracked by an element-based searching scheme. Design-dependent pressure loads are directly applied on interface boundary and are calculated as virtual work equivalent nodal forces in the interface elements based on the finite element formulation. Several numerical examples are presented for topology optimization of mean compliance and compliant mechanisms. The present algorithm is validated through benchmarking with the results in literature and/or full finite element analysis (FEA) results of the optimum compliant mechanism and structure designs.