Effective Strain Energy Density Research Articles

Nonlinear wave propagation in homogenized strain gradient 1D and 2D lattice materials: Applications to hexagonal and triangular networks

AbstractThis work aims to analyze the propagation of fully nonlinear waves, encompassing shear, extension, and bending deformation modes, within homogenized periodic nonlinear hexagonal and triangular networks, successively considering 1D and 2D situations. The wave analysis is conducted from the expression of the effective strain energy density of periodic hexagonal and triangular lattices in the nonlinear regime by a continualization of the discrete lattice equations, considering all forms of energy. We incorporate strain gradient effects into the continuous model to account for the wave‐dispersive nature. The resulting second‐gradient nonlinear continuum exhibits subsonic and supersonic propagation modes. We first examine in a 1D situation the dynamical response of the hexagonal and triangular lattices, considering varying levels of nonlinearity quantified by a single scalar valued parameter. We further evaluate the impact of a fully nonlinear analysis compared to an analysis solely based on the shear energy, regarding both supersonic and subsonic modes. The nonlinear wave propagation analysis is then extended to a 2D situation for the same two lattices. It is shown that the longitudinal mode exhibits a higher frequency at a low degree of nonlinearity; however, as the degree of nonlinearity increases, the shear mode surpasses the longitudinal mode in terms of frequency. As the wavenumber increases, the nonlinearity has a lesser impact on the frequency behavior, and the phase velocity is more influenced by other factors, such as the second gradient contributions of the effective constitutive law. Such a behavior indicates a transition from a highly nonlinear behavior at lower wave numbers to a more linear behavior at higher wave numbers.

Predicting mechanical failure of polycrystalline dual-phase nickel-based alloys by numerical homogenization using a phase field damage model

Brazing of nickel-based alloys plays a major role in the assembly of turbine components, e.g., abradable sealing systems. In a brazed joint of nickel-based alloys a composition of brittle and ductile phases can be formed if the brazing conditions are not ideal. This heterogeneous microstructure is a crucial challenge for predicting the damage behavior of a brazed joint. The initiation and evolution of microdamage inside of the brittle phase of a virtual dual-phase microstructure representing the material in a brazed joint is studied by means of numerical simulations. A phase field approach for brittle damage is employed on the microscale. The simulation approach is capable of depicting phenomena of microcracking like kinking and branching due to heterogeneous stress and strain fields on the microscale. No information regarding the initiation sites and pathways of microcracks is needed a priori. The reliability of calculating the effective critical energy quantities as a microstructure-based criterion for macroscopic damage is assessed. The effective critical strain energy density and the effective critical energy release rate are evaluated for single-phase microstructures, and the approach is transferred to dual-phase microstructures. The local critical strain energy density turns out to be better suited as a model input parameter on the microscale as well as for a microstructure-based prediction of macroscopic damage compared to a model employing the energy release rate. Regarding the uncertainty of the model prediction, using the effective critical energy release rate leads to a standard deviation which is five times larger than the standard deviation in the predicted effective critical strain energy density.

Multiaxial notch fatigue probability modeling based on three-parameter Weibull distribution and effective strain energy density

Multiaxial notch fatigue probability modeling based on three-parameter Weibull distribution and effective strain energy density

Simulation and Experimental Investigation of Balloon Folding and Inserting Performance for Angioplasty: A Comparison of Two Materials, Polyamide-12 and Pebax.

A balloon dilatation catheter is a vital tool in percutaneous transluminal angioplasty. Various factors, including the material used, influence the ability of different types of balloons to navigate through lesions during delivery. Thus far, numerical simulation studies comparing the impacts of different materials on the trackability of balloon catheters has been limited. This project seeks to unveil the underlying patterns more effectively by utilizing a highly realistic balloon-folding simulation method to compare the trackability of balloons made from different materials. Two materials, nylon-12 and Pebax, were examined for their insertion forces via a bench test and a numerical simulation. The simulation built a model identical to the bench test's groove and simulated the balloon's folding process prior to insertion to better replicate the experimental conditions. In the bench test, nylon-12 demonstrated the highest insertion force, peaking at 0.866 N, significantly outstripping the 0.156 N force exhibited by the Pebax balloon. In the simulation, nylon-12 experienced a higher level of stress after folding, while Pebax had demonstrated a higher effective strain and surface energy density. In terms of insertion force, nylon-12 was higher than Pebax in specific areas. nylon-12 exerts greater pressure on the vessel wall in curved pathways when compared to Pebax. The simulated insertion forces of nylon-12 align with the experimental results. However, when using the same friction coefficient, the difference in insertion forces between the two materials is minimal. The numerical simulation method used in this study can be used for relevant research. This method can assess the performance of balloons made from diverse materials navigating curved paths and can yield more precise and detailed data feedback compared to benchtop experiments.

A numerical two-scale approach for nonlinear hyperelastic beams and beam networks

We introduce a numerical framework for modeling hyperelastic slender trusses, oftentimes used as elementary building blocks in architected materials, which accounts for both the geometric nonlinearity inherent in thin structures and the nonlinear constitutive behavior of the base material. Akin to the FE2 method in homogenization, our approach is based on a formal, two-scale expansion. We decompose the three-dimensional (3D) description of a slender structure into a macroscale problem solving for the deformation of the beam center-line (based on an effective strain energy density, which depends on the stretching, bending, and torsional strains of the beam’s center-line) and a series of two-dimensional (2D) microscale boundary value problems (defined over the beam cross-section). We solve a series of 2D problems (covering a range of macroscopic strain combinations) for a given cross-sectional geometry and material distribution in an offline, pre-processing step. Using this pre-computed energy landscape, we solve the macroscopic boundary value problem within a geometrically exact, nonlinear discrete beam framework. We demonstrate the accuracy of this approach through a set of benchmark problems highlighting the nonlinear effects of the cross-sectional geometry and constitutive material, including material heterogeneity and pre-strains. We further illustrate how this technique is applied to truss-based architected materials consisting of networks of slender struts, which are typically made of polymeric base materials and thus allow for large nonlinear elastic deformation.

A unified rule for high-cycle and low-cycle fatigue life prediction in multi-scale framework

This paper develops a unified rule for both low-cycle fatigue and high-cycle fatigue life prediction from macro-scale to micro-scale. In the macro-scale, the unified rule is proposed on the basis of the concept of the effective strain energy density. In the micro-scale, the unified rule represented by a new fatigue indicator parameter is developed by considering the combined effects of the energies in shear and normal directions for individual slip systems. A large number of low-cycle and high-cycle fatigue experimental data of Inconel 718 superalloy at room temperature is used to verify the prediction capacities of the newly unified rule. Compared with the prediction accuracy of the existing fatigue damage models, the unified rule shows more accurately predicted results for both low-cycle and high-cycle fatigue life. Most of the fatigue experimental data falls into a range within a scatter band of ±3 on life.

Exact second moments of strain for composites with isotropic phases

In micromechanical approaches to failure prediction of inhomogeneous microstructures or homogenization of inelastic materials, local field fluctuations beyond the mean field are relevant, but not easily accessible except for resorting to full-field simulations. However, any analytical estimate of the effective strain energy density combined with the Hill–Mandel condition implies some information about the probability distributions of local fields such as stresses and strains. In this paper, analytical equations for the second moments of local fields are derived for particulate composites with isotropic phases. Explicit expressions are provided using the Mori–Tanaka approach, the Hashin–Shtrikman bounds and the singular approximation. The results are compared to full-field simulations of a random sphere assemblage. As the Hashin–Shtrikman covariance is isotropic, it is not well-suited for predicting the behavior under deviatoric loadings. The singular approximation and Mori–Tanaka methods yield sensible approximations for the second moment of matrix strain.

Probabilistic fatigue modelling of metallic materials under notch and size effect using the weakest link theory

Notch fatigue analysis is vital for structural integrity design, while efficient fatigue models coupling notch and size effect are still lacking, which are highly desired for fatigue design of critical components. Accordingly, this study proposes a new probabilistic fatigue model for life assessment by combining the weakest link theory with strain energy concept. Specifically, a novel effective strain energy density concept is developed herein to build the connection between experimental data of smooth specimens and strain energy density of notched ones within the critical damage region. Experimental data of GH4169, TC4 and TC11 alloys with different geometries are utilized for model validation and comparison. Results indicate that the proposed model show higher accuracy than the other four models; in addition, the P–Wa¯–Nf curves describe the experimental data scatter effectively.

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.

A Walker exponent corrected model for estimating fatigue life of metallic materials in loading with mean stress

AbstractMean stress significantly influence the fatigue life predictions of metallic materials. The Walker mean stress equation with its additional material parameter w provides good predictions for a wide range of materials. Unfortunately, additional tests are necessary to determine the Walker exponent w. In order to overcome this shortcoming, for aluminum alloys, the Walker exponent w was correlated linearly with the sum of ultimate tensile strength and true fracture strength. Then, a Walker exponent corrected effective strain energy density criterion was developed by incorporating the Walker mean stress equation into the strain life curve. The capability of fatigue life prediction for the developed model was checked against the tested data of 304 L stainless steel, SAE 1045 steel, 7075‐T651 aluminum alloy, and Incoloy 901 superalloy, and comparisons were also performed by using the Lv's Walker exponent corrected model. The developed model provides more satisfactory results, especially for the considered materials in loading with mean stress.

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.

On the effect of temperature gradient on the stability of circular tubes made of hyperelastic entropic material

Rubber-like materials are very applicable in almost all fields of industries, but due to their large deformation characteristic, they can exhibit a variety of instabilities. Accordingly, many researchers have been motivated to investigate the effects of different parameters on the stability of hyperelastic cylindrical tubes under finite deformation, while the effects of temperature gradient have not been considered. In this paper, the effects of temperature variation on the stability and thermo-mechanical behavior of the cylindrical tubes made of the entropic materials such as rubber-like materials and elastomers are investigated via an effective strain energy density function. To this purpose, an Ogden-type strain energy density with only integer powers is applied in order to determine an analytical solution, not involving the integral form, for the stress distribution through the wall thickness of cylindrical tubes at finite deformation thermoelasticity. This problem is examined in two cases including (i) a thick-walled cylindrical tube under internal pressure and uniform variation of temperature and (ii) a thick-walled cylindrical tube under internal pressure and temperature gradient, simultaneously. It was observed that the positive temperature gradients in comparison with environment temperatures improve the stability of the circular tubes made of the entropic materials.

Wave propagation analysis in 2D nonlinear hexagonal periodic networks based on second order gradient nonlinear constitutive models

We analyze the propagation of nonlinear waves in homogenized periodic nonlinear hexagonal networks, considering successively 1D and 2D situations. Wave analysis is performed on the basis of the construction of the effective strain energy density of periodic hexagonal lattices in the nonlinear regime. The obtained second order gradient nonlinear continuum has two propagation modes: an evanescent subsonic mode that disappears after a certain wavenumber and a supersonic mode characterized by an increase of the frequency with the wavenumber. For a weak nonlinearity, a supersonic mode occurs and the dispersion curves lie above the linear dispersion curve (vp =vp0). For a higher nonlinearity, the wave changes from a supersonic to an evanescent subsonic mode at s=0.7 and the dispersion curves drops below the linear case and vanish for certain values of the wavenumber. An important decrease in the frequency occurs for both subsonic and supersonic modes when the lattice becomes auxetic, and the longitudinal and shear modes become very close to each other. The influence of the lattice geometrical parameters of the lattice on the dispersion relations is analyzed.

Numerical prediction of peri-implant bone adaptation: Comparison of mechanical stimuli and sensitivity to modeling parameters

Long term durability of osseointegrated implants depends on bone adaptation to stress and strain occurring in proximity of the prosthesis. Mechanical overloading, as well as disuse, may reduce the stability of implants by provoking bone resorption. However, an appropriate mechanical environment can improve integration. Several studies have focused on the definition of numerical methods to predict bone peri-implant adaptation to the mechanical environment. Existing adaptation models differ notably in the type of mechanical variable adopted as stimulus but also in the bounds and shape of the adaptation rate equation. However, a general comparison of the different approaches on a common benchmark case is still missing and general guidelines to determine physically sound parameters still need to be developed. This current work addresses these themes in two steps. Firstly, the histograms of effective stress, strain and strain energy density are compared for rat tibiae in physiological (homeostatic) conditions. According to the Mechanostat, the ideal stimulus should present a clearly defined, position and tissue invariant lazy zone in homeostatic conditions. Our results highlight that only the octahedral shear strain presents this characteristic and can thus be considered the optimal choice for implementation of a continuum level bone adaptation model. Secondly, critical modeling parameters such as lazy zone bounds, type of rate equation and bone overloading response are classified depending on their influence on the numerical predictions of bone adaptation. Guidelines are proposed to establish the dominant model parameters based on experimental and simulated data.

Device-related pressure ulcers from a biomechanical perspective

Pressure ulcers (PUs) in the pediatric population are inherently different from those in adults, in their risk factors and etiology, with more than 50% of the cases related to contact with medical equipment at the care setting. The aims of this study were to: (i) Determine the mechanical loads in the scalp of a newborn lying supine, near a wedged encephalogram electrode or wire, which is deforming the scalp at the occiput. (ii) Evaluate the effect of a doughnut-shaped headrest on the mechanical state of tissues at the same site. We used finite element computational modeling to simulate a realistic three-dimensional head of a newborn interacting with the above devices. We examined effective (von Mises) stresses, shear stresses and strain energy density (SED) in the fat and skin tissues at the occipital region. The interfering wire resulted in the worse mechanical conditions in the soft tissues, compared to the lodged electrode and use of a doughnut-shaped headrest, with 345% and 50% increase in effective stresses in skin and fat tissues, respectively. Considering that elevated and localized tissue deformations, stresses and SED indicate a risk for PUs, our simulations suggest that misplaced medical devices, and using a doughnut-shaped headrest, impose an actual risk for developing device-related PUs. We conclude that guidelines for pediatric clinical care should recommend routine inspection of the medical device placement to prevent harmful contact conditions with the patient. Furthermore, improved design of medical equipment for pediatric settings is needed in order to protect these fragile young patients from PUs.

다공성 기지를 갖는 복합재의 이미지 기반 전산 모형화 및 기공 탄성 계수 산출

Poroelastic analyses of fiber-reinforced composites were performed using image-based computational models. The section image of a porous matrix was analyzed in order to investigate the porosity, number of pores, and distribution of pores. The resolution, location, and size of the section image were considered to quantify the effective elastic modulus, poroelastic parameter, and strain energy density using the image-based computational models. The poroelastic parameter was calculated from the effective elastic modulus and pore pressure-induced strain. In addition, the results of the poroelastic analyses were verified through representative volume elements by simplifying various pore configurations and arrangements. Corresponding Author, esshin@jbnu.ac.kr 2014 The Korean Society of Mechanical Engineers C

COMPUTATIONAL HOMOGENIZATION METHOD AND REDUCED DATABASE MODEL FOR HYPERELASTIC HETEROGENEOUS STRUCTURES

A nonconcurrent multiscale homogenization method is proposed to compute the response of structures made of heterogeneous hyperelastic materials. The method uses a database describing the effective strain energy density function (potential) in the macroscopic right Cauchy-Green strain tensor space. Each value of the database is computed numerically by means of the finite element method on a representative volume element, the corresponding macroscopic strains being prescribed as boundary conditions. An interpolation scheme is then introduced to provide a continuous representation of the potential, from which the macroscopic stress and elastic tangent tensors can be derived during macroscopic structures calculations. To efficiently compute the interpolations at the macroscopic scale, the full database is reduced by a tensor product approximation. Several extensions are provided to handle issues related to finite strains. The accuracy of the method is tested through different numerical tests involving composites at finite strains with isotropic or anisotropic microstructures. Second-order accuracy is achieved during the macroscopic Newton-Raphson iterations.

Quantification of embryonic atrioventricular valve biomechanics during morphogenesis

Tissue assembly in the developing embryo is a rapid and complex process. While much research has focused on genetic regulatory machinery, understanding tissue level changes such as biomechanical remodeling remains a challenging experimental enigma. In the particular case of embryonic atrioventricular valves, micro-scale, amorphous cushions rapidly remodel into fibrous leaflets while simultaneously interacting with a demanding mechanical environment. In this study we employ two microscale mechanical measurement systems in conjunction with finite element analysis to quantify valve stiffening during valvulogenesis. The pipette aspiration technique is compared to a uniaxial load deformation, and the analytic expression for a uniaxially loaded bar is used to estimate the nonlinear material parameters of the experimental data. Effective modulus and strain energy density are analyzed as potential metrics for comparing mechanical stiffness. Avian atrioventricular valves from globular Hamburger–Hamilton stages HH25–HH34 were tested via the pipette method, while the planar HH36 leaflets were tested using the deformable post technique. Strain energy density between HH25 and HH34 septal leaflets increased 4.6±1.8 fold (±SD). The strain energy density of the HH36 septal leaflet was four orders of magnitude greater than the HH34 pipette result. Our results establish morphological thresholds for employing the micropipette aspiration and deformable post techniques for measuring uniaxial mechanical properties of embryonic tissues. Quantitative biomechanical analysis is an important and underserved complement to molecular and genetic experimentation of embryonic morphogenesis.

A modification of Smith–Watson–Topper damage parameter for fatigue life prediction under non‐proportional loading

ABSTRACTIn this paper, the shortcomings of the Smith–Watson–Topper (SWT) damage parameter are analysed on the basis of the critical plane concept. It is found that the SWT model usually overestimates the fatigue lives of materials since it only takes into account the fatigue damage caused by the tensile components. To solve this problem, Chen et al. (CXH) modified the SWT model through considering the shear components. However, there are at least two problems present in CXH model: (1) the mean stress is not considered and (2) the different influence of the normal and shear components on fatigue life is not included. Besides, experimental validations show that the modification by Chen et al. usually leads to conservative fatigue life predictions during non‐proportional loading. In order to overcome the shortcomings of SWT and CXH models, a damage parameter as the effective strain energy density (ESED) is proposed. Experimental validations by using eight kinds of materials show that the ESED model can give satisfactory fatigue life predictions under the non‐proportional loading.

A Bionic Approach for Topology Optimization for Tension-only or Compression-only Design

A new bionic approach is presented to find the optimal topologies of a structure with tension-only or compression-only material based on bone remodelling theory. By traditional methods, the computational cost of topology optimization of the structure is high due to material nonlinearity. To improve the efficiency of optimization, the reference-interval with material-replacement method is presented. In the method, firstly, the optimization process of a structure is considered as bone remodelling process under the same loading conditions. A reference interval of Strain Energy Density (SED), corresponding to the dead zone or lazy zone in bone mechanics, is adopted to control the update of the design variables. Secondly, a material-replacement scheme is used to simplify the Finite Element Analysis (FEA) of structure in optimization. In the operation of material-replacement, the original tension-only or compression-only material in design domain is replaced with a new isotropic material and the Effective Strain Energy Density (ESED) of each element can be obtained. Finally, the update of design variables is determined by comparing the local ESED and the current reference interval of SED, e.g., the increment of a relative density is nonzero if the local ESED is out of the current reference interval. Numerical results validate the method.