Strain Energy Density Research Articles

Dynamic compressive properties and constitutive model of waste crumb rubber (CR) modified ultra-high performance engineered cementitious composites (UHP-ECC)

Ultra-high-performance engineered cementitious composites (UHP-ECC) are well-known for their outstanding mechanical properties under static and dynamic loads. However, the effect of modifying UHP-ECC with crumb rubber (CR) on its performance, especially under high strain rates, remains poorly understood. This study systematically examined the dynamic compressive behavior and characteristic failure patterns of CR-modified UHP-ECC under high strain rates using a split Hopkinson pressure bar (SHPB) system. The study focused on evaluating the dynamic compressive strength, stress-strain response, and strain energy density (SED). A dynamic increase factor (DIF) formula was developed based on experimental data, and a nonlinear viscoelastic constitutive model was proposed to predict compressive behavior across various strain rates. Results indicate that while the addition of 10% CR slightly reduces the quasi-static compressive and flexural strength, it significantly enhances tensile ductility. Under high strain rates, CR-modified UHP-ECC shows no substantial effect on peak compressive stress but a marked increase in peak compressive strain. The SED also increased by 20.2%, outperforming other high-performance materials such as UHPC and normal ECC. These findings demonstrate that incorporating 10% CR into UHP-ECC offers considerable advantages, particularly in enhancing structural resilience under dynamic loading conditions. The proposed nonlinear viscoelastic model accurately predicts the compressive behavior of CR-modified UHP-ECC, presenting significant implications for practical engineering applications where high strain rate performance is crucial.

Coupling effects of loading rate and temperature on mode I dynamic fracture characteristics of ductile cast iron

Engineering structures made of ductile cast iron (DCI) have a potential risk of failure due to extreme service environments such as high velocity impacts and sub-zero temperatures. Therefore, it is of great importance to investigate the dynamic fracture behavior of DCI under the coupling effect of rate and temperature. In this paper, two sets of impact velocities (5 m/s and 13.5 m/s), and four sets of temperatures (20 °C, −40 °C, −60 °C, and −80 °C) were specially designed to investigate the coupling effect on the mode I dynamic fracture toughness (DFT). The results show that DFT is positively correlated with impact velocity at 20 °C, −40 °C and −60 °C. However, at −80 °C, the rate effect is reversed. Moreover, DFT decreases with decreasing temperature regardless of impact velocity. With microscopic analysis, the phenomenon of ductile–brittle transition (DBT) was observed in the failure of the material, and it’s verified by dynamic tensile tests. The ductile–brittle transition temperature (DBTT) of DCI is determined as −39.7 °C by comparing the DFT with the strain energy density (SED) characterization method.

AI-driven data fusion modeling for enhanced prediction of mixed-mode I/III fracture toughness

This research evaluates the use of artificial intelligence to enhance the accuracy of predictions for mixed-mode I/III fracture toughness in polymethyl methacrylate . Traditionally, assessing fracture toughness relies heavily on destructive testing methods, specifically using edge-notch disc bend specimens subjected to three-point bending tests. These established methods are not only expensive and time-consuming but also frequently limited by the availability of data. To address these challenges, this study introduces a data fusion source modeling approach. This approach integrates primary fracture toughness test data with secondary predictive data derived from the maximum tangential stress criterion and the local strain energy density criterion. By employing adaptive boosting and general regression neural network algorithms, the models developed in this research demonstrate a marked improvement in predictive performance compared to traditional primary source models. Additionally, a feature importance analysis using Shapley Additive exPlanations values reveals that the mode mixity parameter, specimen thickness, and radius are critical factors influencing fracture toughness. The study highlights that while mode mixity emerges as the most significant factor, a reduction in specimen thickness generally leads to decreased fracture toughness, whereas an increase in radius has a more complex, often negative, effect. The results of this study indicate that AI-powered models using data fusion can overcome limitations related to data scarcity, enabling more accurate predictions in fracture mechanics. Furthermore, this approach provides a pathway for utilizing AI in other engineering domains where data sets are limited.

A unified strain energy-based fatigue life prediction methodology for composite and metal adhesive joints: Effects of adhesive, geometry and, environment

Ensuring the efficiency and safety of bonded structures across diverse engineering sectors requires a robust fatigue model for adhesive joints in both composite and metal substrates. This study presents a unified fatigue life prediction methodology for similar and dissimilar composite and metal adhesive joints. The proposed model integrates hydrostatic and deviatoric strain energy density components with the theory of critical distances. The model has been validated against diverse experimental data from different joint geometries, loading conditions, and different environments. The linear-elastic assumption is sufficient for toughened epoxies, whereas an elastoplastic model is necessary for ductile methacrylate adhesives. Calibration based on a single fatigue test yields the highest accuracy, while quasi-static test calibration is also satisfactory. The methodology addresses diverse loading conditions and environmental factors, showing a strong correlation with experimental results. It advances the performance, design, and reliability of similar and dissimilar composite to metal adhesive joints.

Hybrid-driven probabilistic damage assessment of creep-fatigue-oxidation interaction

This paper presents a hybrid-driven probabilistic damage assessment approach by considering creep-fatigue-oxidation damage interaction (CFO-DI). Based on generalized strain energy density exhaustion (GSEDE) framework, the hybrid-driven concept integrates the strengths of both physics-based models and machine learning, exploring the frontier from deterministic evaluation to probabilistic assessment. Experimental investigations involving generalized creep-fatigue loading tests are conducted to establish a comprehensive dataset in Inconel 718 at 650 °C. Deterministic models for fatigue, creep, and oxidation damages are developed, and their interactions are analyzed using the GSEDE framework. To tackle limited experimental data, a divide-and-conquer strategy employing machine learning models is implemented for data augmentation. Probabilistic assessments are performed incorporating uncertainties from material properties, loading conditions, and model parameters using Monte Carlo simulations and Latin Hypercube Sampling. The results demonstrate accurate life prediction accuracy and reliable probability distributions in the presence of oxidation damage. Finally, a novel three-dimensional probabilistic CFO-DI assessment diagram quantified by the confidence level is developed, providing a technical pathway for safe-life design in high-temperature structural applications.

Ex-situ observation of ferrite grain growth behavior in a welded 9Cr-1Mo-V-Nb steel during aging at 740 °C

It has recently been reported that ferrite grains coarsened to several hundred micrometers were occasionally observed in long-term serviced 9Cr-1Mo-V-Nb steel welds. To clarify the factors that cause ferrite grain growth in martensite during high-temperature exposure, alternating aging heat treatment and observation of the same field of view was performed using a welded 9Cr-1Mo-V-Nb steel. Such ex-situ observations revealed that the rapid grain growth of ferrite by consuming martensite occurred in the weld metal during aging at 740 °C. In the test material used in this study, some δ-ferrite grains were observed in the weld metal near the fusion line. In the region where δ-ferrite grains were observed, the concentrations of austenite-forming elements such as Mn and Ni were locally decreased in the matrix due to dilution by the base metal, promoting δ-ferrite retention after welding. Ex-situ observation indicated that no significant grain growth of δ-ferrite occurred during aging. Therefore, it was suggested that a new ferrite grain formation followed by rapid grain growth consuming martensite occurred during aging. The elastic strain energy density of the dislocations PMdis and the interfacial energy density PMsurf in martensite can affect the driving force for ferrite grain growth by consuming martensite. Based on the evaluation results for PMdis and PMsurf after 500 h of aging, PMsurf was considered to be the main driving force for ferrite grain growth. Although the ferrite-formation process could not be directly observed, it is possible that ferrite was formed by the recrystallization of martensite through the bulging mechanism.

Fracture toughness using strain energy density for aluminum alloys with notched round bars

ABSTRACT This paper introduces a refined method for measuring the fracture toughness of aluminium alloys using notched round bar (NRB) specimens. By incorporating plasticity effects and empirical adjustments, a model based on strain energy density (SED) is proposed. The refined values were validated against experimental data for aluminium 6061-T6 and 2011-T6 alloys. The results demonstrate that the proposed method yields fracture toughness values closely matching the experimental results, underscoring its potential for standardisation and practical applications in material testing and engineering design.

Application of the Effective critical plane approach for the fatigue assessment of ductile cast iron under multiaxial and non-proportional loading conditions

The fatigue assessment of structural components, especially made of ductile cast iron subjected to complex loading conditions, heavily relies on analyzing fatigue damage resulting from stress concentrations induced by geometric irregularities like notches and shrinkage pores. Standard methodologies, encompassing the Theory of Critical Distances (TCD), Strain Energy Density (SED), and Critical Plane (CP), have played pivotal roles in predicting fatigue strength for components featuring such irregularities. In this work, the authors explore the applicability of the Effective Critical Plane (ECP) approach on ductile cast iron notched specimens subjected to multiaxial and non-proportional loading conditions. The method focuses on evaluating the critical plane factor, after averaging the stress and strain field within a given control volume or area (i.e. defined by a control radius), centered on the critical node. The study aims to enhance the accuracy of fatigue life prediction for structural components made of ductile cast iron, thereby contributing to the improvement and practical applicability of fatigue assessment under complex loading conditions. The methodology, integrating the Smith-Watson-Topper and Fatemi-Socie CP factor, was applied to several experimental fatigue data obtained from ductile cast iron notched specimens, tested under multiaxial non-proportional loading conditions. After establishing the control radius associated with the investigated material, the method was utilized to perform a fatigue life forecast analysis on a specimen with porous defects.

Behind Amor Blunt Trauma Lung and Liver Strains From Indenter Loading Via Finite Element Modeling

Abstract To determine behind armor blunt trauma (BABT) injury criteria, experiments have been conducted by launching blunt projectiles at live swine at velocities up to 65 meters per second (m/s) using one type of indenter design. To ensure the generalizability of the developed injury criteria, additional tests with different indenter designs are needed. The objectives of this study were to evaluate the kinematics and injury parameters from two indenter designs using human body finite element modeling. The simulation matrix consisted of chord and cylindrical shape indenter designs with two different masses of 150 and 230 grams. They were used to impact the liver and lung regions at velocities of 30 and 60 m/s using a human body model. Rib and lung strains from lung impacts and rib strain and liver strain energy densities (SED) from liver impacts were used to evaluate the design variables of mass and shape. Both designs played a role in skeletal and organ injury parameters. Analysis revealed an increased susceptibility for skeletal and organ traumas with the high mass indenter during high velocity impacts. The cylindrical indenter may be protective for organ injuries due to the larger area of loading on the ribcage compared to the chord indenter. Results from the chord indenter may serve as a conservative estimate of injury criteria.

Impact resistance performance of 3D woven TZ800H plates with different textile architecture

Two typical methods commonly used to improve the mechanical properties and impact resistance properties of 3D woven composites are studied, namely weave pattern and layered architectures. The mechanical property and impact resistance performance were studied by utilising the quasi-static compressive test, split Hopkinson pressure bar (SHPB) test and ballistic impact test. The compressive responses in warp and weft directions with different strain rates 0.001, 500 and 1300 s-1 were presented and analysed, providing strain rate influence on the material strength of different 3D woven composites. The impact resistance performance including damage mode, ballistic limit and specific energy absorption of three structures were discussed through impact tests. The results reveal that as the strain rate increases, the compressive strength and Young's modulus in both directions of 3D woven composites exhibit a significant increase. The compressive strength and modulus in the warp direction of the composites can be enhanced by using shallow interlocking of the warp tow or layered architectures. However, the two methods degrade the failure strain and weaken the strain rate strengthening effect of compressive strength in the weft direction, resulting in a significant decrease in the average strain energy density. For the ballistic impact case, the crimp of warp tows would decrease its load-bearing capacity, while resisting matrix crack growth under the ballistic impact. The significant reduction in the average strain energy density in the weft direction leads to a decrease in ballistic limit and specific energy absorption capacity under ballistic impact.

Anand Model Constants of Sn-Ag-Cu Solders: What Do They Actually Mean?

Abstract Finite element simulations are broadly utilized to calculate the thermally induced mechanical deformations of interconnecting solder materials in electronics. The Anand unified viscoplasticity model is a prevalent choice for simulating the mechanical responses of solder joints, comprising nine parameters whose individual effects are not fully addressed in the literature. For this reason, this study examines the influence of each Anand parameter on the mechanical response of leadless Tin-Silver-Copper solder alloys with varying silver content through comprehensive nonlinear finite element simulations. Anand model coefficients for different SAC alloys, including SAC105, SAC205, SAC305, and SAC405, were sourced from existing literature and used to establish a systematic study matrix for each parameter. These coefficients were then integrated into thermomechanical simulations to induce inelastic deformations in the solder interconnections. The response of the solder interconnects is analyzed using equivalent inelastic strains and inelastic strain energy density. Results indicated that specific Anand parameters could skew the solder joint stress-strain response towards brittle behavior, while other parameters could lead to a more ductile response. Through statistical factorial analysis, the significance of each parameter is found to vary considerably, ranging from negligible to highly significant. The findings of this study are crucial for understanding the behavior of different SAC solder configurations and their expected thermal fatigue performance based on Anand creep constants. Furthermore, this paper provides foundational insights into the interpretation of Anand coefficients and their influence on the mechanical response of solder materials, a topic not yet explored in the existing literature.

Dynamic response and constitutive model for coal-rock composite material subjected to impact loading

Comprehensive understanding related to the dynamic responses of the composite materials is critical to prevent natural disasters in underground engineering and mining activities. For this reason, experimental studies were undertaken using dynamic impact tests at different strain rates to explore the failure mechanism using split Hopkinson pressure bar (SHPB) system and Digital Image Correlation (DIC) on composite materials; coal-rock (C-R) and rock-coal (R-C). These results suggest a linear correlation between dissipative energy and fractal dimensions. In addition, they suggest cracks that develop predominantly on the coal side have a dominant influence on the failure process that could be attributed to the interface effects. The damage evolution was established assuming elastic conditions using a three-dimensional coupled FDM-DEM numerical simulation system. Anumerical approach is also developed for interpreting failure mechanism of the composite material based on the fabric tensor and strain energy density. Finally, a constitutiverelationshipis established considering strain rate effect and damage evolution using series element studies. There is a good agreement between the experimental and numerical results, providing justification to the proposed constitutive relationship. The numerical approach in this study is promising for assessing the performance of composite materialstaking account of dynamic loading conditions.

Cyclic fatigue responses of double fissure-contained marble: Insights from mechanical responses, energy conversion and hysteresis characteristics

The intermittent fissures are widely distributed in rock mass, rock bridge segment among the fissures are crucial to the mechanical properties and stability of rock mass. The static mechanical behaviors of fissure-contained rock were well understood, yet the fatigue mechanical properties were poorly investigated. This work aims to reveal the influence of rock bridge length on rock mechanical responses, energy dissipation and hysteresis characteristics in lab-scale. Marble samples with rock bridge length (RBL) of 10, 20, 30, and 40 mm were prefabricated as parallel pattern, and they are subjected to multi-level fatigue loading paths. Testing results show that the rock bridge between the fissures resists rock deformation and the terminal volumetric strain decreases with increasing RBL. In addition, the hysteresis energy density and damage evolution are influenced by the rock bridge length. The strain energy density development aligns with deformation pattern results. Deterioration in the rock bridge structure causes reduced capacity and higher energy consumption. More energy is required to drive damage progression and crack connectivity in rocks with a larger RBL. Ultimately, two hysteresis indices are introduced to characterize fatigue damage evolution. These indices exhibit nearly opposing trends throughout the loading process. It is proposed that extending the rock bridge length results in a roughly linear trend for both hysteresis indices.

Mechanical Characterization of Non-degraded Porcine Annulus Fibrosus Material Properties.

Porcine cervical spines are commonly used as a surrogate for human lumbar spines due to their similar anatomic and mechanical characteristics. Despite their use in spinal biomechanics research, porcine annulus fibrosus (AF) yield and ultimate properties have not been fully evaluated. This study sought to provide a novel dataset of elastic, yield, and ultimate properties of the porcine AF loaded in the circumferential direction. AF specimens were dissected from porcine cervical spines (C3/C4-C6/C7) oriented in the circumferential direction. Specimens were uniformly hydrated before being quasi-statically distracted to failure. Linear modulus, yield stress and strain, ultimate stress and strain, and ultimate strain energy density were calculated. Differences between spinal levels, circumferential regions, and radial regions were identified using multifactor ANOVA tests. AF specimens showed a regionally dependent response between outer and inner radial regions, but not between spinal level and circumferential region. The outer region was significantly stronger and stiffer than the inner regions. In both outer and inner tissue, mechanical yield occurred at approximately 80% of their ultimate properties. This study generated a novel dataset of elastic, yield, and ultimate properties of the porcine AF. The data can be used in future research that requires a robust database of healthy, non-degenerated AF mechanical properties, such as the development of future finite-element models.

Elastic bodies with kinematic constraints on many small regions

We study the equilibrium of hyperelastic solids subjected to kinematic constraints on many small regions, which we call perforations. Such constraints on the displacement u are given in the quite general form u(x)∈Fx, where Fx is a closed set, and thus comprise confinement conditions, unilateral constraints, controlled displacement conditions, etc., both in the bulk and on the boundary of the body. The regions in which such conditions are active are assumed to be so small that they do not produce an overall rigid constraint, but still large enough so as to produce a non-trivial effect on the behaviour of the body. Mathematically, this is translated in an asymptotic analysis by introducing two small parameters: ɛ, describing the distance between the elements of the perforation, and δ, the size of the element of the perforation. We find the critical scale at which the effect of the perforation is non-trivial and express it in terms of a Γ-limit in which the constraints are relaxed so that, in their place, a penalization term appears in the form of an integral of a function φ(x,u). This function is determined by a blow-up procedure close to the perforation and depends on the shape of the perforation, the constraint Fx, and the asymptotic behaviour at infinity of the strain energy density σ. We give a concise proof of the mathematical result and numerical studies for some simple yet meaningful geometries.

Nonlinear incompressible shear wave models in hyperelasticity and viscoelasticity frameworks, with applications to Love waves

General equations describing shear displacements in incompressible hyperelastic materials, holding for an arbitrary form of strain energy density function, are presented and applied to the description of nonlinear Love-type waves propagating on an interface between materials with different mechanical properties. The model is valid for a broad class of hyper-viscoelastic materials. For the Murnaghan constitutive model, shear wave equations contain cubic and quintic differential polynomial terms, including viscoelasticity contributions in terms of dispersion terms that include mixed derivatives uxxt of the material displacement. Full (2+1)-dimensional numerical simulations of waves propagating in the bulk of a two-layered solid are undertaken and analysed with respect to the source position and mechanical properties of the layers. Interfacial nonlinear Love waves and free upper surface shear waves are tracked; it is demonstrated that in the fully nonlinear case, the variable wave speed of interface and surface waves generally satisfies the linear Love wave existence condition c1<v<c2, while tending to the larger material wave speed c1 or c2 for large times.

Unlocking Electrostrain in Plastically Deformed Barium Titanate.

Achieving substantial electrostrain alongside a large effective piezoelectric strain coefficient (d33*) in piezoelectric materials remains a formidable challenge for advanced actuator applications. Here, a straightforward approach to enhance these properties by strategically designing the domain structure and controlling the domain switching through the introduction of arrays of ordered {100}<100> dislocations is proposed. This dislocation engineering yields an intrinsic lock-in steady-state electrostrain of 0.69% at a low field of 10kVcm-1 without external stress and an output strain energy density of 5.24Jcm-3 in single-crystal BaTiO3, outperforming the benchmark piezoceramics and relaxor ferroelectric single-crystals. Additionally, applying a compression stress of 6MPa fully unlocks electrostrains exceeding 1%, yielding a remarkable d33* value over 10000pmV-1 and achieving a record-high strain energy density of 11.67Jcm-3. Optical and transmission electron microscopy, paired with laboratory and synchrotron X-ray diffraction, is employed to rationalize the observed electrostrain. Phase-field simulations further elucidate the impact of charged dislocations on domain nucleation and domain switching. These findings present an effective and sustainable strategy for developing high-performance, lead-free piezoelectric materials without the need for additional chemical elements, offering immense potential for actuator technologies.

Bio-inspired space-filling fractal metamaterial

Inspired by mammal cranial sutures with spatiotemporal morphological variation, two-phase space-filling fractal metamaterial was designed. Designs with different levels of complexity are fabricated via multi-material polymer jetting. Mechanical tests and systematic finite element (FE) simulations are conducted to evaluate the mechanical performance of the designs. It is found that the hierarchical number N of two-phase space-filling fractal metamaterial played an important role in their mechanical behaviors. The experimental results show that with increasing the hierarchical number N, these metamaterials show enhanced stiffness, strength, and toughness under tensile tests. From the simulation results, we found by decoupling the strain energy density in two phases, with increasing N, the soft phase has contributed almost the same energy level, however, the hard phase has contributed increasing energy level. Moreover, we found the volume fraction and the stiffness ratio of the hard phase dominate the overall mechanical properties of these two-phase space-filling fractal metamaterial. The bio-inspired mechanical metamaterials have broad applications in engineering materials for dissipating energy dissipation, mitigating impact, and retarding damages.

A peridynamic compensated critical energy density criterion for mixed-mode fracturing in quasi-brittle materials

This study integrates a compensated critical energy density criterion (CCED) into the ordinary state-based peridynamics framework to enable detailed analysis of mode I, mode II, and mixed-mode fracturing. The proposed bond failure criterion builds upon the advantages of the traditional critical energy density (CED) to enable precise calculation of strain energy density. Additionally, the critical bond rotation criterion (CR) is employed to account for shear bonds that the CED might overlook. Using the peridynamic differential operator (PDDO) format in formulating the entire computational framework enhances both accuracy and numerical stability. The model’s validation is conducted through benchmark examples, including the mode I double cantilever beam test, the mode II compact shear test, and a mixed-mode mechanical fracturing test in pre-notched specimens with central borehole loading. Convergence studies, comparative analyses with finite element method simulations and theoretical solutions thoroughly examine the proposed model’s performance and accuracy.

Weight Minimization of Truss Using Strain Energy Density and Genetic Algorithm for Single and Multiple Load Cases

Weight Minimization of Truss Using Strain Energy Density and Genetic Algorithm for Single and Multiple Load Cases