Strain Energy Density Research Articles

A phase field framework for corrosion fatigue of carbon steel

Corrosion fatigue damage occurs when metallic materials are subjected to cyclic loading in a corrosive medium. In this study, a phase field framework is proposed to predict the corrosion fatigue of carbon steels. The coupling effect of fatigue and corrosion is explicitly implemented in the proposed phase field framework by coupling the displacement field, electrochemical field and phase field. A degradation function of the interface free energy density with the consideration of elastic and plastic strain energies is introduced to account for the fatigue damage accumulated during the corrosion fatigue process. The applicability of this framework is validated by accurately capturing the pure fatigue and corrosion fatigue behaviors of compact tension specimens, particularly the acceleration effect of corrosion on the fatigue crack growth. The propagation morphology and rate of the corrosion fatigue crack in single pit and multiple pit models are studied. The distribution of stress state and strain energy density induces the directionality of crack propagation. The influence of loading frequency on the corrosion fatigue process is discussed in detail. Due to the corrosion-fatigue coupling effect, the corrosion rate increases with increasing of the loading frequency, resulting in an accelerated corrosion fatigue process. Moreover, the significance of the plasticity in the prediction of corrosion fatigue is emphasized.

A simplified approach for predicting last ply failure load of VO-notched laminated composite components

In this study, the last ply failure (LPF) load of composite laminates with VO-notches under pure mode I and mixed mode I/II was investigated experimentally, analytically, and numerically. The efficiency of the virtual isotropic material concept (VIMC) in combination with a new combined failure criterion was also evaluated. The J-integral and average strain energy density failure criteria were combined with VIMC. Using analytical expressions for the J-integral and the average strain energy density (ASED), the prediction of the fracture load of notched components was simplified and expedited. The experimental fracture loads of notched specimens were determined for quasi-isotropic and cross-ply laminates under mode I and mixed mode I/II. The fracture loads were predicted using analytical expressions for the J-integral and ASED, combined with VIMC. The VIMC-ASED and VIMC-J-integral failure criteria demonstrated good accuracy in predicting the LPF, with the highest accuracy for mode I, slightly reduced accuracy for mixed mode I/II. VIMC, in combination with energy-based failure criteria, effectively predicts the fracture load. The use of analytical expressions further simplified and expedited the prediction process without losing accuracy, making it a practical approach for engineering applications. Additionally, this approach significantly reduces the time and cost associated with extensive experimental testing.

Re‐derivation and mathematical analysis for linear peridynamics model for arbitrary Poisson ratio's material

AbstractThis paper is concerned with the modeling and mathematical analysis of linear peridynamic model for arbitrary Poisson ratio's material. Based on the fundamental laws of dynamics, we re‐derive the bond‐based peridynamic model for anisotropic materials by relaxing certain assumptions. Through this process, we draw several significant conclusions, such as the relationship between the equivalent strain energy density hypothesis and the convergence of the peridynamic operator to the classical Navier operator. Additionally, the well‐posedness of time‐dependent peridynamic equations of motion is established. Finally, some necessary conditions for the material stability of anisotropic material are given.

Numerical study of steady-state dynamic characteristic of non-pneumatic tire with local structural damage

Non-pneumatic tires (NPTs) fundamentally avoid the risk of tire blowout of traditional pneumatic tires, but the overall and key component performance inevitably degrades due to factors such as high temperature, variable load, variable working conditions and impact in its service process. Once this leads to failure, it will significantly impact on vehicle safety. To this end, this paper studied the influence of local structural damage on the dynamic response of NPTs, which lays the foundation for realizing health monitoring of intelligent non-pneumatic tires (INPTs). Firstly, a three-dimensional nonlinear finite element model of NPTs was established, and the failure locations of NPTs were determined by fracture mechanics and maximum strain energy density; Secondly, the influence of structural damage on the static and dynamic performance of NPTs was analyzed; Finally, the sensitivity of the acceleration signal and sensor position of the tire inner liner to the local structural damage was studied. The research results show that structural damage will cause the stress of the spokes and shear layer to increase, and as the number of broken spokes increases, the shear layer will bear a larger load. Compared with the circumferential and lateral acceleration, the radial acceleration has the highest sensitivity to the damage of NPTs. The sensor closest to the damage location is the most sensitive to the damage. The research results provide a reference for the structural optimization design and health monitoring of INPTs.

Numerical Evaluation of the performance of different composite bridge decks.

Abstract A comparative analysis was conducted to facilitate the selection of composite bridge deck configurations. Composite bridge decks, with their lightweight design, corrosion resistance, prefabrication, reduced construction time, and wide design flexibility, reduce substructure costs and offer durability. Static and buckling analyses were conducted using finite element software to compare their weight, stiffness, composite inverse reverse factor, and buckling resistance. The analyses considered the same boundary conditions, loads, composite layup configuration, and material mechanical properties. Various geometries were selected from different literature to evaluate their performance and identify an appropriate design. This paper presents a numerical investigation into the structural performance of five different composite bridge decks under the same conditions. G. Rectangle emerged as the easiest manufacturing deck, consisting of pultruded rectangle tubes, and the worst deck for deflection and buckling. The G. Triangle has the lowest deflection and the highest strain energy density. The G. Arch distinguished as the lightest bridge deck, also has good performance against buckling and a low strain energy density. Finally, the G. 2Trapezoidal and G. Trapezoidal have better total performance than all other decks as they have the highest inertia.

Biomechanical evaluation of a sheep tracheal scaffold.

Tissue engineering is a set of techniques for producing or reconstructing tissue that primarily aims to restore or improve the function of tissues in the human body. The aim of the present study was to evaluate the mechanical and histological characteristics of decellularized tracheal scaffolds prepared in comparison with fresh trachea for use in tracheal repair. In order to prepare the scaffold, sheep's trachea was prepared and after cleaning the waste tissues, they were decellularized. Then decellularized scaffolds were evaluated histologically and laboratory and numerical study of the nonlinear mechanical behavior of tracheal tissue and scaffold and their comparison. Examining the results of histological evaluations showed that the decellularization of the scaffolds was completely done. These results were confirmed by hematoxylin-eosin staining. Also, the exact hyperelastic properties of tracheal tissue and scaffold were used in biomechanical models, and according to the presented results, the five-term Mooney-Rivlin strain energy density function became a suitable behavioral model for modeling the hyperelastic behavior of trachea and scaffold. In total, the results of this research showed that the scaffolds obtained from decellularization by preserving the main compositions of the desired tissue can be a suitable platform for investigating cell behaviors.

Assessment of fracture criteria for cracking behavior of asphalt concrete using various SCB specimen sizes

AbstractIn this research, five fracture criteria (maximum tangential stress (MTS), generalized maximum tangential stress (GMTS), minimum strain energy density (SED), generalized minimum strain energy density (GSED), and modified maximum tangential stress (MMTS)) were evaluated for asphalt concrete using fracture data from tests conducted under various loading modes and conditions. Tests were performed on semi‐circular bend (SCB) specimens of various sizes. Based on the results, the MTS criterion tends to overestimate fractures, while the GMTS criterion exhibits more precise predictions. The SED and GSED criteria inaccurately predict fractures, especially under certain loading conditions. The MMTS criterion shows superior predictive capability for asphalt concrete fractures. Specimen size influences fracture resistance, with larger specimens exhibiting higher fracture values. Overall, the MMTS criterion closely mirrors the crack growth behavior of asphalt concrete, highlighting its precision in predicting fracture results. The errors between the predictions of the fracture criteria MTS, GMTS, SED, GSED, and MMTS, and the experimental results ranged from 0% to 42.8%, 0% to 22.5%, 0% to 52.7%, 0% to 42.4%, and 0% to 14.5%, respectively.

Topology Optimization Driven Bone-Remodeling Simulation for Lumbar Interbody Fusion.

This study proposes a numerical approach for simulating bone remodeling in lumbar interbody fusion (LIF). It employs a topology optimization method to drive the remodeling process and uses a pixel function to describe the structural topology and bone density distribution. Unlike traditional approaches based on strain energy density or compliance, this study adopts von Mises stress to guide the remodeling of LIF. A novel pixel interpolation scheme associated with stress criteria is applied to the physical properties of the bone, directly addressing the stress shielding effect caused by the implanted cage, which significantly influences the bone remodeling outcome in LIF. Additionally, a boundary inverse approach is utilized to reconstruct a simplified analysis model. To reduce computational cost while maintaining high structural resolution and accuracy, the Scaled Boundary Finite Element Method (SBFEM) is introduced. The proposed numerical approach successfully generates results that closely resemble human lumbar interbody fusion.

Hyperelastic Material Modeling Using First- and Second-Order Differential Operations with Complex Step Derivative Approximation

This study proposes a method for calculating hyperelastic materials’ stress and tangent modulus using a generalized complex-step derivative approximation. This method is accurate and requires only the operational code of the strain energy density function. Moreover, because it uses only standard complex operations in Fortran, no prior preparation is required, making the method convenient and practical. The effectiveness of the method was confirmed by applying it to the implementation of isotropic and anisotropic hyperelastic models.

Fatigue assessment of structural components through the Effective Critical Plane factor

The integrity assessment of structural components under complex loading conditions relies on the evaluation of the fatigue damage typically arising from stress concentrations, such as geometric irregularities, notches, weld beads, grooves etc.. Various methodologies, including the Notch Stress Approach (NSA), the Theory of Critical Distances (TCD), the Strain Energy Density (SED), and the Critical Plane (CP) concept, have been pivotal in assessing fatigue strength for notched and welded components. Recent works combine some of the above mentioned methodologies, while other works propose to vary the embedded parameters accounting for the loading type or the fatigue lives, trying to improve the accuracy of the fatigue assessment process. This paper introduces a novel approach, the Effective Critical Plane (ECP), which is founded on the critical plane concept. The CP factor is, however, calculated starting from an averaged, over a small volume, stress–strain field. The size of the averaging volume is assumed to be a material parameter and is determined by a best fitting procedure over different experimental data sets. The novel approach is illustrated by means of the Fatemi-Socie and the Smith-Watson-Topper CP damage factors. Its potential application to other CP formulations is straightforward, as well. Literature experimental data for low carbon steel specimens possessing different notches and loading conditions are used to validate the method’s capability in accurately determining the fatigue life and to set the radius of the averaging volume for the given material and CP parameter. A spherical volume or circular area are used in case of fully 3D or 2D numerical models, respectively. Results are compared to those of some of already existing methods, namely SED, TCD and the Modified Wöhler Curve Method.

Influence of interstitial fluid pressure, porosity, loading magnitude, and anisotropy in cortical bone adaptation

Adaptive elasticity in cortical bone has traditionally been modeled using Strain Energy Density (SED). Recent studies have highlighted the importance of interstitial fluid in bone adaptation, yet no research has quantified the role of interstitial fluid pressure and its effects, specifically incorporating both SED and interstitial fluid pressure in the adaptation process. This study introduces a novel formulation combining theory of porous media and theory of adaptive elasticity that considers both SED and interstitial fluid’s pressure in cortical bone adaptation. The formulation is solved using ANSYS Fluent and a MATLAB script, and sensitivity analyses were conducted, analyzing various porosities, loading magnitudes, anisotropic properties of cortical bone, and involvement coefficients of interstitial fluid’s pressure. This study reveals that bones with different vascular porosities (PV) tend to achieve similar density distributions under uniform loading over time. This highlights the significant role of interstitial fluid pressure in accelerating the convergence to optimal bone properties, especially in specimens with larger PV porosities. The findings emphasize the importance of fluid pressure in bone remodeling, aligning with previous studies. Furthermore, this study demonstrates that considering transversely isotropic material properties can significantly alter the remodeling configuration compared to isotropic material properties. This highlights the importance of accurately representing the anisotropic nature of cortical bone in models to better predict its adaptive responses. However, aspects such as fluid density variations and bone geometry changes remain unexplored, suggesting directions for future research. Overall, this research enhances the understanding of cortical bone adaptation and its mechanical interactions.

Oxygen-induced decomposition of the body-centered cubic HfNbTaTiZr high-entropy alloy

The factors affecting the stability of an initially single-phase body-centered cubic (BCC) HfNbTaTiZr solid solution are investigated by heat treatments at 900 °C up to 1000 h in different atmospheres. The BCC solid solution is stable but oxygen (O) contamination originating from the aging atmosphere leads to the co-precipitation of an O-enriched hexagonal close-packed (HCP) Hf-Zr-rich phase and a second Nb-Ta-rich BCC phase. As O from the atmosphere diffuses from the surface to the core of the sample at a much faster rate along grain boundaries than through the grain interior, this results in an inhomogeneous distribution of precipitates, which formed at grain boundaries by a monotectoid reaction. In contrast, coincidence-site-lattice boundaries are not affected by this transformation owing to their lower diffusivities and lower capacity for heterogeneous nucleation. Similar phases were observed in a HfNbTaTiZr enriched with 3 at.% O, thus confirming the role of oxygen in phase stability.In both alloys, the orientation relationship (OR) between the HCP and BCC phases is predominantly of Burgers type but our analyses further revealed the presence of two minor ORs. More interestingly, both alloys exhibit rod-shaped HCP precipitates in contrast to the plate-shaped precipitates observed in conventional Ti and Zr-based alloys. This result can be rationalized by calculations of elastic strain energy density, which suggest that the rods should grow along 〈100〉BCC to minimize the elastic strain energy. Deviations of up to 20° from this theoretical direction were observed and are likely related to relaxation by misfit dislocations or by plastic deformation.

Phase field modeling of underloads induced fatigue crack acceleration

This study proposes a novel methodology to model the fatigue crack growth acceleration under underloads using a phase field fracture framework. In this model, fatigue crack growth is characterized by the degradation of fracture toughness, with an emphasis on employing a representative loading strategy instead of explicit cyclic loading, thus accelerating simulations of high-cycle fatigue. The model integrates a zone-based crack acceleration approach that responds to single-cycle underload. Notably, the model adeptly captures the dynamics described by the Paris-Erdogan law. Post-underload crack growth acceleration is simulated by identifying an acceleration zone near the crack tip, inspired by existing models based on the plastic zone. This zone is defined by a strain energy density threshold, and the underload ratio governs the rate of fatigue damage accumulation attenuation within this area. The implementation leverages the UMAT user subroutine in Abaqus, utilizing coupled temperature-displacement elements where temperature analogously represents the phase field parameter. Experimental validation of the model confirms its ability to accurately reflect the loss of fatigue life and the acceleration of crack growth rates in compact tension and middle tension specimens. Additionally, the model shows promise for extension to periodic underloads, highlighting its potential for simulating real-world fatigue scenarios effectively.

Creep-fatigue damage level evaluation based on the relationship between microstructural evolution and mechanical property degradation

Creep-fatigue interaction is identified as a primary failure mode for components operating under high temperatures. As operational durations extend, this interaction not only alters the material's microstructures but also initiates a gradual degradation in mechanical properties, significantly impacting its deformation and damage behaviors. In this work, the dynamic microstructural evolution of GH4169 superalloy during creep-fatigue was elucidated via qualitative characterization, and damage level evaluation method was subsequently developed by bridging microstructure degradation to mechanical property degradation. Creep-fatigue tests were performed at 650 °C with various tensile holding times and were interrupted at lifetime fractions of 10 %, 50 % and 80 % for further analysis and tensile evaluations. Results revealed that the prolonged exposure to holding times induced the coarsening of γ″ precipitates alongside an increase in low-angle grain boundaries, culminating a reduction in creep-fatigue strength. The development of voids and cracks exacerbated the degradation of elongation, leading to a hybrid fracture mode encompassing both intergranular and transgranular cracking paths. Synthesizing microstructural evolutions to qualitatively categorize diverse degradation levels imparted a robust physical basis for damage evaluation. A mapping model was established to correlate the average kernel average misorientation (micro-degradation indicator) with the tensile plastic strain energy density (macro-degradation indicator). The damage level evaluation method was endowed with quantitative metrics utilizing this model, and its generality was additionally validated in P92 steel. This work offers an insight into the quantitative damage evaluations of creep-fatigue-induced degradations in materials, thereby providing a theoretical basis for the development of operation management and inspection plans of components.

Multiaxial fatigue rule applied to disc specimens with variable thickness subjected to an equibiaxial stress state

AbstractAssessing multiaxial fatigue theories requires experimental verification of a failure criterion, posing a significant challenge. This study aims to investigate the validity of a multiaxial fatigue method based on the Lagoda‐Macha‐Sakane (LMS) theory. The LMS theory links the critical strain energy density to the fatigue crack initiation cycles through a non‐linear equation defined by a set of empirical parameters calibrated by a strain‐controlled uniaxial fatigue test. This work adopts the LMS formulation, numerically calibrating the fatigue curve based on strain energy density from uniaxial fatigue experimental data and considering only the LMS theory for the critical strain energy density computation. This method avoids compatibility condition uncertainties and previous identification of material parameters. The study uses bi‐axial bending tests on AISI 316 L steel plate specimens with 3D finite element analyses to support computational assessment. The predictive capability of the model and the effectiveness of the testing method are presented and discussed.

Computational biomechanical study on hybrid implant materials for the femoral component of total knee replacements

Multifunctional materials have been described to meet the diverse requirements of implant materials for femoral components of uncemented total knee replacements. These materials aim to combine the high wear and corrosion resistance of oxide ceramics at the joint surfaces with the osteogenic potential of titanium alloys at the bone-implant interface. Our objective was to evaluate the biomechanical performance of hybrid material-based femoral components regarding mechanical stress within the implant during cementless implantation and stress shielding (evaluated by strain energy density) of the periprosthetic bone during two-legged squat motion using finite element modeling. The hybrid materials consisted of alumina-toughened zirconia (ATZ) ceramic joined with additively manufactured Ti–6Al–4V or Ti–35Nb–6Ta alloys. The titanium component was modeled with or without an open porous surface structure. Monolithic femoral components of ATZ ceramic or Co–28Cr–6Mo alloy were used as reference. The elasticity of the open porous surface structure was determined within experimental compression tests and was significantly higher for Ti–35Nb–6Ta compared to Ti–6Al–4V (5.2 ± 0.2 GPa vs. 8.8 ± 0.8 GPa, p < 0.001). During implantation, the maximum stress within the ATZ femoral component decreased from 1568.9 MPa (monolithic ATZ) to 367.6 MPa (Ti–6Al–4V/ATZ), 560.9 MPa (Ti–6Al–4V/ATZ with an open porous surface), 474.9 MPa (Ti–35Nb–6Ta/ATZ), and 648.4 MPa (Ti–35Nb–6Ta/ATZ with an open porous surface). The strain energy density increased at higher flexion angles for all models during the squat movement. At ∼90° knee flexion, the strain energy density in the anterior region of the distal femur increased by 25.7 % (Ti–6Al–4V/ATZ), 70.3 % (Ti–6Al–4V/ATZ with an open porous surface), 43.7 % (Ti–35Nb–6Ta/ATZ), and 82.5% (Ti–35Nb–6Ta/ATZ with an open porous surface) compared to monolithic ATZ. Thus, the hybrid material-based femoral component decreases the intraoperative fracture risk of the ATZ part and considerably reduces the risk of stress shielding of the periprosthetic bone.

Data-driven continuum damage mechanics with built-in physics

Soft materials such as rubbers and soft tissues often undergo large deformations and experience damage degradation that impairs their function. This energy dissipation mechanism can be described in a thermodynamically consistent framework known as continuum damage mechanics. Recently, data-driven methods have been developed to capture complex material behaviors with unmatched accuracy due to the high flexibility of deep learning architectures. Initial efforts focused on hyperelastic materials, and recent advances now offer the ability to satisfy physics constraints such as polyconvexity of the strain energy density function by default. However, modeling inelastic behavior with deep learning architectures and built-in physics has remained challenging. Here we show that neural ordinary differential equations (NODEs), which we used previously to model arbitrary hyperelastic materials with automatic polyconvexity, can be extended to model energy dissipation in a thermodynamically consistent way by introducing an inelastic potential: a monotonic yield function. We demonstrate the inherent flexibility of our network architecture in terms of different damage models proposed in the literature. Our results suggest that our NODEs re-discover the true damage function from synthetic stress-deformation history data. In addition, they can accurately characterize experimental skin and subcutaneous tissue data.

A multiscale modeling to determine in vitro mechanical responses of different cells at the cell-substrate interface under fluid perfusion.

Investigating the influence of different cellular mechanical and physical properties on cells in vitro is important for assessing cellular activities like differentiation, proliferation, and migration. Evaluating the mechanical response of the cells lodged on a scaffold due to variations in substrate roughness, substrate elasticity, fluid flow, and the shapes of the cells is the main goal of the study. In this comprehensive analysis, a combination of the fluid structure interaction method and the submodeled finite element technique was employed to anticipate the mechanical responses across various cells at the interface between cells and the substrate. Fluid inlet velocity, substrate roughness, and substrate material were varied in this analysis. Different cell shapes were considered along with various components such as cell membrane, cytoplasm, nucleus, and cytoskeletons. This analysis shows the effect of these individual parameters on the elastic strain and strain energy density of cells at the cell-substrate interface. The results highlight that substrate roughness has a more significant impact on the mechanical response of cells at the interface than substrate elasticity. However, effect of the substrate elasticity becomes crucial for extremely soft substrate materials. The results of this research can be applied to identify the optimal parameters for fluid flow and create a suitable condition for cell culture.

Finite Element Analysis of Delamination Initiation in Wind Turbine Blade Spar Caps: Role of Compression, Strain Energy, and Principal Stresses

In wind turbine blades, utilizing the finite element method (FEM) to identify potential failure modes before production provides considerable value and cost savings in contrast to conventional structural tests. This paper focuses on examining the factors leading to delamination in spar caps and exploring the initiation of this delamination using the finite element method. Compression, total strain energy density, and principal stresses are significant among the variables examined to determine the effects on model deformation and crack formation. Examining these variables may contribute to understanding the mechanisms involved in initiating and progressing delamination. This study examined the effects of compression, total strain energy density, and principal stresses on model deformation and crack formation. This study focuses on the effects of principal stresses on displacement, the effects of compression on delamination and crack formation, and the effects of total strain energy density on fracture tendency and failure modes due to delamination.

Rapid screening of creep resistance in additive manufactured Ti-6Al-4V alloy

In this study, rapid screening of creep resistance for additively manufactured (AM) materials is enabled using an accelerated creep test (ACT) called the dynamic negative stepped test (DNST). Additively manufactured components are on the rise in aerospace systems, particularly turbine engine parts. It can be challenging to identify the AM process parameters that correspond to high-temperature strength and creep resistance. Tests are needed that rapidly screen and qualify creep resistance. In DNST, the temperature is held constant, and the specimen is taken through a series of load holds from high to low. The DNST contains two distinct characteristics: (1) the hold duration for each step is “dynamically” controlled to ensure the minimum-creep-strain-rate is achieved at every step, and (2) the “descending” profile maximizes damage accumulation while remaining on the edge of instability. The DNST assesses a material's ability to resist creep damage accumulated across multiple deformation mechanisms. In this study, electron beam melted Ti–6Al–4V, built at six (6) directions: 0°, 30°, 45°, 60°, 90°, and V, is tested using a four-step DNST at 650 °C. Analysis of the minimum-creep-strain-rate (at each step), creep ductility, and rupture were performed. A strain energy density (SED) benchmark metric was established to rapidly screen and classify creep resistance using an “Ashby-like” plot that balanced energy dissipated and energy dissipation rate. The creep resistance was found to be ranked from V, 60, 45, 30, 90, and 0 in descending order. Microstructural and crystallographic characterization revealed grain refinement progressing from the strain-free to the high-strain regions, indicative of dynamic recrystallization (DRX). Further observations revealed that the material's energy capacity increased with the alignment of the grains' long axis and tensile direction.