Energy Density Function Research Articles

An efficient implementation of a phase field model for fatigue crack growth

Recently, phase field modeling of fatigue fracture has gained a lot of attention from many researches and studies, since the fatigue damage of structures is a crucial issue in mechanical design. Differing from traditional phase field fracture models, our approach considers not only the elastic strain energy and crack surface energy, additionally, we introduce a fatigue energy contribution into the regularized energy density function caused by cyclic load. Comparing to other type of fracture phenomenon, fatigue damage occurs only after a large number of load cycles. It requires a large computing effort in a computer simulation. Furthermore, the choice of the cycle number increment is usually determined by a compromise between simulation time and accuracy. In this work, we propose an efficient phase field method for cyclic fatigue propagation that only requires moderate computational cost without sacrificing accuracy. We divide the entire fatigue fracture simulation into three stages and apply different cycle number increments in each damage stage. The basic concept of the algorithm is to associate the cycle number increment with the damage increment of each simulation iteration. Numerical examples show that our method can effectively predict the phenomenon of fatigue crack growth and reproduce fracture patterns.

Spatially-dependent material uncertainties in anisotropic nonlinear elasticity: Stochastic modeling, identification, and propagation

This paper develops a stochastic model for the spatially-dependent material parameters parameterizing anisotropic strain energy density functions. The construction is cast within the framework of information theory, which is invoked to derive a least-informative model while ensuring consistency with theoretical requirements in finite elasticity. Specifically, almost sure polyconvexity and uniform growth conditions are enforced through proper repulsion constraints and regularization, hence making the forward problem of uncertainty propagation well posed. In addition, transformations arising from the linearization procedure are introduced for consistency and induce statistical dependencies in the primary variables. The latter include material moduli, a weight balancing between the isotropic and anisotropic contributions, and the angle defining the structural tensors. The identification of the model is subsequently performed, using an existing database on human arterial walls. Maximum likelihood estimators are obtained and provided for the adventitia, media, and intima layers, which enables the use of the proposed model as a generative surrogate for, e.g., training and classification in data-driven approaches integrating inter-patient variability. Finally, uncertainty propagation on a realistic, patient-specific geometry is conducted to demonstrate the efficiency of the stochastic modeling framework.

An FEA-Assisted Decision-Making Framework for PEMFC Gasket Material Selection

Recent research studies on industrial cyber-physical systems (ICPSs) have witnessed vast patronage with emphasis on data utility for improved design, maintenance, and high-level decision making. The design of proton-exchange membrane fuel cells (PEMFC) is geared towards improving performance and extending life cycles. More often, material selection of PEMFC components contributes a major determining factor for efficiency and durability with the seal/gasket quality being one of the most critical components. Finite element analysis (FEA) offers a simulated alternative to real-life stress analysis of components and has been employed on different rubber-like gasket materials for hydrogen fuel cells for determining an optimal strain energy density function using different hyperelastic models following uniaxial tensile testing. The results show that the Mooney–Rivlin, Ogden, and Yeoh models were the most fitting model with the best stress–strain fit following a weighted error evaluation criteria which returned 18.54%, 19.31%, and 21.96% for 25% displacement, and 22.1%, 21.7%, and 21.17% for 40% displacements, respectively. Further empirical analysis using the multi-metric regression technique for compatibility testing (curve similarity) between the hyperelastic model outputs and the tensile data reveal that the Yeoh model is the most consistent as seen in the marginal error difference amidst increasing displacement while the Arruda–Boyce model is most inconsistent as shown in the high error margin as the displacement increases from 25% to 40%. Lastly, a comparative assessment between different rubber-like materials (RLM) was presented and is expected to contribute to improved decision-making and material selection.

Soft Tissue Hybrid Model for Real-Time Simulations.

In this article, a recent formulation for real-time simulation is developed combining the strain energy density of the Spring Mass Model (SMM) with the equivalent representation of the Strain Energy Density Function (SEDF). The resulting Equivalent Energy Spring Model (EESM) is expected to provide information in real-time about the mechanical response of soft tissue when subjected to uniaxial deformations. The proposed model represents a variation of the SMM and can be used to predict the mechanical behavior of biological tissues not only during loading but also during unloading deformation states. To assess the accuracy achieved by the EESM, experimental data was collected from liver porcine samples via uniaxial loading and unloading tensile tests. Validation of the model through numerical predictions achieved a refresh rate of 31 fps (31.49 ms of computation time for each frame), achieving a coefficient of determination R2 from 93.23% to 99.94% when compared to experimental data. The proposed hybrid formulation to characterize soft tissue mechanical behavior is fast enough for real-time simulation and captures the soft material nonlinear virgin and stress-softened effects with high accuracy.

Mechanical and histological characteristics of aortic dissection tissues

Aims: This study investigated the association between the macroscopic mechanical response of aortic dissection (AoD) flap, its fibre features, and patient physiological features and clinical presentations.Methods: Uniaxial test was performed with tissue strips in both circumferential and longitudinal directions from 35 patients with (AoD:CC) and without (AoD:w/oCC) cerebral/coronary complications, and 19 patients with rheumatic or valve-related heart diseases (RH). A Bayesian inference framework was used to estimate the expectation of material constants (C1, D1, and D2) of the modified Mooney-Rivlin strain energy density function. Histological examination was used to visualise the elastin and collagen in the tissue strips and image processing was performed to quantify their area percentages, fibre misalignment and waviness.Results: The elastin area percentage was negatively associated with age (p = 0.008), while collagen increased about 6% from age 40 to 70 (p = 0.03). Elastin fibre was less dispersed and wavier in old patients and no significant association was found between patient age and collagen fibre dispersion or waviness. Features of fibrous microstructures, either elastin or collagen, were comparable between AoD:CC and AoD:w/oCC group. Elastin and collagen area percentages were positively correlated with C1 and D2, respectively, while the elastin and collagen waviness were negatively correlated with C1 and D2, respectively. Elastin dispersion was negatively correlated to D2. Multivariate analysis showed that D2 was an effective parameter which could differentiate patient groups with different age and clinical presentations, as well as the direction of tissue strip.Conclusion: Fibre dispersion and waviness in the aortic dissection flap changed with patient age and clinical presentations, and these can be captured by the material constants in the strain energy density function. Statement of significanceAortic dissection (AoD) is a severe cardiovascular disease. Understanding the mechanical property of intimal flap is essential for its risk evaluation. In this study, mechanical testing and histology examination were combined to quantify the relationship between mechanical presentations and microstructure features. A Bayesian inference framework was employed to estimate the expectation of the material constants in the modified Mooney-Rivlin constitutive equation. It was found that fibre dispersion and waviness in the AoD flap changed with patient age and clinical presentations, and these could be captured by the material constants. This study firstly demonstrated that the expectation of material constants can be used to characterise tissue microstructures and differentiate patients with different clinical presentations.

Thickness-Stretch Vibration of an Infinite Piezoelectric Plate with Flexoelectricity

In this paper, the thickness-stretch vibration of an infinite piezoelectric plate is studied, with consideration of the flexoelectric effect. The theoretical model developed herein is based on a one-dimensional formulation, with the assumption that the displacement and electric potential vary through the thickness. The Gibbs energy density function and variational principle are adopted to derive the constitutive equation with flexoelectricity, governing equations, and boundary conditions. For the effect of flexoelectricity, the coupling between the strain gradient through the thickness and the electric field is considered. Two electric boundary conditions are used in this work and the corresponding frequency shift due to the flexoelectricity is calculated. The present results demonstrate that the flexoelectric effect decreases the fundamental frequency of the thickness-stretch vibration and has a significant effect on the vibrational frequencies of the thickness-stretch mode of a thin piezoelectric plate. The results also show that the flexoelectric effect has a significant size dependency, and should be taken into consideration for the design and development of next-generation high-precision and high-frequency piezoelectric transducers and resonators in the future.

A comparative study on energy-based fatigue criteria for lifetime calculation of magnetorheological elastomers under multiaxial loading

Predicting fatigue life of magnetorheological elastomers (MREs) containing, different volumes of carbonyl iron particles (CIPs) with various external magnetic fields has been investigated. Unlike many studies on fatigue behavior of MREs, representing experimental models, a novel three-dimensional constitutive magneto-hyper-viscoelastic semi-coupling model based on strain energy density function (3DSED) is derived, analyzing the fatigue of structures enhanced by MRE experimentally and numerically. In this 3DSED model, to obtain the best predicting MREs’ fatigue life, three different hyper-elastic functions, including Neo-Hookean, Mooney-Rivlin, and Yeoh are utilized. Using some uniaxial compression, such as loading phase, relaxation, and loading/unloading phases tests, hyper-elastic, magneto-elastic, and viscoelastic constants are determined. In addition, multi-axial fatigue tests are performed to define the number of cycles to failure. These tests have been induced by the magnetic fields of 0, 0.27, and 0.4 T. The fatigue tests re-performed with the magnetic field of 0.3 T to compare the predicted results using the proposed fatigue models. The results show that the 3DSED model can be used for the prediction of MREs fatigue life when subjected to complex and repetitive loading. For all CIP contents, the Mooney-Rivlin fatigue predictor has more accurate results. The Yeoh fatigue predictor indicates better results for MREs with higher CIP contents, while the Neo-Hookean fatigue predictor doesn’t show acceptable results for any CIP contents.

Cellular Microbiaxial Stretching Assay for Measurement and Characterization of the Anisotropic Mechanical Properties of Micropatterned Cells.

Characterizing the mechanical properties of single cells is important for developing descriptive models of tissue mechanics and improving the understanding of mechanically driven cell processes. Standard methods for measuring single-cell mechanical properties typically provide isotropic mechanical descriptions. However, many cells exhibit specialized geometries in vivo, with anisotropic cytoskeletal architectures reflective of their function, and are exposed to dynamic multiaxial loads, raising the need for more complete descriptions of their anisotropic mechanical properties under complex deformations. Here, we describe the cellular microbiaxial stretching (CμBS) assay in which controlled deformations are applied to micropatterned cells while simultaneously measuring cell stress. CμBS utilizes a set of linear actuators to apply tensile or compressive, short- or long-term deformations to cells micropatterned on a fluorescent bead-doped polyacrylamide gel. Using traction force microscopy principles and the known geometry of the cell and the mechanical properties of the underlying gel, we calculate the stress within the cell to formulate stress-strain curves that can be further used to create mechanical descriptions of the cells, such as strain energy density functions. © 2022 Wiley Periodicals LLC. Basic Protocol 1: Assembly of CμBS stretching constructs Basic Protocol 2: Polymerization of micropatterned, bead-doped polyacrylamide gel on an elastomer membrane Support Protocol: Cell culture and seeding onto CμBS constructs Basic Protocol 3: Implementing CμBS stretching protocols and traction force microscopy Basic Protocol 4: Data analysis and cell stress measurements.

Wormhole solutions in [formula omitted] gravity

In this paper, the traversable wormholes have been explored in the context of f(R,T) theory of gravity with f(R,T)=R+2λT, where λ is an arbitrary constant. The wormhole solutions have been developed in two cases: (i) Using a logarithmic form of shape and redshift functions and (ii) using a power-law of energy density and constant redshift function. In each case, the regions satisfying the energy conditions such as null, weak, strong and dominated energy conditions are determined.

Structural and material electro-mechanical instabilities in microstructured dielectric elastomer plates

In this paper, we investigate the stability of microstructured dielectric elastomer (DE) plates under electro-mechanical finite deformations. We examine the influence of the DE composite microstructure parameters on the stability of the DE composite plate. In particular, we analyze the DE plates made out of laminated ideal dielectric phases. We use an exact solution for the incompressible DE laminates to determine the effective material behavior of the composite plates. The corresponding micromechanics-based energy density functions are used in the subsequent analysis of the DE plate stability. The unstable domains for different materials compositions, volume fractions, and properties of the DE composite phases are determined. Finally, we compare these results for the onset of structural instabilities in DE plates with the material instabilities predicted based on the loss of ellipticity analysis.

Phase Field vs Gradient Enhanced Damage Models: A Comparative Study

A comparison of the two different approaches for modelling damage in material in an infinitesimal strain setting is studied. The first approach is the nonlocal gradient enhanced damage model where the damage variable is taken as an independent variable which will be determined based on the local strain measure. Here, the nonlocal integral form is approximated to an implicit or explicit differential form using the Taylor’s series expansion for simpler numerical implementation. The second approach is the phase field damage model where a Helmholtz free energy density function is considered that includes a new energy degradation function along with a phase field non-conserved order parameter. The first variational principle on this energy density functional with respect to the corresponding order parameter variable will reach a stationarity value resulting in the non-conserved Allen-Cahn equation. The relationship of the order parameter with the damage variable gives the Allen-Cahn evolution equation for damage. A 1D bar example is considered for commenting on the similarities and differences of the two approaches to damage based on the mesh convergence studies based on the results obtained for various meshing profiles, changing length scale parameter values and the obtained damage profiles.

Computational investigation for deformation of lipid membrane by BAR proteins due to electrostatic interaction

Deformation of lipid membrane by peripheral BAR (Bin/Amphiphysin/RVs) proteins due to electrostatic interaction between them has an important implication in field of cell-biological systems. We study the deformation behavior of a lipid membrane sheet subjected to electrostatic traction with the help of a very simple finite element elastic model. The lipid membrane is considered as an elastic sheet subjected to the electrostatic traction derived from the literature for the interaction between the lipid membrane and a single BAR protein. Both the small and large deformation behavior of the lipid membrane is considered. To predict small or infinitesimal deformation of the membrane, we use Hooke’s law and large deformation is predicted by Neo-Hookean strain energy density function. The small deformation behavior obtained from the present finite element model does not support the scaffolding mechanism of a single BAR protein binding with the lipid membrane. However, the large deformation behavior of the membrane predicts very well the scaffolding mechanism for the wide range of BAR proteins involved in cell-biological systems. The present model can be used for understanding deformation of any kind of elastic sheet driven by electrostatic forces.

Efficient and Robust Method for Identifying Material Constant of Hyperelastic Material Model via Complex-step Derivative Approximation

An efficient and robust method for deriving the stress-strain relationship of hyperelastic material model (HM-model) was proposed via complex-step derivative approximation. This method has a capability to calculate the stress-strain relationship with the same accuracy as the analytical solution by coding only the strain energy density function, being applied for the implementation of an identification program of material constants for HM-model using a Microsoft/Excel. The performance evaluation of HM-models (Yeoh and Ogden models) using this method revealed that all models could reproduce the results of the three types of material tests (uniaxial tensile (UT), pure shear (PS) and equibiaxial tensile (BT) tests) of carbon-black-filled SBR. Moreover, only the hyperelastic model composed of the first invariant of the right Cauchy-Green tensor could predict the PS and BT test results from the UT data. As a future prospect, we would like to proceed with studies on other rubber materials and HM models.

Influence of laser power in nanosecond laser texturing for a hydrophobic state on SS316L

The use of lasers in surface engineering has recently made significant progress. The hydrophobic surface is commonly studied because of the application in various fields, including vehicles, aerospace, biomedicine, etc. Since these laser methods require many combination parameters, such as laser power (P), frequency (ƒ), scan speed (ʋ) and laser beam diameter (D), the effect of the parameters must therefore be investigated to produce the hydrophobic condition. This research tries to relate the laser power with the morphological properties and contact angle of the SS316L surfaces. Samples are subjected to laser texturing with different laser power settings. The surface is then characterised by surface roughness, and the contact angle is measured according to a specific time interval. The laser power output and energy density function on the surface and contact angle were investigated in these contexts experimentally. Surface roughness was defined and validated to show that the laser parameters' effect is effective and controllable. This study shows that the laser output intensity significantly contributes to regulating surface roughness and the substrate's wetting activity. The 18W and 24W laser outputs produce a spiked surface with various peaks that cause the surface to become hydrophobic over time because of the air-trap that happens in the valley.

Work‐conjugacy of stress and strain tensors and its role in 3D hyperelastic isotropic material formulation

AbstractIn accordance with the framework of hyperelastic isotropic material description it appears possible to replace work‐conjugate stress and strain quantities by alternative pairs of work‐conjugate stress and strain quantities while other things left unaltered, i.e. based on a retained architecture of the strain energy density function. This replacement results in formally representing different kinds of material response behavior. This may be recognized from the various comparable forms of resulting uniaxial force versus axial stretch behavior in the representative 3D principal axes system. Clarifying and achieving comprehensive understanding of these observations is the main purpose of this paper.

Mechanical-Geometrical Modeling Of The Hyperelastic Materials At Uniaxial Stretching

Hyperelastic materials, such as rubber, occupy an important place in the design and operation of various technological equipment and machines. The article analyzed the deformation behavior of hyperelastic materials using a mechanical-geometric model. The method of mechanical-geometric modeling is a new method for obtaining constitutive relations and strain energy density functions for nonlinear elastic solids. It is based on physically and geometrically consistent prerequisites. The resulting models can describe broad classes of nonlinear elastic materials (both isotropic and anisotropic) depending on the mechanical and geometric properties “embedded” in them at the first stages of design. This paper discusses two basic types of models based on different initial geometry. The mechanical parameters of the models are constants, and the models themselves are considered in a statement corresponding to isotropic hyperelastic materials. The article presents the most common diagrams of deformation of artificial and natural rubbers, as well as steel. Hyperelastic materials, depending on the task, can be described in the nonlinear theory of elasticity as ideal incompressible, or as weakly compressible. Parameters of expressions of strain energy density functions of mechanical-geometric models obtained for cases of incompressible and weakly compressible continuous solids were identified. Stretch diagrams and diagrams of the transverse deformation function of the obtained mechanical-geometric models for the two cases mentioned above are plotted. The extension diagram for the model with parameters corresponding to the classic structural material of the steel type is also shown. Comments are given on the possibility of further paths of developing the method of mechanical-geometric modeling to obtain results not only in the field of nonlinear theory of elasticity, but also viscoelasticity.

Vibration source identification using an energy density method

The localization of shaking forces acting on an operating machine is an important step to identify vibration and noise sources. The forced vibration response of a linearly vibrating structure is assumed to be linear. However, the energy distribution of a linearly vibrating structure contains “coupled terms” in the modal decomposition of the vibration energy density function. These coupled energy terms represent the cross-modal energy density associated with the exciting force of a dynamic structure under forced vibration. In this research, it is proved analytically that the high-order cross-modal energy densities of a linear dynamic structure are highly correlated to the location of the external exciting force. Using this finding, a new force localization index based on the high-order cross-modal energy densities of a dynamic structure is proposed and tested. Numerical tests on uniform and step beam structures under force excitation with different frequencies and locations have been carried out to test the effectiveness of the proposed force localization method. It is found that the proposed force localization method works well on vibrating beam structures. Experiments are carried out to verify the proposed force localization method.

The Valanis–Landel strain energy function Elasticity of incompressible and compressible rubber-like materials

The paper revisits the original article by Valanis and Landel on the strain energy density function W = w(λ1) + w(λ2) + w(λ3), where λi are the stretch ratios. The main new results and findings are: (i) An extension of the above algebraic form of W to compressible rubber-like materials (ii) Proof that the Cauchy-Green stress tensor t, as a function of the Left Stretch tensor V, has the same analytical form as the function λw′(λ), within a hydrostatic pressure. (iii) The determination of the form of the constitutive equation t(V) from a single experiment, the plane strain test. (iv) Comparison with experimental data of several empirical algebraic forms of w′(λ) in the literature. (v) Findings of a study of the Mooney-Rivlin and Ogden models of the Valanis- Landel function, in axial compression.

Exploring the Biomechanical Properties of the Human Cornea In Vivo Based on Corvis ST.

Purpose: The aim of this study was to provide a method to determine corneal nonlinear viscoelastic properties based on the output data of corneal visualization Scheimpflug technology (Corvis ST). Methods: The Corvis ST data from 18 eyes of 12 healthy humans were collected. Based on the air-puff pressure and the corneal displacement from the Corvis ST test of normal human eyes, the work done by the air-puff attaining the whole corneal displacement was obtained. By applying a visco-hyperelastic strain energy density function of the cornea, in which the first-order Prony relaxation function and the first-order Ogden strain energy were employed, the corneal strain energy during the Corvis ST test was calculated. Then the work done by the air-puff attaining the whole corneal displacement was completely regarded as the strain energy of the cornea. The identification of the nonlinear viscoelastic parameters was carried out by optimizing the sum of difference squares of the work and the strain energy using the genetic algorithm. Results: The visco-hyperelastic model gave a good fit to the data of corneal strain energy with time during the Corvis ST test (R 2 > 0.95). The determined Ogden model parameter μ ranged from 0.42 to 0.74 MPa, and α ranged from 32.76 to 55.63. The parameters A and τ in the first-order Prony function were 0.09–0.36 and 1.21–1.95 ms, respectively. Conclusion: It is feasible to determine the corneal nonlinear viscoelastic properties based on the corneal contour information and air-puff pressure of the Corvis ST test.

Flexoelectric effect on thickness-shear vibration of a rectangular piezoelectric crystal plate

Thickness-shear (TSh) vibration of a rectangular piezoelectric crystal plate is studied with the consideration of flexoelectric effect in this paper. The developed theoretical model is based on the assumed displacement function which includes the anti-symmetric mode through thickness and symmetric mode in length. The constitutive equation with flexoelectricity, governing equations and boundary conditions are derived from the Gibbs energy density function and variational principle. For the effect of flexoelectricity, we only consider the shear strain gradient in the thickness direction so as to simply the mathematical model. Thus, two flexoelectric coefficients are used in the present model. The electric potential functions are also obtained for different electric boundary conditions. The present results clearly show that the flexoelectric effect has significant effect on vibration frequencies of thickness-shear modes of thin piezoelectric crystal plate. It is also found that the flexoelectric coefficients and length to thickness ratio have influence on the thickness-shear modes. The results tell that flexoelectricity cannot be neglected for design of small size piezoelectric resonators.