Strain Invariants Research Articles

Three-Dimensional Surface Displacements of the 8 January 2022 Mw6.7 Menyuan Earthquake, China from Sentinel-1 and ALOS-2 SAR Observations

The 8 January 2022 Mw6.7 Menyuan earthquake was generated in the transition zone between the western Lenglongling fault and the eastern Tuolaishan fault, both being part of the Qilian–Haiyuan fault system with an important role in the adjustment of the regional tectonic regime. In this study, four pairs of SAR (synthetic aperture radar) data from Sentinel-1 and ALOS-2 (Advanced Land Observation Satellite-2) satellites were used to derive the surface displacement observations along the satellite line-of-sight (LOS) and azimuth directions using the differential interferometric SAR (InSAR, DInSAR), pixel offset-tracking (POT), multiple aperture InSAR (MAI), and burst overlap InSAR (BOI) methods. An SM-VCE method (i.e., a method for measuring three-dimensional (3D) surface displacements with InSAR based on a strain model and variance component estimation) was employed to combine these derived SAR displacement observations to calculate the 3D co-seismic displacements. Results indicate that the 2022 Menyuan earthquake was dominated by left-lateral slip, and the maximum horizontal and vertical displacements were 1.9 m and 0.6 m, respectively. The relative horizontal surface displacement across the fault was as large as 2–3 m, and the fault-parallel displacement magnitude was larger on the southern side of the fault compared with the northern side. Furthermore, three co-seismic strain invariants were also investigated, revealing that the near-fault area suffered severe deformation, and two obviously expanding and compressed zones were identified. We provide displacements/strains derived in this study in the prevailing geotiff format, which will be useful for the broad community studying this earthquake; in addition, the SM-VCE code used in this study is open to the public so that readers can better understand the method.

Large-deformation constitutive modeling of viscoelastic foams: Application to a closed-cell foam material

Elastomeric foams display a nonlinear viscoelastic mechanical response, which is relevant to impact protection applications. In this paper, we present a methodology for the experimental characterization and constitutive modeling of the rate-dependent, large-deformation mechanical behavior of isotropic, viscoelastic foam materials under quasi-static and moderate loading rates. We conduct large-deformation, homogeneous simple compression/tension experiments under quasi-static and moderate strain rates for a closed-cell elastomeric foam to inform a phenomenological, isotropic constitutive model. The model is based on a multi-mechanism strategy, consisting of a rate-independent, hyperelastic contribution and several rate-dependent, viscoelastic contributions. This structure is motivated by the rheological concept of a hyperelastic spring in parallel with several nonlinear Maxwell elements. The hyperelastic modeling utilizes the invariants of the logarithmic strain and accounts for high compressibility and strong volumetric–distortional coupling. For the rate-dependent mechanisms, we employ the framework of the multiplicative decomposition of the deformation gradient into non-equilibrium elastic and viscous parts. Validation experiments that involve inhomogeneous deformation and deformation-rate fields are conducted, which include spherical indentation, simple-shear-like deformation both without and with a fixed pre-compression, and tension of a specimen with circular holes at different deformation rates. The responses from the validation experiments are compared against model predictions obtained from finite-element-based numerical simulations, demonstrating that the experimental results are accurately captured by the proposed model.

A hyperelastic-damage model based on the strain invariants

We develop a unified framework to describe the hyperelastic and damage behaviors of soft materials. The free energy density consists of both the crosslinked part of Langevin chains and the entanglement part described by a tube model. The micro–macro transition is obtained by using the averaging value over a microsphere. Consequently, the free energy density only depends on the strain invariants. The damage mechanism is then incorporated via the network alteration theory for the crosslinked part, while the deformation can also induce dilation of the tube and subsequent a decrease in the modulus for the entangled part. The hyperelastic model captures the Mooney Rivlin plot with a first decrease in the small deformation regime and then an increase in the large deformation region. The hyperelastic model can also accurately reproduce the stress–strain relationships in the uniaxial and biaxial loading conditions for various elastomer systems. The damage model can describe the stress response of filled rubbers in uniaxial and biaxial loading conditions, which shows a considerable improvement compared with the damage model without the entanglement effect. Since the model only depends on the strain invariants, the model can be easily implemented into the commercial finite element software Abaqus through a UHYPER. The finite element model can accurately reproduce the experimentally measured strain distribution of filled rubbers with complex geometry in the loading process. These results clearly show that the hyperelastic-damage model developed in this work has strong potentials to predict the mechanical responses of soft materials for practical applications.

Complete three-dimensional coseismic displacements due to the 2021 Maduo earthquake in Qinghai Province, China from Sentinel-1 and ALOS-2 SAR images

On 22nd May 2021 (local time), an earthquake of Ms7.4 struck Maduo county in Qinghai Province, China. This was the largest earthquake in China since the 2008 Wenchuan earthquake. In this study, ascending/descending Sentinel-1 and advanced land observation satellite-2 (ALOS-2) synthetic aperture radar (SAR) images were used to derive the three-dimensional (3-D) coseismic displacements of this earthquake. We used the differential interferometric SAR (InSAR, DInSAR), pixel offset-tracking (POT), multiple aperture InSAR (MAI), and burst overlap interferometry (BOI) methods to derive the displacement observations along the line-of-sight (LOS) and azimuth directions. To accurately mitigate the effect of ionospheric delay on the ALOS-2 DInSAR observations, a polynomial fitting method was proposed to optimize range-spectrum-split-derived ionospheric phases. In addition, the 3-D displacement field was obtained by a strain model and variance component estimation (SM-VCE) method based on the high-quality SAR displacement observations. Results indicated that a left-lateral fault slip with the largest horizontal displacement of up to 2.4 m dominated this earthquake, and the small-magnitude vertical displacement with an alternating uplift/subsidence pattern along the fault trace was more concentrated in the near-fault regions. Comparison with the global navigation satellite system data indicated that the SM-VCE method can significantly improve the accuracy of the displacements compared to the classical weighted least squares method, and the incorporation of the BOI displacements can substantially benefit the accuracy of north-south displacement. In addition to the displacements, three coseismic strain invariants calculated based on the strain model parameters were also investigated. It was found that the eastern and western parts of the faults suffered more significant strains compared with the epicenter region.

Parameter-free strain-energy function which automatically and accurately matches benchmark test data for soft elastic solids

PurposeA new and explicit form of the elastic strain-energy function for modeling large strain elastic responses of soft solids is constructed based on Hencky's logarithmic strain tensor.Design/methodology/approachWell-designed invariants of the Hencky strain are introduced for characterizing deformation modes and, furthermore, a new interpolating technique is proposed for combining piecewise splines into a single smooth function.FindingsWith this new form and this new technique, objectives in three respects may be achieved for the first time.Originality/valueFirst, no adjustable parameters need to be treated. Second, large strain responses for three benchmark modes are derivable in a decoupled sense without involving strongly nonlinear coupling effects. Finally, large strain data may be automatically and accurately matched for three benchmark modes, including uniaxial, equi-biaxial and plane-strain extension. Numerical examples are presented and compared with usual approaches.

Role of anisotropic invariants in numerically modeling soft biological tissues as transversely isotropic hyperelastic materials: A comparative study

Fiber-reinforced biological soft tissues are often modeled as anisotropic hyperelastic materials. Four strain invariants can be used to define a strain energy function for soft tissues reinforced with a single fiber family: two isotropic invariants I1,I2 and two anisotropic invariants I4,I5. Invariant I5 is often omitted in the strain energy functions to simplify the problem mathematically. In this study, the implications of using only I4 or the use of both anisotropic invariants I4,I5 in the numerical modeling of soft tissues were analyzed. A simple modification to the Holzapfel–Gasser–Ogden (HGO) model is proposed by adding a term that contains invariant I5. Material parameters were calculated by fitting the models with experimental data of uniaxial traction in the tibialis anterior tendon tissue of rats. General analytical solutions for the simple load scenarios were obtained. Such solutions are taken as a reference point to measure the precision of the numerical results obtained in the finite element simulations. The main differences between the models were observed in the shear behavior. The proposed model predicts three different shear responses (two responses with fiber reinforcement and one isotropic response), while the HGO model predicts two equal isotropic responses and only one with fiber reinforcement. Two sets of simple shear experimental data on a fiber-reinforced elastomer material were used to verify the shear stress prediction of the models. The experimental data show that the three shear behaviors are different; thus, the HGO model offers a limited description of the shear behavior. Furthermore, analytical solutions and experimental data suggest that invariant I5 is related to the elastic energy of the fibers when the material is subjected to shear in a direction parallel to the fibers. The models for non-homogeneous deformations were also compared. For this comparison, an irregular geometry was implemented in ABAQUS, and the traction and shear conditions were simulated by changing the fiber inclination. The most critical difference was found when the shear was parallel to the direction of the fiber and the minor difference when the fiber was at 45° to the load force.

A direct Jacobian total Lagrangian explicit dynamics finite element algorithm for real‐time simulation of hyperelastic materials

AbstractThis article presents a novel direct Jacobian total Lagrangian explicit dynamics (DJ‐TLED) finite element algorithm for real‐time nonlinear mechanics simulation. The nodal force contributions are expressed using only the Jacobian operator, instead of the deformation gradient tensor and finite deformation tensor, for fewer computational operations at run‐time. Owing to this proposed Jacobian formulation, novel expressions are developed for strain invariants and constant components, which are also based on the Jacobian operator. Results show that the proposed DJ‐TLED consumed between 0.70× and 0.88× CPU solution times compared to state‐of‐the‐art TLED and achieved up to 121.72× and 94.26× speed improvements in tetrahedral and hexahedral meshes, respectively, using GPU acceleration. Compared to TLED, the most notable difference is that the notions of stress and strain are not explicitly visible in the proposed DJ‐TLED but embedded implicitly in the formulation of nodal forces. Such a force formulation can be beneficial for fast deformation computation and can be particularly useful if the displacement field is of primary interest, which is demonstrated using a neurosurgical simulation of brain deformations for image‐guided neurosurgery. The present work contributes towards a comprehensive DJ‐TLED algorithm concerning isotropic and anisotropic hyperelastic constitutive models and GPU implementation.

Incorporating fiber recruitment in hyperelastic modeling of vascular tissues by means of kinematic average.

We present a framework for considering the gradual recruitment of collagen fibers in hyperelastic constitutive modeling. An effective stretch, which is a response variable representing the true stretch at the tissue-scale, is introduced. Properties of the effective stretch are discussed in detail. The effective stretch and strain invariants derived from it are used in selected hyperelastic constitutive models to describe the tissue response. This construction is investigated in conjunction with Holzapfel-Gasser-Ogden family strain energy functions. The ensuing models are validated against a large body of uniaxial and bi-axial stress-strain response data from human aortic aneurysm tissues. Both the descriptive and the predictive capabilities are examined. The former is evaluated by the quality of constitutive fitting, and the latter is assessed using finite element simulation. The models significantly improve the quality of fitting, and reproduce the experiment displacement, stress, and strain distributions with high fidelity in the finite element simulation.

A State Dependent Thermo-Hyperelastic Model for Geomaterials

A hyperelastic model for the state dependent and thermo-mechanical coupled nonlinear elastic behavior of geomaterials is developed in this paper. In the model the stress-density state dependent elastic moduli and the elastic instability are predicted by the high order terms of elastic strain invariants in the elastic potential function. The effects of true cohesion on the elastic potential of bonded geomaterials such as bonded sands, natural structured clays and rocks are also taken into account. As a result, unified relations between the density, the degradable cohesion, the confining stress and the elastic moduli can be derived for different geomaterials. Meanwhile, such an approach theoretically results in a state boundary and thus a strength criterion for geomaterials from the stability of elasticity. Based on such a hyperelastic approach, the nonlinear thermo-elastic coupled behavior is further considered by defining an equivalent elastic volumetric strain and taking into account a volumetric elastic thermal expansion coefficient dependent on both the volume fraction and thermal property of bound water absorbed on mineral surfaces.

Multiaxial fatigue experiments for elastomers based on true strain invariants

For decades, multiaxial fatigue tests have been proposed in order to derive models for fatigue life prediction of elastomers. Nevertheless, the importance of quantifying the multiaxiality has not sufficiently been discussed. In this paper, a new method is proposed to handle the multiaxiality in elastomers fatigue testing. It is based on the pair $$(K_2,K_3)$$ , two invariants of the Hencky (true) strain tensor. Thanks to the example of the simultaneous uniaxial tension-torsion fatigue of a Styrene Butadiene Rubber, we demonstrate that it is possible to prescribe a constant multiaxiality level throughout different tests. The method consists in fixing the multiaxiality indicator $$K_3$$ , and then varying the strain level with the intensity of distorsion indicator $$K_2$$ . Results show that the fatigue lives seem to unify with $$K_2$$ regardless of the value of $$K_3$$ . They also demonstrate that the initiation crack angle depends on the multiaxiality indicator $$K_3$$ but not on $$K_2$$ . Finally, the relevance of the approach is acknowledged by comparing the pair $$(K_2,K_3)$$ to the classical largest principal stretch ratio and biaxiality factor.

Development of Hyperelastic Model for Butadiene Rubber Using a Neural Network

A strain energy density function is used to characterize the hyperelasticity of rubber-like materials. Conventional models, such as the Neo-Hookean, Mooney-Rivlin, and Ogden models, are widely used in automotive industries, in which the strain potential is derived from strain invariants or principal stretch ratios. A fitting procedure for experimental data is required to determine material constants for each model. However, due to the complexities of the mathematical expression, these models can only produce an accurate curve fitting in a specified strain range of the material. In this study, a hyper-elastic model for Neodymium Butadiene rubber is developed by using the Artificial Neural Network. Comparing the analytical results to those obtained by conventional models revealed that the proposed model shows better agreement for both uniaxial and equibiaxial test data of the rubber.

Tangent second-order homogenisation estimates for incompressible hyperelastic composites with fibrous microstructures and anisotropic phases

This work presents new estimates for the effective mechanical response of composites with fibrous microstructures and hyperelastic, anisotropic, incompressible phases. The composites are two-phase materials consisting of aligned cylindrical fibres of circular cross section that are randomly distributed in the transverse plane of the matrix phase. The proposed homogenisation is based on the tangent second-order method proposed by Avazmohammadi and Ponte Castañeda (2013), which utilises variational principles and a suitably chosen “linear comparison material” to provide estimates for effective strain energies of nonlinear materials. To take into account the incompressibility of the fibre phase, we employed an appropriate asymptotic expansion of the fibre deformation field. By doing so, a constraint pressure is identified as the consistent limit of emerging indeterminate forms in the dilatational part of the fibre stress tensor. Furthermore, we performed an asymptotic analysis to obtain regular expressions of the underlying equations in the incompressible matrix limit. While we addressed the general case of incompressible and generally anisotropic phases, we put particular emphasis on composites with transversely isotropic phases whose symmetry axes are collinear with the fibre direction. For such materials, we derived estimates for the effective strain energies in terms of macroscopic strain invariants. In addition, for composites that are described by isotropic phase energies augmented with J4-invariant-based anisotropic terms, which is the case for many biological materials, we show that the anisotropic part of the energy can be easily homogenised by means of a simple Voigt-type average. Moreover, we present closed-form results for composites with isotropic Neo-Hookean phases augmented by transversely isotropic energy contributions. Finally, it is shown that the new estimates correspond to certain exact results under longitudinal shear deformation and that they properly linearise for small strains by recovering the corresponding linear-elastic estimates.

Role of Fiber Orientations in the Mechanics of Bioinspired Fiber-Reinforced Elastomers.

Fiber reinforcement is a crucial attribute of soft-bodied muscular hydrostats that have the ability to undergo large deformations and maintain their posture. Helically wound fibers around the cylindrical worm body help control the tube diameter and length. Geometric considerations show that a fiber winding angle of 54.7°, called the magic angle, results in a maximum enclosed volume. Few studies have combined both experimental and theoretical techniques to explore the effects of fiber winding at varied angles on the large deformation mechanics of fiber-reinforced elastomers (FRE). We fabricated FRE materials in transversely isotropic layouts varying from 0° to 90° using a custom filament winding technique and characterized the nonlinear stress-strain relationships using uniaxial and equibiaxial experiments. We used these data within a continuum mechanical framework to propose a novel constitutive model for incompressible FRE materials with embedded extensible fibers. The model includes individual contributions from the matrix and fibers in addition to coupled terms in strain invariants, I1 and I4. The deviatoric stress components show inversion at fiber orientation angles near the magic angle in the FRE composites. These results are useful in soft robotic applications and in the biomechanics of fiber-reinforced tissues such as the myocardium, arteries, and skin.

A compressible anisotropic hyperelastic model with I5 and I7 strain invariants

It is obvious that the mechanical properties of arterial tissue include compressibility, anisotropy, and the fact that the out-of-plane shear modulus is smaller than the shear modulus in the plane of the fibers. However, the last point is rarely considered when it comes to compressible anisotropic hyperelastic models. In order to acquire different shear moduli, we propose a modified hyperelastic model including the influence of strain invariants and The convergence and correctness of this model are verified through the hydrostatic tension test, uniaxial tension test, and shear deformation test. It turns out that our model correctly predicts an anisotropic response and volume change to hydrostatic tensile test and the fact that the out-of-plane shear modulus is always smaller than the shear modulus in the plane of the fibers in shear deformation test. We conclude that the influence of strain invariants and is great, especially in the shear deformation, so that it is necessary to include and in the compressible anisotropic hyperelastic model.

Finite strain parametric HFGMC micromechanics of soft tissues.

A micromechanical analysis is offered for the prediction of the global behavior of biological tissues. The analysis is based on the isotropic-hyperelastic behavior of the individual constituents (Collagen and Elastin), their volume fractions, and takes into account their detailed interactions. The present analysis predicts the instantaneous tensors from which the effective current first tangent tensor is established, thus providing the overall anisotropic constitutive behavior of the composite and the resulting field distribution in the composite. This is in contradistinction with the macroanalysis in which the composite internal energy, which involves unknown functions that depend on several strain invariants, must be proposed. The offered micromechanical analysis forms a generalization to the finite strain high-fidelity generalized method of cells (HFGMC) based on the homogenization technique for periodic composites to the parametric finite strain. This involves an arbitrary discretization of the repeating unit-cell of the periodic composites. Results are given for the response of the human abdominal aorta, which consists of three layered tissues: intima, media, and adventitia, all of which are composed out of the Collagen and Elastin. The isotropic-hyperelastic constituents (Mooney-Rivlin and Yeoh) of the composites are calibrated by utilizing available experimental data which describe the response of the tissue. Validation of the results is performed by comparison of the predicted Cauchy stress and stretches with the experimental measurements. In addition, results are given in the form of Cauchy stress and deformation gradient field distributions in the constituents of several tissues.

Characterization of hyperelastic deformation behavior of rubber-like materials

This paper proposes a new hyperelastic model for rubber-like materials applicable over the entire modes of the large deformations. The whole material parameters have been identified using the mathematical approach combined with the genetic algorithm for the best method for the structure of hyperelastic models applicable to incompressible rubber-like materials. The step-by-step method is used on the restricted form by gradually adding a term of the highest degree of the hyperelastic model when the deformation increases. This combination greatly minimizes the difference between experimental data and analytical solutions. It is about a method that is applied to nonlinear behaviors leading to an optimal solution. The power of the model is demonstrated via comparisons with different phenomenological hyperelastic models using the experimental data from the Treloar and Nunes works. According to the litterature, many authors have proposed a model based on both strain invariants but unable to reproduce the equibiaxial deformation mode. This work offers the constitutive hyperelastic model based on both strain invariants $$I_1$$ and $$I_2$$ with a minimal number of rheological parameters.

A vortex identification method based on strain and enstrophy production invariants

A new vortex identification method is proposed for extracting vortical structures from homogeneous isotropic turbulence. The method is compared with other identification schemes such as the high rotational method ([Formula: see text]), the vorticity magnitude method ([Formula: see text]), the negative eigenvalue method ([Formula: see text]) and the normalized vorticity method ([Formula: see text]). A new normalization method based on the probability distribution function (PDF) of the identification invariants is also introduced. In addition, a modification for the discriminant criterion known as the [Formula: see text] method is carried out and it is denoted as the modified delta method ([Formula: see text]). The velocity of the isotropic turbulent field is simulated using the lattice Boltzmann method with resolution [Formula: see text]. The new identification method depends on the higher-orders of the invariants of the velocity gradient tensor as well as the strain rate and the enstrophy production terms. The elongated tube-like vortices are extracted successfully using the new method and several features of the vortices are demonstrated and compared with the vortical structures that are extracted using the [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] identification methods. The recommended normalization method enabled the justification of the visualization threshold value to be within the order of unity and the threshold value [Formula: see text] is used in all identification methods. A remarkably similar geometrical worm-like vortices are extracted and a high similarity between the identification methods is observed and statistically studied.

An analysis driven construction of distortional-mode-dependent and Hill-Stable elastic potential with application to human brain tissue

We propose an innovative procedure by exploiting the physical meaning of natural strain or Lode invariants with the following salient contributions: 1) Uniaxial data for human brain tissue is used to stipulate the mathematical structure of the potential in terms of the Lode invariant that quantifies the magnitude of distortion along with the modulus term being an unknown function of the Lode angle that quantifies the mode or type of distortion. 2) By a priori analysis using the Baker-Ericksen inequalities, the mathematical form of the modulus function is determined in a novel manner. 3) The derived modulus function is corrected by adding a constant, which in turn is determined using analysis involving sufficient conditions of the stronger Hill inequality. 4) In addition, we also prove that any potential that satisfies Hill inequality also satisfies true-stress-true-strain monotonicity condition in plane stress. Compared to Mihai-Ogden model, besides excellent quantitative agreement with data for human brain tissue (see Mihai et al., 2017), the constructed model also emulates the observed non-linear behavior of shear stress with respect to the amount of shear as opposed to the nearly linear response predicted by the antecedent model. Additionally, when only tension-compression data is available for determining material parameters, the predicted combined tension and shear response associated with the proposed constitutive relation shows monotone decreasing Poynting stress (compressive), while the former predicts an unexpected non-monotone response for certain levels of tension.

Experimental characterization and hyperelastic constitutive modeling of open-cell elastomeric foams

Open-cell elastomeric foams – materials consisting of an elastomeric matrix and a connected pore space – exhibit mechanical behavior marked by high compressibility and strong coupling between the volumetric and distortional responses. In this paper, we present a methodology for the experimental characterization and constitutive modeling of non-localizing, isotropic, open-cell elastomeric foam materials under quasi-static, equilibrium loading. We conduct large-deformation, homogeneous simple compression/tension experiments on three relative densities of a polyurethane-based elastomeric foam to inform a phenomenological, isotropic, hyperelastic constitutive model. The model is based on the invariants of the logarithmic strain and accounts for high compressibility and strong volumetric-distortional coupling. To validate the predictive capability of the model, we consider three types of validation experiments that involve inhomogeneous deformation: spherical and conical indentation, simple-shear-like deformation both without and with a fixed amount of pre-compression, and tension of a specimen with circular holes. We compare load-displacement responses as well as full displacement fields from the validation experiments against corresponding model predictions obtained using finite-element-based numerical simulations and demonstrate that the model is capable of accurately capturing the experimental response.

A microstructurally-based, multi-scale, continuum-mechanical model for the passive behaviour of skeletal muscle tissue

This work presents a novel microstructurally-based, multi-scale model describing the passive behaviour of skeletal muscle tissue. The model is based on the detailed description of the mechanically relevant parts of the microstructure. The effective constitutive material response is obtained by a homogenisation of mechanical energies and stresses from the micro- to the macroscale. The key feature of the new model is that it does not require any constitutive assumptions or calibration on the macroscale. The effective mechanical response is a pure consequence of the stiffness and structural arrangement of microscopic components. In this sense, the model inherits its direction-dependent properties directly from the microstructure. This is achieved by employing a Voigt-type homogenisation and by utilising for the complex collageneous network of the extracellular matrix an angular integration method. For physiologically realistic microscopic model parameters, this model reveals that muscle tissue exhibits a tensile stiffness that is larger transverse to the muscle fibre than in muscle fibre direction. This highlights that muscle tissue in general does not obey a classical fibre-reinforcement solely for tensile stretches of the muscle fibres but rather a general transversely isotropic behaviour. Moreover, the formulation of the effective macroscopic energy is provided in terms of well-known macroscopic strain invariants, which allows for an easy application of the model in standard numerical settings.