Left Cauchy-Green Deformation Tensor Research Articles

An interface preserving and residual-based adaptivity for phase-field modeling of fully Eulerian fluid-structure interaction

In this paper, we present a novel interface preserving and residual-based adaptivity procedure for the phase-field modeling of a fully Eulerian fluid-structure interaction. The proposed adaptive fully Eulerian procedure involves a fixed background unstructured finite element mesh on which the fluid-structure boundary is treated implicitly via a diffused interface description. We employ the stabilized finite element formulation and a partitioned iterative procedure to solve the coupled system of the Allen-Cahn phase-field equation with the unified momentum equation comprising solid and fluid dynamics. To evaluate the solid stresses, the left Cauchy-Green deformation tensor is convected at each time step to trace the evolution of the solid strain in the Eulerian reference frame. We utilize a residual based error indicator and the newest vertex bisection algorithm for the adaptive refinement/coarsening of the unstructured mesh. Through the adaptive finite element method, we aim to preserve the profile of the diffused interface in the presence of convective distortion. We propose an early stopping criterion for the adaptivity of the interface dynamics by combining the effects of the phase-field interface thickness parameter and the mesh resolution. In addition, we control the size of the refined/coarsened elements by introducing a characteristic length scale governed by a weighted harmonic mean of the desired bulk and the interface mesh resolutions. The proposed stopping criterion and the characteristic length scale provide significant reductions in the computational cost while simultaneously ensuring convergence properties of the coupled fluid-structure system. We perform a detailed convergence and accuracy analysis via two benchmark problems namely, the pure solid system and a coupled fluid-solid system with an interface in a rectangular domain. We further provide a comparative study between feature- and residual based adaptive strategies. We systematically assess the performance of the adaptive procedure in terms of conservation properties and computational efficiency. Finally, we demonstrate our fully Eulerian adaptive solver to simulate the contact and bouncing phenomenon between an elastic solid and a rigid wall.

A full-Eulerian approach for simulation of a system of fluid–rigid–elastic structure interaction based on the vorticity-stream function formulation

A monolithic mathematical framework for understanding the fluid–rigid–elastic structure interaction problem is proposed. A numerical method in a secondary formulation of the Navier–Stokes equations accompanying a technique for imposing the rigid boundaries is applied. The one-fluid formulation of the incompressible Navier–Stokes equation, containing the terms governing the elastic structure, is transformed into the vorticity-stream function formulation. The rigid structure is imposed in the flow field based on the velocity–vorticity kinematic relation and harmonic function theorem. The vorticity, level-set function, and left Cauchy–Green deformation tensor are updated utilizing three transport equations to investigate the evolution of the velocity field, elastic structure(s) configuration, and elastic stress tensor. The method is implemented to solve three challenging problems, and the results show its capabilities in proper imposing the rigid structures in the flow field and also the simultaneous modeling the rigid and elastic structure interactions with incompressible fluid flow.

Stretch formulations and the Poynting effect in nonlinear elasticity

The second invariant of the left Cauchy–Green deformation tensor B (or right C) has been argued to play a fundamental role in nonlinear elasticity. Generalized neo-Hookean materials, which depend only on the first invariant, lead to universal relations that conflict with experimental data, fail to display important mechanical behaviors (such as the Poynting effect in simple shear), and may not provide a satisfactory link with the mesoscale. However, the second invariant term is not a higher order strain contribution to the energy, which lead us to reflect on what is incomplete about neo-Hookean materials. Instead of the usual Cauchy–Green elastic formulation, we investigate this matter from the perspective of left stretch V=B and Bell strain EBell=V−I formulations. Invariants of these tensors offer a different interpretation than those of B and are linked to different classes of materials. The main example we adopt is a general isotropic energy quadratic in Bell strains, the quadratic-Biot material. Despite being quadratic in stretch like neo-Hookean, this material presents both the classic and reverse Poynting effect in simple shear, whose direction switches as a function of the constant conjugate to the second invariant of EBell. Its second normal stress also presents a local maximum as a function of the amount of shear, a transition that is not observed in a Mooney–Rivlin solid. Moreover, even the Varga model, linear in Bell strains, presents Poynting in simple shear, which poses the question of why this is not true for a model linear in Green–Lagrange strains. Pure torsion of a solid cylinder is also discussed, particularly how the behavior of the resultant axial force contrasts between the different formulations.

Explicit compatibility condition formula for the Eulerian left Cauchy–Green deformation tensor field

AbstractAn explicit compatibility condition formula is presented for the Eulerian left Cauchy–Green deformation tensor field. It is shown to be the appropriate finite‐strain counterpart of Saint‐Venant's compatibility condition. The difference between the Eulerian problem and the Lagrangian one is highlighted.

Nonlinear interaction of the damped large dynamic deformations of the Mooney–Rivlin hyperelastic plates with the viscoelastic and shear reactions of the supporting substrate

In this research, the interaction between the damped large deformation dynamic responses of the Mooney–Rivlin hyperelastic plates and the viscoelastic and shear characteristics of the supporting substrate (that constitute a visco-hyperelastic system) is investigated for various distributions and various time variations of the transverse loads and different boundary conditions. A viscoelastic Winkler–Pasternak model is chosen for the supporting substrate to account for the dissipative, shearing, and supporting features of the foundation. The governing equations of motion are obtained by using Hamilton’s principle, von Karman assumptions, left Cauchy–Green deformation tensor, and a modified classical plate theory whose accuracy is enhanced through the incorporation of an appropriate high-order incompressibility condition. The combination of the nonlinear and coupled motion equations and boundary conditions is solved by an iterative 2D differential quadrature (DQ) spatial-discretization and Newmark’s time-marching methods. The effects of the constitutive parameters of the hyperelastic material, magnitude and type of the distributed loads, the thickness and aspect ratios of the plate, parameters of the Winkler–Pasternak viscoelastic foundation, and boundary conditions on the dynamic lateral deflections of the plate are studied examined during the parametric studies. Results emphasize the role of the visco-hyperelasticity features interaction of the combined plate-substrate system on the vibration suppression and dynamic performance of the structure in different spatial and time-variation patterns of the distributed loads and various boundary conditions.

Dynamic behavior of heterogeneous neo-Hookean/Mooney–Rivlin plates reinforced nonuniformly by hyperelastic inclusions: Proposing the correct micromechanical model

Time-dependent nonlinear lateral vibrations of plates composed of a hyperelastic matrix and uniformly/nonuniformly distributed hyperelastic reinforcing inclusions are studied. Since the material constants of a hyperelastic must be extracted from the whole slope-varying stress-strain curve rather than a single slope, choosing power/exponential distributions for the material constants or using Voigt’s rule of mixtures is quite wrong. The neo-Hookean and Mooney–Rivlin constitutive models are adopted, and their results are compared. Another hint is incorporating the incompressibility condition. The governing equations of motion are derived by using Hamilton’s principle, a new energy-equivalence-based micromechanical model that can be employed for reinforcing phases with nonlinear constitutive laws, and von Kármán assumptions in the left Cauchy–Green deformation tensor, and solved by incorporation of an updating finite-element and Newmark’s techniques. Not only the displacement but typical stress results are also reported here. Results show that the neo-Hookean model overestimates the rigidity in comparison to the Mooney–Rivlin model, and unlike the elastic plates, the effect of the stiffer phase is more remarkable in the uniform distribution in comparison to the nonuniform distribution of the stiffening materials because the magnitudes of the tensile stresses of the hyperelastic plate are much larger than that of the bending stresses.

An inverted Rivlin-type universal relation for simple shear

The pure shear stress formulation for incompressible isotropic hyperelastic materials is generalised to include an in-plane normal traction. A new inverted Rivlin-type universal relation for simple shear is then obtained in terms of the left Cauchy–Green deformation tensor, the counterpart for the stress formulation of the classical Rivlin relation for simple shear that follows from a displacement formulation of the problem. Bounds are then obtained for the corresponding amount of shear and the triaxial stretches in terms of the applied stress. The general results obtained are then illustrated for the neo-Hookean material, for which it is shown that a compressive normal stress in conjunction with a shearing traction results in a non-monotonic relationship between the amount of shear and the shear stress, with a finite amount of shear predicted in the limit of infinite shear stress. However the relative displacement of the top and bottom faces of the sheared specimen is shown to be a monotonic function of the shear stress, with an infinite relative displacement predicted for an infinite shear stress.

On the central role of the invariant I2 in nonlinear elasticity

The necessity of the inclusion of the second invariant of the left Cauchy-Green deformation tensor B, namely I2, in the strain energy function W of rubber-like materials is analysed. Universal relationships that underline such necessity are revisited, and experimental data are examined to establish the trends in the variation of ∂W/∂I2. Corroborated by the established experimental trends, we consider (meso)structural arguments to devise a plausible approach for incorporation of I2 into the W function. On the basis of the molecular theory of rubbers and considering the entanglements as a topological tube constraint, our analysis confers that a first approximation of W(I1,I2) is of the form W(I1,I2)=f(I1)+g(I2). The f(I1) contribution may be that of any classical generalised neo-Hookean model, and the functional form of g(I2) is directly deduced from the tube model of entangled molecules. An additional logarithmic functional form of I2 is also devised based on the rational approximation of the response function β−1. The ensuing additive-type W(I1,I2) models are then compared with experimental datasets. While this additive consideration may not be sufficient to account for all aspects of the mechanics of rubber-like materials, the fitting results demonstrate an eminent improvement in the predictions of the additive-type models compared with generalised neo-Hookean models having the same number of constitutive parameters. These analyses underline the central role of I2 in modelling the finite deformation of rubber-like materials.

Small superimposed radial oscillations for a class of damaged limited elastic tubes

Small superimposed radial oscillations for a class of damaged limited elastic, incompressible, isotropic and homogeneous thick-walled circular cylindrical tube are examined. Both the damage function and the considered limited elastic class depend only on the first invariant $$ I_{1} $$ of the left Cauchy–Green deformation tensor. The present work puts some light in the context of the surgical procedure of angioplasty, used especially to treat cardiovascular diseases. Our simplified isotropic assumption allows for a much faster and analytical computation of comparably accurate results. The examples of three limited elastic material models are presented, namely the Arruda–Boyce, Gent, and Demiray. The radial oscillations are further investigated for a damaged limited elastic thin membrane, and the examples above are explored. Universal relations are found for the three material membranes. The obtained results are matched with the clinical data of the existing literature. It appears that the pressure ratio for the virgin and damaged arterial tissue is higher before angioplasty and vice versa. A similar trend for the frequency ratio also follows.

Alignment and rotation of spheroids in unsteady vortex flow

Preferential orientations of inertialess nonspherical particles are examined through three qualitatively different stages of a time-evolving Taylor–Green vortex flow. Despite an unexpected decorrelation between the vorticity vector and the direction of Lagrangian stretching, experienced by material fluid elements over a substantial time interval, prolate spheroids aligned with the Lagrangian stretching direction, whereas oblate spheroids aligned with the Lagrangian compression direction. We, therefore, infer that spheroidal tracers orient themselves relative to the Lagrangian history of the velocity gradients, defined by the left Cauchy–Green deformation tensor, rather than with the fluid vorticity vector. This preferential alignment persists all throughout the statistically unsteady flow field and even in the inviscid and nonturbulent early stage of the time-dependent vortex flow. This explains the observed preferential spinning of rods and tumbling of disks, similar to that in homogeneous isotropic turbulence, even at the early stage when the flow is anisotropic and laminar. These preferred modes of particle rotation prevail all through the evolving flow despite a surprisingly long time interval, during which the fluid vorticity decorrelates from the direction of Lagrangian stretching.

An anisotropic stress-driven growth model for soft tissue based on Eulerian deformation tensor and growth potential

An anisotropic stress-driven growth model of living tissue is presented for tissue engineering based on Eulerian deformation tensor and growth potential. The evolution of growth is simply defined by the homeostatic pressure and Cauchy stress caused by the left Cauchy–Green deformation tensor that is an Eulerian deformation measure. For more generality, anisotropy is considered by proposing the concept of growth flow and growth potential that are inspired by the flow rule of plasticity theory. This method is then able to make three-dimensional anisotropic growth modeling. The Eulerian tensor-based characteristic enables easy implementation into finite element method (FEM). The presented model was implemented into an FEM code and validated by a theoretical solution of thick-walled hollow tubes of soft tissue in homeostatic state, and aortic stenting simulation was also performed for biomedical engineering application. The simulation results show good agreement with the references. The effect of the anisotropic function is also discussed with one element tension simulation. Finally, it is shown that the proposed model can capture experimental data of a growing tumor tissue, and a future work is discussed.

Finite deformation analysis of isotropic magnetoactive elastomers

In this study, the large deformation analysis of the magnetoactive elastomers based on continuum mechanics approach has been conducted. First, the governing differential equations for the spatial configuration are presented. Stored energy density function defined with respect to the invariants of the right or left Cauchy–Green deformation tensor and the magnetic field induction vector is adopted to develop a material model for finite deformation of isotropic magnetoactive elastomers (MAEs). An isotropic magnetoactive sample with 15% iron particle volume fraction is then fabricated, and a test setup has been designed to measure its magnetic permeability. Using an advanced magnetorheometer, quasi-static tests are then carried out on MAE circular cylindrical samples to find their torque-twist response under various magnetic fields. The experimental results are then effectively utilized to identify the unknown parameters in the proposed material model. The accuracy of the proposed constitutive model in predicting the response behavior of the MAE is then demonstrated through comparison of theoretical results with those obtained experimentally.

Universal relations in coupled electro-magneto-elasticity

In the present work, we develop a class of the coupled universal relations with the possible forms of electro-magneto-elastic (EME) deformation families in smart materials. In line with that, we adopt a classical continuum mechanics-based approach following the second law of thermodynamics. More precisely, we first formulate the deformation of an EME continua through the fundamental laws of physics with an amended form of energy function. This amended energy function successfully resolves the physical interpretation of the Maxwell stress tensor under large deformations. Next, we develop the EME coupling type of universal relations through a new inequality Tb−bT≠0 for a class of an EME material parallel to an equation Tb−bT=0 for an isotropic elastic material existing in the literature. Wherein, T and b denote the total Cauchy stress tensor and left Cauchy-Green deformation tensor, respectively. Further, we propose the possible forms of EME deformation families in smart materials for some standard experimental arrangements. At last, we also apply the above findings to a magnetostriction phenomenon in order to check the practical feasibility of the same and a good agreement is achieved successfully.

Linearly polarized waves of finite amplitude in pre-strained elastic materials

We study the propagation of linearly polarized transverse waves in a pre-strained incompressible isotropic elastic solid. Both finite and small-but-finite amplitude waves are examined. Irrespective of the magnitude of the wave amplitude, these waves may propagate only if the (unit) normal to the plane spanned by the directions of propagation and polarization is a principal direction of the left Cauchy-Green deformation tensor associated with the pre-strained state. A rigorous asymptotic analysis of the equations governing the propagation of waves of small but finite amplitude reveals that the time scale over which the nonlinear effects become significant depends on the direction along which the wave travels. Moreover, we design theoretically an experimental procedure to determine the Landau constants of the fourth-order weakly nonlinear theory of elasticity.

Effect of damage on the free radial oscillations of an incompressible isotropic tube

The effect of damage on the finite-amplitude, free radial oscillations of an arbitrary incompressible, isotropic, homogeneous cylindrical tube is investigated. Pressure is applied to the inner and outer surfaces of the damaged cylinder and constrained from both ends. A purely radial motion is observed when the pressure is removed. The corresponding equation of motion is obtained, incorporating the effect of damage. A simple exponential front factor damage function is introduced in the tube problem. The damage function is a function of the first invariant of the left Cauchy–Green deformation tensor and is dependent on its maximum previous ever value. It is found that the period of oscillation for a thin-walled neo-Hookean membrane varies with the damage function. In contrast, the respective period for an undamaged neo-Hookean membrane is a constant. The described study may help in the surgical procedure of angioplasty, performed during inflammatory diseases such as atherosclerosis. During angioplasty, owing to inflation of the balloon, the arterial wall is damaged; this study may help to gain more insight on the surgical procedure. Both thick- and thin-walled analysis of the damaged cylindrical tube are performed and compared with the undamaged case. Several results are inferred and illustrated graphically for two types of parent material model, namely Gent and neo-Hookean.

Nonlinear elastic effects in phase field crystal and amplitude equations: Comparison toab initiosimulations of bcc metals and graphene

We investigate non-linear elastic deformations in the phase field crystal model and derived amplitude equations formulations. Two sources of non-linearity are found, one of them based on geometric non-linearity expressed through a finite strain tensor. It reflects the Eulerian structure of the continuum models and correctly describes the strain dependence of the stiffness. In general, the relevant strain tensor is related to the left Cauchy-Green deformation tensor. In isotropic one- and two-dimensional situations the elastic energy can be expressed equivalently through the right deformation tensor. The predicted isotropic low temperature non-linear elastic effects are directly related to the Birch-Murnaghan equation of state with bulk modulus derivative $K'=4$ for bcc. A two-dimensional generalization suggests $K'_{2D}=5$. These predictions are in agreement with ab initio results for large strain bulk deformations of various bcc elements and graphene. Physical non-linearity arises if the strain dependence of the density wave amplitudes is taken into account and leads to elastic weakening. For anisotropic deformations the magnitudes of the amplitudes depend on their relative orientation to the applied strain.

A vorticity based approach to handle the fluid-structure interaction problems

A vorticity based approach for the numerical solution of the fluid-structure interaction problems is introduced in which the fluid and structure(s) can be viewed as a continuum. Retrieving the vorticity field and recalculating a solenoidal velocity field, specially at the fluid-structure interface, are the kernel of the proposed algorithm. In the suggested method, a variety of constitutive equations as a function of left Cauchy–Green deformation tensor can be applied for modeling the structure domain. A nonlinear Mooney–Rivlin and Saint Venant–Kirchhoff model are expressed in terms of the left Cauchy–Green deformation tensor and the presented method is able to model the behavior of a visco-hyperelastic structure in the incompressible flow. Some numerical experiments, with considering the neo-Hookean model for structure domain, are executed and the results are validated via the available results from literature.

An Invariant-Based Ogden-Type Model for Incompressible Isotropic Hyperelastic Materials

The Ogden model for an incompressible isotropic hyperelastic material is versatile enough to match complicated data for rubber-like materials at large deformations. However, the tensorial expression for the Cauchy stress in the Ogden model requires determination of the eigenvalues and eigenvectors of the left Cauchy-Green deformation tensor \(\mathbf{B}\). The objective of this paper is to propose an invariant-based Ogden-type model for isotropic incompressible hyperelastic materials. The strain energy function in this new model depends on classical invariants of \(\mathbf{B}\) and the Cauchy stress tensor can be expressed directly in terms of the tensor \(\mathbf{B}\) without need for its spectral form. Examples show that this new Ogden-type model retains the versatility of the original Ogden model in characterizing material response.

Identifying hyperelastic and isotropic materials by examining the variation of principal direction of left Cauchy–Green deformation tensor in uniaxial loading

Many bodies of biological and engineering interest have a fibrous and layered structure. Hence, it is believed that these bodies are inhomogeneous and made up of anisotropic material. Classical mechanical experiments used to find the required material symmetry in the constitutive relation cannot distinguish inhomogeneous bodies made of isotropic material and homogeneous bodies made of anisotropic material. Therefore, it is of interest to find an alternative hypothesis so that inhomogeneity and anisotropy can be determined independent of the other. This study finds that the principal (or eigen) direction of the left Cauchy–Green deformation tensor, B does not vary with the magnitude of the applied uniaxial load at a given location whenever the body – homogeneous or inhomogeneous – is made of isotropic and hyperelastic material and the deformations are measured from a stress free reference configuration. In general, the principal direction of the left Cauchy–Green deformation tensor varies with the magnitude of the uniaxial load when the body is made up of anisotropic material. Thus, it is concluded that if the variation in the principal direction of B with the magnitude of the applied uniaxial load is experimentally investigated then one could ascertain whether the body is made up of isotropic or anisotropic material.

Finite-amplitude Love waves in a pre-stressed compressible elastic half-space with a double surface layer

The dispersive behavior of finite-amplitude time-harmonic Love waves propagating in a pre-stressed compressible elastic half-space overlaid with two compressible elastic surface layers of finite thickness is investigated. The half-space and layers are made of different pre-stressed compressible neo-Hookean materials. The dispersion relation which relates wave speed and wavenumber is obtained in explicit form. Results for the energy density and energy flux of the waves are also presented. The special case where the interfaces between the layers and the half-space are principal planes of the left Cauchy–Green deformation tensor is also investigated. Numerical results are presented showing the variation of the Love wave speed with the pre-stress and the propagation angle.