Cauchy-Green Strain Tensor Research Articles

MODIFICATION OF THE EXTENDED TUBE MODEL (METM) FOR THE CHARACTERIZATION OF FILLED VULCANIZATES

ABSTRACT The aim of this study is to develop a material model for filled vulcanizates that is physically justifiable. This model builds upon the established extended tube model and is incorporated into a finite element program. The research demonstrates that the intrinsic deformation concept is inadequate for describing nonlinear deformation behavior under the assumption of incompressible, isotropic materials. Consequently, an alternative approach is proposed, employing a strain function rather than direct use of principal strains, to characterize reinforcement behavior. This strain function aligns with the first invariant of the right Cauchy-Green strain tensor over a wide deformation range. At minor deformations, the entanglements’ contribution is considered through an additional reinforcement term. The novel reinforcement function is depicted as a sum of three elements, each representing reinforcement at different strain levels: low, medium, and high. Experimental comparisons show that the Modified Extended Tube Model (METM) effectively captures the stress-strain response of filled systems across all deformation levels. Furthermore, the reinforcement function parameters, derived from fitting the METM to experimental data, offer a quantitative assessment of the fillers’ reinforcing effects, while the extended tube model parameters reflect the network characteristics.

Vibration of solid and thin-walled slender structures made of soft materials by high-order beam finite elements

In this work, high-order beam (1D) finite element models for the modal analysis of structures made of compressible and nearly-incompressible hyperelastic soft materials are presented in the well-established framework of Carrera Unified Formulation (CUF). In this investigation, the modal behavior of soft structures subjected to progressively increasing loads can be correctly predicted by higher-order structural theories, and the influence of pre-stress conditions applied on the modal response of structures is investigated. The mathematical formulation of hyperelastic isotropic materials is presented in terms of invariants of the right Cauchy–Green strain tensor, obtaining the most general expression of the Piola–Kirchoff 2 stress tensor and tangent elasticity tensor, both independent of the model adopted for the material strain energy function. Governing equations in matrix forms for the static nonlinear analysis and subsequent vibration problem around non-trivial equilibrium states are derived through the Principle of Virtual Displacements (PVD) under a total Lagrangian formulation, defining the fundamental nuclei of stiffness matrices and internal and external forces vector, all independent of expansion theories and kinematic models adopted in the mathematical modeling of finite elements. Actual numerical results are obtained by an iterative Newton–Raphson linearized scheme coupled with line-search algorithms, and they are compared with results obtained by the commercial code ABAQUS. Our proposed models are tested with large strain problems involving hyperelastic slender and thin-walled structures, for which mode aberration such as crossing or bearing are observed.

Analysis of transversely isotropic compressible and nearly-incompressible soft material structures by high order unified finite elements

This work proposes high-order beam (1D) and plate (2D) finite element models for the large strain analysis of compressible and incompressible transversely isotropic hyperelastic media, defined within the Carrera Unified Formulation (CUF) framework. The strain energy density function adopted in fiber-reinforced hyperelastic materials modeling is presented and expressed in terms of invariants and pseudo-invariants of the right Cauchy-Green strain tensor. The explicit expression of the tangent elasticity tensor is derived through the assumption of coupled formulation of strain energy functions. Refined fully nonlinear beam and plate models are defined in a total Lagrangian formulation, deriving the governing equations of the nonlinear static analysis through the Principle of Virtual Displacements in terms of fundamental nuclei, in resulting expressions of internal and external forces vectors, and tangent stiffness matrix independent of kinematic models and approximation theories adopted. The iterative Newton-Raphson linearization scheme coupled with the arc-length constraint is adopted to obtain actual numerical solutions. Different benchmark analyses in hyperelasticity are performed to assess the capabilities of our proposed model, analyzing the three-dimensional stress field for moderate to large strain states and comparing actual numerical results with exact closed-form solutions or results available in the literature, demonstrating the capabilities and reliability of CUF models in the analysis of fiber-reinforced soft materials and structures.

Pure torsion for incompressible hyperelastic materials of Valanis–Landel form

The stretch-based formulation of basic problems for incompressible isotropic hyperelastic materials has been the subject of renewed recent attention largely motivated by application to modelling the mechanical response of soft tissues. Here, we are concerned with this formulation of the classic problem of simple torsion of a circular cylinder that has been primarily in the past investigated by employing the classical approach of Rivlin in terms of invariants of the Cauchy–Green strain tensor. Attention is focused on examining the resultant axial force necessary to sustain pure torsion, thereby investigating the Poynting effect, and the resultant moment needed to generate the twist. General results for the resultant force and moment are obtained for all hyperelastic materials that can be modelled in the celebrated separable Valanis–Landel form. These expressions are specialized for the one-term Ogden model to provide explicit results valid for all strain-stiffening exponents in this model. Some particular examples are provided, mainly for even values of the exponent. A notable result obtained for the Ogden model is that for values of the hardening exponent n in that model for which n ≤ 6 , one obtains the classical Poynting effect, whereas for n > 6 , a reverse Poynting effect always occurs for sufficiently small twist.

Mapping the shape and dimension of three-dimensional Lagrangian coherent structures and invariant manifolds

We introduce maps of Cauchy–Green strain tensor eigenvalues to barycentric coordinates to quantify and visualize the full geometry of three-dimensional deformation in stationary and non-stationary fluid flows. As a natural extension of Lagrangian coherent structure diagnostics, which provide separate scalar fields and a one-dimensional quantification of fluid deformation, our barycentric mapping visualizes the role of all three Cauchy–Green eigenvalues (or rates of stretching) in a single plot through a novel stretching coordinate system. The coordinate system is based on the distance from three distinct limiting states of deformation that correspond with the dimension of the underlying invariant manifolds. One-dimensional axisymmetric deformation (sphere to rod deformation) corresponds to one-dimensional unstable manifolds, two-dimensional axisymmetric deformation (sphere to disk deformation) corresponds to two-dimensional unstable manifolds and the rare three-dimensional isometric case (sphere to sphere translation and rotation) corresponds to shear-free elliptic Lagrangian coherent structures (LCSs). We provide methods to visualize the degree to which fluid deformation approximates these limiting states, and tools to quantify differences between flows based on the compositional geometry of invariant manifolds in the flow. We also develop a simple analogue for bilinearly representing and plotting both rates of stretching and rotation as a single vector. As with other LCS techniques, these diagnostics define frame-indifferent material features in the flow. We provide multiple computed examples of LCS and momentum transport barriers, and show advantages over other coherent structure diagnostics.

Mechanics of pressurized planar hyperelastic membranes.

The paper examines the mechanics of inflation of incompressible planar hyperelastic membranes that are rigidly fixed at their boundary and subjected to a uniform pressure. Strain energy functions characterized by the neo-Hookean, Mooney-Rivlin and the Ogden forms are used. Fixity is provided along either circular or elliptical boundaries. The computational results indicate that the strain energy function has a significant influence on the pressure versus inflated volume response of the deformed membrane. When the strain energy function corresponds to a Mooney-Rivlin form, the circular membrane displays no tendency to develop any instability. The equivalent circular membranes composed of both the neo-Hookean and Ogden-type strain energy functions developed an initial 'Wrinkling Instability'. For planar membranes with an elliptical planform, the wrinkling instability is more pronounced; membranes composed of hyperelastic materials with a Mooney-Rivlin form of the strain energy function continue to deform without the development of an initial instability point, whereas membranes composed of both the neo-Hookean and Ogden materials exhibit wrinkling behaviour at critical locations at the interior of the fixed boundary region. The dependency of the strain energy function on the second invariant of the Cauchy-Green strain tensor has an influence in the suppression of hyperelastic effects. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

Three-field mixed finite element methods for nonlinear elasticity

In this paper, we extend the tangential-displacement normal–normal-stress continuous (TDNNS) method from Pechstein and Schöberl (2011) to nonlinear elasticity. By means of the Hu–Washizu principle, the distributional derivatives of the displacement vector are lifted to a regular strain tensor. We introduce three different methods, where either the deformation gradient, the Cauchy–Green strain tensor, or both of them are used as independent variables. Within the linear sub-problems, all stress and strain variables can be locally eliminated leading to an equation system in displacement variables, only. The good performance and accuracy of the presented methods are demonstrated by means of several numerical examples.

A nonlinear elastic description of cell preferential orientations over a stretched substrate

The active response of cells to mechanical cues due to their interaction with the environment has been of increasing interest, since it is involved in many physiological phenomena, pathologies, and in tissue engineering. In particular, several experiments have shown that, if a substrate with overlying cells is cyclically stretched, they will reorient to reach a well-defined angle between their major axis and the main stretching direction. Recent experimental findings, also supported by a linear elastic model, indicated that the minimization of an elastic energy might drive this reorientation process. Motivated by the fact that a similar behaviour is observed even for high strains, in this paper we address the problem in the framework of finite elasticity, in order to study the presence of nonlinear effects. We find that, for a very large class of constitutive orthotropic models and with very general assumptions, there is a single linear relationship between a parameter describing the biaxial deformation and cos ^2theta _{mathrm{eq}}, where theta _{mathrm{eq}} is the orientation angle of the cell, with the slope of the line depending on a specific combination of four parameters that characterize the nonlinear constitutive equation. We also study the effect of introducing a further dependence of the energy on the anisotropic invariants related to the square of the Cauchy–Green strain tensor. This leads to departures from the linear relationship mentioned above, that are again critically compared with experimental data.

КОНЕЧНЫЕ ДЕФОРМАЦИИ ТОРОИДАЛЬНОЙ ОБОЛОЧКИ

The stress-strain state of a nonlinear elastic shell exposed to the internal pressure is considered. A surface of the shell is toroidal in shape in the initial state. The Lagrangian coordinates of the shell are assigned to a cylindrical system. The kinematic characteristics of the process are shown: a law of the motion of points, vectors of a material basis, a strain affinor and its polar decomposition, the Cauchy-Green strain measure and tensor, the Finger measure, and the “left” and the“right” Hencky strain tensors. Neglecting the shear components of the stress tensor, a constitutive relation is obtained as a quasilinear relation between true stress tensor and the Hencky corotation tensor. A system of equilibrium equations is presented in terms of physical components of the true stress tensor in the Lagrangian coordinates. Using the equilibrium equations and the incompressibility condition, a closed system of nonlinear ordinary differential equations is obtained to determine six unknown functions, depending on the angle indicating a position of the points along the cross-section in the initial state. The method of successive approximations is applied to estimate stress tensor components and to derive logarithms of the elongations of material fibers.

Universal upper estimate for prediction errors under moderate model uncertainty.

We derive universal upper estimates for model prediction error under moderate but otherwise unknown model uncertainty. Our estimates give upper bounds on the leading-order trajectory uncertainty arising along model trajectories, solely as functions of the invariants of the known Cauchy-Green strain tensor of the model. Our bounds turn out to be optimal, which means that they cannot be improved for general systems. The quantity relating the leading-order trajectory-uncertainty to the model uncertainty is the model sensitivity (MS), which we find to be a useful tool for a quick global assessment of the impact of modeling uncertainties in various domains of the phase space. By examining the expectation that finite-time Lyapunov exponents capture sensitivity to modeling errors, we show that this does not generally follow. However, we find that certain important features of the finite-time Lyapunov exponent persist in the MS field.

Kearsley-type instabilities in finite deformations of transversely isotropic and incompressible hyperelastic materials

We assume that the strain energy density, W, for a transversely isotropic and incompressible hyperelastic solid is a complete quadratic function of components of the right Cauchy-Green strain tensor, C. We first discuss restrictions on the seven material parameters appearing in the expression for W. It is shown that for the reference configuration to be stress-free, both terms linear in (I4 − 1) and (I5 − 1) must simultaneously appear in the expression for W where I4 = A • CA, I5 = A • C2A, and A is a unit vector along the axis of transverse isotropy. Subsequently, we show that for a prismatic body comprised of this material deformed either in dead-load or displacement-controlled uniaxial tension/compression along A, depending upon the sign of the material parameter h14 multiplying the term (I1 − 3)(I4 − 1) in the expression for W, the two lateral stretches may bifurcate from being equal to being unequal for a stable solution defined as the one that has lower free energy. Here I1 = tr(C) where tr is the trace operator. This is similar to Kearsley's instability in an isotropic square Mooney-Rivlin membrane deformed with equal biaxial dead loads on the edges for which the stable deformed configuration shifts from having equal to unequal biaxial stretches.

Soft-grain compression: Beyond the jamming point.

We present the experimental studies of highly strained soft bidisperse granular systems made of hyperelastic and plastic particles. We explore the behavior of granular matter deep in the jammed state from local field measurement from the grain scale to the global scale. By means of a dedicated digital image correlation code and an accurate image recording method, we measure for each compression step the evolution of the particle geometries and their right Cauchy-Green strain tensor fields. We analyze the evolution of the usual macroscopic observables (stress, packing fraction, coordination, fraction of nonrattlers, etc.) along the compression process through the jamming point and far beyond. Analyzing the evolution of the local strain statistics, we evidence a crossover in the material behavior deep in the jammed state for both sorts of particles. We show that this crossover is due to a competition between material compression, dilation, and shear, so its position depends on the particle material. We argue that the strain field is a reliable observable to describe the evolution of a granular system through the jamming transition and deep in the dense packing state whatever the material behavior.

Higher-Order Modules in the Equations of Dynamics of a Prestressed Elastic Solid

In terms of the reference configuration, the linearization of the equations of nonlinear mechanics of an initially isotropic elastic solid (IIES) is carried out in the vicinity of a. certain prestressed equilibrium state. In this case, we used a. special representation of the elastic potential in terms of algebraic invariants of the Cauchy-Green strain tensor containing moduli of higher orders. Linearized defining relations and equations of motion are constructed that take into account the nonlinear effect of the initial deformations on the properties of the IIES.

Preferential orientation of tracer spheroids in turbulent channel flow

ABSTRACT Axis-symmetric spheroids, such as rod-like and disk-like particles, have been found to orient preferentially in near-wall turbulence by both experiment and numerical simulation. In current work we examined the orientation of inertialess spheroids in a turbulent channel flow at medium friction Reynolds number R eτ = 1000 given based on the half of channel height. Both elongated prolate spheroid and flat oblate spheroid are considered and further compared with the reference case of spherical particle. The statistical results show that in near wall region the prolate spheroids tend to align in the streamwise direction while the oblate spheroids prefer to orient in the wall-normal direction, which are consistent with earlier observation in low Reynolds number (R eτ = 180) wall turbulence. Around the channel center we found that the orientation of spheroids is not fully isotropic, even though the fluid vorticity are almost isotropic. The mechanism that gives rise to such particle orientations in wall-turbulence has been found to be related to fluid Lagrangian stretching and compression (Zhao and Andersson 2016). Therefore, we computed the left Cauchy-Green strain tensor along Lagrangian trajectories of tracer spheroids in current flow field and analyzed the fluid Lagrangian stretching and compression. The results indicated that, similar to the earlier observations, the directions of the Lagrangian stretching and compression in near-wall region are in the streamwise and wall-normal directions, respectively. Furthermore, cross over the channel the prolate spheroids aligned with the direction of Lagrangian stretching but oblate spheroids oriented with the direction of Lagrangian compression. The weak anisotropy of orientations of fluid Lagrangian stretching and compression observed at the channel center could be the reason for the aforementioned modest anisotropic orientation of spheroids in channel central region.

Practical concerns of implementing a finite-time Lyapunov exponent analysis with under-resolved data

Using Lagrangian techniques to find transport barriers in complex, aperiodic flows necessitates a careful consideration of the available dimensional support (3D versus 2D) and temporal resolution of the data to be analyzed, a particular challenge in experimental data acquisition. To illustrate and diagnose the detrimental effects that can manifest in the computed Lagrangian flow maps and Cauchy–Green strain tensor that are calculated as part of most Lagrangian coherent structure analyses, planar finite-time Lyapunov exponent (FTLE) fields are computed from analytically defined, experimentally collected, and numerically simulated velocity fields. The FTLE fields calculated using three-component, three-dimensional velocity information (3D FTLE) are compared with calculations using two-dimensional data considering only the in-plane velocities (2D FTLE), data that are typically gathered during fluid dynamics experiments. In some regions, where the vortex rotation axis is perpendicular to the plane of interest, the 2D FTLE may perform well. However, in regions where the vortex rotation axis has a non-zero component parallel to the plane of interest, whole structures can fail to be captured by the 2D FTLE. A quantitative analysis of the error in the 2D FTLE field as it relates to instantaneous vorticity deviation core angle is conducted using Hill’s spherical vortex and the wake of a bioinspired pitching panel. The effect of decreasing temporal resolution is studied using simulated 3D experiments of a fully turbulent channel flow, where the time resolution of the velocity data is artificially degraded. The resultant 3D FTLE fields progressively worsen with degrading velocity field temporal resolution by the visible elongation of coherent structures in the streamwise direction, indicative of the poorly resolved intermediate velocity fields. This effect can be mitigated with a simple method that invokes Taylor’s frozen eddy hypothesis. Both dimensional support and temporal resolution problems in experimental velocity fields can cause major errors in the resulting FTLE fields. With fundamental understanding about the flow field of interest, such as local vortex orientation or relevant length and time scales, some of the pitfalls may be avoided.

Polyconvexity inspired frameworks and structure‐preserving integrators for multi‐field problems

AbstractA new approach for the design of energy‐momentum (EM) consistent time integrators for nonlinear coupled problems is proposed. Polyconvexity inspired internal or Helmholtz free energy functions are obtained by using the rediscovered tensor cross product (see [1]) which is basically applied on the cofactor and the determinant of the right Cauchy‐Green strain tensor and greatly simplifies the algebra (see [1,2]). On this basis multi‐field problems concerning non‐linear thermo‐elastodynamics (see [3]) and electro‐elastodynamics (see [4]) are considered. For the former a temperature based weak form is employed which facilitates the design of a structure‐preserving time‐stepping scheme for coupled thermo‐elastic problems. For the latter a four‐field internal energy‐based dynamic formulation is applied. In both cases, the polyconvexity‐based framework facilitates the design of EM consistent time integrators. In particular algorithmic stress formulas are employed which show a remarkably simple structure when compared to traditional, elaborate projection‐based formulas. The spatial discretisation relies on finite element interpolations for the unknown fields. Eventually, the superior robustness of the proposed formulations is shown in two numerical examples.

A new efficient hyperelastic finite element model for graphene and its application to carbon nanotubes and nanocones

A new hyperelastic material model is proposed for graphene-based structures, such as graphene, carbon nanotubes (CNTs) and carbon nanocones (CNC). The proposed model is based on a set of invariants obtained from the right surface Cauchy-Green strain tensor and a structural tensor. The model is fully nonlinear and can simulate buckling and postbuckling behavior. It is calibrated from existing quantum data. It is implemented within a rotation-free isogeometric shell formulation. The speedup of the model is 1.5 relative to the finite element model of Ghaffari et al. [1], which is based on the logarithmic strain formulation of Kumar and Parks [2]. The material behavior is verified by testing uniaxial tension and pure shear. The performance of the material model is illustrated by several numerical examples. The examples include bending, twisting, and wall contact of CNTs and CNCs. The wall contact is modeled with a coarse grained contact model based on the Lennard-Jones potential. The buckling and post-buckling behavior is captured in the examples. The results are compared with reference results from the literature and there is good agreement.

Numerical investigation of microstructural damage during kneading of wheat dough

Abstract The focus of this study is the kneading process of wheat dough. As an idealized kneader geometry, the eccentric cylinder benchmark geometry was considered. This work presents the first computational study in which the dough model developed by Tanner's group (R.I. Tanner, F. Qi, S.-C. Dai, Bread dough rheology and recoil: I. Rheology. J. Cereal Sci. 148 (1) (2008) 33–40) was solved for inhomogeneous flow. The values of the model parameter were obtained from small amplitude oscillatory shear, stress relaxation, start-up of simple shear, and compression experiments. To computationally evaluate the damage function, a time evolution equation had to be implemented for the Cauchy–Green strain tensor. The flow problem was numerically solved using a finite volume procedure available in the software OpenFOAM package rheoTool. As a result of the rapid decrease in accumulated strain, filaments of low damage were formed near the outer diverging part of the cylinder, convected downstream and further extended, and eventually broken down. Planar extension had the most destroying effect on the microstructure. In the future, we will investigate three-dimensional free-surface flows in industrial kneading geometries.

A model for magneto-elastic behaviour

In this paper, a coupled magnetoelastic model for isotropic ferromagnetic materialsis presented. The constitutive equations are written on the basis of the total energy in whichthe right Cauchy-Green strain tensor and the Lagrangian form of the magnetic eld strengthare used as the basic state variables. It is also applied to ferromacnetic electric steel for whichthe material parameters are calibrated.

Shear waves in pressurized poroelastic media

Shear wave elastography measures shear wave speed in soft tissues for diagnostic purposes. In Rotemberg et al. [J. Biomech. 46(11), (2013), pp. 1875-1881] and Rotemberg et al. [Phys. Med. Biol. 57(2), (2012), pp. 329-341], shear wave speed measurements were shown to depend on prestrain, but not necessarily prestress, in a perfused canine liver. We model this phenomenon by examining incremental waves in a pressurized poroelastic medium with incompressible phases. For a poroelastic material with strain energy function W, which due to pressurization undergoes a volume expansion of Δ, we find the following general expression for the shear wave speed: c2/c02=(W1(Δ) + W2(Δ) Δ2/3)/Δ2/3 (W1(1) + W2(1)). Here, c0 is the shear wave speed in the unpressurized material, Wj = ∂ W/∂ Ij, and Ij is an invariant of the Cauchy-Green strain tensor. We also find important restrictions on the form of the strain energy function, which are typically not satisfied by strain energy functions commonly assumed for soft tissues.