Lagrangian Strain Tensor Research Articles

Nonlinear dynamic behavior of carbon nanotubes incorporating size effects

This research endeavors to investigate the geometrically nonlinear dynamic characteristics of Carbon Nanotubes (CNTs) while incorporating size effects based on the Second Strain Gradient (SSG) elasticity theory. To this end, the nonlinear governing equations and its corresponding boundary conditions are deduced in alignment with the Green–Lagrange strain tensor and Hamilton’s principle. Concurrently, the weak form is expounded through the utilization of the C3 continuum Hermite interpolation functions, which ensure the continuity and smoothness of higher-order strain and displacement fields. Subsequently, a perturbation methodology is introduced, incorporating nonlinear phenomena into the context of linear wave propagation within the framework of periodic structures theory. The wave propagation characteristics manifest a pronounced disparity between the models based on linear SSG theory and those through nonlinear SSG theory with stiffness hardening phenomenon. In contrast to the zigzag CNTs, armchair CNTs evince an elevated aptitude for wave control. The application of nonlinear SSG theory in combination with the wave finite element method is significant for comprehending the wave propagation characteristics of complex CNTs.

Homogenization of plane ring mesh under static loading

AbstractPlane ring meshes can be produced in large dimensions for various applications, for example, in architecture or protective systems. The numerical simulation of all individual rings of ring meshes is very costly, which is why methods to abstract the structure have been sought in the past. In this paper, a method for determining the strain parameters of a representative ring mesh section under in‐plane tensile loads is presented. Methods of rigid body analysis and representative volume elements with periodic boundary conditions are used. Green Lagrangian strain tensors are determined for both an exemplary representative ring mesh section under different tensile load cases and an experimental ring mesh under one load case. The comparison of the Green Lagrangian strain tensors of the ring meshes between numerical simulations and experiments shows a very good agreement.

Modelling intralaminar damage mechanisms in fibre reinforced polymers at finite strains

An approach to predict intralaminar failure mechanisms in laminate fibre-reinforced composites, at finite strains, is proposed. Transverse matrix cracking is described by a smeared crack model, activated by an invariant-based failure criterion, where an embedded cohesive crack is considered in the continuum description. A deformation gradient decomposition based on homogenisation considerations is developed for a more objective finite strain kinematic description of transverse cracks. The local equilibrium problem is established in the reference configuration to naturally include re-orientation of the fracture plane, and the employed cohesive law is able to deal with non-monotonic loading. Moreover, a continuum damage model formulated in terms of the Green–Lagrange strain tensor is used for longitudinal failure. The simulation of an open-hole multi-directional specimen subjected to tension illustrates the ability of the proposed constitutive equations to capture experimental observations.

On the pressure–deflection relations and instability of carbon-based composite nonlinear pipes

In this paper, for the first time, the nonlinear buckling of curved pipes is investigated for different distributions of single-walled carbon nanotubes (SWCNTs) in the matrix. The curved pipe is subjected to uniformly distributed transverse pressure and experiences uniform temperature rise. Additionally, the pipe is in one-sided contact with an elastic nonlinear medium. Defined thickness-dependent and temperature-dependent material properties of SWCNTs are taken from the results of molecular dynamics (MD) simulations, well-established in the literature on the subject. Five technologically justified types of CNT distributions are considered. Shen’s rule of the mixture is applied to determine the effective properties of CNT-based composites. To avoid determining shear correction factor, the higher-order shear deformation theory is used. The nonlinear components of the Green–Lagrange strain tensor and the principle of virtual displacements are applied to derive the nonlinear system of partial differential equations of motion of the curved pipe. The solution to formulated boundary value problem is obtained by the perturbation technique for both simply-supported and clamped-clamped boundary conditions. The pressure–deflection curves of the curved pipe are obtained and discussed. The investigation of snap-buckling intensity and upper/lower limit load of the curved pipe is presented and new insights for its nonlinear stability are presented. For instance, it is shown that the V-type of CNT distribution profile affects on lower deflection of the curved pipe while the higher deflection exists for the Λ-type profile.

Shear strain stabilized high-temperature metallic monoclinic VO2 variants with symmetry permission

In this work, temperature-dependent in situ x-ray diffraction reciprocal space mappings are employed to reveal three-dimensional lattice shear deformation in epitaxial VO2 variants, and linear Lagrangian strain tensors of epitaxial lattices are deduced by metric tensors of the reciprocal space at various temperatures. An equilibrium modulated-monoclinic phase is identified above the critical temperature, which is permitted in a translationengleiche subgroup from high symmetry with an index of 4. Being different from the conventional low-temperature monoclinic phase, the high-temperature monoclinic phase presents a specific metallic feature, which is ascribed to the Mott–Hubbard and charge density wave mechanisms. Under the proposed general procedure, which precisely identifies the normal and shear strain status of deformed lattices, it is demonstrated that the structural symmetry reduction under shear deformation unambiguously acts as the origin of unexpected metallic modulated-monoclinic VO2 at high temperatures. Our results highlight the significance of precise detection and contribution of shear deformation in various fields of physics.

Three-dimensional vibration analysis in extensible pipes conveying fluid with different initial geometrical configurations

In this paper, three-dimensional vibration of fluid-conveying pipes with different initial configurations is investigated. With Green–Lagrange strain tensor, extensible theory, and Plug-flow, governing equations are built by applying Hamiltons principle. The governing partial differential equations are discretized into ordinary differential equations via the differential quadrature method (DQM) and solved numerically to obtain the dynamical character of the different initial configurations. Numerical results show static deflection under different initial geometrical configurations and variation of in-plane and out-of-plane frequencies with increasing fluid velocity. Natural frequency and mode shape are explored under the variation of flow velocity.

Refined Beam Theory for Geometrically Nonlinear Pre-Twisted Structures

This paper proposes a novel fully nonlinear refined beam element for pre-twisted structures undergoing large deformation and finite untwisting. The present model is constructed in the twisted basis to account for the effects of geometrical nonlinearity and initial twist. Cross-sectional deformation is allowed by introducing Lagrange polynomials in the framework of a Carrera unified formulation. The principle of virtual work is applied to obtain the Green–Lagrange strain tensor and second Piola–Kirchhoff stress tensor. In the nonlinear governing formulation, expressions are given for secant and tangent matrices with linear, nonlinear, and geometrically stiffening contributions. The developed beam model could detect the coupled axial, torsional, and flexure deformations, as well as the local deformations around the point of application of the force. The maximum difference between the present deformation results and those of shell/solid finite element simulations is 6%. Compared to traditional beam theories and finite element models, the proposed method significantly reduces the computational complexity and cost by implementing constant beam elements in the twisted basis.

Nonlinear and linearized vibration analysis of plates and shells subjected to compressive loading

The present work provides a numerical model for carrying out virtual Vibration Correlation Technique (VCT) for computing the buckling load, identifying the natural frequencies variation with progressive higher applied load, and providing an efficient means for the verification of the experimental VCT results. The presented nonlinear approach is based on the Carrera Unified Formulation (CUF). Since far nonlinear regimes are investigated, the full Green–Lagrange strain tensor is adopted. Furthermore, geometrical nonlinear equations are written in a total Lagrangian framework and solved with an opportune Newton–Raphson method. For a robustness assessment of the virtual VCT, different flat panel and shell structures are studied and compared with results found in the available literature. The results prove that the proposed approach provides results with an excellent correlation with the experimental ones, allowing to predict the buckling load and the natural frequencies variation in the nonlinear regime with high reliability.

Computational structural dynamics general solution procedure using finite volumes

A method for the solution of the three-dimensional structural dynamics equations with large strains using a finite volume technique is presented. The proposed solution procedure is second order accurate in space and employs a second-order accurate dual time-stepping scheme. The momentum conservation equations are written in terms of the Piola-Kirchhoff stresses. The stress tensor is related to the Lagrangian strain tensor through the St. Venant-Kirchhoff constitutive relationship. The structural solver presented is verified through two test cases. The first test case is a three-dimensional cantilever beam subject to a gravitational load that is verified using theory and two-dimensional simulations reported in literature. The second test case is a three-dimensional highly deformable cantilever plate subject to a gravitational load. The results of this case are verified through a comparison with the modal response calculated by commercially available software. The focus of the current effort is the development and verification of the structural dynamics portion of a future fully coupled monolithic fluid-thermal-structure interaction code package.

Bayesian Regularized Strain Imaging for Assessment of Murine Cardiac Function In vivo.

A cardiac strain imaging framework with adaptive Bayesian regularization (ABR) is proposed for in vivo assessment of murine cardiac function. The framework uses ultrasound (US) radio-frequency data collected with a high frequency (fc = 30MHz) imaging system and a multi-level block matching algorithm with ABR to derive inter-frame cardiac displacements. Lagrangian cardiac strain (radial, er and longitudinal, el) tensors were derived by segmenting the myocardial wall starting at the ECG R-wave and accumulating interframe deformations over a cardiac cycle. In vivo feasibility was investigated through a longitudinal study with two mice (one ischemia-perfusion (IR) injury and one sham) imaged at five sessions (pre-surgery (BL) and 1,2,7 and 14 days post-surgery). End-systole (ES) strain images and segmental strain curves were derived for quantitative evaluation. Both mice showed periodic variation of er and el strain at BL with segmental synchroneity. Infarcted regions of IR mouse at Day 14 were associated with reduced or sign reversed ES er and el values while the sham mouse had similar or higher strain than at BL. Infarcted regions identified in vivo were associated with increased collagen content confirmed with Masson's Trichrome stained ex vivo heart sections.Clinical Relevance-Higher quality cardiac strain images derived with RF data and Bayesian regularization can potentially improve the sensitivity and accuracy of non-invasive assessment of cardiovascular disease models.

Shell finite element formulation for geometrically nonlinear analysis of straight thin-walled pipes

A family of new geometrically nonlinear finite elements is formulated for the simulation of the structural response in the elastic regime of straight pipes with circular cross-sections under various loading conditions. The first Piola–Kirchhoff stress tensor is employed within the principle of virtual work framework in conjunction with a total Lagrangian approach. The Green–Lagrange strain tensor is adopted to capture the finite deformation-small strain effects. The formulations are based on the kinematic assumptions of the thin shell theory and capture the follower effects due to pressure load. Three schemes are proposed to interpolate the displacement fields in the circumferential direction: (1) a Fourier series expansion, (2) a quartic spline interpolation, and (3) a mixed interpolation combining Fourier series and splines. The performance and prediction accuracy of the elements are assessed through comparisons with finite element models based on shell and elbow elements in ABAQUS under various loading conditions. The results demonstrate the ability of the elements to predict the displacement and stress fields. In particular, the element based on Fourier series interpolation is shown to provide accurate predictions.

ANCF Multiplicative-Decomposition Thermoelastic Approach for Arbitrary Geometry

The classical approach for the thermal analysis of solids and fluids uses the strain additive decomposition to account for the effect of the temperature. This strain-based approach does not properly capture the effect of complex stress-free reference-configuration geometry, is applicable only to small deformation problems, and leads to simplified expression for the Green–Lagrange strain tensor. In view of the geometric description of the absolute nodal coordinate formulation (ANCF), a new ANCF gradient-based approach is proposed. This approach employs a multiplicative decomposition of the matrix of position-gradient vectors in the stress-free reference configuration into two position-gradient matrices. One matrix is associated with the reference-configuration geometry before the application of the thermal load, and the other accounts for the volumetric change due to the change in temperature. A numerical study demonstrated the implementation of the proposed gradient-based approach.

Stress States in Highly Flexible Thin-Walled Composite Structures by Unified Shell Model

This work deals with highly flexible laminated shell structures. The main focus is to provide an advanced model for the accurate prediction of the interlaminar three-dimensional stress state of shells subjected to large displacements/rotations, buckling, and snap-through phenomena. In this context, a two-dimensional shell finite element based on the Carrera unified formulation (CUF) is formulated in an orthogonal curvilinear reference system. Thanks to CUF, the governing equations are expressed in terms of fundamental nuclei, which are invariant of the shell theory approximation order. Thus, classical theories of structures to layerwise approaches can be implemented with ease and in a unified manner. The full Green–Lagrange strain tensor is employed because far nonlinear regimes are investigated. Furthermore, the geometrical nonlinear equations are written in a total Lagrangian framework and solved with an opportune Newton–Raphson method along with a path-following approach based on the arc-length constraint. The results demonstrate that classical theories may bring to wrong and unconservative stress predictions, especially in nonlinear equilibrium states, where the use of advanced and layerwise approaches shall be recommended.

Use of a rotation-invariant linear strain measure for linear elastic crack propagation calculations

The classical stress intensity factor concept requires linear elastic calculations based on infinitesimal strains in order to create the correct stress and strain singularity at the crack tip. Infinitesimal strains, however, are not rotation-invariant and cannot be used in structures undergoing large rotations even if the stretch is small. Therefore, a new strain measure is proposed based on a linearization of the Lagrange strain tensor written in terms of the right stretch tensor instead of the deformation gradient. In that way the local rotation is removed from the linear strain measure. An example of crack propagation of a rotating beam shows the effectiveness of the method.

A high-precision curvature constrained Bernoulli–Euler planar beam element for geometrically nonlinear analysis

In this work, a high-precision curvature constrained Bernoulli–Euler beam element based on the gradient deficient beam elements (GDBE) of the absolute nodal coordinate formulation (ANCF) is developed for precisely and efficiently tackling the geometrically nonlinear analysis of straight and strongly curved planar thin beam structures. Firstly, this research proposes a vector interpolation scheme named “curvature constrained interpolation method” (CCIM) which can present comprehensive and accurate curve properties including Frenet frame, position, gradient and especially, curvature at base points. Since the correctness and superiority of the CCIM are proven, it is therefore utilized as the shape function of the proposed element, which ensures the second-order accuracy. With the accurate Frenet frame provided by the CCIM, the strain energy including axial strain and bending strain of a beam element is rigorously derived by applying the definition of the Green–Lagrange strain tensor in continuum mechanics with the Bernoulli–Euler beam assumption. The sign of curvature in the bending strain term is determined by a referenced vector. This revision makes the proposed element feasible when solving the configuration which presents different orientations of concavity or the deformed configuration where the orientation of concavity reverses with comparison to the initial configuration. Due to the constant global reference coordinate, the elastic force and its Jacobian matrix are expressed in an elegant form, which can be calculated by the Gauss quadrature integration conveniently. The efficiency and high accuracy of the proposed element are validated by several nonlinear benchmark problems, which is especially reflected in considerably fewer numbers of elements and continuous, exact strain result between adjacent elements with comparison to relevant examples from literature and ABAQUS. After comprehensive demonstrations, it can be concluded that the proposed element is accurate and effective enough for modelling planar thin beam structures experiencing large deformation and large rotation.

Family of Continuous Strain-Consistent Convective Tensor Rates and Its Application in Hooke-Like Isotropic Hypoelasticity

This paper presents a new family of objective (Lagrangian and Eulerian) continuous strain-consistent convective (corotational and non-corotational) tensor rates. Objective strain-consistent convective tensor rates are defined as having the property that there exist objective (Lagrangian and Eulerian) strain tensors from the Hill family such that the considered rates of these strain tensors are equal to the rotated/standard (Lagrangian/Eulerian) stretching (strain rate) tensors. The family of such Eulerian strain-consistent convective tensor rates was introduced by Bruhns et al. (Proc. R. Soc. Lond. A 460:909–928, 2004). On the one hand, the family of continuous strain-consistent convective tensor rates presented in this paper extends the Bruhns et al. family by including Lagrangian tensor rates and, on the other hand, it is narrower than the Bruhns et al. family due to the continuity requirement imposed on tensor rates, which is necessary for applications. The theorem that any strain tensors only from the Doyle–Ericksen family (which is a subfamily of the Hill family) provide sufficient continuity conditions for strain-consistent convective tensor rates was formulated and proved. The expressions obtained by proving this theorem for convector tensors show that each tensor from the Doyle–Ericksen family is associated with a single strain-consistent convective tensor rate (in the Bruhns et al. family, any strain tensor from the Hill family generates infinitely many convector tensors associated with infinitely many strain-consistent convective tensor rates). In addition, a new family of Hooke-like isotropic hypoelastic material models based on objective continuous strain-consistent convective rates of the rotated/standard (Lagrangian and Eulerian) Kirchhoff stress tensors was constructed. The theorem that any material model from this family is self-consistent provided that the Lame parameter $\lambda =0$ and/or the deformation of the body is isochoric was formulated and proved. By self-consistent hypoelastic material models are meant those models for which constitutive hypoelastic relations are counterparts of constitutive relations for Cauchy/Green elasticity. Some material models from the new family were tested by solving the simple shear problem. Both new and well-known solutions of this problem for material models were obtained using an approach that takes into account the self-consistency property of material models from the new family.

Direct Eulerian Formulation of Anisotropic Hyperelasticity

Abstract Many soft materials and biological tissues comprise isotropic matrix reinforced by fibers in the characteristic directions. Hyperelastic constitutive equations for such materials are usually formulated in terms of a Lagrangian strain tensor referred to the initial configuration and Lagrangian structure tensors defining characteristic directions of anisotropy. Such equations are “pushed forward” to the current configuration. Obtained in this way, Eulerian constitutive equations are often favorable from both theoretical and computational standpoints. In the present note, we show that the described two-step procedure is not necessary, and anisotropic hyperelasticity can be introduced directly in terms of an Eulerian strain tensor and Eulerian structure tensors referring to the current configuration. The newly developed constitutive equation is applied to the particular case of the transverse isotropy for the sake of illustration.

First-principles calculation of higher-order elastic constants using exact deformation-gradient tensors

We propose a general and easy-to-use method of the ab initio calculation of the higher-order elastic constants, which is based on the analytical formulas for the deformation-gradient tensors as functions of the Lagrangian strain. The method allows for elimination of the truncation errors in the Taylor expansion series of the deformation gradients and is particularly useful to calculate the fourth-order elastic constants, where large strains have to be applied. It also facilitates the calculation of the Lagrangian stress, which is helpful in determination of the strain-stress relations. To demonstrate the application of our approach, we derive the analytic formulas for the deformation gradients as functions of the Lagrangian strain tensors, which are used in calculations of the third-order elastic constants in trigonal crystals and the fourth-order elastic constants in cubic crystals. Then, we perform the ab initio calculations of the fourth-order elastic constants in face-centered-cubic aluminum. We discuss the results obtained using the strain-energy and strain-stress relations and analyze the errors of the fourth-order elastic constants which would be incurred when approximating the deformation gradients by the Taylor polynomials. We show that the relatively small truncation errors in the Taylor expansion series of the deformation gradients can cause significant deviations of the fourth-order elastic constants. This effect is larger for the strain-energy method than for the strain-stress approach. We find that in both methods, the deviations are particularly significant for the ${C}_{1155}$, ${C}_{1266}$, ${C}_{4455}$, ${C}_{1255}$, and ${C}_{1456}$ elastic constants and are mainly caused by the truncation errors in the nondiagonal elements of the Taylor expansion series of the deformation gradients.

An advanced shell model for the analysis of geometrical and material nonlinear shells

In this paper layered shells subjected to static loading are considered. The displacements of the Reissner–Mindlin theory are enriched by a an additional part. These so-called fluctuation displacements include warping displacements and thickness changes. They lead to additional terms for the material deformation gradient and the Green–Lagrangian strain tensor. Within a nonlinear multi-field variational formulation the weak form of the boundary value problem accounts for the equilibrium of stress resultants and couple resultants, the local equilibrium of stresses, the geometrical field equations and the constitutive equations. For the independent shell strains an ansatz with quadratic shape functions is chosen. This leads to a significant improved convergence behaviour especially for distorted meshes. Elimination of a set of parameters on element level by static condensation yields an element stiffness matrix and residual vector of a quadrilateral shell element with the usual 5 or 6 nodal degrees of freedom. The developed model yields the complicated three-dimensional stress state in layered shells for elasticity and elasto-plasticity considering geometrical nonlinearity. In comparison with fully 3D solutions present 2D shell model requires only a fractional amount of computing time.

A total Lagrangian Timoshenko beam formulation for geometrically nonlinear isogeometric analysis of spatial beam structures

This paper concerns a novel isogeometric Timoshenko beam formulation for a geometrically nonlinear analysis of spatial beams using the total Lagrangian description. Constitutive laws for hyperelastic materials, whose behavior varies with their deformations, are widely defined by using strain energy density functions that are written in terms of the Green–Lagrange strain tensor. Many finite element beam formulations for geometrically nonlinear analyses of spatial beams are developed using the Green–Lagrange strain tensor and its energy conjugate, the second Piola–Kirchhoff stress tensor. Unfortunately, there virtually exist no isogeometric Timoshenko beam formulations for this type of analysis that are derived by using this energy conjugate pair. To allow the possibility of considering hyperelastic materials, the present isogeometric beam formulation is developed in the total Lagrangian description using the Green–Lagrange strain tensor and the second Piola–Kirchhoff stress tensor. The proposed formulation is capable of simulating beam structures that are subjected to large displacements and rotations, without any restriction in magnitude. Three-dimensional beam configurations are reduced into one-dimensional structures using the beam axis and director vectors of the cross sections. The cross-sectional rotation along the beam axis is represented by an orthogonal tensor, which is parameterized by a vector-like parameter. Updating the cross-sectional rotations is performed purely by natural exponentiation and superposition of relevant rotational quantities. To show the accuracy and efficiency of the proposed beam formulation, some benchmark and well-established numerical examples with various types of beam, i.e., straight, curved, pre-twisted beams, are analyzed. The obtained results are compared with those results in the literature, obtained from both analytical and numerical methods.