Lagrangian Strain Tensor Research Articles

Comparative study of two geometrically non-linear beam models with different strain measures

In this work, two geometrically non-linear beam models which use different strain measures are investigated in problems with large rotations. The first is the Simo–Reissner model. The second is the Green–Lagrange model. The novel aspects of this work are: (1) the 1-D formulation of the second beam model which uses the Green–Lagrange strain tensor components so that is comparable to the Simo–Reissner beam model, (2) the demonstration of the numerical behaviour of this 1-D model after incorporating an invariant update procedure described on the rotation manifold SO(3), and (3) the comparison of the different strain measures of the two beam models through the use of three numerical examples.

Numerical Model for the Determination of Erythrocyte Mechanical Properties and Wall Shear Stress in vivo From Intravital Microscopy.

The mechanical properties and deformability of Red Blood Cells (RBCs) are important determinants of blood rheology and microvascular hemodynamics. The objective of this study is to quantify the mechanical properties and wall shear stress experienced by the RBC membrane during capillary plug flow in vivo utilizing high speed video recording from intravital microscopy, biomechanical modeling, and computational methods. Capillaries were imaged in the rat cremaster muscle pre- and post-RBC transfusion of stored RBCs for 2-weeks. RBC membrane contours were extracted utilizing image processing and parametrized. RBC parameterizations were used to determine updated deformation gradient and Lagrangian Green strain tensors for each point along the parametrization and for each frame during plug flow. The updated Lagrangian Green strain and Displacement Gradient tensors were numerically fit to the Navier-Lame equations along the parameterized boundary to determined Lame's constants. Mechanical properties and wall shear stress were determined before and transfusion, were grouped in three populations of erythrocytes: native cells (NC) or circulating cells before transfusion, and two distinct population of cells after transfusion with stored cells (SC1 and SC2). The distinction, between the heterogeneous populations of cells present after the transfusion, SC1 and SC2, was obtained through principle component analysis (PCA) of the mechanical properties along the membrane. Cells with the first two principle components within 3 standard deviations of the mean, were labeled as SC1, and those with the first two principle components greater than 3 standard deviations from the mean were labeled as SC2. The calculated shear modulus average was 1.1±0.2, 0.90±0.15, and 12 ± 8 MPa for NC, SC1, and SC2, respectively. The calculated young's modulus average was 3.3±0.6, 2.6±0.4, and 32±20 MPa for NC, SC1, and SC2, respectively. o our knowledge, the methods presented here are the first estimation of the erythrocyte mechanical properties and shear stress in vivo during capillary plug flow. In summary, the methods introduced in this study may provide a new avenue of investigation of erythrocyte mechanics in the context of hematologic conditions that adversely affect erythrocyte mechanical properties.

Physiological Motion Reduction Using Lagrangian Tracking for Electrode Displacement Elastography

Minimally invasive treatments such as microwave ablation (MWA) have been growing in popularity for extending liver cancer survival rates in patients, when surgery is not an option. As a non-ionizing, real-time alternative to contrast-enhanced computed tomography, electrode displacement elastography (EDE) has shown promise as an imaging modality for MWA. Despite imaging efficacy, motion artifacts caused by physiological motion result in unintended speckle pattern variance, thereby inhibiting consistent and accurate ablated region visualization. To combat these unavoidable motion artifacts, a Lagrangian deformation tracking (LDT) approach based on freehand EDE was developed to track tissue movement and better define tissue properties. For validating LDT efficacy, a spherical inclusion phantom as well as seven in vivo data sets were processed, and strain tensor images were compared with identical time sampled images estimated using a traditional Eulerian approach. In vivo results revealed greater consistency among visualized LDT strain tensor images, with segmented ablated regions exhibiting standard deviation reductions of up to 98% when compared with Eulerian strain tensor images. Additionally, Lagrangian strain tensor images provided Dice coefficient improvements up to 25%, and success rates improved from approximately 50% to nearly 100% for ablated region visualization.

Attempt to Assess the Scope of Applicability of a Hyperelastic Murnaghan’s Material Model in the Case of Elastomers

In this work, we use the MCMV slightly compressible hyperelastic constitutive model of rubber-like materials that was proposed in the authors’ work and implemented in the MES ABAQUS / Standard program. The stored energy function of the MCMV model is approximated by polynomials relative to the Lagrange strain tensor, attempting to evaluate the scope of applicability of Murnaghan’s models in the boundary-value problem of statics. We analyze in detail the homogeneous deformation problems. The point is to determine the approximate range of deformation, in which we get similar results of the problems mentioned, using the MCMV material model and its approximation.

Thermo-Chemo-Poromechanical Modeling of the Anode Mixture During the Baking Process: Constitutive Laws and Governing Equations

Abstract Aluminum is reduced from alumina by the Hall–Héroult electrolysis process in which the anode is utilized as the positive electrode. The quality of the prebaked anode plays a crucial rule in the efficiency of the aluminum electrolysis process. To produce high-quality anodes in the aluminum industry, the anode baking process calls for a deep understanding of mechanisms that govern the evolution of the anode mixture properties under the high-temperature condition. Therefore, the aim of this paper is to establish a thermo-chemo-poromechanical model for the baking anode by using the theory of reactive porous media based on the theory of mixtures within the thermodynamic framework. For this purpose, an internal state variable called “shrinking index” is defined to characterize the chemical progress of the pitch pyrolysis in the anode skeleton, and the Clausius–Duhem inequality is developed according to the Lagrangian formalism. By introducing a reduced Green–Lagrange strain tensor, a Lagrangian free energy is formulated to found a set of state equations. Then, the thermodynamic dissipation for this pyrolyzing solid–gas mixture is derived, and a constitutive model linking the chemical pyrolysis with the mechanical behavior is achieved. A dissipation potential is consistently defined to ensure the non-negativeness of the thermodynamic dissipation and to obtain the constitutive laws for viscous behaviors. Field equations governing the volatile diffusion and the heat transfer through the draining mixture body are derived from the entropy balance.

Analysis of the Elastoplastic Response in the Torsion Test Applied to a Cylindrical Sample

This work presents an experimental and numerical analysis of the mechanical behavior of a fixed-end SAE 1045 steel cylindrical specimen during the torsion test. To this end, an iterative numerical–experimental methodology is firstly proposed to assess the material response in the tensile test using a large strain elastoplasticity-based model solved in the context of the finite element method. Then, a 3D numerical simulation of the deformation process of the torsion test is tackled with this previously characterized model that proves to be able to predict the development of a high and localized triaxial stress and strain fields caused by the presence of high levels of angular deformation. Finally, the obtained numerical results are analytically studied with the cylindrical components of the Green–Lagrange strain tensor and experimentally validated with the measurements of shear strains via Digital Image Correlation (DIC) and the corresponding torque – twist angle curve.

Hierarchical Motion Estimation With Bayesian Regularization in Cardiac Elastography: Simulation and In Vivo Validation.

Cardiac elastography (CE) is an ultrasound-based technique utilizing radio-frequency (RF) signals for assessing global and regional myocardial function. In this work, a complete strain estimation pipeline for incorporating a Bayesian regularization-based hierarchical block-matching algorithm, with Lagrangian motion description and myocardial polar strain estimation is presented. The proposed regularization approach is validated using finite-element analysis (FEA) simulations of a canine cardiac deformation model that is incorporated into an ultrasound simulation program. Interframe displacements are initially estimated using a hierarchical motion estimation framework. Incremental displacements are then accumulated under a Lagrangian description of cardiac motion from end-diastole (ED) to end-systole (ES). In-plane Lagrangian finite strain tensors are then derived from the accumulated displacements. Cartesian to cardiac coordinate transformation is utilized to calculate radial and longitudinal strains for ease of interpretation. Benefits of regularization are demonstrated by comparing the same hierarchical block-matching algorithm with and without regularization. Application of Bayesian regularization in the canine FEA model provided improved ES radial and longitudinal strain estimation with statistically significant ( ) error reduction of 48.88% and 50.16%, respectively. Bayesian regularization also improved the quality of temporal radial and longitudinal strain curves with error reductions of 78.38% and 86.67% ( ), respectively. Qualitative and quantitative improvements were also visualized for in vivo results on a healthy murine model after Bayesian regularization. Radial strain elastographic signal-to-noise ratio (SNRe) increased from 3.83 to 4.76 dB, while longitudinal strain SNRe increased from 2.29 to 4.58 dB with regularization.

Evaluation of Various Geometrical Nonlinearities in the Response of Beams and Shells

This paper investigates the consistency and compatibility of various assumptions and strain measurements in a large displacements/geometric nonlinear analysis of beams and thin-walled structures. For this purpose, a refined beam model with enhanced three-dimensional accuracy is employed in a total Lagrangian scenario. This model is developed in the domain of the Carrera unified formulation, which allows expression of the nonlinear governing equations in terms of fundamental nuclei. These nuclei are independent of the theory approximation order; thus, low- to high-order theories of structures can be implemented with ease. Various numerical problems are addressed, and solutions are provided by using a classical finite element method and a Newton–Raphson linearization scheme. Given the intrinsic scalable nature of Carrera unified formulation, investigating the effects of the various nonlinear strain components is straightforward. It is demonstrated that the full Green–Lagrange strain tensor produces good approximation in case of large rotations, postbuckling, and nonlinear couplings. In contrast, approximations may be reasonable as moderate displacements and simpler problems (for example, slender beams under flexure) are considered.

Constructing continuum-like measures based on a nonlocal lattice particle model: Deformation gradient, strain and stress tensors

Explicit expressions for continuum-like measures of deformation gradient, strain and stress tensors were constructed using the techniques of weighted least squares and energy conjugate. Although presented based on a nonlocal lattice particle model, the formulations of these measures are general in nature and can be applied to other numerical models. Deformation gradient was formulated based on the deformation states of neighboring material points. Within the small strain limit, the infinitesimal strain tensor was derived from normal strains of bonds connecting material point of interest with all its neighbors. The second Piola–Kirchhoff stress tensor was constructed using the energy conjugate concept with respect to Green–Lagrangian strain tensor. The first Piola–Kirchhoff and Cauchy stress tensors were also formulated based on the derived second Piola–Kirchhoff stress tensor. For problems of homogeneous and inhomogeneous deformations, all three constructed measures yield very good predictions comparing to local continuum mechanics solutions. However, the second Piola–Kirchhoff formulation predicts less accurate results in surface regions due to incomplete neighbor list of material points with these regions.

Shortening Effect on Buckling Behavior of Reddy Plates and Prismatic Plate Structures

A closed-form solution based on the Reddy third-order shear deformation plate theory is proposed for the buckling of both flat and stiffened plates, simply supported on two opposite edges. The effect of the nonlinear strain–displacement terms, usually neglected under the von Kármán hypothesis, on the buckling of thick plates is investigated, and the equations governing the critical behavior considering the full Green–Lagrange strain tensor and the second Piola–Kirchhoff stress tensor are derived using the principle of minimum potential energy. The general Levy-type approach is employed, and the accuracy and effectiveness of the proposed formulation is validated through direct comparison with analytical and numerical results available in the literature. The parametric analyses performed for different geometrical ratios show that the von Kármán hypothesis holds only for thin flat plates whereas it can significantly overestimate buckling loads for stiffened plates, for which the buckling mode entails comparable in-plane and out-of-plane displacements.

Analytical Study of Coupling Effects for Vibrations of Cable-Harnessed Beam Structures

This paper presents a distributed parameter model to study the effects of the harnessing cables on the dynamics of a host structure motivated by space structures applications. The structure is modeled using both Euler–Bernoulli and Timoshenko beam theories (TBT). The presented model studies the effects of coupling between various coordinates of vibrations due to the addition of the cable. The effects of the cable's offset position, pretension, and radius are studied on the natural frequencies of the system. Strain and kinetic energy expressions using linear displacement field assumptions and Green–Lagrange strain tensor are developed. The governing coupled partial differential equations for the cable-harnessed beam that includes the effects of the cable pretension are found using Hamilton's principle. The natural frequencies from the coupled Euler, Bernoulli, Timoshenko and decoupled analytical models are found and compared to the results of the finite element analysis (FEA).

Procedure for 2D fluid–structure interaction simulation

A numerical technique for the solution of the structural dynamics equations of motion is presented. The structural dynamics mass and momentum conservation equations are solved using a control volume technique which is second-order accurate in space along with a dual time-step scheme that is second order accurate in time. The momentum conservation equation is written in terms of the Piola–Kirchoff stresses and the displacement velocity components. The stress tensor is related to the Lagrangian strain and displacement tensors using the St. Venant–Kirchoff constitutive relationship. Source terms are included to account for surface pressure and body forces. Verification of the structural dynamics solution procedure is presented for a two-dimensional vibrating cantilever beam. In addition, the structural dynamics solution procedure has been implemented into a general purpose two dimensional conjugate heat transfer solution procedure that uses a similar dual time-step control volume technique to solve the fluid mass, energy, and Navier–Stokes equations as well as the structural energy heat conduction equation. The resulting overall solution procedure allows for solutions to fluid/structure, fluid/thermal, or fluid/thermal/structure interaction problems. Verification of the multidisciplinary procedure is performed using a cylinder with a flexible solid protruding downstream that mimics a cylinder-flag configuration. The approach is a proof of concept for compressible flow with continuum based solids. The methods are currently being extended to 3D flow fields and solids.

A UNIFIED INELASTICITY AND DAMAGE MECHANICS FOR RUBBER-LIKE MATERIALS

The nonlinearities observed in the behavior of rubber and polymeric materials are influenced not only by the far-field stress conditions but also by the morphology, microstructures, and changes at the meso-level such as kinks, crosslinking, and micro tearing. Considering constants temperature applications, the occurrence of microtearing has a significant contribution on the performance of polymers, soft tissues and other rubber-like materials. In this paper, a unified damage mechanics and nonlinear elasticity approach is presented to model nonlinear behavior of rubber-like materials under constant temperatures. The theory is cast within the general framework of the internal variable theory of thermodynamics with large deformations where the Clausius-Duhem inequality is provoked to develop general damage potential. The strain energy density function is formulated in terms of an effective Lagrangian strain tensor that evolves with cumulative damage as cracking and micro-tearing take places. Piola-Kirchhoff (PK) stress tensor is presented and a new form of the damage response tensor is proposed. The model prediction is illustrated against experimental results with good agreement.

Objective Symmetrically Physical Strain Tensors, Conjugate Stress Tensors, and Hill’s Linear Isotropic Hyperelastic Material Models

We introduce a new family of strain tensors—a family of symmetrically physical (SP) strain tensors—which is also a subfamily of the well-known Hill family of strain tensors. For the further analysis, five scale functions are chosen which generate strain tensors belonging to the families of strain tensors previously introduced by other authors (i.e., the Doyle–Ericksen, Curnier–Rakotomanana, Curnier–Zysset, Itskov, and Darijani–Naghdabadi families) and to the new family of SP strain tensors. In particular, these five scale functions include the scale function generating the Lagrangian and Eulerian Hencky strain tensors. We introduce the family of SPH models of isotropic hyperelastic materials (with Hill’s linear relations) which are generated by SP strain tensors and work-conjugate stress tensors based on Hill’s natural generalization of Hooke’s law. Five SPH models of isotropic hyperelastic materials are generated on the basis of chosen SP strain tensors and work-conjugate stress tensors. These models are tested by solving two problems with homogeneous strain and stress tensors fields: the simple elongation and simple shear problems. Analysis of these solutions shows that the solutions of both problems for the Hencky isotropic hyperelastic material model (one of the five generated SPH models of isotropic hyperelastic materials) are qualitatively different from the solutions for the remaining four material models. That is, the solutions using the Hencky isotropic hyperelastic material model are of yielding nature typical of inelastic deformation of metals whereas the solutions for the other four material models reproduce strain diagrams typical of rubber.

GPU Accelerated Multilevel Lagrangian Carotid Strain Imaging.

A multilevel Lagrangian carotid strain imaging algorithm is analyzed to identify computational bottlenecks for implementation on a graphics processing unit (GPU). Displacement tracking including regularization was found to be the most computationally expensive aspect of this strain imaging algorithm taking about 2.2 h for an entire cardiac cycle. This intensive displacement tracking was essential to obtain Lagrangian strain tensors. However, most of the computational techniques used for displacement tracking are parallelizable, and hence GPU implementation is expected to be beneficial. A new scheme for subsample displacement estimation referred to as a multilevel global peak finder was also developed since the Nelder-Mead simplex optimization technique used in the CPU implementation was not suitable for GPU implementation. GPU optimizations to minimize thread divergence and utilization of shared and texture memories were also implemented. This enables efficient use of the GPU computational hardware and memory bandwidth. Overall, an application speedup of was obtained enabling the algorithm to finish in about 50 s for a cardiac cycle. Last, comparison of GPU and CPU implementations demonstrated no significant difference in the quality of displacement vector and strain tensor estimation with the two implementations up to a 5% interframe deformation. Hence, a GPU implementation is feasible for clinical adoption and opens opportunity for other computationally intensive techniques.

Three-dimensional vibration of cantilevered fluid-conveying micropipes—Types of periodic motions and small-scale effect

A new theoretical model is developed for the three-dimensional (3D) nonlinear vibration analysis of fluid-conveying cantilevered micropipes. Particular attention is given on the derivation and analysis of the reduced equations, and the small-scale effect on the periodic motions. Based on the modified couple stress theory (MCST), the governing equations are derived by using Hamilton’s principle. The material length scale parameter and large-deflection-induced geometric nonlinearities given by the Lagrangian strain tensor are incorporated into the governing equations. Utilizing the center manifold theory, normal form method and O(2) symmetry, the original governing equations can be rigorously reduced to a two-degree-of-freedom (2DOF) dynamical system. Then two possible types of periodic motions, i.e. planar periodic and spatial periodic motions, together with their stabilities are investigated by means of averaging methods and numerical simulations. Results show that the larger the dimensionless material length scale parameter is, the wider the region of mass ratio for stable planar periodic motion is. Particularly, the presence of small length scale parameter makes micropipes be more likely to oscillate in a plane. It is also shown that for mass ratio corresponding to the hysteresis of the curves of critical flow velocity versus mass ratio, the stabilities for bifurcating periodic motions at lower, moderate and higher critical flow velocities may be different.

The Importance of Cell Shape Sampling To Accurately Predict Flexibility in Metal-Organic Frameworks.

In this work, the influence of cell shape sampling on the predicted stability of the different metastable phases in flexible metal-organic frameworks at finite temperatures is investigated. The influence on the free energy by neglecting cell shape sampling is quantified for the prototypical MIL-53(Al) and the topical DUT-49(Cu). This goal is achieved by constructing free energy profiles in ensembles either in which the phase space associated with the cell shape is sampled explicitly or in which the cell shape is kept fixed. When neglecting cell shape sampling, thermodynamic integration of the hydrostatic pressure yields unreliable free energy profiles that depend on the choice of the fixed cell shape. In this work, we extend the thermodynamic integration procedure via the introduction of a generalized pressure, derived from the Lagrangian strain tensor and the second Piola-Kirchhoff tensor. Using this generalized procedure, the dependence on the unit cell shape can be eliminated, and the inaccuracy in free energy stemming from the lack of cell shape sampling can be uniquely quantified. Finally, it is shown that the inaccuracy in free energy when fixing the cell shape at room temperature stems mainly from entropic contributions for both MIL-53(Al) and DUT-49(Cu).

Basis-free expressions for families of objective strain tensors, their rates, and conjugate stress tensors

The objective (Lagrangian and Eulerian) strain tensors, their rates, and conjugate stress tensors used in continuum mechanics equations are considered. These tensors are represented in the eigenprojections of the right and left stretch tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . The novelty of the research lies in the simultaneous derivation of expressions for Lagrangian versions of tensors and their Eulerian counterparts with decomposition of the obtained expressions into components coaxial and orthogonal to the tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . We consider the Lagrangian and Eulerian Doyle–Ericksen families of strain tensors, which, in turn, are subfamilies of the well-known Hill family of strain tensors. Basis-free expressions are obtained for the material rates of Lagrangian tensors from the previously introduced family of generalized strain tensors, whose subfamilies are the Lagrangian and Eulerian Hill families of strain tensors, and similar expressions are obtained for the Green–Naghdi rates of Eulerian strain tensors, which are Eulerian counterparts of the material rates of Lagrangian strain tensors. Basis-free expressions for stress tensors are derived from the classical definitions of the conjugacy of stress and strain tensors. Expressions are found for the Lagrangian and Eulerian symmetric Atluri stress tensors that do not have conjugate strain tensors from the Hill family, and the obtained expressions are decomposed into components coaxial and orthogonal to the tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . The ranges of allowed ratios of different principal stretches at which the Lagrangian and Eulerian Hencky and Atluri stress tensors approximate the rotated/standard Kirchhoff stress tensors are determined. It is shown that the range of allowed ratios for the Hencky stress tensors is wider than that for the Atluri stress tensors.

Buckling analysis of geometrically nonlinear curved beams

The equation of motion of curved beams is derived in polar coordinate system which represents exactly the geometry of the beam. The displacements of the beam in radial and circumferential directions are expressed by assuming Bernoulli–Euler’s theory. The nonlinear strain–displacement relations are obtained from the Green–Lagrange strain tensor written in cylindrical coordinate system, but only the components related with radial and circumferential displacements are used. The equation of motion is derived by the principle of virtual work and it is discretized into a system of ordinary differential equations by Ritz method. Static analysis is performed in parametrical domain, assuming the magnitude of the applied force as parameter, and stability of the solution is determined. The nonlinear system of equations is solved by Newton–Raphson’s method. Prediction for the next point from the force–displacement curve is defined by the arc-length continuation method. Bifurcation points are found and the corresponding secondary branches with the deformed shapes are obtained and presented.

Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam

ABSTRACTIn this article, we present the nonlocal, nonlinear finite element formulations for the case of nonuniform rotating laminated nano cantilever beams using the Timoshenko beam theory. The surface stress effects are also taken into consideration. Nonlocal stress resultants are obtained by employing Eringen’s nonlocal differential model. Geometric nonlinearity is taken into account by using the Green Lagrange strain tensor. Numerical solutions of nonlinear bending and free vibration are presented. Parametric studies have been carried out to understand the effect of nonlocal parameter and surface stresses on bending and vibration behavior of cantilever beams. Also, the effects of angular velocity and hub radius on the vibration behavior of the cantilever beam are studied.