Small Strain Assumption Research Articles

Flexible thin rods with large bending

In order to describe the large bending of a flexible thin rod under the assumption of small strain, a new theory is developed based on the Cosserat elasticity. Two dimensionless parameters ωαρ (α= 1, 2) are presented to distinguish between large and small bends, where ωα are the bending curvatures and ρ is the characteristic radius of cross-sections. The bending is defined as small if ωαρ ≪ 1. On the contrary, the bending should be treated as a large one. In the case of large bending, torsion and bending are coupled. The equations given in the theory can be reduced to the Kirchhoff’ form when the bending becomes small. In a sense, the new theory can be seen as an extension of Kirchhoff's rod theory. The size effect is accounted for.

Modelling damage in multidirectional laminates subjected to multi-axial loading: Ply thickness effects and model assessment

A meso-scale damage modelling framework for composite laminates is assessed against multi-axial experimental data obtained from specimens subjected to combined tension/compression and shear loading using a modified Arcan fixture (MAF). Two laminates made from unidirectional carbon fibre-reinforced epoxy plies of different thickness are considered to investigate the ply thickness effect on the damage behaviour of modified open-hole specimens. Intra-laminar damage is predicted using a continuum damage model combined with a fibre-aligned mesh, while cohesive zones are used to account for inter-laminar damage. The multi-axial open-hole specimen strength is predicted with a mean relative error of 15% across all the investigated lay-up and loading configurations, while the predicted damage maps show encouraging agreement with full-field measurements obtained using digital image correlation (DIC). However, the assessment reveals limitations of the assumption of small strains and rotations in the formulation of the model that lead to spurious model behaviour for shear-dominated damage modes.

A finite strain framework for steady-state problems: Hyperelasto-viscoplasticity

A numerical framework for analyzing steady-state elastic–plastic material deformation at finite strains is developed and demonstrated in the present work. The framework is an extension of the original method by Dean and Hutchinson (1980) develop to analyze steady-state crack propagation, under a small strain assumption, in which the history-dependent material response is captured through streamline integration. Steady-state problems are encountered in numerous engineering processes, and the original studies of crack growth have already been extended to include rolling and drawing, though within a small strain framework. However, the investigation of such manufacturing processes, where strains greater than 10% easily develop, requires a finite strain formulation to provide accurate results. The framework proposed in the present work offers an efficient and accurate method to extract the steady-state solution at finite strains without encountering the numerical issues related to traditional Lagrangian procedures. Furthermore, the framework also accounts for elastic unloading compared to many existing numerical steady-state schemes as they are often restricted to rigid plasticity. The new numerical framework employs a hyperelastic material model, in terms of a Neo-Hookean material, combined with an isotropic viscoplastic material behavior. However, the framework is not limited to any specific hyperelastic material model, nor any specific model for the plastic behavior. The finite strain steady-state framework has been verified through comparison to a traditional Lagrangian analysis conducted in ANSYS. The benchmark case constitutes a plane strain drawing process where the thickness of the specimen is reduced by drawing it between two circular cylindrical tools.

Topology optimization of ultra high resolution shell structures

This work presents a high performance computing framework for ultra large scale, shell-element based topology optimization. The shell elements are formulated using a linear elastic, small strain assumption and are of the solid type, meaning that each quadrilateral shell element is extruded and assigned 24 degrees of freedom. The resulting linear system is solved using a fully parallelized multigrid preconditioned Krylov method, tailored specifically for unstructured quadrilateral shell meshes. The multigrid approach is shown to have good parallel scaling properties and is able to efficiently handle the ill-conditioning arising from the ‘Solid Interpolation of Material Properties’ (SIMP) method. For the optimization, the classical minimum compliance design problem with multiple load cases, prescribed minimum length scale and a local volume constraint is investigated. The latter is implemented through efficient PDE-filtering in contrast to usual local image filtering based implementations. Finally, the framework is demonstrated on two idealized examples from civil and aerospace engineering, solving shell optimization problems with up to 11 million shell elements on 800 cores. As an example, this resolution corresponds to a minimum feature size of 1.5 cm on a high-riser of height 80 m.

Undrained cylindrical cavity expansion in modified cam-clay soil: A semi-analytical solution considering biaxial in-situ stresses

This paper presents a semi-analytical solution for undrained cylindrical cavity expansion in modified Cam-clay soils under biaxial in-situ stresses based on the small strain and large strain assumptions in the elastic and plastic regions, respectively. The cylindrical cavity expansion problem is treated as a boundary value problem and formulated by solving a system of first-order differential equations with four stress components as basic unknown variables. The validity of the proposed solution is examined by comparing with the existing solution, where the in-situ stress is uniform. Extensive parametric studies are conducted to investigate the effects of biaxial in-situ stresses on the stress distribution, the expansion process and the shape and size of elastic-plastic boundary. The results indicate that the elastic-plastic boundary is in the shape of an ellipse due to the shear stress induced by biaxial in-situ stresses. The major axis of the elliptical elastic-plastic boundary is in accordance with the direction of the larger in-situ stress. The present solution is free of the limitation that the cavity expands under the uniform in-situ stress, therefore it is expected to provide a more general method for practical geotechnical and petroleum problems.

Shape-shifting panel from 3D printed undulated ribbon lattice

Materials that change their shape in response to external stimuli open up new prospects for efficient and versatile design and shaping of three-dimensional objects. Here, we present a novel class of micro-structures exhibiting an extension-bending coupling (EBC) effect, that can be harnessed as an elementary building block for shape-shifting panels. They are built with a single material as a network of undulated ribbons. The deformations mechanisms of both single and connected undulated ribbons are analysed using the finite element method to explain the main features of the EBC mechanism. For a particular micro-structure of the proposed class, the elastic response is investigated both under small strain assumption combining two-scale homogenization with Kirchhoff–Love plate theory, and at finite strains relying on numerical analysis. The range of achievable EBC ratio is then assessed with respect to the geometric parameters of the unit cell. Patterned specimens are manufactured using a commercial FFF Ultimaker 3D printer and are mechanically tested at finite strain up to 20%. The displacement measured by point tracking match the predictions from the finite element simulations and indicate that the structure maintain its properties at finite strain. Moreover, a tensile test load with point-like boundary is proposed to highlight exceptional out of plane displacement. The proposed ribbon based architectures can be combined with active materials for the actuation of shape shifting structures, like soft robots, control systems and power devices.

Electric-field induced deformation and bending in nematic elastomer strips with orientation gradient

Swollen nematic elastomers can deform under an applied electric field due to the electro-opto-mechanical coupling effect. This unique coupling mechanism causes the nematic elastomer to exhibit larger deformations under weak electric fields as compared to many conventional dielectric elastomers. Further, the electric-field induced deformations can be controlled by proper design of the initial liquid crystal director. Based on a simple electro-mechanical-director coupled model under small strain assumption, finite element simulations are conducted for beam-shaped thin strip samples with orientation gradient initially. Approximate analytical solutions are obtained for the electric potential, director reorientations, stress, and strain under weak electric fields. Using the first- order shear strain beam theory, the analytical solutions for the bending deflections are determined for beams with various boundary conditions, and they are only governed by the spontaneous curvature and the average transversal spontaneous shear resulting from the spontaneous strains induced by the applied electric fields. Optimization of the electric-field induced bending curvature leads to linearly-varying director distributions, which have much larger bending curvatures than that exhibited by the most commonly studied splay-bend distribution. Moreover, we demonstrate that bending shapes strongly depend on the orientational distribution of the director. Thus, a proper design can be used to effectively control the electric-field induced deformation and bending. Overall, we believe that our study is potentially useful to create desired and optimized bending for electrically-triggered sensors and actuators.

Geometric non-linearity in a multi-fiber displacement-based finite element beam model – an enhanced local formulation under torsion

This paper deals with a geometrically nonlinear finite element formulation for the analysis of torsional behaviour of RC members. Using the corotational framework, the formulation is developed for the inclusion of nonlinear geometry effects in a multi-fiber finite element beam model. The assumption of small strains but large displacements and rotations is adopted. The principle is an element-independent algorithm, where the element formulation is computed in a local reference frame which is uncoupled from the rigid body motions (translations and rotations) of the reference frame. In the corotational based frame, strains and stresses are measured from corotated to current, while base configuration is maintained as reference to measure rigid body motions. Corresponding to the requirement of corotational based, in the local frame, taking into account the torsional effect conducts to nonlinear strain assumption, thus require some specific development using a new kinematic model. Second order strain is accounted in the axial term, however lateral buckling is neglected, therefore this formulation is recommended to use in case of solid cross-section with arbitrarily large finite motions, but small strains and elastic material behaviour, such as slender of long-span reinforced concrete beam-column under flexion-torsional effect following serviceability limit state design. The enhanced formulation is validated in linear and nonlinear material range by several examples concerning beams of rectangular cross-section.

3D Beam Element for the Analysis of Torsional Problems of Steel-Structures in Fire

The paper presents the development of a three-dimensional (3D) beam element for the analysis of steel structures in fire that properly accounts for the degradation of the torsional strength and stiffness owing to thermal exposure. The element is well-suited for the analysis of members that are subjected to significant torsional effects, as steel members with open cross-sections subjected to torsion, lateral-torsional buckling, and torsional buckling. The element is based on a corotational formulation and small strains assumption, while the local formulation was developed according to both the Timoshenko and Bernoulli beam theories. The stress-strain relationship and thermal expansion of steel were implemented according to Eurocode EN 1993-1-2 (CEN 2005). A comprehensive numerical analysis was performed and results for several case studies are presented in the paper to assess the performance of the 3D beam element. The outcomes indicated very good agreement when compared with shell-based models, highlighting promising capabilities of modeling steel structures in fire. Moreover, the proposed element showed more accurate results with respect to beam elements included into two commercial software, i.e., ABAQUS and SAFIR.

Finite element implementation of viscoelastic and viscoplastic models

PurposeThe purpose of this study is to present a finite element (FE) implementation of phenomenological three-dimensional viscoelastic and viscoplastic constitutive models for long term behaviour prediction of polymers.Design/methodology/approachThe method is based on the small strain assumption but is extended to large deformation for materials in which the stress-strain relation is nonlinear and the concept of incompressibility is governing. An empirical approach is used for determining material parameters in the constitutive equations, based on measured material properties. The modelling process uses a spring and dash-pot and a power-law approximation function method for viscoelastic and viscoplastic nonlinear behaviour, respectively. The model improvement for long term behaviour prediction is done by modifying the material parameters in such a way that they account for the current test time. The determination of material properties is based on the non-separable type of relations for nonlinear materials in which the material properties change with stress coupled with time.FindingsThe proposed viscoelastic and viscoplastic models are implemented in a user material algorithm of the FE general-purpose program ABAQUS and the validity of the models is assessed by comparisons with experimental observations from tests on high-density polyethylene samples in one-dimensional tensile loading. Comparisons show that the proposed constitutive model can satisfactorily represent the time-dependent mechanical behaviour of polymers even for long term predictions.Originality/valueThe study provides a new approach in long term investigation of material behaviour using FE analysis.

Gibbs free energy based representation formula within the context of implicit constitutive relations for elastic solids

We derive a representation formula for a class of solids described by implicit constitutive relations between the Cauchy stress tensor and the Hencky strain tensor. Using a thermodynamic framework, we show that the Hencky strain tensor can be obtained as the derivative of the specific Gibbs free energy with respect to a stress tensor related to the Cauchy stress tensor. Unlike previous studies that have considered implicit relations between the Cauchy stress tensor and the Hencky strain we work with quantities that allow us to split the deformation into two parts. One part is connected to deformations that change the volume and the other to deformations where volume is preserved. Such a decomposition allows us to clearly characterise the interplay between the corresponding parts of the stress tensor, and to identify additional restrictions regarding the admissible formulae for the Gibbs free energy. We also show that if the constitutive relations of this type are linearised under the small strain assumption, then one can transparently obtain linearised models with density/pressure/stress dependent elastic moduli in a natural manner.

Analytical modelling of a tunnel accounting for elastoplastic unloading and reloading with reverse yielding and plastic flow

Abstract Applying a high internal pressure inside an underground cavity can increase the extent or intensity of the Excavation Damaged Zone (EDZ). This issue arises in many situations: the geological disposal of radioactive waste where different swelling materials can be encountered (engineered barriers, …), in-situ pressuremeter tests, gas storage projects, etc. This paper presents an analytical approach of the elastoplastic behaviour of a tunnel subjected to a simplified 2-stage loading under the assumption of small strains. In stage-1 (unloading), its excavation is represented in a classical way by cancelling the fictitious internal pressure initially applied to the wall of the future tunnel. In stage-2 (reloading), the internal pressure is increased up to relatively high values. The mechanical behaviour considered is elastic-perfectly-plastic, obeying Tresca's yield criterion and its associated flow rule. In both stages, the elastic phase preceding yielding is considered. During stage-2, the reversed direction of the stress path compared to stage-1 induces reverse yielding, reversing the direction of plastic strain rates. A fully analytical solution is presented, where all field quantities can be assessed by entirely explicit equations. Notably, the critical values of the internal pressure for which yielding resumes, then further extends, are presented.

Analysis of 1D large strain consolidation of structured marine soft clays

Structured soft clays are widely distributed in the coastal regions of China. The characteristics of the natural structure greatly influence the compressibility and permeability of marine soft clays and should be considered in the theory of their consolidation. When a large surcharge load is applied to a structured clay deposit, large strains can be induced in the clay layer due to the high compressibility, where the consolidation process follows the large strain assumption. However, there are few published theories of consolidation in which both the natural structure of marine soft clays and the large strain assumption can be considered simultaneously. In this study, a novel large strain consolidation model for structured marine soft clays was developed by considering the variation of structural yield stress with depth and using different calculation methods for initial effective stress of structured clay deposits in the Lagrangian coordinate system. The corresponding solution was derived by the finite difference method. Finally, the influences of the natural structure of soft clays and different geometric assumptions on consolidation behavior were investigated. The results show that the dissipation rate of excess pore water pressure of structured clays under the large strain assumption is expected to be faster than that under the small strain assumption, and the difference in consolidation behavior between the two assumptions increases with the strain level of natural structured clays. If the strain level in the clay layer is more than 15%, the difference in consolidation behavior between the large and small strain assumptions must be considered.

Strongly Nonlinear Vibrations of a Hyperelastic Thin-walled Cylindrical Shell Based on the Modified Lindstedt–Poincaré Method

Some significant behaviors on strongly nonlinear vibrations are examined for a thin-walled cylindrical shell composed of the classical incompressible Mooney–Rivlin material and subjected to a single radial harmonic excitation at the inner surface. First, with the aid of Donnell’s nonlinear shallow-shell theory, Lagrange’s equations and the assumption of small strains, a nonlinear system of differential equations for the large deflection vibration of a thin-walled shell is obtained. Second, based on the condensation method, the nonlinear system of differential equations is reduced to a strongly nonlinear Duffing equation with a large parameter. Finally, by the appropriate parameter transformation and modified Lindstedt–Poincar[Formula: see text] method, the response curves for the amplitude-frequency and phase-frequency relations are presented. Numerical results demonstrate that the geometrically nonlinear characteristic of the shell undergoing large vibrations shows a hardening behavior, while the nonlinearity of the hyperelastic material should weak the hardening behavior to some extent.

A novel hybrid shell element formulation (QUAD+ and TRIA+): A benchmarking and comparative study

This paper introduces a novel hybrid finite element (FE) formulation of shell element to enable assembly process simulation of compliant sheet-metal parts with higher efficiency and flexibility. Efficiency was achieved by developing both new hybrid quadrilateral and triangular elements. Quadrilateral element (QUAD+) was formulated by combining area geometric quadrilateral 6 (AGQ6) nodes and mixed interpolated tensorial components (MITC) to model membrane and bending/shear component respectively. Triangular element (TRIA+) was formulated by merging assumed natural deviatoric strain (ANDES) for membrane and MITC for bending/shear component. Flexibility was addressed by developing an open-source C++ code, enhanced by the OpenMP interface for multiprocessing programming. Tests and benchmarks were compiled and executed within Matlab using the MEX API interface. Extensive benchmark studies were accomplished to evaluate the performance of the proposed hybrid formulation and the shell formulations used in three FEM packages - ABAQUS, ANSYS and COMSOL- under static linear elastic condition with small strain assumption. It was observed that the proposed QUAD+ and TRIA+ elements performed better amongst the FE packages, especially when there was in-plane mesh distortion, with errors below 3%. It was also identified that the best efficiency is obtained by adopting dominant QUAD+ elements compared to the TRIA+ when working on complex geometries. This paper also contributes to present a wide set of benchmark studies required to verify new release of FE packages using shell element or evaluate the performance of new shell formulations.

The Characterization of the Stress Fields Near a Crack Tip for a Compact Specimen for Elastic-Plastic Materials Dominated by the Plane Strain State

The paper presents a comprehensive analysis of the stress field near a crack tip for a compact specimen dominated by the plane strain state using the finite element method. The analysis also includes the calculation of some parameters of in-plane constraints, both for small and large strain assumptions. It discusses the influence of the material characteristic, relative crack length and external load for the stress field, and the in-plane constraint parameter. The approximation formulas for some in-plane constraint parameters are presented.

Multi-Scale Numerical Analysis of the Effect of Microstructural Features on the Mechanical Behavior of Polycrystalline Ti-6Al-4V Alloy

The present work aims to model the influence of microstructural features of Ti-6Al-4V titanium alloy on its mechanical behavior. A multi-scale approach based on crystal plasticity is considered. The elasto-viscoplastic constitutive equations of Meric-Cailletaud are modified to take into consideration the effect of the grain size by introducing the Hall-Petch relationship at the local scale. This modified model is coupled with finite element calculations under small strain assumption to simulate the monotonic mechanical behavior of Ti-6A-4V at local and global scales. It is shown that the mechanical behavior of Ti-6Al-4V is drastically dependent upon the material features. Strong crystallographic texture can result in the formation of hardened and less hardened areas. Moreover, by increasing the grain size scattering, the heterogeneously deformed areas are multiplied. By decreasing the average grain size, the yield strength increases. It is observed that the effects of grain size, grain size scattering and crystallographic texture are coupled.

New hybrid cycle jump approach for predicting low-cycle fatigue behavior by a micromechanical model with the damage induced anisotropy concept

In this work, a new hybrid cycle jump approach was proposed to predict the low-cycle fatigue (LCF) response using a well-established micromechanical model. The damage activation/deactivation phenomenon was formulated at the macroscale. Using the concept of rate independent plasticity with the small strains assumption, the damage deactivation effect on the LCF response and its lifetime was accurately described by the model under simple and complex cyclic loading paths. Three algorithms namely, Taylor 1st and 2nd order and predictor-corrector were utilized for estimating the model response after each cycle jump. An optimum precision factor (ρ) value was defined and then tested via a suitable compromise between computation times and accuracy. In order to numerically determine the cyclic jump length (ΔN), a new hybrid approach was developed based on the overall von Mises stress and the overall damage together with monitoring the accumulated slip. Then, a comparative study revealed that the predictor-corrector algorithm was opted due to its robustness compared to the other algorithms. The numerical study was carried out using two different cyclic loading configurations. One was under simple tension-compression and the other under complex tension-torsion with 90° out-of-phase angle. The model response was described under two distinct states: (i) cycle-by-cycle and (ii) cycle jump concept. For each numerical state, the overall and local responses were recorded. With the new hybrid cycle jump approach, the model response highlights that the greater, the number of cycles, the greater, the gain in computation time but the less, the accuracy. For example, in TC for Nf = 10,000 cycles, a recorded gain in computation time was attained 77% (against 95% by means of cycles number) with a relative error of 4%; whereas in TT90 of Nf = 2124, the computation gain became 43.4% (and 73.4% with respect to number of jumped cycles) with a relative error of 3%.

Nonclassical mathematical model for the mechanical analysis of additive manufacturing viscoelastic materials on rotating cylindrical substrates

Technological processes of additive manufacturing viscoelastic aging axisymmetric material layers on the inner surface of a rotating rigid cylindrical substrate by means of deposition of supplementary material plies are modelled. The added plies may be preliminary tensed or compressed in the circumferential direction. A mathematical model based on general approaches of mathematical theory of accreted solids is proposed for describing the deformation process of the being formed material layer under the action of centrifugal forces in assumption of small plain strain and in quasistatic approximation. The corresponding closed-form expressions containing quadratures are constructed for describing the technological stresses evolution in the manufactured layer during and after the process of its accretion.

A Robust-to-Noise Deconvolution Algorithm to Enhance Displacement and Strain Maps Obtained with Local DIC and LSA

Digital Image Correlation (DIC) and Localized Spectrum Analysis (LSA) are two techniques available to extract displacement fields from images of deformed surfaces marked with contrasted patterns. Both techniques consist in minimizing the optical residual. DIC performs this minimization iteratively in the real domain on random patterns such as speckles. LSA performs this minimization nearly straightforwardly in the Fourier domain on periodic patterns such as grids or checkerboards. The particular case of local DIC performed pixelwise is considered here. In this case and regardless of noise, local DIC and LSA both provide displacement fields equal to the actual one convolved by a kernel known a priori. The kernel corresponds indeed to the Savitzky-Golay filter in local DIC, and to the analysis window of the windowed Fourier transform used in LSA. Convolution reduces the noise level, but it also causes actual details in displacement and strain maps to be returned with a damped amplitude, thus with a systematic error. In this paper, a deconvolution method is proposed to retrieve the actual displacement and strain fields from their counterparts given by local DIC or LSA. The proposed algorithm can be considered as an extension of Van Cittert deconvolution, based on the small strain assumption. It is demonstrated that it allows enhancing fine details in displacement and strain maps, while improving the spatial resolution. Even though noise is amplified after deconvolution, the present procedure can be considered as robust to noise, in the sense that off-the-shelf deconvolution algorithms do not converge in the presence of classic levels of noise observed in strain maps. The sum of the random and systematic errors is also lower after deconvolution, which means that the proposed procedure improves the compromise between spatial resolution and measurement resolution. Numerical and real examples considering deformed speckle images (for DIC) and checkerboard images (for LSA) illustrate the efficiency of the proposed approach.