Model For Small Deformations Research Articles

Investigation of wave-driven hydroelastic interactions using numerical and physical modelling approaches

Wave-driven hydroelasticity is of great importance to a wide range of applications within offshore and coastal engineering. Harnessing the benefits of hydroelasticity or minimising its impacts, depending on the application, has recently led to substantial investment in research effort in this field. However, the complex and strongly-coupled nature of the problem generally make the impacts very case specific, highlighting the importance of accurate numerical tools for assessing the impact on a case-by-case basis. Therefore, this study aims to provide novel experimental data to assist with the development of a coupled numerical methodology for simulating fully nonlinear hydroelastic interactions with highly-flexible floating structures. Novel physical data from a laboratory campaign conducted at the University of Plymouth is presented, and used as a reference for assessing the capabilities of an existing coupled numerical approach. The numerical model is a partitioned approach based within the open-source computational fluid dynamics software OpenFOAM and consisting of a two-phase fluid solver; a linear solid model for small deformations solved via the block-coupled method; and strongly-coupled through the Dirichlet–Neumann method with dynamic Aitken under-relaxation. The numerical model is shown to capture well the wave-induced deformation, and the qualitative differences between structures of varying dimensions. However, the high computational cost limits the scope of this work to 2-D, and future work should focus on optimising the approach to allow for application in 3-D problems.

Modification of the Mechanical Properties of Core‐Shell Liquid Gallium Nanoparticles by Thermal Oxidation at Low Temperature

AbstractGallium nanoparticles (Ga NPs) are attracting increasing attention because of their appealing physical‐chemical properties. In particular, their mechanical properties play a key role in the implementation of these core‐shell structures on certain applications, such as soft and stretchable electronics. Thus, efforts are being addressed to modulate them mainly by chemical means. In contrast, this study investigates how the mechanical properties of the outer gallium thin oxide shell change when its thickness is increased through a thermal oxidation strategy. Specifically, as‐deposited Ga NPs, as well as those subjected to thermal oxidation at 300 °C for three different times, are studied by performing single‐particle indentations by atomic force microscopy over a wide range of NP radius. This analysis helps to confirm that the Reissner's thin‐shell model for small deformations within the elastic regime is obeyed. From these data, the dependence of the shell stiffness and the Young's modulus of the gallium oxide on the thermal treatment is obtained. It is found that the shell stiffness increases with the annealing time, even by a factor of 50 under prolonged thermal oxidation, while the gallium oxide Young's modulus, close to 30 GPa, does not change significantly.

Interfacial Tension Measurements in Microfluidic Quasi-Static Extensional Flows.

Droplet microfluidics provides a versatile tool for measuring interfacial tensions between two immiscible fluids owing to its abilities of fast response, enhanced throughput, portability and easy manipulations of fluid compositions, comparing to conventional techniques. Purely homogeneous extension in the microfluidic device is desirable to measure the interfacial tension because the flow field enables symmetric droplet deformation along the outflow direction. To do so, we designed a microfluidic device consisting of a droplet production region to first generate emulsion droplets at a flow-focusing area. The droplets are then trapped at a stagnation point in the cross junction area, subsequently being stretched along the outflow direction under the extensional flow. These droplets in the device are either confined or unconfined in the channel walls depending on the channel height, which yields different droplet deformations. To calculate the interfacial tension for confined and unconfined droplet cases, quasi-static 2D Darcy approximation model and quasi-static 3D small deformation model are used. For the confined droplet case under the extensional flow, an effective viscosity of the two immiscible fluids, accounting for the viscosity ratio of continuous and dispersed phases, captures the droplet deformation well. However, the 2D model is limited to the case where the droplet is confined in the channel walls and deforms two-dimensionally. For the unconfined droplet case, the 3D model provides more robust estimates than the 2D model. We demonstrate that both 2D and 3D models provide good interfacial tension measurements under quasi-static extensional flows in comparison with the conventional pendant drop method.

Analytical model for coupled consolidation and diffusion of organic contaminant transport in triple landfill liners

A triple-layer composite liner consisting of a geomembrane liner (GMB), a geosynthetic clay liner (GCL) and a compacted clay liner (CCL) is commonly used at the landfill bottom liner system to isolate the contaminated leachates. In this paper, one-dimensional quasi-steady-state small deformation model (SDSS) was developed to investigate the behavior of organic chemicals transport in landfill composite liner system considering coupled effect of consolidation, diffusion and degradation. The first and second type bottom boundary conditions are used to derive the analytical solutions. The generalized integral transform technique (GITT) is adopted to derive the analytical solutions. The effect of consolidation on the performance of GMB/GCL/CCL with intact or leaking GMB is investigated. The triple liner under double drainage boundary condition (DDBC) has better performance compared to the case under single drainage boundary condition (SDBC). This is because the velocity induced by consolidation under DDBC is lower than that under SDBC. The effect of GCL consolidation shows an opposite trend compared to CCL consolidation. Considering GCL consolidation can increase the breakthrough time. The effective diffusion coefficient of GCL can be two magnitude orders smaller after consolidation, which provides a better diffusion barrier for the chemical transport. The effects of adsorption and degradation have been analyzed as well. Increasing the adsorption capacity of a deforming composite liner can increase the steady-state bottom flux, which shows the opposite tendency compared to the case without considering consolidation. This is due to the fact that for the case of a deforming composite liner, the advection induced by consolidation includes a new term due to the solid velocity. This velocity will result in the increase the mass of chemical migration through the composite liner.

Torsion control in structures isolated with the triple friction pendulum system

A parametric study was carried out to investigate the torsional response of structures isolated with the Triple Friction Pendulum (TFP) system with building properties such as uncoupled torsional to lateral frequency ratio, Ωs, building slenderness, and mass and stiffness eccentricities. A simplified model of a parametric multistory structure and a small deformation model of the TFP, but with the ability to capture uplift, were used to perform parametric 3D non-linear dynamic analyses. The input ground motions were 20 Chilean records selected using a conditional mean spectrum. Response history results showed that uplift in the devices occurs in structures with large mass eccentricity, and in some extreme cases, normal force values in the TFPs may be as large as 6 times the nominal average gravitational load in each isolator. Torsional envelope response results also show that mass eccentric systems may undergo larger normalized edge amplification factors relative to stiffness eccentric systems, with values 2.5 and 1.8, respectively, and that torsional balance cannot be achieved naturally. Although rotations are small in magnitude due to seismic isolation, displacement values in resisting planes of the superstructure equidistant from the geometric center of the plan will be different. Moreover, structures with large Ωs present correlation values between response histories of torsion and translation measured at the geometric center of the plan near one, as well as values above zero for different Ωs and plan eccentricity, which implies no torsional control in the weak sense. It is concluded that parameter Ωs strongly influences the torsional amplification values computed herein.

A gradient-extended two-surface damage-plasticity model for large deformations

This work is concerned with the theoretical development, algorithmic implementation and numerical investigation of a novel thermodynamically consistent ‘two-surface’ gradient-extended damage-plasticity model for arbitrarily large deformations. It can be considered the geometrically nonlinear version of a corresponding model for small deformations which was presented recently (Brepols et al., (2017; 2018)). The terminology ‘two-surface’ stems from the fact that damage and plasticity are treated in the model as truly distinct but coupled dissipative mechanisms, meaning that individual damage loading and plastic yield criteria as well as appropriate loading/unloading conditions are used in the formulation, respectively. Such an approach makes the model flexibly adaptable to various situations in which the material under consideration shows either a more brittle- or ductile-like damaging behavior. Nonlinear Armstrong-Frederick kinematic hardening, nonlinear Voce isotropic hardening as well as nonlinear damage hardening are accounted for in the presented formulation that relies exclusively on symmetric internal variables, being beneficial, e.g., from the computational point of view. The model is formulated in the framework of Continuum Damage Mechanics and its gradient-extension is derived on the basis of the micromorphic approach according to Forest (2009, 2016). As such, it allows for the computation of mesh-insensitive results in finite element simulations involving material softening which is practically confirmed by means of several structural example problems. Furthermore, many other interesting aspects are highlighted, as, e.g., the model's implementation into finite elements, the computation of the algorithmically consistent tangent operators necessary to retain a quadratic rate of convergence in a global Newton-Raphson scheme, or a suitable time-integration of the model's evolution equations at the integration point level.

Development of Geometrically Accurate Continuum-Based Tire Models for Virtual Testing

AbstractIn this paper, an approach based on the integration of computer-aided design and analysis (I-CAD-A) is used to develop new continuum-based finite element (FE) tire models for the small and large deformation analyses. Based on given tire specifications, the mechanics-based geometry/analysis absolute nodal coordinate formulation (ANCF) is used to define the tire geometry with the same degree of accuracy as B-splines and nonuniform rational B-spline (NURBS), widely used in the computer-aided design (CAD) systems. In the case of large deformations, the ANCF geometry can be used directly as the analysis mesh without the need for conversion or adjustments. In order to define the material parameters that characterize the ANCF tire composite structure, a virtual test rig is developed, and the tire calibration process is performed according to the standards defined by the Society of Automotive Engineers (SAE). In order to develop small-deformation models that can be used in the prediction of the tire frequencies and mode shapes, the ANCF position vector gradients are consistently written in terms of rotation parameters, leading to geometrically accurate floating frame of reference (FFR) finite elements, referred to as ANCF/FFR elements. Using this mechanics-based geometry/analysis approach, new geometrically accurate reduced-order tire models are systematically developed and used to define vibration equations for the prediction of the tire frequencies, which are verified using a commercial FE software. The element stiffness matrix is calculated using the general continuum mechanics approach (GCM), and the effectiveness of the strain split method (SSM) for locking alleviation is tested. The results obtained in this investigation show that the I-CAD-A tire modeling approach can be used to develop geometrically accurate tire models suited for the large-deformation multibody system (MBS) problems as well as for the prediction of the tire frequencies and mode shapes.

Research on deformation law and mechanism for milling micro thin wall with mixed boundaries of titanium alloy in mesoscale

At mesoscale, the blade of micro impeller with Ti-6Al-4V titanium alloy (micro thin wall with mixed boundaries) is a thin-walled structure, characterized by small size, low stiffness, high surface precision, complex boundaries, and difficult-to-control machining deformation. To control and reduce the deformation of micro thin wall, it is necessary to study the micro-deformation mechanisms during milling micro thin-wall parts. In side milling of micro thin wall with mixed boundaries, the micro thin wall suffers from dynamic alternating forces, and the deformation modes are complex, so the direct and accurate theoretical modelling is difficult to make. Since the feed speed in milling micro thin wall is less, the small deformation of micro thin wall can be described based on the flexure deformation model of the thin plate subjected to elastic loadings. Firstly, accompanied by the reciprocal theorem of work for a curved thin plate, Kirchhoff-Love small deformation model of micro thin wall is used to establish the deformation equation and boundary conditions of micro thin wall with mixed boundaries under concentrated milling forces. Thus, the obtained boundary conditions are mainly applied to the subsequent finite element simulation of three-dimensional milling deformation of micro thin wall. Then a three-dimensional deformation simulation of the micro thin wall under different milling parameters is carried out by considering the micro-walled structure, material elasto-plastic constitutive model, the stiffness and geometric structure of micro milling tools in finite element method. In the workpiece constitutive modelling, considering the strain gradient plastic model in micro milling of Ti-6Al-4V titanium alloy, the workpiece plastic constitutive model is modified by introducing the intrinsic characteristic length of the material to describe the size effects in mesoscale micro milling. Finally, a series of the corresponding experiments on the milling of titanium alloy micro thin walls are carried out. By comparing and analyzing the experimental values and finite element numerical results of the micro thin-walled deformation, the micro deformation mechanisms in milling of thin wall are revealed. It also verifies the accuracy and effectiveness of this established milling finite element model of thin wall, and provides theoretical basis and technical support for controlling the milling deformation of micro thin-walled parts.

A flexible two‐surface gradient‐extended damage‐plasticity model for large deformations with nonlinear kinematic hardening

AbstractA novel ‘two‐surface’ gradient‐extended damage‐plasticity model for large deformations is discussed which can be considered the geometrically nonlinear version of a corresponding model for small deformations developed recently [1, 2]. The ‘two‐surface’ approach makes the model flexible in the sense that it can be adapted to various situations in which the considered material's behavior is either (quasi‐)brittle‐like, ductile‐like or possibly something in between. Nonlinear kinematic and isotropic hardening as well as nonlinear damage hardening are accounted for by the model which is finally formulated in terms of symmetric internal variables only, being advantageous not only from the computational point of view. Selected theoretical and numerical aspects are briefly discussed and a numerical example practically illustrates the model's functionality.

Two-scale simulation of the hot stamping process based on a Hashin–Shtrikman type mean field model

The hot stamping process of sheet metals into a w-shape profile is considered in experiments and numerical simulation using the FEM tool ABAQUS. The influence of the processing parameters of the hot stamping process on the final shape and the residual stress state are investigated. Tensile tests of thermomechanically treated specimens serve as a starting point for the modeling of the thermomechanical behavior of the steel grade 22MnB5. Based on the thermo-micromechanical (TMM) model for small deformations suggested by Neumann and Böhlke (2016), the updated Lagrangian formulation is used to render the finite deformation process.The model captures in a two-scale framework the temperature dependent elasto-plastic effect of the different phases, the phase transformation effect, and the phase transformation accompanying transformation strain occurring during the quenching of the austenitized sheet metal. Additionally, a model for the transformation induced plasticity (TRIP) effect in the context of the two-scale approach is suggested. The homogenization and localization of the thermomechanical properties is performed with a HashinöShtrikman (HS) type homogenization scheme. In this work, a reduced formulation for isotropic elastic behavior of the different phases is found reducing the amount of numerical costs. Furthermore, a self-consistent scheme based on the HS type method is introduced.Finally, the suggested TMM model is validated by considering the time-temperature-transformation- (TTT) and continuous-cooling-time-curves (CCT), dilatation tests under external load, and the hot stamping of w-shape profiles. The TMM model is compared to both experimental results and a phenomenological reference model usually used for the simulation of hot stamping processes.It is found that the suggested TMM model fits the final shape and the residual stress state of hot stamped parts better than the phenomenological reference model.

Electric Field Deformation of Protein-Coated Droplets in Thin Channels.

High-strength droplet interfaces are attractive for many applications, specifically in cases where droplets are channeled through fluidic devices and manipulated by electromagnetic fields. Using models and experiments, we study the deformation of droplets and capsules with protein interfaces in an electric field in thin and wide electrode gaps. Proteins are chosen from candidates expected to display qualitatively different interfacial interactions and strengths: a globular protein (bovine serum albumin), a reversible cross-linking peptide (AFD4), and a hydrophobin (cerato ulmin). Dilute protein additives can lead to over 1 order of magnitude stronger oil-water interfaces than those stabilized by small surfactants. We develop small deformation models to evaluate a protein membrane's interfacial elasticity, notably accounting for the electric field perturbation encountered in a gap and a careful treatment of a generalized elastic interface with both surface tension and interfacial elasticity. Results indicate that globular proteins, which typically have comparable surface tension and interfacial elasticity, can be modeled well by this generalized elastic interface. We further find that when in a gap, droplets and capsules migrate toward one electrode, deform asymmetrically, exhibit polar spreading on the electrode, and predictably stretch more than in the infinite gap scenario at constant field strength.

Small and large deformation models of post-buckled beams under lateral constraints

This study aims at theoretically and experimentally investigating the buckling behavior of bilaterally constrained beams with respect to different geometric parameters and conditions. The theoretical models are developed based on small and large deformation theories, respectively. The nonlinear Euler–Bernoulli beam theory is used to form the governing equations. An energy method is introduced to solve the equilibrium beams by minimizing the total potential energy with respect to the weight coefficients of the buckling modes. The theoretical models are compared with experiments. Good agreements are obtained with respect to the force–displacement relationship and deformed beam shape configuration. This study indicates that the small deformation model is insufficient in predicting beam end shortening since the longitudinal displacement is negligible in the model. The large deformation model effectively predicts severe deflection of beams in terms of end shortening and rotation. Parametric studies are carried out to indicate the applicability of the presented models. In particular, the small deformation model is defined as “more applicable” when the difference of the post-buckling response between the small and large deformation models is less than 5% (Diff < 5%), given that its computational cost is generally smaller than the large model. In contrast, when the difference is greater than 5%, the large deformation model is suggested. In the end, a polynomial function is fitted to define the relationship between the ratio of net gap-to-beam length η and highest achievable buckling mode Φ. The presented small and large deformation models are effective in understanding and predicting the post-buckling responses of laterally confined beams under different conditions.

Small deformations of Helfrich energy minimising surfaces with applications to biomembranes

In this paper, we introduce a mathematical model for small deformations induced by external forces of closed surfaces that are minimisers of Helfrich-type energies. Our model is suitable for the study of deformations of cell membranes induced by the cytoskeleton. We describe the deformation of the surface as a graph over the undeformed surface. A new Lagrangian and the associated Euler–Lagrange equations for the height function of the graph are derived. This is the natural generalisation of the well-known linearisation in the Monge gauge for initially flat surfaces. We discuss energy perturbations of point constraints and point forces acting on the surface. We establish existence and uniqueness results for weak solutions on spheres and on tori. Algorithms for the computation of numerical solutions in the general setting are provided. We present numerical examples which highlight the behaviour of the surface deformations in different settings at the end of the paper.

A continuum membrane model for small deformations of a spider orb-web

In this paper we propose a continuum membrane model for the infinitesimal deformation of a spider web. The model is derived in the simple context of axially-symmetric webs formed by radial threads connected with circumferential threads belonging to concentric circles. Under suitable assumption on the tensile pre-stress acting in the referential configuration, the out-of-plane static equilibrium and the free transverse and in-plane vibration of a supported circular orb-web are studied in detail. The accuracy of the model in describing a discrete spider web is numerically investigated.

Finite plasticity in $$\varvec{P}^\top \! \varvec{P}$$ P ⊤ P . Part I: constitutive model

We address a finite-plasticity model based on the symmetric tensor \(\varvec{P}^\top \! \varvec{P}\) instead of the classical plastic strain \(\varvec{P}\). Such a structure arises by assuming that the material behavior is invariant with respect to frame transformations of the intermediate configuration. The resulting variational model is lower dimensional, symmetric and based solely on the reference configuration. We discuss the existence of energetic solutions at the material-point level as well as the convergence of time discretizations. The linearization of the model for small deformations is ascertained via a rigorous evolution-\(\Gamma \)-convergence argument. The constitutive model is combined with the equilibrium system in Part II where we prove the existence of quasistatic evolutions and ascertain the linearization limit (Grandi and Stefanelli in 2016).

Phase field modeling of lithium diffusion, finite deformation, stress evolution and crack propagation in lithium ion battery

Mechanical degradation under cyclic charging and discharging has been the one of the most important bottlenecks for the developments of lithium ion batteries. Understanding of stress evolution process of anode during cycling is a way to deeply understand the mechanical degradation. In this paper, a phase field model (PFM) coupling lithium diffusion, finite deformation, stress evolution with crack propagation is established. Then the model is applied to spherical silicon particle to explore the stress evolution with taking account of finite deformation, elastic softening and plastic flow. The numerical results show that small deformation model and finite deformation model provide different stress states especially when the anode undergoes a large volumetric expansion. Elastic softening has not a significant effect on stress evolution in spherical particle. Plastic flow decreases the magnitude of stress, and deeply changes the stress evolution, making the hoop stress near the particle surface to become positive during charge process, indicating the particle surface is subjected to tensile hoop stress.

Magnetic resonance elastography of slow and fast shear waves illuminates differences in shear and tensile moduli in anisotropic tissue

Mechanical anisotropy is an important property of fibrous tissues; for example, the anisotropic mechanical properties of brain white matter may play a key role in the mechanics of traumatic brain injury (TBI). The simplest anisotropic material model for small deformations of soft tissue is a nearly incompressible, transversely isotropic (ITI) material characterized by three parameters: minimum shear modulus (µ), shear anisotropy (ϕ=µ1µ−1) and tensile anisotropy (ζ=E1E2−1). These parameters can be determined using magnetic resonance elastography (MRE) to visualize shear waves, if the angle between the shear-wave propagation direction and fiber direction is known. Most MRE studies assume isotropic material models with a single shear (µ) or tensile (E) modulus. In this study, two types of shear waves, “fast” and “slow”, were analyzed for a given propagation direction to estimate anisotropic parameters µ, ϕ, and ζ in two fibrous soft materials: turkey breast ex vivo and aligned fibrin gels. As expected, the speed of slow shear waves depended on the angle between fiber direction and propagation direction. Fast shear waves were observed when the deformations due to wave motion induced stretch in the fiber direction. Finally, MRE estimates of anisotropic mechanical properties in turkey breast were compared to estimates from direct mechanical tests.

Regularization of non-convex strain energy function for non-monotonic stress–strain relation in the Hencky elastic–plastic model

The boundary value problem of small static deformation range is formulated as the variational problem for the displacement. The general local conditions of convexity and rank-one-convexity are obtained for smooth elastic–plastic potentials depending on two first convex invariants of the Cauchy strain tensor. It is demonstrated that all types of convexity are equivalent for the Hencky elastic–plastic potential, but, for example, the strain energy function of continuum fracture can have every type of convexity for different relationships between material parameters. The classical regularization method is applied for the construction of the lower convex envelope of the Hencky strain energy potential for the experimental non-monotonic stress–strain relation with falling part. For this relation, the concept of the Maxwell line is used by analogy with the Van der Waals’ gas theory. The results of 1-D and 2-D numerical examples show that this regularization is principally necessary for the convergence of the finite-difference methods that are usually used for numerical computation within the framework of the small deformation models with non-monotonic stress–strain relations.

Three-dimensional chemo-thermomechanically coupled simulation of curing adhesives including viscoplasticity and chemical shrinkage

Based on the one-dimensional material model developed by Liebl et al. (Arch Appl Mech, 2011) a three-dimensional viscoelastic-viscoplastic material model for small deformations of curing adhesives on the basis of continuum mechanics is proposed in this contribution. The model describes the most relevant phenomena which occur during curing processes in the automotive industry and includes the effects of temperature and degree of cure on the mechanical properties of the material. Thermal expansion as well as chemical shrinkage are also contained. The yield stress for the viscoplastic part of the model goes back to the work of Schlimmer and Mahnken (Int J Numer Meth Eng 63:1461---1477, 2005), but is formulated in reference to the degree of cure and the temperature. Therefore this model considers chemo-thermomechanical coupling and extends the plasticity approach of Schlimmer and Mahnken, which is devised for cured adhesives, to the whole curing range, from the uncured to the fully cured adhesive. A peculiar focus is hereby laid on epoxy resins used in the automotive industry as structural adhesives.

On internal elastic membrane with strong damping

In this work we study existence and uniqueness of solutions to the abstract framework related to the Kirchhoff model for small deformations of a membrane \[ u'' +M ( \| u \| ^{2} ) Au+ F(u)+Au'=0, \] when $M$ is degenerate and $A,$ $F$ are operators. Furthermore, we obtain the asymptotic behavior, as $t \rightarrow\infty,$ of solutions in a concrete case.