arXiv subjects

Hui-Hui Dai

Publications and source records attributed to Hui-Hui Dai.

14 recordsLinked to original sources

arXiv2021

On propagation of waves in pressurized fiber-reinforced hyperelastic tubes based on a reduced model

A refined dynamic finite-strain shell theory for incompressible hyperelastic materials was developed by the authors recently. In this paper, we first derive the associated linearized incremental theory, and then use it to investigate wave propagation in a fiber-reinforced hyperelastic tube that is subjected to an axial pre-stretch and internal pressure. We obtain the dispersion relations for both axisymmetric and non-axisymmetric waves and discuss their accuracy by comparing them with the exact dispersion relations. The bending effect is also examined by comparing the dispersion curves based on the present theory and membrane theory. It is shown that the present theory is more accurate than the membrane theory in studying wave propagation and the bending effect plays an important role in some wave modes for relatively large wavenumbers. The effects of the pressure, axial pre-stretch and fiber angle on the dispersion relations are displayed. These results provide the necessary validation for using the refined dynamic finite-strain shell theory and wave propagation to determine arterial properties.

physics.bio-ph↗

arXiv2020

A refined dynamic finite-strain shell theory for incompressible hyperelastic materials: equations and two-dimensional shell virtual work principle

Based on previous work for the static problem, in this paper we first derive one form of dynamic finite-strain shell equations for incompressible hyperelastic materials that involve three shell constitutive relations. In order to single out the bending effect as well as to reduce the number of shell constitutive relations, a further refinement is performed, which leads to a refined dynamic finite-strain shell theory with only two shell constitutive relations (deducible from the given three-dimensional (3D) strain energy function) and some new insights are also deduced. By using the weak formulation of the shell equations and the variation of the 3D Lagrange functional, boundary conditions and the two-dimensional (2D) shell virtual work principle are derived. As a benchmark problem, we consider the extension and inflation of an arterial segment. The good agreement between the asymptotic solution based on the shell equations and that from the 3D exact one gives verification of the former. The refined shell theory is also applied to study the plane-strain vibrations of a pressurized artery, and the effects of the axial pre-stretch, pressure and fibre angle on the vibration frequencies are investigated in detail.

cs.CE↗

arXiv2020

Bending-induced director reorientation in a nematic liquid crystal elastomer bonded to a hyperelastic substrate

In this paper, the two-dimensional pure bending of a hyperelastic substrate coated by a nematic liquid crystal elastomer (abbreviated as NLCE) is studied within the framework of nonlinear elasticity. The governing system, arising from the deformational momentum balance, the orientational momentum balance and the mechanical constraint, is formulated, and the corresponding exact solution is derived for a given constitutive model. It is found that there exist two different bending solutions. In order to determine which the preferred one is, we compare the total potential energy for both solutions and find that the two energy curves may have an intersection point at a critical value of the bending angle $α_c$ for some material parameters. In particular, the director $\bm n$ abruptly rotates $\dfracπ{2}$ from one solution to another at $α_c$, which indicates a director reorientation (or jump). Furthermore, the effects of different material and geometric parameters on the bending deformation and the transition angle $α_c$ can be revealed using the obtained bending solutions. Meanwhile, the exact solution can offer a benchmark problem for validating the accuracy of approximated plate models for liquid crystal elastomers.

cond-mat.soft↗

arXiv2018

Pattern Transitions in a Soft Cylindrical Shell

Instability patterns of rolling up a sleeve appear more intricate than the ones of walking over a rug on floor, both characterized as uniaxially compressed soft-film/stiff-substrate systems. This can be explained by curvature effects. To investigate pattern transitions on a curved surface, we study a soft shell sliding on a rigid cylinder by experiments, computations and theoretical analyses. We reveal a novel post-buckling phenomenon involving multiple successive bifurcations: smooth-wrinkle-ridge-sagging transitions. The shell initially buckles into periodic axisymmetric wrinkles at the threshold and then a wrinkle-to-ridge transition occurs upon further axial compression. When the load increases to the third bifurcation, the amplitude of the ridge reaches its limit and the symmetry is broken with the ridge sagging into a recumbent fold. It is identified that hysteresis loops and the Maxwell equal-energy conditions are associated with the co-existence of wrinkle/ridge or ridge/sagging patterns. Such a bifurcation scheme is inherently general and independent of material constitutive models.

cond-mat.soft↗

arXiv2017

Exact solution for growth-induced large bending deformation of a hyperelastic plate

In this paper, the growth-induced bending deformation of a thin hyperelastic plate is studied. For a plane-strain problem, the governing PDE system is formulated, which is composed of the mechanical equilibrium equations, the constraint equation and the boundary conditions. By adopting a gradient growth field with the growth value changes linearly along the thickness direction, the exact solution to the governing PDE system can be derived. With the obtained solution, some important features of the bending deformation in the plate can be found and the effects of the different growth parameters can be revealed. This exact solution can serve as a benchmark one for testing the correctness of numerical schemes and approximate plate models in growth theory.

physics.class-ph↗

arXiv2017

Wave Patterns in A Nonclassic Nonlinearly-Elastic Bar Under Riemann Data

Recently there has been interest in studying a new class of elastic materials, which is described by implicit constitutive relations. Under some basic assumption for elasticity constants, the system of governing equations of motion for this elastic material is strictly hyperbolic but without the convexity property. In this paper, all wave patterns for the nonclassic nonlinearly elastic materials under Riemann data are established completely by separating the phase plane into twelve disjoint regions and by using a nonnegative dissipation rate assumption and the maximally dissipative kinetics at any stress discontinuity. Depending on the initial data, a variety of wave patterns can arise, and in particular there exist composite waves composed of a rarefaction wave and a shock wave. The solutions for a physically realizable case are presented in detail, which may be used to test whether the material belongs to the class of classical elastic bodies or the one wherein the stretch is expressed as a function of the stress.

math.AP↗

arXiv2014

Pathways, Scaling Laws and Analytical Solutions for Crease Formations in a Gel Layer

An analytical study on crease formations in a swelling gel layer is conducted. By exploring the smallness of the layer thickness and using a method of coupled series-asymptotic expansions, the original nonlinear eigenvalue problem of partial differential equations is reduced to one of ordinary differential equations. The latter problem is then solved analytically to obtain closed-form solutions for all the post-bifurcation branches. With the available analytical results, a number of deep insights on crease formations are provided, including the unveiling of three pathways to crease (depending on the layer thickness), determination of the bifurcation type, establishment of a lower bound for mode numbers and two scaling laws. Also, a number of experimental results are captured, which are then nicely interpreted based on the analytical solutions. In particular, it is shown that some critical physical quantities are invariant with respect to the thickness at the moment of crease formation. It appears that the present work offers a comprehensive understanding on crease formation, a widely-spread phenomenon.

cond-mat.soft↗

arXiv2014

On a consistent finite-strain plate theory based on 3-D energy principle

This paper derives a finite-strain plate theory consistent with the principle of stationary three-dimensional (3-D) potential energy under general loadings with a third-order error. Staring from the 3-D nonlinear elasticity (with both geometrical and material nonlinearity) and by a series expansion, we deduce a vector plate equation with three unknowns, which exhibits the local force-balance structure. The success relies on using the 3-D field equations and bottom traction condition to derive exact recursion relations for the coefficients. Associated weak formulations are considered, leading to a 2-D virtual work principle. An alternative approach based on a 2-D truncated energy is also provided, which is less consistent than the first plate theory but has the advantage of the existence of a 2-D energy function. As an example, we consider the pure bending problem of a hyperelastic block. The comparison between the analytical plate solution and available exact one shows that the plate theory gives second-order correct results. Comparing with existing plate theories, it appears that the present one has a number of advantages, including the consistency, order of correctness, generality of the loadings, applicability to finite-strain problems and no involvement of unphysical quantities.

physics.class-ph↗

arXiv2013

Critical Thickness Ratio for Buckled and Wrinkled Fruits and Vegetables

Fruits and vegetables are usually composed of exocarp and sarcocarp and they take a variety of shapes when they are ripe. Buckled and wrinkled fruits and vegetables are often observed. This work aims at establishing the geometrical constraint for buckled and wrinkled shapes based on a mechanical model. The mismatch of expansion rate between the exocarp and sarcocarp can produce a compressive stress on the exocarp. We model a fruit/vegetable with exocarp and sarcocarp as a hyperelastic layer-substrate structure subjected to uniaxial compression. The derived bifurcation condition contains both geometrical and material constants. However, a careful analysis on this condition leads to the finding of a critical thickness ratio which separates the buckling and wrinkling modes, and remarkably, which is independent of the material stiffnesses. More specifically, it is found that if the thickness ratio is smaller than this critical value a fruit/vegetable should be in a buckling mode (under a sufficient stress); if a fruit/vegetable in a wrinkled shape the thickness ratio is always larger than this critical value. To verify the theoretical prediction, we consider four types of buckled fruits/vegetables and four types of wrinkled fruits/vegetables with three samples in each type. The geometrical parameters for the 24 samples are measured and it is found that indeed all the data fall into the theoretically predicted buckling or wrinkling domains. Some practical applications based on this critical thickness ratio are briefly discussed.

cond-mat.soft↗

arXiv2011

Analytical Solutions for the Equilibrium states of a Swollen Hydrogel Shell and an Extended Method of Matched Asymptotics

A polymer network can imbibe water, forming an aggregate called hydrogel, and undergo large and inhomogeneous deformation with external mechanical constraint. Due to the large deformation, nonlinearity plays a crucial role, which also causes the mathematical difficulty for obtaining analytical solutions. Based on an existing model for equilibrium states of a swollen hydrogel with a core-shell structure, this paper seeks analytical solutions of the deformations by perturbation methods for three cases, i.e. free-swelling, nearly free-swelling and general inhomogeneous swelling. Particularly for the general inhomogeneous swelling, we introduce an extended method of matched asymptotics to construct the analytical solution of the governing nonlinear second-order variable-coefficient differential equation. The analytical solution captures the boundary layer behavior of the deformation. Also, analytical formulas for the radial and hoop stretches and stresses are obtained at the two boundary surfaces of the shell, making the influence of the parameters explicit. An interesting finding is that the deformation is characterized by a single material parameter (called the hydrogel deformation constant), although the free-energy function for the hydrogel contains two material parameters. Comparisons with numerical solutions are also made and good agreements are found.

physics.bio-ph↗

arXiv2010

On the Post-Peak Structural Response due to Softening with Localization

An analytical study is taken to investigate the relationship between material softening and structural softening through the use of a model problem in one dimension. With general nonlinear assumptions on the constitutive relations, it turns out that the governing equations can be viewed as a system of parametric equations, which couple the size effect and the nonlinear effect. Compared with the bilinear assumptions in previous literature, we find that the nonlinear assumptions herein capture more details in the post-peak structural response. After doing standard mathematical analysis to the nonlinear equations, we manage to derive necessary and sufficient conditions for the occurrence of four important post-peak cases, which are often observed in experiments. In particular, our analysis reveals that the mechanism of the snap-through phenomenon is due to the convexity change of the constitutive curve of the softening part. Mathematical examples are also given to illustrate the proposed procedures.

physics.bio-ph↗

arXiv2009

An Analytical Study on the Instability Phenomena During the Phase Transitions in a Thin Strip under Uniaxial Tension

In the experiments on stress-induced phase transitions in SMA strips, several interesting instability phenomena have been observed, including a necking-type instability, a shear-type instability and an orientation instability. By using the smallness of the maximum strain, the thickness and width of the strip, we use a methodology, which combines series expansions and asymptotic expansions, to derive the asymptotic normal form equations, which can yield the leading-order behavior of the original three-dimensional field equations. Our analytical results reveal that the inclination of the phase front is a phenomenon of localization-induced buckling (or phase-transition-induced buckling as the localization is caused by the phase transition). Due to the similarities between the development of the Luders band in a mild steel and the stress-induced transformations in a SMA, the present results give a strong analytical evidence that the former is also caused by macroscopic effects instead of microscopic effects. Our analytical results also reveal more explicitly the important roles played by the geometrical parameters.

physics.class-ph↗

arXiv2009

Instabilities Induced by Phase Fronts Coalescence During the Phase Transitions in a Thin Sma Layer: Mechanism and Analytical Descriptions

Systematic experiments on stress-induced phase transitions in thin SMA structures in literature have revealed two interesting instability phenomena: the coalescence of two martensite-austenite fronts leads to a sudden stress drop and that of two austenite-martensite fronts leads to a sudden stress jump. In order to get an insight into these two phenomena, in this work we carry out an analytical study on the stress-induced phase transitions in a thin SMA layer (a simple structure in which the two coalescence processes can happen). We derive a quasi-2D model with a non-convex effective strain energy function while taking into account the rate-independent dissipation effect. By using a coupled series-asymptotic expansion method, we manage to express the total energy dissipation in terms of the leading-order term of the axial strain. The equilibrium equations are obtained by maximizing the total energy dissipation, which are then solved analytically under suitable boundary conditions. The analytical results reveal that the mechanism for such instabilities is the presence of "limited points", which cause the switch of nontrivial solution modes to trivial solution modes. Descriptions for the whole coalescence processes of two fronts are also provided based on the analytical solutions, which also capture the morphology varies of the specimen. It is also revealed the key role played by the thickness-length ratio on these instabilities: the zero limit of which can lead to the smooth switch of nontrivial modes to trivial modes with no stress drop or stress jump.

physics.class-ph↗

arXiv2008

On Constructing the Asymptotic Solutions for Phase Transitions in a Slender Cylinder Composed of a Compressible Hyperelastic Material with Clamped End Conditions

In this paper, we study phase transitions in a slender circular cylinder composed of a compressible hyperelastic material with a non-convex strain energy function. We aim to construct the asymptotic solutions based on an axisymmetrical three-dimensional setting and use the results to describe the key features (in particular, instability phenomena) observed in the experiments by others. The difficult problem of the solution bifurcations of the governing nonlinear partial differential equations (PDE's) is solved through a novel approach. By using a methodology involving coupled series-asymptotic expansions, we derive the normal form equation of the original complicated system of nonlinear PDE's. By writing the normal form equation into a first-order dynamical system and with a phase-plane analysis, we manage to deduce the global bifurcation properties and to solve the boundary-value problem analytically. The asymptotic solutions (including post-bifurcation solutions) in terms of integrals are obtained. The engineering stress-strain curve plotted from the asymptotic solutions can capture the key features of the curve measured in a few experiments (e.g., the stress drop, the stress plateau, and the small stress valley). It appears that the asymptotic solutions obtained shed certain light on the instability phenomena associated with phase transitions in a cylinder, in particular the role played by the radius-length ratio. Also, an important feature of this work is that we consider the clamped end conditions, which are more practical but rarely used in literature for phase transition problems.

physics.class-ph↗