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Abstract Under a variety of external stimuli, hydrogels can undergo coupled solid deformation and fluid diffusion and exhibit large volume changes. The numerical analysis of this process can be complicated by numerical instabilities when using mixed formulations due to the violation of the inf-sup condition. In addition, the large deformations produce complex instability patterns causing singularities in the underlying set of equations. For these reasons, the experimentally observed complex patterns remain elusive and poorly understood. Furthermore, a stability criterion suitable to detect critical conditions and predict post-instability patterns is lacking for hydrogel simulations. Here we investigate the stability criterion for coupled problems with a saddle point nature and propose a generic framework to study diffusion-driven swelling-induced instabilities of hydrogels. Adopting a numerically stable subdivision-based mixed isogeometric analysis, we show that the proposed framework for stability analysis accurately captures instability points during the transient swelling of hydrogels. The influence of geometrical and material parameters on the critical conditions are also presented in stability diagrams for two useful problems involving the buckling of hydrogel rods and the wrinkling on the surface of hydrogel bilayers. The results show that the short-time response of hydrogels immersed in water are highly unstable. We believe that this generic scheme provides a theoretical and computational foundation to study the morphogenesis in nature, and it also paves the way to create functional materials and design novel hydrogel devices through stability diagrams. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a study of the mechanical response of hexachiral honeycombs with transversally curved ligaments under planar uniaxial tensile, compressive and shear loads. The impact of the chiral cell design parameters on the resulting macroscopic behaviour is assessed utilising finite elements calculations. It is shown that the presence of ligament curvature permits to attain mechanical responses which are not achievable through conventional chiral honeycomb designs. In addition, the resulting responses exhibit, for all considered load cases, significant tunability through the investigated geometrical design parameters. Two chiral lattices with identical geometries, only differing in their ligament curvature, were manufactured and experimentally tested to validate the finite elements predictions. A connection and assembly strategy is presented and utilised, offering a fast and robust approach to build larger finite lattice structures through 3D printed single basic cells. The hexachiral lattices were tested in tension, compression and in-plane shear, showing good agreement with the numerical predictions. [ABSTRACT FROM AUTHOR]

Protein microtubules (MTs) are one of the important intercellular components and have a vital role in the stability and strength of the cells. Due to applied external loads, protein microtubules may be involved buckling phenomenon. Due to impact of protein microtubules in cell reactions, it is important to determine their critical buckling load. Considering nature of protein microtubules, various parameters are effective on microtubules buckling. The small size of microtubules and also lack of uniformity of MTs properties in different directions caused the necessity of accuracy in the analysis of these bio-structure. In fact, microtubules must be considered as a size dependent cylinder, which behave as an orthotropic material. Hence, in the present work using first-order shear deformation model (FSDT), the buckling equations of anisotropic MTs are derived based on new modified couple stress theory (NMCST). After solving the stability equations, the influences of various parameters are measured on the MTs critical buckling load. [ABSTRACT FROM AUTHOR]

In present paper, a unified size-dependent high-order beam model which contains various higher-order shear deformation beam models as well as Euler–Bernoulli and Timoshenko beam models is developed to study the simultaneous effects of nonlocal stress and strain gradient on the bending and buckling behaviors of nanobeams by using the nonlocal strain gradient theory. For this objective, a nonlocal parameter is introduced to capture the nonlocal effect and a material length scale parameter is involved to evaluate the strain gradient effect. The governing equations and the associated boundary conditions are formulated by using Hamilton's principle. Navier's method is utilized to obtain the analytical solutions for bending and buckling of a simply supported nanobeam. The present model is validated by comparing the obtained results with those available in literature. The influences of nonlocal parameter, material length scale parameter, slenderness ratio and shear deformation on the bending and buckling behaviors of the nanobeam are examined in detail. Results reveal that within the framework of nonlocal strain gradient theory, results predicted by Timoshenko beam model and various higher-order beam models are almost same with some negligible differences. Moreover, it is found that the nanobeam could exhibit either stiffness-softening effect or stiffness-hardening effect, which depends on the relative magnitude of the nonlocal parameter and the material length scale parameter. [ABSTRACT FROM AUTHOR]

Metamaterials are composed of structural elements and derive their properties mainly from the inner structure of the elements, rather than the properties of their constituent material. By designing an unstable structural element as the building block of a metamaterial, many interesting effective material properties can be obtained. The deformation and dissipation mechanisms of such a material built from unstable structural elements is studied in detail. To do so a combination of analytical, semi-analytical, and numerical models are applied to a single buckling element, a periodic cell, and finite size combinations of buckling elements including gradients in the properties of the building blocks. This not only provides insight into the micromechanics and the resulting effective behavior of such metamaterials, but also makes them accessible on the different relevant length scales. A metamaterial built from these building blocks shows programmable or switchable properties and can display energy dissipation with fully reversible deformation, distinguishing it from plastic materials, and timescale independent behavior, distinguishing it from viscoelastic materials. [ABSTRACT FROM AUTHOR]

The post-buckling behavior of a beam subjected to lateral constraints is of practical importance in a variety of applications, such as stent procedures, filopodia growth in living cells, endoscopic examination of internal organs, and deep drilling. Even though in reality the constraining surfaces are often deformable, the literature has focused mainly on rigid and fixed constraints. In this paper, we make a first step to bridge this gap through a theoretical and experimental examination of the post-buckling behavior of a beam constrained by a fixed wall and a springy wall, i.e. one that moves laterally against a spring. The response exhibited by the proposed system is much richer compared to that of the fixed-wall system, and can be tuned by choosing the spring stiffness. Based on small-deformation analysis, we obtained closed-form analytical solutions and quantitative insights. The accuracy of these results was examined by comparison to large-deformation analysis. We concluded that the closed-form solution of the small-deformation analysis provides an excellent approximation, except in the highest attainable mode. There, the system exhibits non-intuitive behavior and non-monotonous force–displacement relations that can only be captured by large-deformation theories. Although closed-form solutions cannot be derived for the large-deformation analysis, we were able to reveal general properties of the solution. In the last part of the paper, we present experimental results that demonstrate various features obtained from the theoretical analysis. [ABSTRACT FROM AUTHOR]

Thermo-mechanical size-dependent buckling analysis of embedded functionally graded (FG) microbeams is performed based on sinusoidal shear deformation beam and modified couple stress theories. It is assumed that material properties vary smoothly and continuously throughout the thickness. Winkler elastic foundation model is used to simulate the interaction between FG microbeam and elastic medium. The governing equations and corresponding boundary conditions are obtained with the aid of minimum total potential energy principle. The buckling characteristics of simply supported embedded FG microbeams in thermal environment are investigated. The obtained results are compared with the results of simple beam theory with no shear deformation effects and classical theory. Influences of thickness-to-material length scale parameter ratio, material property gradient index, slenderness ratio, temperature change and Winkler parameter on critical buckling loads of embedded FG microbeams are discussed in detail. [ABSTRACT FROM AUTHOR]

The prediction of the critical buckling conditions of shell structures is plagued by imperfection sensitivity. Non-destructive testing through point-load probing has been recently proposed to map the stability landscape of cylindrical shells. However, the counterpart procedure for spherical shells is still debatable. Here, we focus on the mechanical response of pressurized spherical shells containing a single dimple–like defect to a point probe. Combining experiments, finite element modeling, and existing results from classic shell theory, we characterize the nonlinear force-indentation response of imperfect shells at different pressurization levels. From these curves, we seek to identify the critical buckling pressure of the shell. In particular, the indentation angle is varied systematically to examine its effect on the probing efficacy. We find that the critical buckling point can be inferred non-destructively by tracking the maxima of the indentation force–displacement curves, if the probe is implemented sufficiently close to the defect. When probing further away from the defect, the test fails in predicting the onset of buckling since the deformation due to indentation remains localized in the vicinity of the probe. Using FEM simulations and shallow shell theory, we quantify the characteristic length associated with this localized deformation, both in the linear and nonlinear regimes. Our results demonstrate the limiting conditions of applicability for the usage of probing as a non-destructive technique to assess the stability of spherical shells. [Display omitted] • We investigate a probing technique to assess the buckling of spherical shells. • We combine precision experiments, finite element modeling, and classic shell theory. • The response of imperfect shells to a probe is affected by the indentation location. • The deformation of the spherical shell is localized near the point of indentation. • The neighborhood of indentation is characterized by the length-scale, R h. [ABSTRACT FROM AUTHOR]

Abstract: We present a three-dimensional continuum model for layered crystalline materials made out of weakly interacting two-dimensional crystalline sheets. We specialize the model to multilayer graphene materials, including multi-walled carbon nanotubes (MWCNTs). We view the material as a foliation, partitioning of space into a continuous stack of leaves, thus loosing track of the location of the individual graphene layers. The constitutive model for the bulk is derived from the atomistic interactions by appropriate kinematic assumptions, adapted to the foliation structure and mechanics. In particular, the elastic energy along the leaves of the foliation results from the bonded interactions, while the interaction energy between the walls, resulting from van der Waals forces, is parametrized with a stretch transversal to the foliation. The resulting theory is distinct from conventional anisotropic models, and can be readily discretized with finite elements. The discretization is not tied to the individual walls and allows us to coarse-grain the system in all directions. Furthermore, the evaluation of the non-bonded interactions becomes local. We test the accuracy of the foliation model against a previously proposed atomistic-based continuum model that explicitly describes each and every wall. We find that the new model is very efficient and accurate. Furthermore, it allows us to rationalize the rippling deformation modes characteristic of thick MWCNTs, highlighting the role of the van der Waals forces and the sliding between the walls. By exercising the model with very large systems of hollow MWCNTs and suspended multilayer graphene, containing up to 109 atoms, we find new complex post-buckling deformation patterns. [Copyright &y& Elsevier]

Abstract: This paper presents a nonlocal sinusoidal shear deformation beam theory for the bending, buckling, and vibration of nanobeams. The present model is capable of capturing both small scale effect and transverse shear deformation effects of nanobeams, and does not require shear correction factors. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion as well as the boundary conditions of the beam are derived using Hamilton’s principle. Analytical solutions for the deflection, buckling load, and natural frequency are presented for a simply supported beam, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory. The comparison firmly establishes that the present beam theory can accurately predict the bending, buckling, and vibration responses of short nanobeams where the small scale and transverse shear deformation effects are significant. [Copyright &y& Elsevier]

Abstract: A nonlocal shear deformation beam theory is proposed for bending, buckling, and vibration of nanobeams using the nonlocal differential constitutive relations of Eringen. The theory, which does not require shear correction factor, accounts for both small scale effects and quadratic variation of shear strains and consequently shear stresses through the thickness of the beam. In addition, it has strong similarities with nonlocal Euler–Bernoulli beam theory in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The equations of motion are derived from Hamilton’s principle. Analytical solutions of deflection, buckling load, and natural frequency are presented for a simply supported beam, and the obtained results compare well with those predicted by the nonlocal Timoshenko and Reddy beam theories. [Copyright &y& Elsevier]

Abstract: We carry out axisymmetric, finite deformation finite element analyses of the uniaxial compression of cylindrical bundles of vertically aligned carbon nanotubes (VACNTs) firmly attached to a Si substrate. A compressible elastic–viscoplastic constitutive relation with a piecewise, linear hardening–softening–hardening flow strength is used to model the material. Calculations are performed for VACNTs both with uniform properties and with axially graded properties. We show that, with uniform properties, sequential buckling initiates at the substrate and propagates away from it, in agreement with previous experimental findings. We investigate the dependence of the magnitude and wavelength of the buckles on characteristics of the function defining the flow strength. When a property gradient giving a more compliant response at the end opposite to the substrate is specified, we find that sequential buckling initiates at that end and propagates toward the substrate. Results of the analyses are compared with the experimental observations and capture many of the experimentally obtained stress–strain and morphological features. The proposed model serves as a promising foundation for capturing the underlying energy absorption mechanisms in these systems. Comparison of the model predictions with the experimental results also suggests directions for model improvement. [Copyright &y& Elsevier]

Abstract: Mechanics of living tissues focusing on the relationships between growth, morphology and function is not only of theoretical interest but can also be useful for diagnosis of certain diseases. In this paper, we model the surface wrinkling morphology of mucosa, the moist tissue that commonly lines organs and cavities throughout the body, induced by either physiological or pathological volumetric growth. A theoretical framework of finite deformation is adopted to analyze the deformation of a cylindrical cavity covered by mucosal and submucosal layers. It is shown that compressive residual stresses induced by the confined growth of mucosa can destabilize the tissue into various surface wrinkling patterns. A linear stability analysis of the critical condition and characteristic buckling patterns indicates that the wrinkling mode is sensitive to the thicknesses of the mucosal and submucosal layers, as well as the properties of the tissues. The thinner the mucosal layer and the lower its elastic modulus, the shorter the buckling wavelength. A series of finite element simulations are performed to validate the theoretical predictions and to study local wrinkling or non-uniform patterns associated with inhomogeneous growth. Our postbuckling analysis shows that the surface pattern may evolve towards a period-doubling morphology due to continuous growth of mucosa or submucosa beyond the critical state. Finally, the theoretical predictions and numerical simulations are compared to experimental observations. [Copyright &y& Elsevier]

Abstract: The multiple complexities inherent to living objects have motivated the search for abiotic substitutes, able to mimic some of their relevant physical properties. Hydrogels provide a highly monitorable counterpart and have thus found many applications in medicine and bioengineering. Recently, it has been recognized that their ability to swell could be used to unravel some of the universal physical processes at work during biological growth. However, it is yet unknown how the microscopic distinctions between swelling and biological growth affect macroscopic changes (shape, stresses) induced by volume variations. To answer this question, we focus on a clinically motivated example of growth. Some solid tumors such as melanoma or glioblastoma undergo a shape transition during their evolution. This bifurcation appears when growth is confined at the periphery of the tumor and is concomitant with the transition from the avascular to the vascular stage of the tumor evolution. To model this phenomenon, we consider in this paper the deformation of an elastic ring enclosing a core of different stiffness. When the volume of the outer ring increases, the system develops a periodic instability. We consider two possible descriptions of the volume variation process: either by imposing a homogeneous volumetric strain (biological growth) or through migration of solvent molecules inside a solid network (swelling). For thin rings, both theories are in qualitative agreement. When the interior is soft, we predict the emergence of a large wavelength buckling. Upon increasing the stiffness of the inner disc, the wavelength of the instability decreases until a condensation of the buckles occurs at the free boundary. This short wavelength pattern is independent of the stiffness of the disc and is only limited by the presence of surface tension. For thicker rings, two scenarios emerge. When a volumetric strain is prescribed, compressive stresses accumulate in the vicinity of the core and the deformation localizes itself at the boundary between the disc and the ring. On the other hand, swelling being an instance of stress-modulated growth, elastic stretches near the core saturate and the instability occurs primarily at the free boundary. Besides its implications for the mechanical stability of avascular tumors, this work provides important results concerning layered tissues growth and the role of hydrogels as biological tissues substitutes. [Copyright &y& Elsevier]

Abstract: We formulate a multiscale modeling framework to investigate the deformation morphologies and energetics of covalently bridged multi-walled carbon nanotubes (MWCNTs). The formulation involves extending a previously established quasi-continuum model by incorporating the inter-wall bridging energy density function into the constitutive relations via message passing from fully atomistic simulations. Using the extended numerical model, we studied the mechanical responses of the 10-walled MWCNT with varying inter-wall bridge densities under torsion, bending, and uniaxial compression. Our simulation results show that the presence of inter-wall covalent bridges not only enhances the post-buckling rigidities of the MWCNTs, but also modifies the deformation morphologies and morphology pathways. For bending and uniaxial compression, we constructed in the space of bridge density and applied strain the deformation morphology phase diagram, where three phases, uniformly deformed phase, rippling pattern, and diamond-shaped pattern, are identified and separated by linear phase boundaries. We attribute the deformation phase transitions to the interplay of inter-wall and intra-wall interaction energies. The multiple shape transitions of MWCNTs and the elastic nature of the deformation suggest that MWCNTs can be designed as shape-memory nanodevices with tunable stabilities. [ABSTRACT FROM AUTHOR]

Buckling of stiff thin films on compliant substrates has many important applications ranging from stretchable electronics to precision metrology and sensors. Mechanics plays an indispensable role in the fundamental understanding of such systems. Some existing mechanics models assume plane-strain deformation, which do not agree with experimental observations for narrow thin films. Systematic experimental and analytical studies are presented in this paper for finite-width stiff thin films buckling on compliant substrates. Both experiments and analytical solution show that the buckling amplitude and wavelength increase with the film width. The analytical solution agrees very well with experiments and therefore provides valuable guide to the precise design and control of the buckling profile in many applications. The effect of film spacing is studied via the analytical solutions for two thin films and for periodic thin films. [Copyright &y& Elsevier]

Abstract: If a body with a stiffer surface layer is loaded in compression, a surface wrinkling instability may be developed. A bifurcation analysis is presented for determining the critical load for the onset of wrinkling and the associated wavelength for materials in which the elastic modulus is an arbitrary function of depth. The analysis leads to an eigenvalue problem involving a pair of linear ordinary differential equations with variable coefficients which are discretized and solved using the finite element method. The method is validated by comparison with classical results for a uniform layer on a dissimilar substrate. Results are then given for materials with exponential and error-function gradation of elastic modulus and for a homogeneous body with thermoelastically induced compressive stresses. [Copyright &y& Elsevier]

Abstract: The lateral forces exerted on a substrate by a layer of end-grafted polymer molecules are calculated on the basis of simple scaling arguments. The results are cast in terms of an equilibrium surface stress and an elastic constant, which describes the rate of change of the surface stress upon deformation of the substrate. This allows for straightforward integration of the present results into a continuum framework describing the response of a compliant structure, which facilitates device design and analysis. The results are illustrated with calculations for end-grafted poly(styrene) and poly(ethylene oxide), and the implications for building micromechanical devices based on adsorption-induced deformation are discussed. [Copyright &y& Elsevier]

Abstract: An investigation of the mechanics of bending and buckling is carried out for a class of nonlinear fiber composite rods composed of embedded unidirectional fibers parallel to the rod axis. The specific class of composite considered is one in which the fibers interact with the matrix through a nonlinear Needleman-type cohesive zone [Needleman, A., 1987. A continuum model for void nucleation by inclusion debonding. ASME J. Appl. Mech. 54, 525–531; Needleman, A., 1992. Micromechanical modelling of interfacial decohesion. Ultramicroscopy 40, 203–214]. The primary decohesive mechanism active in bending and buckling of these composite rods is shear slip along the fiber–matrix interfaces allowing the use of a previously developed constitutive relation for antiplane shear response [Levy, A.J., 2000b. The fiber composite with nonlinear interface—part II: antiplane shear. ASME J. Appl. Mech. 67, 733–739]. The formulation requires the specification of a potential interface force–slip law that is assumed to permit interface failure in shear. Four cases of the bending and shearing of beams (concentrated or uniform load on a cantilever or a simply supported beam) are analyzed, each of which exhibits qualitatively distinct response. For certain values of interface parameters, the beam deflection or its gradient at a fixed location can change discontinuously with load. Furthermore, for interface parameter values within a certain range, singular surfaces will exist in uniformly loaded beams where there is a non-uniform distribution of shear stress along the beam length. These singular surfaces divide the beam into regions of maximal and minimal fiber slip and propagate with a rate that varies inversely as the square of the applied load. For other parameter values, singular surfaces will not exist and fiber slip will be diffuse. For the class of nonlinear composite considered, bifurcation and imperfection buckling of pinned–pinned columns is analyzed. For bifurcation buckling, a nonlinear eigenvalue problem is derived and the solution is obtained by Galerkin''s method. It is demonstrated that critical loads are influenced by the initial slope, and hence the linear portion, of the interface force–slip relation but the post-buckling response, which in some sense resembles that of plastic buckling, is affected by the entire interface constitutive relation. Imperfection buckling is analyzed in a similar manner by assuming a slight initial curvature of the rod. Sensitivity of the response to imperfection magnitude is discussed as well. [Copyright &y& Elsevier]

We analyse here the problem of large deformation of dielectric elastomeric membranes under coupled electromechanical loading. Extremely large deformations (enclosed volume changes of 100 times and greater) of a toroidal membrane are studied by the use of a variational formulation that accounts for the total energy due to mechanical and electrical fields. A modified shooting method is adopted to solve the resulting system of coupled and highly nonlinear ordinary differential equations. We demonstrate the occurrence of limit point, wrinkling, and symmetry-breaking buckling instabilities in the solution of this problem. Onset of each of these "reversible" instabilities depends significantly on the ratio of the mechanical load to the electric load, thereby providing a control mechanism for state switching. [ABSTRACT FROM AUTHOR]

Mechanical metamaterials have gained increasing interest owing to its unique properties and promising applications. However, most developed mechanical metamaterials feature a single-step pathway and/or a single deformation mode, which limits their multi-task applications. In this paper, a multi-step metamaterial (MSM) is introduced. Under compression, the MSM can translate global loading into multiple pathways of deformation, giving rise to multi-stress plateaus in the stress-strain curves. The underlying mechanism is the combination of sequential snap-through and Euler buckling at microscopic unit cell level. Theoretical models are developed to quantify the multi-step deformation feature of the MSM and are validated by experiments and finite element simulations. By varying the geometric parameters of the constituent components of the MSM, its deformation can be programmed. The concept of the developed MSM provides a new perspective for designing metamaterials with multiple tasks, e.g., an energy absorber suitable for both light and heavy impacts, logic gates for manipulating mechanical signals in mechanical computing, among others. [ABSTRACT FROM AUTHOR]