Critical Stretch Research Articles

Post-buckling of an elastic half-space coated by double layers

We investigate the buckling and post-buckling properties of a hyperelastic half-space coated by two hyperelastic layers when the composite structure is subjected to a uniaxial compression. In the case of a half-space coated with a single layer, it is known that when the shear modulus of the layer is larger than the shear modulus of the half-space, a linear analysis predicts the existence of a critical stretch and wave number, whereas a weakly nonlinear analysis predicts the existence of a threshold value of the modulus ratio below which the buckling is super-critical and above which the buckling is sub-critical. It is shown that when another layer is added, a larger variety of behaviour can be observed. For instance, buckling can occur at a preferred wavenumber super-critically even if both layers are softer than the half-space although the top layer would need to be harder than the bottom layer. When the shear modulus of the bottom layer lies in a certain interval, the super-critical to sub-critical transition can happen a number of times as the shear modulus of the top layer is increased gradually. Thus, an extra layer imparts more flexibility in producing wrinkling patterns with desired properties, and our weakly nonlinear analysis provides a road map on the parameter regimes where this can be achieved.

Macro- and micro-instabilities in incompressible bioinspired composite materials with nacre-like microstructure

In this work, we study the macroscopic and microscopic instabilities in 2D incompressible nacre-like composite materials induced by uniaxial loadings. The interchange between macro- and micro-instabilities in bioinspired composite materials has been investigated by examining different combinations of platelets volume fraction, aspect ratio, and shear modulus contrast between the stiff (platelets) and soft (matrix) phases. It has been highlighted that the critical instability mode shapes, together with their critical load factors and wavelengths, are highly influenced by the microscopic geometrical arrangement and the material composition. For a wide range of geometrical and material parameters, the instabilities are found to be global with an in-phase mode shape. It has been shown that the out-of-phase instability mode shapes occur at a significantly higher magnitude of the homogenized energy compared to the in-phase ones. The results indicate that, adopting a small unit cell assembly, the microscopic stability analysis provides in most of cases strong underestimates of the critical stretch ratios, and that an accurate and efficient instability prediction can be instead obtained based on the loss of strong ellipticity condition of the homogenized incremental moduli tensor, except for few cases in which local modes occur.

Instability of compressible soft electroactive plates

In this paper, we investigate the formation of two-dimensional surface wrinkles in a pre-deformed compressible soft electroactive (SEA) plate. Two common electrical loading protocols are considered: one is the so-called voltage-control where the plate is applied with a constant voltage across the thickness; the other is named as charge-control where the plate is activated by uniformly spraying electrical charges on its main faces. By developing the surface impedance matrix method with the Stroh formulation for compressible SEA materials, we decouple the resulting dispersion equations into symmetric and antisymmetric modes and obtain the associated closed-form expressions, respectively. These analytical expressions of the critical stretch are crucial to the quantitative understanding of wrinkling in the SEA plate. For illustration, we treat examples of compressible Gent and neo-Hookean ideal dielectric elastomers. Results show that the symmetric wrinkles may appear prior to the antisymmetric ones in a sufficiently stretched compressible SEA plate. This phenomenon is in principle caused by the release of incompressibility constraint, upon which the variation of the material chain extension limit only mildly affects the magnitude of the critical stretch. Especially, for charge-controlled plates, a compressibility-driven competition mechanism is revealed that with the increase of compressibility, the symmetric wrinkling is promoted while the antisymmetric one is suppressed. Moreover, enhancing compressibility significantly stabilizes the intrinsic instability of charge-controlled plate while makes the voltage-controlled plate more unstable. These findings, for the first time, systematically demonstrate the great significance of material compressibility on the nonlinear response and stability performance of SEA elastomers and can serve a quantitative guidance for the industrial design of new-generation SEA-based devices by exploring the material compressibility.

A continuum-kinematics-inspired peridynamic model of anisotropic continua: Elasticity, damage, and fracture

In this work, linearized elastic isotropic continuum-kinematics-inspired-peridynamics (CPD) is further generalized to study the elastic, damage, and fracture behavior of anisotropic continua, focusing on 3D transversely isotropic and 2D orthotropic body. The anisotropy is generated by changing the material properties, including the micromodulus, critical stretch, and critical micropotential energy of basic finite kinematic elements with their direction in the principal material axes in terms of eight-order double Fourier expansion. It is critically proven that the volumetric micromodulus of finite volume elements in the 3D case is independent of their direction, which is similarly applied to the areal micromodulus in the 2D case. For this reason, the 2D orthotropic CPD formulation characterized by three independent elastic moduli and the 3D transversely isotropic CPD formulation characterized by four independent elastic moduli are distinguished. The accuracy of the computational models applied to elastic analysis is assessed by comparing the predicted displacement fields with the results from finite element analysis. Quantitative simulation of an orthotropic lamina with a central rectangular hole under a tensile load, eccentric three-point bending test for orthotropic lamina, and compact tension test for cortical bone are performed to verify the ability of the proposed model to describe the damage and fracture behavior of anisotropic materials.

A new peridynamic mixed-mode bond failure model for interface delamination and homogeneous materials fracture analysis

A new peridynamic bond failure model is proposed for mixed-mode crack fracture analysis in material interface and homogeneous isotropic materials, which utilize bond failure criteria presented for mixed-mode peridynamic bonds using the angle-dependent formations of critical stretch (CS) or critical energy density (CED). Then, three tests of asymmetric double cantilevered beams (ADCB), single edge-notched plate (SENP) and three-point bend (TPB) tests are investigated for model verification and application, and the numerical predicted fracture characteristics (e.g., crack path, load–displacement curve and critical load) from the proposed model are compared to those obtained from the finite element method (FEM) result, theoretical solution and experimental data. The close comparisons demonstrate that the proposed peridynamic bond failure model can successfully capture mixed-mode fracture behaviors of interface and homogeneous materials fracture.

Intersonic shear crack propagation using peridynamic theory

Dynamic crack propagation of mode-II cracks is simulated using bond-based Peridynamic Theory (PD) implemented in finite element analysis software ABAQUS. The specimen is a bonded homogeneous Homalite plate with a pre-notch that is subjected to impact shear loading simulating the experiments of Rosakis et al. (1999). The PD bonds at the bonding interface are utilized with a scalar critical stretch value that corresponds to mode-II fracture toughness of the interface. The crack initiation and propagation are naturally captured in the bond-based PD simulations by modifying the original prototype microelastic brittle law formulation introduced by Silling and Askari (2005). Impact loading is introduced at the specimen as a pulse speed field boundary condition. Using bond-based PD, sub-Rayleigh and intersonic regimes of crack growth are obtained as a function of fracture toughness ( $$G_{II}$$ ) and impact speed ( $$V_i$$ ) values. The intersonic crack growth is discerned from the sub-Rayleigh crack growth by the existence of shear Mach waves in the particle velocity magnitude contours. For critical values of $$G_{II}$$ and $$V_i$$ , a crack growing at a speed just below the Rayleigh wave speed is observed to transition to an intersonic speed with a Burridge-Andrews mechanism. The sustained intersonic crack tip speed is found to be between $$1.57c_S$$ ( $$c_S$$ is the shear wave speed) and $$c_D$$ ( $$c_D$$ is the dilatational wave speed). For a reduced impact pulse duration, an intersonic crack is found to approach the theoretical value of $$\sqrt{2}c_S$$ , which, however is not maintained. The results are in qualitative agreement with the experiments of Rosakis et al. (1999) and previous simulations in the literature.

A micropolar peridynamic model with non-uniform horizon for static damage of solids considering different nonlocal enhancements

Peridynamic models typically adopt regular point distributions and uniform horizons, limiting their flexibility and engineering applicability. In this work, a micropolar peridynamics approach with a non-uniform horizon (NHPD) is proposed. This approach is implemented in a conventional finite element framework using element-based discretization. Through modification of the dual-horizon approach in the preprocessing step, a point-dependent horizon and non-uniform beam-like bonds are built. By a domain correction strategy, the equivalence of the strain energy density is assured. Then, a novel energy–density-based failure criterion that directly relates the critical stretch to the mechanical strength is presented. The numerical results indicate the weak mesh dependency of NHPD and the effectiveness of the new failure criterion for Brazilian disk tests. Moreover, damage to solids with different nonlocal effects is shown to yield similar results through adjustment of only the mechanical strength.

Bond-Based Peridynamics with Stretch and Rotation Kinematics for Opening and Shearing Modes of Fracture

This study presents a new bond-based peridynamic approach for modeling the elastic deformation of isotropic materials with bond stretch and rotation, thus removing the constraint on the Poisson’s ratio. The resulting PD equilibrium equations derived under the assumption of small deformation are solved by employing implicit techniques. The bond constants associated with stretch and rotation kinematic are directly related to the constitutive relations of stress and strain components in continuum mechanics. Also, the expressions for the critical stretch and critical relative rotation are derived in terms of mode I and mode II critical energy release rates, respectively. Lastly, it does not require a surface correction procedure, and the displacement and traction type boundary conditions are directly imposed without introducing fictitious regions in the domain. The capability of this approach is first demonstrated by capturing the correct deformation of plate type structures under general loading conditions. Subsequently, its capability for failure prediction is established by simulating the response of a double cantilever beam (DCB) under mode I type loading and compact shear specimen under mode II type loading.

Static and dynamic stability of dielectric elastomer fiber composites

Abstract The capability of Dielectric Elastomers (DEs) undergoing large deformation is limited by electromechanical instability arising because of the positive feedback between the applied electric field and the reduction in thickness of the elastomer. This paper theoretically analyzes the stability of a dielectric elastomer fiber composite made up of two soft incompressible DE phases in the quasi-static and dynamic modes of actuation. An energy method, which relies on the energy balance at the position of maximum overshoot in an oscillation cycle, is used for extracting the critical electric field and stretch in the dynamic mode. The material behaviors of the fiber and matrix phases are characterized by neo-Hookean energy density function of the elastomer. The results demonstrate that DE composite with higher fiber volume fraction exhibits higher electric field at the onset of instability in both static and DC dynamic modes of actuation. The results of the present investigation can find potential applications in the development and design of the soft electroactive actuators subjected to dynamic loading.

Role of expansion joint in the study of seismic analysis of a multi-storied building

In this paper, the study on the multi-storied building with and without expansion joint when it is subjected to seismic forces in both x and y direction are taken into consideration. By maintain same floor area of different plans the expansion joint is provided along the weaker column forces stretch and stiffed the building in order to make more resistant when compared to the plans without expansion joints. The analysis is done using ETABS 2016. The parameters such as shear force and bending moment at the critical column are considered and compared for with expansion joint and without expansion joint. It is concluded that the L shaped building shear force is being decreased from the bottom to top storey of the building when compared to without expansion joint. The effect of expansion joint is shown effectively on U-shaped building in terms of reduction of shear force and bending moment when compared to L-shaped and T-shaped, it is because of providing two expansion joints since there are two critical and weaker columns stretch. When it comes to economy of the structure, the cost of the construction of the structure with expansion joint is relatively high but it can resistant far better in case of seismic forces. The future scope of the study can be the type of filler material to be used such that the reduction of cost and to maintain structure integrity.

Stability analysis at the micro- and macro-scales in periodic bioinspired composites

This work deals with the onset of instabilities at the microscopic and macroscopic scales in uniaxially compressed bioinspired composite materials with periodic microstructure. particular, the interchange between macro- and micro-instabilities in incompressible nacre-like composite materials has been investigated for different combinations of platelets aspect ratio, platelets volume fraction and shear modulus contrast between platelets and soft matrix. It has been demonstrated that the critical stretch ratios and the critical mode shapes are highly influenced by both the geometrical and materials parameters. The stability analyses highlighted that the microscopic stability analysis provides in most cases strong underestimates of the critical stretch ratios (adopting a small unit cell assembly), and that, by performing a macroscopic stability analysis based on the strong ellipticity condition of the homogenized tangent moduli tensor, a more accurate evaluation of the load associated with the primary instability can be instead obtained with less computational efforts. In addition, for a wide range of the investigated parameters, the instabilities are found to be characterized by long wavelengths with an in-phase critical mode shape.

Neural cell injury pathology due to high-rate mechanical loading

Successful detection and prevention of brain injuries relies on the quantitative identification of cellular injury thresholds associated with the underlying pathology. Here, by combining a recently developed inertial microcavitation rheology technique with a 3D in vitro neural tissue model, we quantify and resolve the structural pathology and critical injury strain thresholds of neural cells occurring at high loading rates such as encountered in blast, cavitation or directed energy exposures. We find that neuronal dendritic spines characterized by MAP2 displayed the lowest physical failure strain at 7.3%, whereas microtubules and filamentous actin were able to tolerate appreciably higher strains (14%) prior to injury. Interestingly, while these critical injury thresholds were similar to previous literature values reported for moderate and lower strain rates (<100 1/s), the pathology of primary injury reported here was distinctly different by being purely physical in nature as compared to biochemical activation during apoptosis or necrosis. Statement of SignificanceMitigation and prevention of cellular injury is challenging in part due to the lack of quantitative correlation between mechanical insult and cellular pathology, especially at high deformation rates (>104 s−1) that occur in blast and directed energy related brain injury, or laser and sonic-based medical procedures. By utilizing a recently developed inertial microcavitation rheology technique for generating high-rate deformations in a 3D in vitro neural tissue model, we quantitatively correlate critical stretch, strain and stress-based injury criteria to observed cell pathology. These quantitative experimental measurements provide unprecedented new detail into the cellular pathology of neural tissues affected by high-rate injury including the first quantitative high-rate injury threshold metrics.

A stable non‐ordinary state‐based peridynamic model for laminated composite materials

AbstractNon‐ordinary state‐based peridynamics (NOSBPD) has instability issue due to zero‐energy modes during nodal integration. A zero‐energy mode controlling scheme of NOSBPD for laminated composite materials is derived by using the linearized bond‐based peridynamics, which forms a stable force state formulation corresponding to the non‐uniform deformation. The proposed controlling scheme does not include any controlling parameter to avoid the complex parametric adjustment. A critical stretch continuously varying with the angle between fiber and bond directions is further proposed for the failure analysis of laminated composite materials with arbitrary fiber angles. The improved NOSBPD model adopts the explicit integration method to solve static problems. Several numerical examples are conducted to validate the proposed scheme suppressing the oscillation caused by zero‐energy modes.

An in-depth investigation of critical stretch based failure criterion in ordinary state-based peridynamics

This study presents an in-depth investigation of the critical stretch based failure criterion in ordinary state-based peridynamics for both static and dynamic conditions. Seven different cases are investigated to determine the effect of the failure parameter on peridynamic forces between material points and dilatation. Based on crack opening displacement (COD) results from both peridynamics and finite element analysis, it was found that one of the seven cases provides the best agreement between the two approaches. This particular case is further investigated by considering the influence of the discretisation and the horizon sizes on COD and crack propagation speeds. Moreover, PD predictions of COD for PMMA material is analysed with the theory of dynamic fracture mechanics and compared with the fracture experiments. It is shown that the peridynamic model can correctly model, simulate and predict the behaviour of the crack under different loading conditions. Furthermore, the presented PD models capture accurate fracture phenomena, specifically the crack path, branching angles and crack propagation speeds, which are in good agreement with experimental results.

Wrinkling of a compressed hyperelastic half-space with localized surface imperfections

We consider a variant of the classical Biot problem concerning the wrinkling of a compressed hyperelastic half-space. The traction-free surface is no longer flat but has a localized ridge or trench that is invariant in the x1-direction along which the wrinkling pattern is assumed to be periodic. With the x2-axis aligned with the depth direction, the localized imperfection is assumed to be slowly varying and localized in the x3-direction, and an asymptotic analysis is conducted to assess the effect of the imperfection on the critical stretch for wrinkling. The imperfection introduces a length scale so that the critical stretch is now weakly dependent on the wave number. It is shown that the imperfection increases the critical stretch (and hence reduces the critical strain) whether the imperfection is a ridge or trench, and the amount of increase is proportional to the square of the maximum gradient of the surface profile.

Flammability and critical parameters at low oxygen extinction limit of stagnation-point diffusion flames with varied stretch rates

Recently, to simulate microgravity flames in a terrestrial environment, low stretched stagnation-point diffusion flames have aroused wide interest, while the extinction behaviors of varied stretched flames at reduced oxygen concentration were scarcely investigated. This issue was accomplished by an investigation in this paper. Firstly, local and total extinction modes with ‘U’-shape boundaries were found to enlarge the unstable area in the flammability map. Furthermore, gas-phase radiation inversely proportional to stretch rate contributed to decreased critical flame temperature with dropped stretch rate. Two different power laws of critical flame standoff distance versus stretch rate existed at quenching and blowoff due to the coupling effects of boundary layer and critical oxygen concentration. Surface energy balance to evaluate the mechanisms of quenching and blowoff demonstrated that, both solid-phase conduction and surface radiation dominated at quenching, while they can be ignored at blowoff. Finally, critical burning rate was obtained from the surface energy balance and found to decrease with declined stretch rate. The similar changing tendencies of flammability and burning characteristics versus stretch rate in this work to the microgravity results implied a low-cost earth-based research method to simulate microgravity flames, and some new findings can provide instructions for fire protection in space.

Bifurcation behavior of compressible elastic half-space under plane deformations

A finitely deformed elastic half-space subject to compressive stresses will experience a geometric instability at a critical level and exhibit bifurcation. While the bifurcation of an incompressible elastic half-space is commonly studied, the bifurcation behavior of a compressible elastic half-space remains elusive and poorly understood to date. The main objective of this manuscript is to study the bifurcation of a neo-Hookean compressible elastic half-space against the well-established incompressible case. The formulation of the problem requires a novel description for a non-linear Poisson’s ratio, since the commonly accepted definitions prove insufficient for the current analysis. To investigate the stability of the domain and the possibility of bifurcation, an incremental analysis is carried out. The incremental analysis describes a small departure from an equilibrium configuration at a finite deformation. It is shown that at the incompressibility limit, our results obtained for a compressible elastic half-space recover their incompressible counterparts. Another key feature of this contribution is that we verify the analytical solution of this problem with computational simulations using the finite element method via an eigenvalue analysis. The main outcome of this work is an analytical expression for the critical stretch where bifurcation arises. We demonstrate the utility of our model and its excellent agreement with the numerical results ranging from fully compressible to incompressible elasticity. Moving forward, this approach can be used to comprehend and harness the instabilities in bilayer systems, particularly for compressible ones.

Analysis of Damage and Crack Propagation in Unidirectional Composite Laminates with a Peridynamic Model

A continuous function was applied to establish a peridynamics model to study the damage of unidirectional composite laminate, a kind of transverse isotropic material with three-dimensional (3D) structure. To express the interaction of nonadjacent layers and the influence of thickness on elastic deformation and damage, this paper employed a spherical material horizon. The spherical harmonics function was applied to express the relationship of the micro-elastic modulus and the bond-fiber angle, as well as critical bond stretch and the bond-fiber angle, respectively. Spatial stress or strain conditions were employed to calculate the strain energy density and solve the bond stiffness constant. The model was verified with unidirectional multilayer composite laminate with orifice. The damage mode and the crack propagation process of laminates with orifice were also consistent with that of experimental.

On energy release rates in Peridynamics

This work is concerned with the calculation of fracture energy release rates in Peridynamics. The existing crack growth criterion and the related damage model in Peridynamics, e.g. the critical bond stretch in the prototype micro-elastic brittle (PMB) material model, are based on the energy release rate of an idealized infinite crack model. Therefore, applications of such damage model in Peridynamics to arbitrary fracture patterns, e.g. cracks with finite increment, kinked cracks, and penny-shaped cracks etc., are problematic, because that the idealized crack model assumes an infinitely large crack surface, and the crack growth direction is always assumed being along the direction of the old crack surface.In all engineering applications, we only encounter cracks of finite size. In this work, based on the celebrated Griffith theory that interprets the energy release rate as the surface energy density, we have systematically investigated the fracture energy release rate in Peridynamics for a number of finite crack models: two-dimensional (2D) plane strain finite size horizontal crack, 2D edge crack in a plate, 2D kinked crack, and three-dimensional (3D) penny-shaped crack.The highlights of this work are a set of elegant mathematical solutions: (1) We have proved that for 2D horizontal finite cracks, the energy release rate is exactly the same as that of 2D infinite large crack, i.e. Silling’s original result; (2) We have derived the exact expressions of the energy release rates as well as the related critical bond stretch formulas for 2D finite edge cracks and kinked cracks, and (3) We have found an asymptotic solution of the energy release rate for 3D penny-shaped cracks of finite size in Peridynamics, showing that Silling’s energy release formula for 3D crack model is only the zero-th order approximation of the exact solution.In addition, we have presented some numerical solutions of finite increment crack growths in Peridynamics, and discussed their implications, ramifications, and applications in nonlocal continuum fracture mechanics.

On the experimental measurement of fracture toughness in SENT rubber specimens

This work provides a direct comparison of several experimental approaches used in the literature to measure fracture toughness of rubber of rubber using single edge notched in tension (SENT) specimens, with the final aim to provide guidelines for an optimal testing procedure. Digital image correlation measurements were used to get new insights into the fracture process. SENT is experimentally advantageous because of the simple preparation from laboratory plates and the small amount of material required. The most common experimental approaches to measure fracture toughness of rubber rely on the energy release rate, measured by the tearing energy or the J-integral parameters. This work points out the importance of experimental conditions and test procedures: long specimens and short notches are preferred, identification of fracture initiation from the front view is necessary, strain energy density should not be evaluated from un-notched specimens at the critical stretch level, rather alternative strategies are shown in this work.