In this paper, a key differential equation is proposed to formulate fatigue damage evolution in metallic alloys under multiaxial, multiblock, proportional loadings in high cycle fatigue (HCF) and very high cycle fatigue (VHCF) regimes. This differential equation possesses two main components: one is a stress function to accommodate the adopted fatigue criterion and the other one is a characteristic damage function that serves to capture the HCF response of alloys. Two distinct characteristic damage functions with three different multiaxial fatigue criteria, namely Sines, Crossland, and Dang Van criteria, are examined to develop six (out of many possible) variants of the presented damage accumulation model. As a validation measure, Chaboche’s HCF damage model is retrieved as a specific case of the developed formalism. For model parameters identification, an ad hoc two-level identification scheme is designed and numerically verified. It is demonstrated that endurance limit, which is determined from fully reversed HCF tests (i.e., R = −1), can be identified from fatigue tests with positive stress ratio (R > 0), thus making our development quite suitable for specimens prone to buckling under compression. Another salient feature of the devised identification scheme is its capability in extracting model parameters from noisy data.

To determine the impact of cohesive law shapes on the modeling of interfacial debonding in lithium-ion battery electrodes, analytical methods based on different cohesive models for the debonding process have been developed individually. Three different cohesive laws, namely, triangular, trapezoidal, and rectangular laws, have been employed. To ensure comparability, the cohesive strength and the fracture toughness have been set to be identical for different cohesive laws. The evaluation of debonding onset has suggested that the cohesive law shape affects the modeling results only when the interface is ductile. The largest possible difference for the triangular law and the rectangular law on the debonding onset has been estimated. A discussion for specific electrodes has also been provided.

This paper presents an analysis of void growth and coalescence in isotropic, elastoplastic materials exhibiting sigmoidal hardening using unit cell calculations and micromechanics-based damage modeling. Axisymmetric finite element unit cell calculations are carried out under tensile loading with constant nominal stress triaxiality conditions. These calculations reveal the characteristic role of material hardening in the evolution of the effective response of the porous solid. The local heterogeneous flow hardening around the void plays an important role, which manifests in the stress–strain response, porosity evolution, void aspect ratio evolution, and the coalescence characteristics that are qualitatively different from those of a conventional power-law hardening porous solid. A homogenization-based damage model based on the micromechanics of void growth and coalescence is presented with two simple, heuristic modifications that account for this effect. The model is calibrated to a small number of unit cell results with initially spherical voids, and its efficacy is demonstrated for a range of porosity fractions, hardening characteristics, and void aspect ratios.

Curved shell structures are known for their excellent load-carrying capability and are commonly used in thin-walled constructions. Although theoretically able to withstand greater buckling loads than flat structures, shell structures are notoriously sensitive to imperfections owing to their postbuckling behavior often being governed by subcritical bifurcations. Thus, shell structures often buckle at significantly lower loads than those predicted numerically and the ensuing dynamic snap to another equilibrium can lead to permanent damage. Furthermore, the strong sensitivity to initial imperfections, as well as their stochastic nature, limits the predictive capability of current stability analyses. Our objective here is to convert the subcritical nature of the buckling event to a supercritical one, thereby improving the reliability of numerical predictions and mitigating the possibility of catastrophic failure. We explore the elastically nonlinear postbuckling response of axially compressed cylindrical panels using numerical continuation techniques. These analyses show that axially compressed panels exhibit a highly nonlinear and complex postbuckling behavior with many entangled postbuckled equilibrium curves. We unveil isolated regions of stable equilibria in otherwise unstable postbuckled regimes, which often possess greater load-carrying capacity. By modifying the initial geometry of the panel in a targeted—rather than stochastic—and imperceptible manner, the postbuckling behavior of these shells can be tailored without a significant increase in mass. These findings provide new insight into the buckling and postbuckling behavior of shell structures and opportunities for modifying and controlling their postbuckling response for enhanced efficiency and functionality.

Most tough hydrogels suffer accumulated damages under cyclic loads. The damages may stem from breakage of covalent bonds, unzipping of ionic crosslinks, or desorption of polymer chains from nanoparticle surfaces. Recent experiments report that when a tough hydrogel is subject to cyclic loads, the stress–stretch curves of tough hydrogels change cycle by cycle and approach a steady-state after thousands of cycles, denoted as the shakedown phenomenon. In this paper, we develop a phenomenological model to describe the shakedown of tough hydrogels under prolonged cyclic loads for the first time. We specify a new evolution law of damage variable in multiple cycles, motivated by the experimental observations. We synthesize nanocomposite hydrogels and conduct the cyclic tests. Our model fits the experimental data remarkably well, including the features of Mullins effect, residual stretch and shakedown. Our model is capable of predicting the stress–stretch behavior of subsequent thousands of cycles by using the fitting parameters from the first and second cycle. We further apply the model to polyacrylamide (PAAM)/poly(2-acrylanmido-2-methyl-1-propanesulfonic acid) (PAMPS) and PAAM/alginate double-network hydrogels. Good agreement between theoretical prediction and experimental data is also achieved. The model is hoped to serve as a tool to probe the complex nature of tough hydrogels, through cyclic loads.

If an elastomeric material is subjected to sufficiently large deformations, it eventually fractures. There are two typical micromechanisms of failure in such materials: chain scission and crosslink failure. The chain scission failure mode is mainly observed in polymers with strong covalent crosslinks, while the crosslink failure mode is observed in polymers with weak crosslinks. In two recent papers, we have proposed a theory for progressive damage and rupture of polymers with strong covalent crosslinks. In this paper, we extend our previous framework and formulate a theory for modeling failure of elastomeric materials with weak crosslinks. We first introduce a model for the deformation of a single chain with weak crosslinks at each of its two ends using statistical mechanics arguments, and then upscale the model from a single chain to the continuum level for a polymer network. Finally, we introduce a damage variable to describe the progressive damage and failure of polymer networks. A central feature of our theory is the recognition that the free energy of elastomers is not entirely entropic in nature; there is also an energetic contribution from the deformation of the backbone bonds in a chain and/or the crosslinks. For polymers with weak crosslinks, this energetic contribution is mainly from the deformation of the crosslinks. It is this energetic part of the free energy which is the driving force for progressive damage and fracture of elastomeric materials. Moreover, we show that for elastomeric materials in which fracture occurs by crosslink stretching and scission, the classical Lake–Thomas scaling—that the toughness G c of an elastomeric material is proportional to 1 / G 0 , with G 0 = N k B ϑ the ground-state shear modulus of the material—does not hold. A new scaling is proposed, and some important consequences of this scaling are remarked upon.

A series of stress-controlled uniaxial cyclic tension-unloading tests are discussed to investigate the ratchetting of a filled rubber at room temperature. It is shown that obvious ratchetting occurs and depends apparently on the applied stress level, stress rate, and stress history. Based on the experimental observations, a damage-coupled hyper-viscoelastic-plastic constitutive model is then developed to describe the ratchetting of the filled rubber, which consists of three branches in parallel, i.e., a hyperelastic, a viscoelastic, and a plastic one. The damage is assumed to act equally on three branches and consists of two parts, i.e., the Mullins-type damage caused by the initial tensile deformation and the accumulated damage occurred during the cyclic deformation. The developed model is validated by comparing the predicted results with the experimental data.

The skeleton of many natural and artificial soft materials can be abstracted as networks of fibers/polymers interacting in a nonlinear fashion. Here, we present a numerical model for networks of nonlinear, elastic polymer chains with rate-dependent crosslinkers similar to what is found in gels. The model combines the worm-like chain (WLC) at the polymer level with the transition state theory for crosslinker bond dynamics. We study the damage evolution and the force—displacement response of these networks under uniaxial stretching for different loading rates, network topology, and crosslinking density. Our results suggest a complex nonmonotonic response as the loading rate or the crosslinking density increases. We discuss this in terms of the microscopic deformation mechanisms and suggest a novel framework for increasing toughness and ductility of polymer networks using a bio-inspired sacrificial bonds and hidden length (SBHL) mechanism. This work highlights the role of local network characteristics on macroscopic mechanical observables and opens new pathways for designing tough polymer networks.

An efficient and novel micromechanical computational platform for progressive failure analysis of fiber-reinforced composites is presented. The numerical framework is based on a recently developed micromechanical platform built using a class of refined beam models called Carrera unified formulation (CUF), a generalized hierarchical formulation which yields a refined structural theory via variable kinematic description. The crack band theory is implemented in the framework to capture the damage propagation within the constituents of composite materials. The initiation and orientation of the crack band in the matrix are determined using the maximum principal stress state and the traction-separation law governing the crack band growth is related to the fracture toughness of the matrix. A representative volume element (RVE) containing randomly distributed fibers is modeled using the component-wise (CW) approach, an extension of CUF beam model based on Lagrange type polynomials. The efficiency of the proposed numerical framework is achieved through the ability of the CUF models to provide accurate three-dimensional (3D) displacement and stress fields at a reduced computational cost. The numerical results are compared against experimental data available in the literature and an analogous 3D finite element model with the same constitutive crack band model. The applicability of CUF beam models as a novel micromechanical platform for progressive failure analysis as well as the multifold efficiency of CUF models in terms of CPU time are highlighted.

This paper investigates the effect of strain rate on the scaling behavior of dynamic tensile strength of quasibrittle structures. The theoretical framework is anchored by a rate-dependent finite weakest link model. The model involves a rate-dependent length scale, which captures the transition from localized damage to diffused damage with an increasing strain rate. As a result, the model predicts a rate- and size-dependent probability distribution function of the nominal tensile strength. The transitional behavior of the strength distribution directly leads to the rate and size effects on the mean and standard deviation of the tensile strength. The model is verified by a series of stochastic discrete element simulations of dynamic fracture of aluminum nitride specimens. The simulations involve a set of geometrically similar specimens of various sizes subjected to a number of different strain rates. Both random microstructure geometry and fracture properties are considered in these simulations. The simulated damage pattern indicates that an increase in the strain rate results in a more diffusive cracking pattern, which supports the theoretical formulation. The simulated rate and size effects on the mean and standard deviation of the nominal tensile strength agree well with the predictions by the rate-dependent finite weakest link model.

A phase-field damage model for orthotropic materials is proposed and used to simulate delamination of orthotropic laminated composites. Using the deviatoric and hydrostatic tensile components of the stress tensor for elastic orthotropic materials, a degraded elastic free energy that can accommodate damage is derived. The governing equations follow from the principle of virtual power and the resulting damage model, by its construction, conforms with the physical relevant condition of no matter interpenetration along the crack faces. The model also dispenses with the traction separation law, an extraneous hypothesis conventionally brought in to model the interlaminar zones. The model is assessed through numerical simulations on delaminations in mode I, mode II, and another such problem with multiple initial notches. The present method is able to reproduce nearly all the features of the experimental load displacement curves, allowing only for small deviations in the softening regime. Numerical results also show forth a superior performance of the proposed method over existing approaches based on a cohesive law.

Double and triple network (TN) elastomers can be made by infusing monomers into a single network (SN) polymer, causing it to swell, and then polymerizing and cross-linking the monomers. The result is a double network (DN) elastomer in which one network is stretched and the other is in hydrostatic compression. TN systems are made by repeating the process starting with the DN material. The multinetwork (MN) elastomers exhibit a Mullins effect in which softening occurs upon a first cycle of loading, with the elastomer stiffness recovered above the previous maximum strain. The Mullins effect is attributed to rupture of the stretched network, eliminating the constraint on the compressed network, thereby motivating straining at the lower stiffness of the remaining material. A model for this process is developed, based on the previous work of Horgan et al. (2004, “A Theory of Stress Softening of Elastomers Based on Finite Chain Extensibility,” Proc. R. Soc. A, 460 (2046), pp. 1737–1754). In the proposed model, a composite stiffness for the MN system is developed and a damage process introduced to degrade the contribution of the stretched network. The damage model is designed to account for the progressive elimination of chains that are most highly loaded in the stretched network, so that the undamaged stiffness is restored when the strain rises above levels previously experienced. The proposed model reproduces the behavior of the Mullins effect in the MN system.

A cohesive zone modeling (CZM) approach with a bilinear traction-separation relation is used to study the peeling of a thin overhanging plate from the edge of an incompressible elastomeric layer bonded firmly to a stationary rigid base. The deformations are approximated as plane strain and the materials are assumed to be linearly elastic, homogeneous, and isotropic. Furthermore, governing equations for the elastomer deformations are simplified using the lubrication theory approximations, and those of the plate with the Kirchhoff–Love theory. It is found that the peeling is governed by a single nondimensional number defined in terms of the interfacial strength, the interface fracture energy, the plate bending rigidity, the elastomer shear modulus, and the elastomeric layer thickness. An increase in this nondimensional number monotonically increases the CZ size ahead of the debond tip, and the pull-off force transitions from a fracture energy to strength dominated regime. This finding is supported by the results of the boundary value problem numerically studied using the finite element method. Results reported herein could guide elastomeric adhesive design for load capacity and may help ascertain test configurations for extracting the strength and the fracture energy of an interface from test data.

By directly solving the prescribed differential equations, an analytical method based on the cohesive model has been developed to investigate the interfacial debonding process induced by lithiation in an axisymmetric thin film electrode where an elastic active layer is bonded on a rigid substrate. The assumption of rigid substrate has been proved acceptable for high-modulus substrates such as copper and aluminum which are common materials for current collectors in lithium-ion batteries. For the case where the weak interface is assumed and the radial concentration gradient is neglected, an extremely simplified solution has been obtained. The simplified solution which has acceptable accuracy provides a good guidance for understanding and predicting the interfacial debonding.

The polymer network of a nanocomposite (NC) hydrogel is physically crosslinked by nanoclay. Recently reported high toughness of nanocomposite (NC) hydrogels highlights the importance of their dissipative properties. The desorption of polymer chains from clay surface may contribute mostly to the hysteresis of NC hydrogels. Here, we proposed a mechanistically motivated pseudoelastic model capable of characterizing the hysteresis of NC hydrogels. The two parameters in the proposed damage variable can be determined by the experiments. We applied the model to the uniaxial tension and reproduced the ideal Mullins effect of NC hydrogels. Furthermore, we considered two nonideal effects: residual deformation and nonideal reloading in multicycle test, using newly proposed damage parameters. A power law with the order of 1/3 is established between the residual fraction of the stretch and the re-adsorption ratio of polymer chains. Finally, we demonstrated the dissipative properties of various NC hydrogels with the model.

The spectral approach is used to examine the wave dispersion in linearized bond-based and state-based peridynamics in one and two dimensions, and comparisons with the classical nonlocal models for damage are made. Similar to the classical nonlocal models, the peridynamic dispersion of elastic waves occurs for high frequencies. It is shown to be stronger in the state-based than in the bond-based version, with multiple wavelengths giving a vanishing phase velocity, one of them longer than the horizon. In the bond-based and state-based, the nonlocality of elastic and inelastic behaviors is coupled, i.e., the dispersion of elastic and inelastic waves cannot be independently controlled. In consequence, the difference between: (1) the nonlocality due to material characteristic length for softening damage, which ensures stability of softening damage and serves as the localization limiter, and (2) the nonlocality due to material heterogeneity cannot be distinguished. This coupling of both kinds of dispersion is unrealistic and similar to the original 1984 nonlocal model for damage which was in 1987 abandoned and improved to be nondispersive or mildly dispersive for elasticity but strongly dispersive for damage. With the same regular grid of nodes, the convergence rates for both the bond-based and state-based versions are found to be slower than for the finite difference methods. It is shown that there exists a limit case of peridynamics, with a micromodulus in the form of a Delta function spiking at the horizon. This limit case is equivalent to the unstabilized imbricate continuum and exhibits zero-energy periodic modes of instability. Finally, it is emphasized that the node-skipping force interactions, a salient feature of peridynamics, are physically unjustified (except on the atomic scale) because in reality the forces get transmitted to the second and farther neighboring particles (or nodes) through the displacements and rotations of the intermediate particles, rather than by some potential permeating particles as on the atomic scale.

A material model is presented that includes the following deformation mechanisms: the instantaneous response of ice due to distortion of crystal lattices, creep, the formation of microcrack nuclei due to creep, the formation of microcracks, and deformation due to microcracks. The new material model has a strict foundation on deformation mechanisms. This constitutive equation was applied to sea ice for engineering applications through implementation in the Abaqus explicit code by writing a VUMAT subroutine. The computed results show that the model correctly predicts the uniaxial tensile and the uniaxial compressive strengths of ice. The computed compressive strength versus strain-rate relation takes an almost linear relation when expressed in the log–log coordinates, which fits well with the data obtained from the literature. The material model shows the Hall–Petch type of strength dependency on the grain size.

Hypervelocity impact (HVI) is a scenario involving an impacting velocity in excess of 1 km/s. Ubiquitous in outer space, paradigms of HVI are typified by the collision between orbital debris and spacecraft. HVI features transient, localized, and extreme material deformation under which the induced acoustic emission (AE) signals present unique yet complex features. A dedicated modeling and numerical simulation approach, based on the three-dimensional smooth-particle hydrodynamics (SPH), was developed to gain an insight into characteristics of HVI-induced AE propagation. With the approach, both normal and oblique HVI scenarios were interrogated, and material failure in both cases was predicted. The coincidence in results between simulation and HVI experiment, as observed at a qualitative degree, has demonstrated the effectiveness of the modeling. Signal analysis shows that the shock wave converts to Lamb wave quickly as propagation from HVI spot, with the zeroth-order symmetric wave mode (S 0 ) (i.e., the first-arrival wave) dominating wave signal energy. S 0 is observed dispersive in a wide frequency range with majority of it below 1 MHz. In comparison, the antisymmetric wave mode distributes in a range below 200 kHz with a peak value at 30 kHz. S 0 was employed to pinpoint the location of HVI, using an enhanced delay-and-sum-based diagnostic imaging algorithm, which was validated by locating orbital debris-induced orifice in space structures, showing precise identification results.

This work presents a collection of advanced computational methods, and their coupling, that enable prediction of fatigue-damage evolution in full-scale composite blades of wind turbines operating at realistic wind and rotor speeds. The numerical methodology involves: (1) a recently developed and validated fatigue-damage model for multilayer fiber-reinforced composites; (2) a validated coupled fluid–structure interaction (FSI) framework, wherein the 3D time-dependent aerodynamics based on the Navier–Stokes equations of incompressible flows is computed using a finite-element-based arbitrary Lagrangian–Eulerian–variational multiscale (ALE–VMS) technique, and the blade structures are modeled as rotation-free isogeometric shells; and (3) coupling of the FSI and fatigue-damage models. The coupled FSI and fatigue-damage formulations are deployed on the Micon 13M wind turbine equipped with the Sandia CX-100 blades. Damage initiation, damage progression, and eventual failure of the blades are reported.

This paper presents a new crack band model (CBM) for probabilistic analysis of quasibrittle fracture. The model is anchored by a probabilistic treatment of damage initiation, localization, and propagation. This model regularizes the energy dissipation of a single material element for the transition between damage initiation and localization. Meanwhile, the model also takes into account the probabilistic onset of damage localization inside the finite element (FE) for the case where the element size is larger than the crack band width. The random location of the localization band is related to the random material strength, whose statistics is described by a finite weakest link model. The present model is applied to simulate the probability distributions of the nominal strength of different quasibrittle structures. It is shown that for quasibrittle structures direct application of the conventional CBM for stochastic FE simulations would lead to mesh-sensitive results. To mitigate such mesh dependence, it is essential to incorporate the strain localization mechanism into the formulation of the sampling distribution functions of material constitutive parameters.