Theoretical and computational hierarchical nanomechanics of protein materials: Deformation and fracture

Proteins constitute the building blocks of biological materials such as tendon, bone, skin, spider silk or cells. An important trait of these materials is that they display highly characteristic hierarchical structures, across multiple scales, from nano to macro. Protein materials are intriguing examples of materials that balance multiple tasks, representing some of the most sustainable material solutions that integrate structure and function. Here we review progress in understanding the deformation and fracture mechanisms of hierarchical protein materials by using a materials science approach to develop structure-process-property relations, an effort defined as materiomics. Deformation processes begin with an erratic motion of individual atoms around flaws or defects that quickly evolve into formation of macroscopic fractures as chemical bonds rupture rapidly, eventually compromising the integrity of the structure or the biological system leading to failure. The combination of large-scale atomistic simulation, multi-scale modeling methods, theoretical analyses combined with experimental validation provides a powerful approach in studying deformation and failure phenomena in protein materials. Here we review studies focused on the molecular origin of deformation and fracture processes of three types of protein materials. The review includes studies of collagen - Nature’s super-glue; beta-sheet rich protein structures as found in spider silk - a natural fiber that can reach the strength of a steel cable; as well as intermediate filaments - a class of alpha-helix based structural proteins responsible for the mechanical integrity of eukaryotic cells. The article concludes with a discussion of the significance of universally found structural patterns such as the staggered collagen fibril architecture or the alpha-helical protein motif.     

Multiscale Analysis of Fracture of Carbon Nanotubes Embedded in Composites

Due to the enormous difference in the scales involved in correlating the macroscopic properties with the micro- and nano-physical mechanisms of carbon nanotube-reinforced composites, multiscale mechanics analysis is of considerable interest. A hybrid atomistic/continuum mechanics method is established in the present paper to study the deformation and fracture behaviors of carbon nanotubes (CNTs) in composites. The unit cell containing a CNT embedded in a matrix is divided in three regions, which are simulated by the atomic-potential method, the continuum method based on the modified Cauchy–Born rule, and the classical continuum mechanics, respectively. The effect of CNT interaction is taken into account via the Mori–Tanaka effective field method of micromechanics. This method not only can predict the formation of Stone–Wales (5-7-7-5) defects, but also simulate the subsequent deformation and fracture process of CNTs. It is found that the critical strain of defect nucleation in a CNT is sensitive to its chiral angle but not to its diameter. The critical strain of Stone–Wales defect formation of zigzag CNTs is nearly twice that of armchair CNTs. Due to the constraint effect of matrix, the CNTs embedded in a composite are easier to fracture in comparison with those not embedded. With the increase in the Young’s modulus of the matrix, the critical breaking strain of CNTs decreases.     

Continuum metrics for deformation and microrotation from atomistic simulations: Application to grain boundaries

Motivated by a desire to incorporate micro- and nanoscale deformation mechanisms into continuum mechanical models of material behavior, we apply recently developed volume-averaged metrics to the results of atomistic simulations to investigate deformation and microrotation in the vicinity of grain boundaries. Three-dimensional bicrystalline structures are employed to study the inelastic deformation behavior under uniaxial tension and simple shear at a temperature of 10 K. Each bicrystal is constructed by molecular statics followed by thermal equilibration under NPT using an embedded atom method potential for copper. Strain is imposed in each simulation cell at a constant 10⁹ s⁻¹ strain rate applied perpendicular and parallel to the grain boundary plane for tension and shear, respectively. A variety of grain boundary deformation mechanisms arise and the resulting deformation and microrotation fields are examined. We also include an analysis showing how microrotation varies as a function of distance from the grain boundary with increasing strain for different grain boundary deformation mechanisms. This work demonstrates that critical interface behavior can be extracted from atomistic simulations using volume-averaged metrics, offering a potential avenue for translating fundamental information to continuum theories of grain boundary deformation in polycrystalline materials.     

A robust nano-mechanics approach for tensile and modal analysis using atomistic–continuum mechanics method

Nanoscale engineering has been developing rapidly. However, experimental investigations at the nanoscale level are very difficult to conduct. This research seeks to employ the same model to investigate an atomic-scale structure for tensile and modal analyses, based on atomistic–continuum mechanics (ACM) and a finite element method (FEM). The ACM transfers an originally discrete atomic structure into an equilibrium continuum model using atomistic–continuum transfer elements. All interatomic forces, described by the empirical potential functions, can be transferred into springs to form the atomic structure. The spring network models were also widely utilized in FEM based nano-structure studies. Thus, this paper attempts to explore ACM using three examples including silicon, carbon nanotube, and copper. All of the results are validated by bulk properties or literature.     

A bridging domain and strain computation method for coupled atomistic–continuum modelling of solids

We present a multiscale method that couples atomistic models with continuum mechanics. The method is based on an overlapping domain‐decomposition scheme. Constraints are imposed by a Lagrange multiplier method to enforce displacement compatibility in the overlapping subdomain in which atomistic and continuum representations overlap. An efficient version of the method is developed for cases where the continuum can be modelled as a linear elastic material. An iterative scheme is utilized to optimize the coupled configuration. Conditions for the regularity of the constrained matrices are determined. A method for computing strain in atomistic models and handshake domains is formulated based on a moving least‐square approximation which includes both extensional and angle‐bending terms. It is shown that this method exactly computes the linear strain field. Applications to the fracture of defected single‐layer atomic sheets and nanotubes are given. Copyright © 2006 John Wiley & Sons, Ltd.     

Buckling analysis of carbon nanotubes by a mixed atomistic and continuum model

A mixed atomistic and continuum model is applied to carbon nanotubes, in order to study their buckling behavior. Herein, the term “atomistic” refers to the underlying constitutive model that is formulated on the basis of interatomic potentials, whereas “continuum” means the application of the Cauchy–Born rule, which links the bond vectors before and after deformation via the deformation gradient of the continuum. Because the bond vectors are not infinitesimal and the continuum is modeled as surface, the Cauchy–Born rule has to be appropriately adapted to crystalline sheets. This is done via an exponential mapping in a new and surprisingly simple form such that in the analysis the current configuration has never to be left. The numerical buckling analysis of carbon nanotubes using the mixed atomistic and continuum model is carried out by means of the finite element method. For this purpose, the linearization of the equilibrium equations is provided.     

Tensile deformation behaviour of high isotactic polypropylenes

Abstract

The tensile flow behaviour and deformation processes during uniaxial loading of high isotactic (high crystallinity) polypropylenes have been investigated at various strain rates. The strain rate sensitivity index indicated that long chain high isotactic (high crystallinity) polypropylene (iPP-L) is relatively less sensitive to strain rate than short chain high isotactic (high crystallinity) polypropylene (iPP-S). Deformation bands formed at low strain contributed to the separation of fibrils/microfibrils, and on subsequent increase in tensile strain, the deformation bands developed into an array of closely spaced crazes that multiplied with an increase in strain and strain rate, and grew inwards. Three fracture morphologies were identified in iPP-L: brittle fracture, crazing - tearing, and brittle fracture with ductile pulling of fibrils/microfibrils (quasi-cleavage). In iPP-S, the predominant mode of failure at all displacement rates was brittle fracture. However, at lower displacement rates, smooth and circular voids associated with nucleating particle were randomly present on the brittle fracture surface of iPP-S. Occasional wedges were also observed on the tensile deformed surface of iPP-S. The deformation processes have been depicted in terms of strain rate - strain deformation diagrams.     

Finite element implementation of a kinetic model of dynamic strain aging in aluminum–magnesium alloys

Soare and Curtin (Acta Mater. 2008; 56 :4091–4101, 4046–4061) have recently developed a model of dynamic strain aging in solute‐strengthened alloys. Their constitutive law describes time‐dependent solute strengthening using rate equations that can be calibrated using atomistic simulations. In this paper, their material model is incorporated into a continuum finite element simulation, with a view to completing a multi‐scale method for predicting the formability of solute‐strengthened alloys. The Soare–Curtin model is first re‐formulated as a state‐variable constitutive law, which is suitable for finite element computations. An efficient numerical procedure is then developed to track the strength distribution of aging mobile and forest dislocations in the solid during deformation. The method is tested by simulating the behavior of a 3D aluminum–magnesium alloy tensile specimen subjected to uniaxial loading at constant nominal strain rate. The model predicts the influence of strain rate on the steady‐state flow stress of Al–Mg alloys, but no Portevin–Le Châtelier bands or serrated flow were observed in any of our simulations, and the influence of strain rate on tensile ductility is not predicted correctly. The reasons for this behavior and possible resolutions are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Tensile behaviour of nonwoven structures: comparison with experimental results

Nonwoven structures have been recently explored for numerous novel applications ranging from composites to scaffolds. The tensile property of nonwovens is a pre-requisite and indeed, one of the main parameters to determine their performance for such applications. In the first part, a modified micromechanical model describing the tensile behaviour of thermally bonded nonwovens was proposed by incorporating the effect of fibre re-orientation during the deformation (Rawal et al., J Mater Sci 45:2274, 2010). In this study, an attempt has been made to compare the theoretical and experimental stress–strain curves of thermally bonded and spunbonded nonwoven structures. These theoretical findings have been obtained from the most popular analytical tensile models of nonwovens available in the literature in addition to our modified tensile model. Poisson’s ratio has also been determined experimentally in order to predict the stress–strain behaviour of nonwoven, and its relationship with longitudinal strain has clearly distinguished between the randomly and preferentially orientated types of structures. In thermally bonded nonwovens, the tensile strength in various test directions is computed through pull-out stress and a comparison is made with the experimental results.     

Void coalescence processes quantified through atomistic and multiscale simulation

Simulation of ductile fracture at the atomic scale reveals many aspects of the fracture process including specific mechanisms associated with void nucleation and growth as a precursor to fracture and the plastic deformation of the material surrounding the voids and cracks. Recently we have studied void coalescence in ductile metals using large-scale atomistic and continuum simulations. Here we review that work and present some related investigations. The atomistic simulations involve three-dimensional strain-controlled multi-million atom molecular dynamics simulations of copper. The correlated growth of two voids during the coalescence process leading to fracture is investigated, both in terms of its onset and the ensuing dynamical interactions. Void interactions are quantified through the rate of reduction of the distance between the voids, through the correlated directional growth of the voids, and through correlated shape evolution of the voids. The critical inter-void ligament distance marking the onset of coalescence is shown to be approximately one void radius based on the quantification measurements used, independent of the initial separation distance between the voids and the strain-rate of the expansion of the system. No pronounced shear flow is found in the coalescence process. We also discuss a technique for optimizing the calculation of fine-scale information on the fly for use in a coarse-scale simulation, and discuss the specific case of a fine-scale model that calculates void growth explicitly feeding into a coarse-scale mechanics model to study damage localization. The U.S. Government’s right to retain a non-exclusive, royalty-free license in and to any copyright is acknowledged.     

Solid-state deformation of polyethylene and nylon and its effects on their structure and morphology

Solid-state deformation processes have been used to achieve polymers of high tensile moduli. The processes are based on achieving continuity for the oriented high-strength covalent bonds of the polymer chain. This is accomplished by the pulling out of chain folds and the subsequent extension of the long polymer chains which run both through and between crystal lamellae. Three major processes are used in solid-state deformation of semicrystalline polymers. In common cold drawing, the polymer is stretched at or below the crystalline melting point. In cold extrusion, a plug of solid polymer is forced by a ram through an orifice of smaller cross-sectional area to achieve draw. The third process, hydrostatic extrusion, is similar to cold extrusion except that the solid plug is surrounded by a pressure transmitting fluid which exerts a hydrostatic pressure on the plug that forces it through an orifice. Each method has distinct advantages and disadvantages that are described. Proposed molecular models for the three solid-state deformation processes generally consider the breakup of crystalline lamellae, their orientation in the deformation direction, and the pulling out of folded chains. These unfolded chains form tie-molecules between and among the disrupted lamellae. At highest deformation, fibril formation is observed which involves the partially extended tiemolecules. It is these chain-extended tie-molecules which are responsible for the unusually high tensile properties for drawn semicrystalline thermoplastics in the orientation direction.     

Strain‐Gradient Elasticity for Bridging Continuum and Atomistic Estimates of Stiffness of Binary Lennard‐Jones Crystals

Lagrangian variational approach is employed to derive the equations of equilibrium of strain‐gradient elasticity. For a periodic lamellar‐morphology strain‐gradient medium, we present an exact formula for the overall, system stiffness. We compare the formula with direct atomistic estimates of stiffness of binary Lennard‐Jones crystals. The comparison reveals that the strain‐gradient formula remains fairly accurate for all the crystals studied, including those with order of unity atoms in the crystal unit cell. Thus, one can surmise that the strain‐gradient correction alone can already be sufficient to extend the scope of validity of continuum‐level elasticity to near atomistic length scales.     

Effect of strain rate on quasistatic tensile flow behaviour of solution annealed 304 austenitic stainless steel at room temperature

Room temperature tensile test results of solution annealed 304 stainless steel at strain rates ranging between 5 × 10⁻⁴ and 1 × 10⁻¹ s⁻¹ reveal that with increase in strain rate yield strength increases and tensile strength decreases, both maintaining power–law relationships with strain rate. The decrease in tensile strength with increasing strain rate is attributed to the lesser amount of deformation-induced martensite formation and greater role of thermal softening over work hardening at higher strain rates. Tensile deformation of the steel is found to occur in three stages. The deformation transition strains are found to depend on strain rate in such a manner that Stage-I deformation (planar slip) is favoured at lower strain rate. A continuously decreasing linear function of strain rate sensitivity with true strain has been observed. Reasonably good estimation for the stress exponent relating dislocation velocity and stress has been made. The linear plot of reciprocal of strain rate sensitivity with true strain suggests that after some critical amount of deformation the increased dislocation density in austenite due to the formation of some critical amount of deformation-induced martensite plays important role in carrying out the imposed strain rate.     

Very large scale simulations of materials failure

With present-day supercomputers, simulation is becoming a powerful tool for providing immediate insights into the nature of fracture dynamics. Atomistic simulations yield ab initio information about crack-tip formation and deformation at length-scales unattainable by experimental measurement and unpredictable by continuum elasticity theory. We will describe several atomistic dynamics studies concerning brittle fracture and ductile deformation.     

Progressive Failure Modeling for Dynamic Loading of Woven Composites

A three parameter constitutive model was developed for representing tensile progressive damage of the nonlinear large-deformation rate-dependent behavior of polymer-based composite materials, which was characterized using off-axis composite specimens. A strain based failure criterion was proposed that reduces data for different loading directions and strain rates to a single representation. A method of combining the nonlinear constitutive theory and the failure strain methodology for different strain rates is suggested. The strength of the material was successfully represented with a single material constant, for all strain rates and tensile loading directions.     

Application of the higher‐order Cauchy–Born rule in mesh‐free continuum and multiscale simulation of carbon nanotubes

This paper investigates the application of a recently proposed higher‐order Cauchy–Born rule in the continuum simulation and multiscale analysis of carbon nanotubes (CNTs). A mesh‐free computational framework is developed to implement the numerical computation of the hyper‐elastic constitutive model that is derived from the higher‐order Cauchy–Born rule. The numerical computation reveals that the buckling pattern of a single‐walled carbon nanotube (SWCNT) can be accurately displayed by taking into consideration the second‐order deformation gradient, and fewer mesh‐free nodes can provide a good simulation of homogeneous deformation. The bridging domain method is employed to couple the developed mesh‐free method and the atomistic simulation. The coupling method is used to simulate the bending buckling of an SWCNT and the tensile failure of an SWCNT with a single‐atom vacancy defect, and good computational results are obtained. Copyright © 2008 John Wiley & Sons, Ltd.     

Pre-yield tensile set of a semi-crystalline polymer,its blend and composite

Tensile set was studied at low strains on polypropylene, aliphatic polyketone, rubber toughened blends and CaCO₃ particle toughened composites. The rubber in the rubber toughened blends had a particle size of 0.7 μm. The CaCO₃ particles had a size of 0.7 μm and had been coated with stearic acid. Step-cyclic loading was applied in 1% strain incrementals at a strain rate of 10⁻² s⁻¹. The maximum strain applied was 20%. The temperature of the test bar was studied with an infra-red camera. Pre-yield deformation is normally assumed to take place in a nonlinear elastic manner. However, for polypropylene and polyketone elastoplastic deformation starts at low strains. For PP the onset of tensile set is at very low strains and increases with strain. The tensile set at the yield point was only 50% and at the drawing strain 100%. Polyketone had a similar tensile set development but shifted to slightly higher strains. Here too the tensile set at the yield point was about 50% and at the drawing strain 100%. The temperature of the non yielded material was found to rise in polyketone a 7 °C. The rubber toughened blends had at low strains a higher tensile set, but after the yield strain the set was similar to the base polymer. At 5% strain the tensile set increased with rubber content. The sub micron CaCO₃ particle toughened composites increased the tensile set too. The tensile set is a simple technique for studying the pre-yield behaviour of multi phase systems.     

Deformation response and constitutive modeling of vinyl ester polymer including strain rate and temperature effects

The deformation behavior of vinyl ester polymer under monotonic tensile loading is characterized and modeled. The Standard Linear Solid model, which is a physical model, was used and modified to represent the stress–strain behavior of this polymer over a wide range of strain rates and temperatures. This model was also used to predict the stress-relaxation and short-term creep behavior of this material. The comparisons between the predictions and experimental data from tensile and relaxation tests demonstrate that this model can represent the deformation behavior of the material reasonably well.