Plane Strain Conditions Research Articles

Plane vibrational modes and localized nonlinear excitations in carbon nanotube bundle

Bulk nanomaterials made of carbon nanotubes (CNTs) possess unique physical and mechanical properties that are used in a number of applications, for example, for creating mass or force sensors, protection against shock and vibration. One of the specific implementations of such materials is a CNT bundle in which aligned CNTs are held together due to van der Waals interactions. Linear and nonlinear dynamic excitations in such materials are still poorly understood. Here we study plane waves and spatially localized nonlinear dynamic modes in a CNT bundle under plane strain conditions, when the CNT cross sections completely determine the deformed state of the bundle. In particular, we analyze vibration spectrum of single CNT, dispersion curves of phonons propagating along the close packed direction of CNT bundle, compressive solitons and discrete breathers in the form of single CNT oscillating at large amplitude in the CNT bundle. It is shown that compressive solitons can propagate long distances in the bundles of CNTs having radius no more than about 0.8 nm. Discrete breathers can exist in bundles of CNTs of any radius. The results obtained revealed new channels of energy localization and transport in CNT bundles, which is important for their use in various technologies.

A suitable mixed mode I/II test specimen for fracture toughness study of polyurethane foam with different cell densities

Rigid foam materials are among the low strength structural components that can be failed catastrophically under mechanical loads. Low fracture toughness is therefore a major failure mode in these materials and researchers try to find suitable testing methods for determining the fracture toughness of foam materials. In this research a suitable test specimen that is an edge notch disc bend specimen subjected to asymmetric three-point bending and called Asymmetric Edge Notch Disc Bend (AENDB) is proposed for conducting mixed mode I/II fracture experiments on polyurethane (PUR) foam with different cell densities. The geometry factors of the specimen that are required for determining the stress intensity factors were determined by means of the finite element analyses for different mode mixities. The mixed mode I/II fracture toughness envelope and fracture initiation angle were obtained for the (AENDB) specimen made of three foam densities ranging from 100 to 300 kg/m3. For example, KIc and KIIc values were varied in the ranges 0.08–0.31 MPa.m0.5 and 0.09–0.34 MPa.m0.5, respectively. It was observed that the fracture toughness data are increased for all mode mixities by increasing the foam density. However, the mixed mode fracture curve and fracture initiation directions obtained from the AENDB specimen were lower than the corresponding curves of other mixed mode I/II test samples such as asymmetric semi-circular bend (ASCB) specimen. This can be attributed to the plane strain condition that is obtained at the mid-section of the AENDB specimen, since the crack front length in the AENDB specimen is significantly greater than the ASCB specimen. Finally, the experimental results obtained from the AENDB specimen were predicted using two fracture criterions. The results of this research indicate that the suggested AENDB specimen is a good candidate mixed mode I/II fracture configuration for investigating foam and other similar cellular materials.

Investigation on in-plane and out-of-plane constraint effects for plates with I-II mixed mode cracks under biaxial compressive loading

Crack sizes, geometrical configurations of specimens and structures, as well as loading conditions have significant effect on in-plane and out-of-plane constraint effects. Safety assessment with constraint effects can effectively improve the accuracy. There are prominent discrepancies between compressive fracture and tensile fracture. Most of engineering structures are usually under biaxial loading conditions, so that the influence of biaxial compressive loading on fracture analysis with constraint effects is worth paying attention. The size effect factor (relative crack length a/W, thickness to width ratio t/W and crack inclination angle β) and load factor (lateral pressure coefficient λ) of central cracked plate (CCP) specimens with I-II mixed mode crack on constraint effects under biaxial compressive loading have been investigated through theoretical analysis and finite element analysis. In-plane non-singular T-stress (T11) and its components (Tx, Ty, and Txy), out-of-plane non-singular T-stress (T33), out-of-plane constraint parameter Tz and stress triaxiality factor h have been adopted to quantify constraint effects. It demonstrates that λ has remarkable influence on SIF, T11 and T33 under biaxial compressive loading. The effect of higher a/W (long cracks) on B11 is more noticeable when I-II mixed mode crack is closer to pure mode II crack or pure mode I crack. The proportion of shear fracture increases under biaxial compression with λ. With the increase of β, B33 decreases sharply and then increases rapidly with higher λ. Tz can directly reflect stress states and adequately describe the loss of out-of-plane constraint. The stress state of crack tip is more likely to asymptotic plane-strain condition ahead of crack tip under compression. There are crucial differences in variation range and tendency of Tz subjected to biaxial tensile and biaxial compressive loading. Moreover, Tz is insensitive to λ. Stress triaxiality factor h can illustrate in-plane and out-of-plane constraint effects including size effect and load factor (sensitive to a/W, t/W, β and λ), especially can be adopted as a unified constraint parameter for long cracks under biaxial compressive loading. The in-plane T-stress components can be directly calculated by T11 under biaxial compression. The empirical equations of KII and T11 containing size effect and load factor simultaneously are effective and convenient for applications.

A study of crack initiation in a low alloy steel

Taking advantage of in-situ fracture testing method inside a transmission electron microscope (TEM), crack evolution in a low alloy steel under low triaxiality conditions is studied and the interaction between cementite particles and the crack is revealed. It is found that the ferrite matrix is the major void initiation site due to the low stress triaxiality in the thin TEM sample (plane stress condition), which contrasts the behavior under plane strain conditions in bulk specimens, where voids are typically found to initiate by decohesion at the particle/matrix interface. This work reveals that fracture behavior proceeds differently under low triaxiality conditions, such as the shear lip region of fractured bulk sample, and demonstrates the possibility to avoid interface decohesion and thereby to enhance ductility in steels.

Adhesive Boundary Element Method Using Virtual Crack Closure Technique

In this study, a new adhesive contact model is built upon a boundary element method (BEM) model developed by Pohrt and Popov (2015). The strain energy release rate (SERR) on the edge of the bonding interface is evaluated using Virtual Crack Closure Technique (VCCT) which is shown to have better accuracy and weaker mesh-size dependency than the closed-form SERR formula derived by Pohrt and Popov. A composite delamination criterion is proposed for crack nucleation and propagation. Numerical results predicted by the present model are in good agreement with the analytical solutions of two classic problems, namely, the axisymmetric parabolic contact and the sinusoidal waviness contact in the plane strain condition. The model of Pohrt and Popov can achieve a similar accuracy for the axisymmetric parabolic contact where the mesh grid is non-conforming to the crack front. Once the conforming mesh grid is used, the accuracy of their model is significantly deteriorated, especially at high work of adhesion and high mesh density. In both BEM models, however, the crack nucleation is found to be mesh-dependent which may be solved by introducing an upper limit for the tensile normal traction.

Plastic zone analysis of SS316L and Ti6Al4V materials under mixed mode loading conditions

In present work, plastic zone size and shapes are investigated to predict failure of hip implant materials Ti6Al4V and SS316L (titanium and stainless steel, respectively). Generally, they fracture due to crack propagation in presence of wear which initiates the crack. Plastic zone analysis of SS316L and Ti6Al4V are carried out under standard fracture modes of failure that are first and mixed mode (tensile and shear loading). Crack inclination angle is one of the important parameters to simulate mixed-mode loading conditions. Analytical formulations of plastic zone analysis validated by finite element modeling based on von-Mises criteria. Further, isolines are plotted which explains the variation of plastic zone and geometry near the crack tip. Results obtained under plane strain and plane stress conditions through analytical formulations are in close relevance with FEM. Finally, it is observed that SS316L has large plastic zone which results in slower crack propagation and better life as compared to Ti6Al4V.

Mixed mode equilibrium analysis of asymptotic fields around stationary sharp V-notches in polymer gels: A linear poroelastic study

In this paper, the equilibrium asymptotic stress and solvent concentration fields around stationary sharp V-notches are extracted for in-plane mixed mode loadings employing the linear poroelasticity theory. It is shown that at thermodynamic equilibrium, mechanical and chemical equilibrium equations are uncoupled and thus can be solved independently. The mechanical equilibrium equations are then solved using the Airy stress function technique. This technique replaces the coupled mechanical equilibrium relations with the compatibility biharmonic equation which can be solved using the separation of variables method. Applying traction free boundary conditions on the notch edges leads to an eigenvalue problem which gives both mode I and mode II eigenvalues. These eigenvalues can be real or complex numbers depending on the notch opening angle. Since there are infinite eigenvalues for both modes of deformation, series solutions will be obtained for the equilibrium fields. From these asymptotic solutions, it is found that the obtained stress fields are similar to their corresponding linear elasticity solution. Furthermore, from the solutions, it can be observed that the opening and shear deformations near the notch tip lead to cosine and sine variations of the solvent concentration field with respect to the angular coordinates, respectively. The accuracy of these asymptotic results is finally verified using finite element analyses of a single-edge notched (SEN) specimen subjected to far field applied small displacements. The numerical analyses are performed for two notch opening angles of 30° and 60° for both plane stress and plane strain conditions. The comparative study between the finite element results and the asymptotic solution proves that the present solution can accurately capture the near notch tip stress and solvent concentration fields by considering only the first few terms of the series solutions.

Three-dimensional stress state during dynamic shear rupture propagation along frictional interfaces in elastic plates

The state of stress in plates, where one geometric dimensions is much smaller than the others, is often assumed to be of plane stress. This assumption is justified by the fact that the out-of-plane stress components are zero on the free-surfaces of a plate to satisfy the boundary conditions, and have little chance to develop in the bulk, due to the small thickness of the plate. At the same time, it is known that the static stress field associated with cracks approximates plane-strain conditions in the near-crack tip. However, it is not clear how the pre-existing plane-stress field of a plate is modified by a propagating dynamic shear rupture. Here we study the particle velocities and stress fields of dynamic shear ruptures in mode II propagating along the predefined frictional interface of two plates of an elastic material, loaded in compression and shear, using three-dimensional finite element modeling. The numerical simulations show the rapid development of out-of-plane stresses in the interior of the specimen, between the free surfaces. The out-of-plane normal stress is characterized by an anti-symmetric pattern with lobes of alternating polarity, in planes parallel to the free-surfaces. On the interface plane, the out-of-plane stress has a complex pattern exhibiting an initial sudden variation over the plane-stress conditions followed by crisscrossing features, behind the rupture tip. This study shows how plane-stress conditions, defining the state of stress before rupture arrival, are suddenly altered during dynamic rupture propagation. The out-of-plane stress rapidly deviates from the free-surface condition and a state of equivalent plane-strain in the stress-changes field is attained at the rupture tip, while behind the rupture tip a fully three-dimensional stress state is established. The three-dimensional finite element simulations presented here help interpret and explain previous experiments of dynamic shear ruptures by showing the complex particle velocity and stress fields in the interior of the specimen and along the interface plane, which are currently not accessible to full-field experimental measurements.

Dynamic Analysis of a Deeply Buried Tunnel Influenced by a Newly-built Adjacent Cavity with a Special Emphasis on the Minimum Seismically Safe Tunnel Distance

Contemporary life streams, more often than ever, impose the necessity for construction of new underground structures in the vicinity of existing tunnels, with an aim to accommodate transportation systems and utility networks. A previously uninvestigated case, in which a newly-constructed tunnel opening is closely positioned behind an existing tunnel, referred to as the tunnel–cavity configuration, has been considered in this study. An exact analytical solution is derived considering a pair of parallel circular cylindrical structures of infinite length, with the horizontal alignment, embedded in a boundless homogeneous, isotropic, elastic medium and excited by time-harmonic plane SV-waves under the plane-strain conditions. The Helmholtz decomposition theorem, the wave functions expansion method, the translational addition theorem for bi-cylindrical coordinates, and the pertinent boundary conditions are jointly employed in order to develop a closed-form solution of the corresponding boundary value problem. The primary goal of the present study is to examine the increase in dynamic stresses at an existing tunnel structure due to the presence of a closely driven unlined cavity, as well as in a localized region around the tunnel (at the position of the cavity in close proximity), under incident SV-waves. A new quantity called dynamic stress alteration factor is introduced and the aspect of the minimum seismically safe distance between the two structures is particularly considered.

A novel centroid-enriched edge-based smoothed radial point interpolation method for upper bound limit analysis

This paper presents a novel centroid-enriched edge-based smoothed radial point interpolation method (CE-ESRPIM) to overcome the volumetric locking problem and increase the numerical accuracy for upper bound limit analysis. Based on the edge-based smoothed radial point interpolation method (ESRPIM), the centroid of the triangle element is used as a field node to discretize the problem domain. Simultaneously, the local support node selection scheme, which is established based on a modified centroid-enriched triangle mesh (CE-TScheme), is proposed to realize local radial point interpolation (RPIM). Using the Mohr–Coulomb yield criterion and associated flow rule, the upper bound theorem in the plane strain condition is discretized into a standard second-order cone programming (SOCP) problem through the CE-ESRPIM, which is solved using the primal–dual interior-point algorithm (PDIP). The proposed approach with the enriched centroid considered as the field node is straightforward and efficient and can directly inherit the formulation framework of the traditional ESRPIM. Moreover, this method has been proven to be effective in eliminating the volumetric locking problem under incompressible conditions.

Validity of small-scale yielding regime in notched-cracked geometries

This paper proposes a procedure based on crack tip opening displacement (CTOD) to define the boundary of the small-scale yielding (SSY) regime. Numerical modelling was applied to notched samples with different notch radii, under both plane stress and plane strain conditions, with and without crack flank contact. Results indicate that an increase in notch radius promotes the validity of SSY. The increase in crack length promotes a progressive decrease of SSY's validaty, particularly in notched specimens. Most of the situations studied fall outside the SSY regime, which reinforces the importance of verifying the applicability of LEFM to fatigue crack growth rate calculations.

Investigations of granular specimen size effect in interface shear box test using a micro‐polar continuum description

AbstractParametric studies were carried out to investigate the effects of specimen size and interface friction angle on the mechanical behaviour of a cohesionless granular material in the interface shear box test under constant normal load and plane strain conditions. For a description of the pressure and density dependent properties of granular materials such as sand, a micro‐polar hypoplastic continuum model is used. Numerical simulations were performed to obtain the evolution of the deformations and the shear resistance for three different specimen lengths and interface friction angles using the finite element method in the updated Lagrange frame. Due to the presence of lateral rigid boundaries of the shear box, the deformations and the stresses became significantly inhomogeneous within the granular specimen. Correspondingly, the mobilised interface shear resistance and the deformation field within the specimen were not uniform along the interface. In the case of higher interface friction angles, the evolution of the mechanical quantities along the interface can also be influenced by the specimen size. It is shown that the calculated average friction angle mobilised along a rough interface is influenced by the shear box length and it can be lower than the prescribed one. In addition, the influence of sidewall friction inside the shear box frame was explored in terms of the response of the specimen. The results of the micro‐polar hypoplastic model were also compared with those obtained from the corresponding non‐polar version of the model to clarify the role of polar effects within the granular body. The numerical results provide new insights into the microstructural effects of the granular material on the mobilisation of the interface friction angle and indicate the complexities and difficulties to precisely determine and interpret the interface friction angle, obtained from the classical interface shear box experiment.

Refractive changes of a new asymmetric intracorneal ring segment with variable thickness and base width: A 2D finite-element model.

PurposeTo evaluate the local geometric effects of a unilateral intrastromal ring segment with a combined variation of ring thickness and base width in a finite element simulation, and to compare it against the isolated effect of thickness or base width variation alone.MethodsA two-dimensional finite-element model of a transversely isotropic cornea was created assuming either axisymmetric stress or plane strain condition. The model geometry was composed of a three-layered corneal tissue (epithelium, anterior and posterior stroma) fixed at the limbus. The implantation of a triangular-shape asymmetric ring segment with varying ring thickness (150 to 300 μm) and base width (600 to 800 μm) was simulated. Also, changes induced by thickness or base width alone were studied and compared their combined effect in the asymmetric ring segment. Geometrical deformation of the simulated cornea and sagittal curvature were the main parameters of study.ResultsIncreasing ring thickness and base width along the arc of the asymmetric ring segment produced a more pronounced flattening in this part of the ring. The asymmetric design did find a good balance between maximizing corneal flattening at one end and minimizing it at the other end, compared to the isolated effect of ring thickness and width. Ring thickness was the most robust parameter in flattening both, the central and peripheral cornea.ConclusionThe finite-element model permitted a theoretical study of corneal deformation undergoing implantation of realistic and hypothetical ring geometries. Intracorneal asymmetric ring segments with varying thickness and base width can be a good alternative in corneas with asymmetric keratoconus phenotypes.

Characteristic parameters of soil failure criteria for plane strain conditions – experimental and semi-theoretical study

Abstract The paper concerns the characteristic parameters of the selected isotropic failure criteria, i.e. Mohr–Coulomb, Drucker–Prager, Matsuoka–Nakai and Lade–Duncan. The parameters are determined directly from the failure criteria and stress measurements or by semi-theoretical approach, assuming that the soil obeys the associated flow rule and using the plane strain condition. In the latter case, the parameters can be expressed as functions of the plane strain internal friction angle, which is determined from measurements. The principal stress tensor components, corresponding to the soil peak strength and necessary to obtain the failure criteria parameters, are measured in a series of true triaxial, plane strain tests, on coarse Skarpa sand samples of different initial relative density, subjected to various confining pressures.

SAE 4340 Fracture Toughness Evaluation Following Wang’s Formulation for Spiral Notch Torsion Test Specimens

Abstract The fracture toughness parameter (fracture toughness in plane strain-state [KIC], fracture toughness near the onset of crack growth, or crack tip open displacement) is essential information for understanding fracture behavior in mechanical and structural components. To determine this property, several methods have been developed; these require specimens with sufficient thickness to ensure a plane-strain condition at the crack front, which depending on materials mechanical properties, may lead to the need of large amounts of material. To overcome this problem, the spiral notch torsion test (SNTT) was proposed by Wang in 2000, and over these years, methodologies for testing many different materials have been presented, as well as the elastic compliance equations for assessing crack size (essential for introducing the fatigue precrack), also presented in 2012. Although the SNTT presented advantages and potentialities, its application is still quite restricted, and information on its use is limited. Implementation of the SNTT is a challenge, mainly because of the inexistence of a specific standard, thus producing ambiguities when dealing with the application of the test, methodology, and limitations because the SNTT implementation is principally based on previously published theoretical and experimental studies. The study presented herein aims at organizing already existing information based on work conducted by Wang while presenting in detail a test chart and the necessary techniques for obtaining a fracture toughness parameter, in mode I loading and in the domain of a plane strain-state. Determining KIC using the SNTT was performed on a quenched and tempered AISI 4340 steel (45 HRC) in air at room temperature. The obtained results proved the great potential of SNTT methodology for KIC determination. The KIC values obtained by the SNTT methodology exhibited a maximum error of 3.5 % and were statistically equal to those provided by the conventional method compact tension (C(T)) that follows ASTM E399-20, Standard Test Method for Linear-Elastic Plane-Strain Fracture Toughness KIC of Metallic Materials. The fracture surfaces observed in SNTT and C(T) specimens showed similar fractographies and mechanisms.

Hypoplastic Interface Model considering Plane Strain Condition and Surface Roughness

The soil-structure interface problem is an important part of soil-structure interaction research. These problems are mostly three-dimensional space problems, which is more complex to solve. In this paper, reduced stress and strain rate vectors are incorporated into the explicitly granular hypoplastic model by considering the plane strain state precisely. In addition, considering the important influence of roughness on the mechanical properties of contact surface, an improved hypoplastic model is established by incorporating the influence of roughness into the hypoplastic model, and the applicability of the new improved model is validated by comparing with the simulation results of the Mohr–Coulomb model, the explicitly granular hypoplastic models, and the experimental data. The results indicate that the improved model can be utilized to reflect the nonlinearity of the mechanical properties of the contact surface, which is in good agreement with the experimental data.

Groundwater effect on bearing capacity of shallow strip footings

A rigorous and practical solution aimed at accounting for the effect of groundwater in the assessment of bearing capacity of shallow strip footings in plane strain conditions has been developed using the method of characteristics.A steady water table in the foundation soil is considered and a bearing capacity factor Nγw to be used in the classical trinomial bearing capacity equation is introduced. The proposed bearing capacity factor depends on the angle of shear strength, on the footing roughness and on the water table depth and is also influenced by the soil unit weight.Normalizing Nγw with respect to the bearing capacity factor Nγ computed for the case of dry foundation soil, a corrective coefficient has been introduced for which best-fit empirical relationships have been proposed for both smooth and rough foundations. The size of the plastic mechanism under the footing has also been investigated and equations allowing to evaluate its relevant dimensions have been provided.The proposed solutions have been checked against results obtained via finite element analyses, finding a perfect agreement in terms of both corrective coefficients and shape and size of the soil plastic volume. Finally, the actual bearing capacity problem was solved showing that the trinomial formula, even in presence of a steady water table, still yields conservative values of the bearing capacity.

Further development of the gamma exponent model for assessment of flaws in oil and gas pipelines

The Gamma Exponent Model (“GEM”) is a recently developed methodology for failure pressure prediction of flaws in oil and gas pipelines. The model has previously demonstrated applicability for the assessment of axial surface breaking (or “part wall”) cracks and localized metal loss/corrosion. This work extends the applicability of the model to circumferential surface cracks. In addition, the GEM is shown to be consistent and complementary to the log-secant model for assessment of axial through wall cracks, developed by the NG18 Committee of the Battelle Institute in the early 1970s. The ongoing development of the GEM raises some important theoretical considerations related to understanding of the stress state at the flaw tip during the failure process. The current industry models are not explicit regarding the plane stress or plane strain conditions at the flaw tip, though this has a significant effect on model development. Analysts often assume plane stress conditions, due to oil and gas pipelines typically being considered “thin wall” pipe. However, this work confirms the theory that surface cracks are subject to plane strain conditions, and the selection of fracture toughness correlations from impact energy data becomes very important. The model is validated against available laboratory, hydrotest and in-service failure data. Statistical analyses demonstrate the mathematical form of the new model is valid. Failure pressure predictions are shown to be improved over current industry models. This can have a significant effect on the reliability pipeline integrity programs and improved efficiency in the form of cost savings.

Frictionless contact mechanics of an orthotropic coating/isotropic substrate system

This study has been performed to investigate the receding contact problem of a homogeneous orthotropic coating that is not bonded to a homogeneous isotropic substrate without any interfacial defects. The isotropic substrate is supported on a Winkler foundation. The problem is solved assuming that the contact between the rigid punch and orthotropic coating, and that between the orthotropic coating and isotropic substrate, are frictionless. Additionally, the effect of the body forces is neglected, and only compressive normal tractions can be transmitted through the interfaces. The contact analysis of the orthotropic coating, which is subjected to a contact load using a rigid cylindrical punch, is performed under plane strain conditions. The governing equations are analytically found using the theory of elasticity and Fourier integral transformation techniques. Subsequently, the governing equations are reduced to a system of two singular equations, wherein the unknowns are the contact stresses and contact widths. To numerically solve the resulting singular integral equations, Gauss-Chebyshev integration formulas are employed. It is analyzed the influence of the following parameters on the contact stresses and contact widths: orthotropicmaterial properties, punch radius, load ratio, Winkler foundation stiffness.

Phase-Field Modeling Fracture in Anisotropic Materials

The phase-field method is a widely used technique to simulate crack initiation, propagation, and coalescence without the need to trace the fracture surface. In the phase-field theory, the energy to create a fracture surface per unit area is equal to the critical energy release rate. Therefore, the precise definition of the crack-driving part is the key to simulate crack propagation. In this work, we propose a modified phase-field model to capture the complex crack propagation, in which the elastic strain energy is decomposed into volumetric-deviatoric energy parts. Because of the volumetric-deviatoric energy split, we introduce a novel form of the crack-driving energy to simulate mixed-mode fracture. Furthermore, a new degradation function is proposed to simulate crack processes in brittle materials with different degradation rates. The proposed model is implemented by a staggered algorithm and to validate the performance of the phase-field modelling, and several numerical examples are constructed under plane strain condition. All the presented examples demonstrate the capability of the proposed approach in solving problems of brittle fracture propagation.