Stress Discontinuities Research Articles

New fatigue life design approach for metal sheets with discontinuities

The existing references of Wöhler data curves (S-Nf fatigue life) for fully reversed cycling loading fail to incorporate the discontinuities effect on metal sheets fatigue life. Experimental results show that the discontinuities, i.e., holes and notches weaken metal sheets and reduce fatigue life. The notch fatigue strength reduction factor (Kf) cannot incorporate the discontinuity effect into the fatigue life prediction models. This paper investigates a new method to study the discontinuity effect on metals fatigue life by applying the Discontinuity Stress Triaxiality (DST) factor to fatigue life prediction models. Three discontinuity geometries were studied in order to investigate the different effect of different radius. A simulation using ABAQUS FEA was used to export the stress triaxiality at the notch root element. The results show that this new method can help in predicting fatigue life for high-strength metals and metals with sharp notches, unlike the Kf factor. Also, the theoretical results show good correlation with the experimental results.

Swelling response of functionally graded temperature-sensitive hydrogel valves: Analytic solution and finite element method

Temperature-sensitive hydrogels are classified as high stretchable materials undertaking large recoverable deformation under thermomechanical loadings. Due to vast applications of these materials in microfluidic valves and stress/deformation discontinuity observed in multi-layered structures, understanding of thermomechanical behavior of functionally graded hydrogels is of great practical interest. In this article, the swelling of these materials is analyzed considering various types of micro-valves including solid cylinder, hollow cylinder, and Jacket-pillar cylinder. Accordingly, analytical solutions are developed for swelling behavior of diverse types of cylinders composed of functionally graded temperature-sensitive hydrogel. The distribution of the cross-link density alters along the radial direction in exponential and linear trends. Besides, finite element analysis is presented to evaluate the proposed analytical solution in different case studies. Then, to account for more realistic environmental conditions, heat transfer equation is also solved along with the mechanical equilibrium equation to propose a novel analytical solution validated by finite element method. Clarifying the micro-valves mechanism and illustrating the capability and accuracy of the analytical method, the effects of material and environmental variables are studied in detail; continuous stress and deformation fields are observed opposed to the multi-layered structures which normally involve stress discontinuity.

Estimation of Errors in Stress Distributions Computed in Finite Element Simulations of Polycrystals

The accuracy of the stresses predicted from crystal plasticity-based finite element formulation depends on estimation and control of the errors associated with the discretization. In the current work, the errors in the stress distribution are estimated in virtual polycrystalline samples of α-phase titanium (hexagonal close-packed phase of Ti–6Al–4V). To estimate the error, the stress field, which does not possess inter-element continuity, is smoothed over a grain using an $$L_2$$ projection, thereby providing continuous stress distributions with inter-element continuity. The differences between the continuous (smooth) and discontinuous (raw) stress fields are calculated at individual Gauss quadrature points and used to estimate errors for corresponding elements and grains. Error estimations are performed for a Voronoi-tessellated microstructure, an equiaxed microstructure, and two microstructures with varying grain sizes for tensile loading extending into the fully plastic regime ($$\approx $$ 5% extension). Magnitudes of the errors are found to depend on microstructural characteristics, particularly the shape and size of grains. Samples having variations in grain size or having less spherical grains exhibited higher errors than samples with uniformly sized, equiaxed grains, with the size variations having a more pronounced effect. Errors correlate with proximity to grain boundaries at small (elastic) strains and with deformation-induced features (deformation bands) at large strains.

Validation of an improved contact method for multi-material Eulerian hydrocodes in three-dimensions

Realistic and accurate modeling of contact for problems involving large deformations and severe distortions presents a host of computational challenges. Due to their natural description of surfaces, Lagrangian finite element methods are traditionally used for problems involving sliding contact. However, problems such as those involving ballistic penetrations, blast-structure interactions, and vehicular crash dynamics, can result in elements developing large aspect ratios, twisting, or even inverting. For this reason, Eulerian, and by extension Arbitrary Lagrangian-Eulerian (ALE), methods have become popular. However, additional complexities arise when these methods permit multiple materials to occupy a single finite element.Multi-material Eulerian formulations in computational structural mechanics are traditionally approached using mixed-element thermodynamic and constitutive models. These traditional approaches treat discontinuous pressure and stress fields that exist in elements with material interfaces by using a single approximated pressure and stress field. However, this approximation often has little basis in the physics taking place at the contact boundary and can easily lead to unphysical behavior. This work presents a significant departure from traditional Eulerian contact models by solving the conservation equations separately for each material within each computational element and then imposing inequality constraints associated with contact to the solutions for each material with the appropriate tractions included. The advantages of this method have been demonstrated with several computational examples. These examples are limited to contact between two materials. Future work will examine contact between three or more materials within an element. This work concludes by drawing a comparison between the method put forth in this work and traditional treatment of multi-material contact in Eulerian methods.

Experimental and Numerical Investigations on the Thermomechanical Behavior of 304 Stainless Steel/Q345R Composite Plate Weld Joint.

In this study, the welding process of 304 stainless steel/Q345R low alloy steel composite plate is modeled by experimental and finite element methods to study the complex thermomechanical behavior. The residual stress and microstructure evolution of composite plate in the welding process are also investigated. The welding thermal cycle curve and residual stress distribution at the joint are obtained by using thermocouple and blind-hole methods. Optical microscopy, scanning electron microscopy, and energy-dispersive X-ray spectroscopy were used to investigate the evolution of microstructure, morphology, and element diffusion of the joint. The results show that the maximum von Mises welding residual stress is 312 MPa, which is located in the bottom of the start point of the weld zone. The residual stress gradually decreases and tends to be stable along the direction from the weld to the base metal. In addition, a residual stress discontinuity is found at the interface between the bimetal. It is also found that the closer it is to the weld joint, the more uniform is the austenite distribution and the smaller are the grain sizes.

Optimization of the stress discontinuity value at the interface of a cylindrical stainless steel substrate and electroless Ni-P coating

Electroless coating and electrolytic coating can be easily differentiated by the use of electricity in the respective processes. The former does not require any electricity for its operation. The advantages include uniform deposition of coating material on the substrate. Besides, nonconductive materials can also be coated by this process. The most important and challenging work in the coating industry nowadays is to apply the electroless plating process on different type of material at a wide range of static and dynamic loading conditions. This literature mainly focuses on the stress distribution along a radial path at the coating and substrate interface of a cylindrical pressure vessel coated with the basic Ni-P coating at static loading condition. There is a total of nine models prepared in Ansys and the Von Mises stress distribution is obtained consecutively. For obtaining the stress distribution, a separate path is created radially and the stress values are plotted for each model separately. In each case, the cylinder thickness is kept constant (2 mm) but the coating thickness and the internal pressure are varied. In each case, it is observed that there is a sharp discontinuity in stress value at the coating and substrate interface but at the same time, the discontinuity is very much dependent on the internal pressure and coating thickness. Optimization has been done on the response of this stress discontinuity with the applied pressure and the coating thickness as the input parameters. The coating thickness of 45 μm and pressure 50 kPa shows the least stress discontinuity as well as the most optimized result.

A mesh-free method for interface problems using the deep learning approach

In this paper, we propose a mesh-free method to solve interface problems using the deep learning approach. Two types of PDEs are considered. The first one is an elliptic PDE with a discontinuous and high-contrast coefficient. While the second one is a linear elasticity equation with discontinuous stress tensor. In both cases, we represent the solutions of the PDEs using the deep neural networks (DNNs) and formulate the PDEs into variational problems, which can be solved via the deep learning approach. To deal with inhomogeneous boundary conditions, we use a shallow neural network to approximate the boundary conditions. Instead of using an adaptive mesh refinement method or specially designed basis functions or numerical schemes to compute the PDE solutions, the proposed method has the advantages that it is easy to implement and is mesh-free. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method for interface problems.

Homogenization and Localization of Ratcheting Behavior of Composite Materials and Structures with the Thermal Residual Stress Effect.

In this contribution, the ratcheting behavior and local field distribution of unidirectional metal matrix composites are investigated under cyclic loading. To that end, we extended the finite-volume direct averaging micromechanics (FVDAM) theory by incorporating the rule of nonlinear kinematic hardening. The proposed method enables efficient and accurate simulation of the ratcheting behavior of unidirectional composites. The local satisfaction of equilibrium equations of the FVDAM theory facilitates quick and rapid convergence during the plastic iterations. To verify the proposed theory, a finite-element (FE) based unit cell model is constructed with the same mesh discretization. The remarkable correlation of the transverse response and local field distribution generated by the FVDAM and FE techniques demonstrates the effectiveness and accuracy of the proposed models. The stress discontinuities along the fiber/matrix interface that are generic to the finite-element theory are absent in the FVDAM prediction. The effects of thermal residual stresses induced during the consolidation process, as well as fiber orientations, are revealed. The generated results indicate that the FVDAM is well suited for simulating the elastic-plastic ratcheting behavior of metal matrix composites, which will provide the conventional finite-element based technique with an attractive alternative.

Imperfection Sensitivity of Stiffened Cylindrical Shells Using a Mixed‐Hybrid Finite Element Formulation

AbstractThe imperfection sensitivity of stiffened cylinders is analysed on the basis of a critical perturbation associated with a minimum of strain energy. To identify worst perturbations including displacements and stresses, the perturbation energy concept is used, which is developed for mixed finite elements. Disadvantages in modelling branched structures due to stress discontinuities are avoided by transferring the concept to a mixed‐hybrid formulation. Beside an evaluation of common design recommendations, the identification of non‐initial perturbations lead to an insight into the quite complex buckling behaviour of stiffened shells.

An immersed interface method for discrete surfaces

Fluid-structure systems occur in a range of scientific and engineering applications. The immersed boundary (IB) method is a widely recognized and effective modeling paradigm for simulating fluid-structure interaction (FSI) in such systems, but a difficulty of the IB formulation of these problems is that the pressure and viscous stress are generally discontinuous at fluid-structure interfaces. The conventional IB method regularizes these discontinuities, which typically yields low-order accuracy at these interfaces. The immersed interface method (IIM) is an IB-like approach to FSI that sharply imposes stress jump conditions, enabling higher-order accuracy, but prior applications of the IIM have been largely restricted to numerical methods that rely on smooth representations of the interface geometry. This paper introduces an immersed interface formulation that uses only a C0 representation of the immersed interface, such as those provided by standard nodal Lagrangian finite element methods. Verification examples for models with prescribed interface motion demonstrate that the method sharply resolves stress discontinuities along immersed boundaries while avoiding the need for analytic information about the interface geometry. Our results also demonstrate that only the lowest-order jump conditions for the pressure and velocity gradient are required to realize global second-order accuracy. Specifically, we demonstrate second-order global convergence rates along with nearly second-order local convergence in the Eulerian velocity field, and between first- and second-order global convergence rates along with approximately first-order local convergence for the Eulerian pressure field. We also demonstrate approximately second-order local convergence in the interfacial displacement and velocity along with first-order local convergence in the fluid traction along the interface. As a demonstration of the method's ability to tackle more complex geometries, the present approach is also used to simulate flow in a patient-averaged anatomical model of the inferior vena cava, which is the large vein that carries deoxygenated blood from the lower and middle body back to the heart. Comparisons of the general hemodynamics and wall shear stress obtained by the present IIM and a body-fitted discretization approach show that the present method yields results that are in good agreement with those obtained by the body-fitted approach.

Thermoelastic fracture modelling in 2D by an adaptive cracking particle method without enrichment functions

Thermoelastic fracture of brittle materials is a key concern for many types of engineering structures such as aerospace components and pressure vessels. These problems present difficulties in modelling as they involve coupling of the effects of thermal and mechanical loadings, and usually material behaviours are strongly discontinuous at crack locations, and highly nonlinear during crack propagation. Most current numerical methods applied to these problems use enrichment functions to model discontinuities of temperature and stresses at cracks, however, these enrichments bring extra unknowns into the model and can lead to numerical difficulties in attempting solutions. In this paper, an adaptive cracking particle method is developed for thermoelastic fracture. The method is meshless, so crack discontinuities can be introduced by modifying the influence domains of particles, rather than through external enrichments. Another benefit is an easy implementation of h-adaptivity, since no transition is required for different densities of particles, and it is easy to build dense groups of particles around crack tips. The results demonstrate that this new method can provide the same level of accuracy as others that employ enrichment functions, but using fewer degrees of freedom. A number of examples show the flexibility of the new method to model a range of thermoelastic problems in 2D including multiple cracks and crack propagation.

Block coordinate descent energy minimization for dynamic cohesive fracture

A block coordinate descent algorithm within a discontinuous Galerkin finite element formulation is proposed for both explicit and implicit energy minimization in solid bodies in which rigid-cohesive fractures initiate and propagate. The crack opening dependent Barenblatt type surface cohesive energy function is non-differentiable at the origin. In each iteration or each explicit update, the algorithm minimizes the potential with respect to each crack opening displacement unknown and with respect to the block of deformation unknowns, sequentially. This decomposes minimization of the full non-differentiable problem into a number of “small” non-differentiable sub-problems that must be solved locally at the inter-element boundaries of the finite element mesh and an “easy” differentiable sub-problem that characterizes global equilibrium. As a result, non-differentiability is effectively treated locally and the algorithm can be easily incorporated into standard finite element codes. On the basis of a convexity analysis of the proposed non-differentiable energy functional, we obtain a minimum cohesive process zone resolution criterion, known empirically in the previous literature as a requirement for capturing the amount of dissipated fracture energy correctly. The method is free of any regularization parameters and preserves the discrete nature of fracture without introducing a stress discontinuity at initiation of decohesion. Robustness of the method is shown through several numerical examples of fragmentation and branching and through comparisons with existing numerical and experimental results.

Surface Effect on Static Bending of Functionally Graded Porous Nanobeams Based on Reddy’s Beam Theory

In this paper, the surface effect on the static bending behavior of functionally graded porous (FGP) nanobeams subjected to a concentrated transverse load is studied by using Reddy’s higher-order beam theory. Three types of porosity distributions are considered for the nanobeam, i.e. uniform porosity distribution, symmetric and asymmetric non-uniform porosity distributions. With the consideration of the surface effect, the nanobeams can be abstracted as a composite beam composed of a surface layer and a bulk volume. According to the generalized Young–Laplace equation, the normal stress discontinuity across a surface due to the effect of surface stress is taken into consideration. The analytical solutions of the static bending problem of FGP nanobeams are obtained for the beams with hinged-hinged, clamped-clamped and clamped-free boundary conditions. The effects of the residual surface stress, porosity distribution type, porosity coefficient and length-to-thickness ratio on the transverse displacement of the FGP beams are discussed.

A mixed DOF collocation method for elastic problems in heterogeneous structure

SummaryIn this paper, a collocation method with mixed degrees of freedom (DOFs) is proposed for heterogeneous structures. Local tractions of the outer and interface boundaries are introduced as DOFs in the mixed collocation scheme. Then, the equilibrium equations of all the nodes and the outer boundary conditions are discretized and assembled into the global stiffness matrix. A local force equilibrium equation for modeling the stress discontinuity through the interface is developed and added into the global stiffness matrix as well. With those contributions, a statically determined stiffness matrix is obtained. Numerical examples show that the present method is superior to the classical mixed collocation method in the heterogeneous structure because it improves the accuracy and the convergence and remains the efficiency. Besides, almost constant convergence rates of displacements and stresses are found in all the examples, even for three‐dimensional problems.

Hybrid SAFE-GMM approach for predictive modeling of guided wave propagation in layered media

Dispersion curve, and displacement modeshapes of multilayer structures can be obtained using Semi-Analytical Finite Element (SAFE) method. Stress mode shapes calculated from SAFE were discontinuous at the interface, because SAFE method formulation does not apply stress continuity at the interface. A case study of 1 mm aluminum-1 mm steel double-layer plate is presented, showing the SAFE stress predictions at the interface. The results observed were discontinuity of out-of-plane stress modeshapes at the interface. Mesh refinement was performed at the interface to study the convergence of stress mode shapes at the interface, yet the stress discontinuity problem was not solved. A fine mesh at the interface was created by using variable mesh techniques. This also did not provide continuous stress mode shape at the interface. Therefore, in this paper, a novel hybrid SAFE-GMM (Global Matrix Method) approach was used to obtain stress mode shapes accurately and efficiently. GMM develops the displacement and stress equations for individual layers in a multilayered structure and assembles a global matrix by applying the boundary and interface continuity conditions. A case of practical importance, CFRP strengthened concrete structure was analyzed using the SAFE-GMM approach. The drawback of the SAFE technique to this case is also presented, and it shows that the hybrid SAFE-GMM approach gave the interface stresses accurately.

Damped interfacial oscillation of a particle-embedded viscous drop

This article describes how a drop with an embedded particle exhibits interfacial waves with transient decay due to the interplay between capillary and viscous effects. To reveal the damped oscillation of the system properly, the deformation and pressure fields inside the domain are described in terms of complete sets of basis functions. Such representation leads to a matrix formulation which enforces no-slip condition at the solid-liquid interface and ensures correct discontinuity in normal stress due to surface tension at the drop periphery. The resulting characteristic equation involving the natural frequencies and the decay constants is solved numerically to determine these quantities. The matrix expression implies a block-diagonalized structure with two uncoupled blocks corresponding to two distinctly different dynamics. The first of these is related to pure rotational velocities on spherical surfaces which monotonically decay in time without any fluctuation in the absence of any peripheral deformation. By contrast, the second block is associated with the undulation in shape. Due to the restoring features of surface tension, the latter can exhibit underdamped oscillatory modes, if the capillary number Ca is below a critical value. However, even these waves would become overdamped if the critical number is exceeded. These values of Ca for a few most relevant modes are plotted in this paper as functions of particle-to-drop size ratio. Also, the natural frequencies for the underdamped cases as well as the damping constants for all considered modes are presented for different size ratios and capillary numbers. The findings are verified by matching the computed results to a novel boundary layer theory under low capillary number limit. Under the limiting condition, both sets of independent calculations yield the same decay constants and natural frequencies providing mutual validations.

Mathematical modeling of the propagation of precursors of the front edges of cracks as spatial curves on the fronts of waves of a strong discontinuity of rates and stresses

Crack propagation occurs at the macro level with finite elastic displacements, deformations and stresses far from the front edge of the crack. In the neighborhood of the front spatial edge of a crack during its development, the behavior of the material is really inelastic, when the changes take place at the micro and nano levels.In this paper with the use of the Bingham model of an elasto viscoplastic (EVP) material, a small δ-neighborhood is chosen around the front spatial edge of a crack. The neighborhood has the shape of a cylinder with a curvilinear axis. It is shown that the wave fronts of plastic deformation are waves of longitudinal or transverse deformation and bear the front edges of the precursors these are longitudinal shear cracks and transverse shear cracks or detachment cracks.Ordinary differential equations of transfer of the intensity of tangential shear stresses and normal detachment stresses behind the precursor of the crack edge depending on the path traveled by the front edge of the crack are constructed. Exact solutions are obtained in the case of a non-stressed material. The finiteness of the length of a growing crack in an EVP material has made it possible to clarify the classical criteria for brittle fracture.

Effect of interface mechanical discontinuities on scaffold-cartilage integration.

A consistent lack of lateral integration between scaffolds and adjacent articular cartilage has been exhibited in vitro and in vivo. Given the mismatch in mechanical properties between scaffolds and articular cartilage, the mechanical discontinuity that occurs at the interface has been implicated as a key factor, but remains inadequately studied. Our objective was to investigate how the mechanical environment within a mechanically loaded scaffold-cartilage construct might affect integration. We hypothesized that the magnitude of the mechanical discontinuity at the scaffold-cartilage interface would be related to decreased integration. To test this hypothesis, chondrocyte seeded scaffolds were embedded into cartilage explants, pre-cultured for 14 days, and then mechanically loaded for 28 days at either 1N or 6N of applied load. Constructs were kept either peripherally confined or unconfined throughout the duration of the experiment. Stress, strain, fluid flow, and relative displacements at the cartilage-scaffold interface and within the scaffold were quantified using biphasic, inhomogeneous finite element models (bFEMs). The bFEMs indicated compressive and shear stress discontinuities occurred at the scaffold-cartilage interface for the confined and unconfined groups. The mechanical strength of the scaffold-cartilage interface and scaffold GAG content were higher in the radially confined 1N loaded groups. Multivariate regression analyses identified the strength of the interface prior to the commencement of loading and fluid flow within the scaffold as the main factors associated with scaffold-cartilage integration. Our study suggests a minimum level of scaffold-cartilage integration is needed prior to the commencement of loading, although the exact threshold has yet to be identified. © 2019 Orthopaedic Research Society. Published by Wiley Periodicals, Inc. J Orthop Res.

Explicit multistep time integration for discontinuous elastic stress wave propagation in heterogeneous solids

SummaryA multistep explicit time integration algorithm is presented for tracking the propagation of discontinuous stress waves in heterogeneous solids whose subdomain‐to‐subdomain critical time step ratios range from tens to thousands. The present multistep algorithm offers efficient and accurate computations for tracking discontinuous waves propagating through such heterogeneous solids. The present algorithm, first, employs the partitioned formulation for representing each subdomain, whose interface compatibility is enforced via the method of the localized Lagrange multipliers. Second, for each subdomain, the governing equations of motion are decomposed into the extensional and shear components so that tracking of waves of different propagation speeds is treated with different critical step sizes to significantly reduce the computational dispersion errors. Stability and accuracy analysis of the present multistep time integration is performed with one‐dimensional heterogeneous bar. Analyses of the present algorithm are also demonstrated as applied to the stress wave propagation in one‐dimensional heterogeneous bar and in heterogeneous plain strain problems.

Mechanical Characterization and Modeling of Direct Metal Laser Sintered Stainless Steel GP1

The additive manufacturing (AM) process is unique in that it can facilitate anisotropy because of the layer-by-layer deposition technique intrinsic to this process. In order to develop a component for a desired application, it is necessary to understand the mechanics that facilitate this material behavior. This study investigates how build orientation affects the mechanical performance of as-built direct metal laser sintered (DMLS) stainless steel (SS) GP1 (also referred to as 17-4PH) through strain-controlled monotonic tension and completely reversed low-cycle fatigue (LCF) testing. The anisotropic behavior of DMLS SS GP1 is assessed for samples built along the horizontal plane. Fracture surfaces were found to exhibit ductile responses that were consistent with the σ–ε curves. Constitutive models (i.e., Ramberg–Osgood, Hahn) based upon linear elasticity and nonlinear plasticity are presented and used to simulate the monotonic discontinuous stress–strain yielding response of this material, which are found to be in agreement with the experimental data. A collection of low-cycle fatigue tests reveals initial strain hardening to stabilization, followed by softening to fracture. Tensile and fatigue material constants determined from experimental findings are also presented in this study. Plasticity effects on the life of varying build orientations are explored.