In-plane Stress Field Research Articles

A semi-analytical solution for the in-plane stress fields in isotropic convex finite solids with circular holes reinforced with cylindrically orthotropic rings

In this paper a semi-analytical solution for the stress distribution in a finite convex plate with a reinforced hole and subjected to a general loading condition is presented. The method described accounts for the geometrical parameters of the plate, as well as material properties and loading conditions, which can be applied internally to the hole boundary and externally to the outside boundaries of the plate. An extensive validation of the proposed solution is carried out comparing the theoretical predictions with results from various finite element analyses, showing an excellent agreement.

A Double Coated Circular Inhomogeneity Neutral to an Arbitrary Uniform in-Plane Stress Field

A Double Coated Circular Inhomogeneity Neutral to an Arbitrary Uniform in-Plane Stress Field

Exact in-plane stress field solution for isotropic plates with circular holes reinforced with cylindrically orthotropic rings

In this paper an analytical solution for the stress distribution in an infinite plate with a concentrically reinforced hole under a general loading condition is derived. The developed solution explicitly accounts for the elastic properties of the plate and of the reinforcing annulus, for the hole radius to reinforcing ring thickness ratio, as well as the loading conditions, including loads applied onto the hole boundary and on the plate, far away from the hole. A careful validation of the proposed framework is carried out by comparing the newly developed solution with the results from a number of finite element analyses, documenting a very satisfactory agreement.The solution derived represents a useful tool toward the understanding of the stress fields in polymeric components with reinforced holes and made with fused deposition modelling technologies.

Observation of in-plane shear stress fields in off-axis SiC wafers by birefringence imaging.

For the nondestructive characterization of SiC wafers for power device application, birefringence imaging is one of the promising methods. In the present study, it is demonstrated that birefringence image contrast variation in off-axis SiC wafers corresponds to the in-plane shear stress under conditions slightly deviating from crossed Nicols according to both theoretical consideration and experimental observation. The current results indicate that the characterization of defects in SiC wafers is possible to achieve by birefringence imaging.

On the prediction of the stress field in adhesive joints using a combined analytical-numerical method

In this paper, the stress field in adhesively bonded joints is predicted by employing an analytical approach together with the finite element method (FEM). To accomplish this, the adhesive joint is considered as a bi-material notch. Then, a unified analytical formulation for the in-plane stress and displacement field of bi-material notches is presented in the form of asymptotic series. Subsequently, a numerical procedure based on an over-determined system of equations is used to obtain the coefficients of the series solution for various types of adhesive joints including single lap joint (SLJ), single fillet lap joint (SFLJ), butt joint (BJ), scarf joint (SJ), and single lap joint with an inside taper and adhesive fillet (SLJ-ITAF). By comparing the results obtained from the analytical stress field with the numerical results, the accuracy of the calculated coefficients is evaluated. It is shown that by using the proposed approach one can obtain the stress field within the adhesive layer by using the stress or displacement fields of the adherend. The obtained results also show that neglecting the higher-order terms can result in significant errors while considering the first three non-singular stress terms besides the singular term reveals very good results.

New refined higher-order shear deformation theories for functionally graded plates conforming to graded variations of material properties

This paper proposes new refined higher-order shear deformation theories (RHSDTs) for functionally graded (FG) plates conforming to graded variations of material properties. The theoretical formulation starts with initial displacement descriptions based on general higher-order theories, determines in-plane stress fields using geometrical and constitutive relations, and further achieves transverse shear stress fields though equilibrium equations. By imposing tangential stress-free conditions and defining new variables, the transverse shear stress expressions without differential variables are obtained. Then, the uniform and quadratic distributions of the transverse displacement field are considered, respectively, and the final displacement expressions of the present 2D and quasi-3D RHSDTs are further formulated by integrating the associated shear strain expressions. This novel formulation approach first introduces gradient parameters of FG materials into the shear strain shape functions. Compared with traditional HSDTs, the present refined theories fully incorporate the influences of the graded variations of material properties and can better predict mechanical responses for FG plates. Moreover, the governing equations and associated analytical solutions are presented. Numerical examples demonstrate high accuracy of the present RHSDTs and reflect their obvious accuracy advantages over traditional HSDTs especially when predicting the transverse shear stresses for FG plates with arbitrary gradients. Furthermore, it is also validated that the present RHSDTs with more shear expansion terms have higher accuracy for the prediction of transverse shear stress fields.

Buckling analysis of symmetrically laminated composite plates including the effect of variable pre-stress field using the Ritz method

A buckling solution capable of accounting for prebuckling stress field heterogeneities is proposed. Oftentimes the same structural elements that restrict the out-of-plane displacement of the edges of a plate may also constrain their in-plane movements and cause heterogeneous prebuckling stress fields. The effect of these constraints and the consequent prebuckling stress field heterogeneity is vastly overlooked in the context of approximate analytical buckling solutions. The novelty of this work lies in basing the eigenvalue buckling analysis on a calculated prebuckling in-plane stress field. The in-plane stress field is determined by conducting a displacement-based linear elasticity analysis. In both elasticity and eigenvalue buckling analyses, the deformations of the anisotropic laminate were approximated using a first-order shear deformation theory. The equilibrium and instability equations were derived using the principle of stationary total potential energy and by employing the Ritz method. The derived formulation along with arbitrarily selected sets of boundary conditions was incorporated into a Matlab program and used for demonstrating the effect of in-plane constraints on prebuckling and buckling behavior of plates. Through a parametric study, the effect of the geometric configuration of plates on their buckling strength for selected symmetrical laminates was investigated.

A novel computational framework for the analysis of bistable composite beam structures

Over the last few years, several studies have shown the potential of multi-stability in engineering applications such as shape-adaptive structures and energy harvesting devices. Due to their highly non-linear mechanics, multi-stable structures are inherently more complex to model than conventional engineering structures. Especially when composite materials are involved, two-dimensional or three-dimensional finite elements solutions are often required in order to have a thorough understanding of the stress field evolution within the structure throughout the snap-through phenomenon. Nevertheless, such modelling strategies yield time-consuming design and optimisation procedures. In order to exploit multi-stability in industrial applications, an enhancement of the existing modelling capabilities is needed. Starting from the state-of-the-art methodologies available in the literature, a novel computational framework is proposed in this study for the analysis of bistable composite beam structures. An advanced structural modelling approach based on Carrera’s Unified Formulation (CUF) is adopted. By harnessing the potential of CUF, a full in-plane stress field (including axial, transverse shear and transverse normal stress) can be accurately yet efficiently predicted with no need to resort to two-dimensional or three-dimensional FEM solutions. Furthermore, the equilibrium paths of slender as well as thicker bistable beams are investigated, showing that refined predictions of the stable geometries and snap-through loads in bistability analyses can be obtained via the proposed advanced one-dimensional formulation. Validation of the proposed modelling approach towards reference solutions and commercial software finite elements is provided.

Boundary shape design of planar membrane structures based on an airy stress model

Maintaining surface accuracy is crucial to realizing the required performance of membrane structure. However, compressive stress inevitably leads to wrinkling, and the accurate analysis of the stresses becomes significant. The present study analyzed membrane structures with discrete loads. First, a simple method was proposed and the stress field was accurately described using an Airy stress model. The first and second principal stresses were obtained, and regions of wrinkling with negative stress were predicted. Second, for a square membrane structure, vertical tensions around the boundary were applied such that in-plane stresses were more evenly distributed. Finally, to reduce the wrinkling areas more effectively, an arc-edge structure was designed by removing some areas of low-stress. Simulation results showed that the in-plane stress field distribution was described accurately and the wrinkling regions were diminished effectively with the proposed method.

Prebuckling and buckling analysis of moderately thick variable angle tow composite plates considering the extension-shear coupling

In this paper, an analytical model based on a generalized Rayleigh-Ritz method is developed to deal with both the prebuckling and buckling problems of moderately thick VAT composite plates under a more general in-plane boundary condition. The non-uniform in-plane stress fields of VAT composite plates under various in-plane boundary conditions is firstly determined based on the minimum complementary energy principle combined with the Airy’s stress function. Afterwards, the governing equation of buckling problem of VAT composite plates under non-uniform in-plane stress fields is derived based on third-order shear deformation theory of Reddy’s type. The innovation of this paper lies in that a linear fitting method combined with Chebyshev-Gauss-Labotto point distribution is introduced to retrieve the boundary stress coefficients from the stress constraint conditions generated by the Lagrangian multiplier method, and meanwhile an integral expression of work done by the unknown boundary force along the prescribed in-plane displacement is adopted to satisfy the displacement constraint condition along the boundary line. In so doing, three kinds of in-plane boundary conditions are taken into account, that is, pure stress boundary condition, pure displacement boundary condition and mixed stress and displacement boundary condition. In addition, no assumption is made on the extension-shear coupling terms and thus the proposed analytical model is suitable to the unbalanced layup. The accuracy and robustness of the proposed analytical model is validated by finite element analysis and with prior results. Effects of the in-plane boundary condition, extension-shear coupling and varying fibre orientation angles on both in-plane stress and buckling responses of VAT composite plates are studied by numerical examples. It is shown that for the balanced panels under uniform compression, the benign load redistribution mechanism offered by the VAT layups remain in effect even if there exists the non-uniform shear constraint along the loading edges. However, the buckling load of the panel can be reduced by the shear constraint along the loading edges to a certain degree. Furthermore, it is found that for some displacement-controlled shear loading cases, the extension-shear coupling plays a positive role in the load redistribution, which results in a sight improvement in the longitudinal load of the panel when critical buckling occurs.

Layerwise Theories of Laminated Composite Structures and Their Applications: A Review

It is very challenging to accurately analyze the displacement and stress fields in the laminated composite structures due to the transverse anisotropy and higher transverse flexibilities. The equivalent single-layer theory (EST) is first developed to simplify this complex 3D problem to a pure 2D problem based on a displacement assumption in the thickness direction. The EST gives good results for the global responses of the very thin laminates with minimum computation cost, but poor results for the local responses at the interfaces and can not presents the zig-zag distribution of in-plane displacements. The elasticity solutions based on the 3D displacement-based finite element method (FEM) can present the accurate displacement and stress fields, but requires huge computational cost. As a quasi 3D method, the layerwise theories (LWT) is more accurate than most of the ESTs, and its computational cost is less than that of the 3D-based displacement FEMs, so it is attracting more and more attention with the rapid development of computer technology. This paper presents an overall review for the LWTs of the laminated composite structures and their applications. In general, there are two basic schemes employed to establish a LWT according to the construction of the displacements fields in the thickness direction. In the first scheme, the laminated composite structures are described as an assembly of individual layers, and these individual layers are combined by the interlaminar relationships to keep the displacement continuity and stress equilibrium. All of the individual layers are simulated by using the EST. In the second scheme, the displacement and/or stress fields along the thickness direction are constructed by a 1D interpolation functions and the in-plane displacement and/or stress fields are discretized by the 2D finite elements. The review in this paper is organized by the this classification.

Micro-Raman and X-ray diffraction stress analysis of residual stresses in fatigue loaded leached polycrystalline diamond discs

Polycrystalline diamond (PCD) cylindrical tool-bits used in oil well drilling are susceptible to fracture due to the hostile environment of randomly occurring high impact loads. These tool-bits generally comprise of a PCD layer sintered onto a Co-cemented tungsten carbide substrate. The cobalt metallic phase primarily aids the formation of the diamond to diamond bonds, however during application the same cobalt expands much quicker than the diamond, breaking the very same bonds it helped to form in the first place, leading to premature failure of the tool bits. As the PCD is virtually a two-phase material comprised of cobalt and diamond, substantial volumes of the metallic phase can be removed through a leaching process without compromising the cohesiveness of the diamond matrix, with reported improved wear resistance and thermal stability. X-ray diffraction and Raman spectroscopy techniques were used to investigate residual stresses in leached polycrystalline diamond disc samples. A systematic investigation and evaluation of the average in-plane residual stress fields using the Raman technique showed a progressive shift of the residual stress state from a compressive stress state to an average tensile stress state as a function of increasing number of loading cycles. In contrast the X-ray diffraction method recorded compressive stresses for all the measurements even at the highest number of loading cycles. The apparent disagreement between the two sets of results were satisfactorily explained by considering the probing beam size and sampling depth for the two different but complementary techniques.

The effect of out-of-plane constraint on the stress fields near the front of a crack in a thin ductile plate

Classical plane solutions based on the elastic-plastic fracture mechanics are applied widely in practical engineering. These plane solutions follow the plane stress or plane strain assumption. However, in the case of a thin ductile plate with a through-thickness crack under tension, plane stress conditions exist at a distance of about one half of the plate thickness ahead of the crack front cross the thickness of the plate. What is the stress state in the region where both plane strain and plane stress conditions cannot be met? In the current paper, a semi-analytical method is presented to investigate the problem. Three dimensional Maxwell stress functions, the minimum complementary potential energy principle and three dimensional J-integrals are employed to obtain solutions for crack front fields in a thin ductile plate. Three-dimensional finite element (FE) analyses are carried out to verify the current solutions. FE results reveal that the out-of-plane constraint level Tz increases with increasing remote loading near the crack front. FE results also show that the in-plane stress fields near the crack front can be characterized by the current J-Tz solutions. Both FE and theoretical results illustrate that opening stresses decrease gradually to the corresponding plane stress HRR-field solutions with increasing radial distance r from the crack front in the mid-plane when remote loading is large enough.

Elastic stress analysis of blunt V-notches under mixed mode loading by considering higher order terms

This paper presents the in-plane asymptotic displacement and stress fields for blunt V-notched components based on Kolosov–Muskhelishvili's approach. In the first part, the displacement and stress components in the polar coordinate system are determined by choosing appropriate complex potential functions. In order to construct the notch geometry, the Neuber's mapping relation is utilized. Then, the notch boundary conditions are imposed to calculate the free parameters of the stress distribution. Eventually, the stress and displacement components are calculated in the Cartesian and polar coordinates in the forms of series expansion. In the second part, the coefficients of series expansions are computed by using the least square method (LSM). The blunt V-notched Brazilian disk (BV-BD) specimen under mixed mode loading is used as an example to verify the proposed procedure. The stress components in arbitrary distances and directions are determined for different blunt V-notches in order to evaluate the accuracy of the calculated stress series solutions and their associated coefficients. The numerical results indicate that a single-term solution can lead to considerable errors, and to achieve good accuracy in the stress field calculation, one should take account of at least three terms in the stress series solution.

Magnetic frustration and spontaneous rotational symmetry breaking in PdCrO2

In the triangular layered magnet PdCrO2 the intralayer magnetic interactions are strong, however the lattice structure frustrates interlayer interactions. In spite of this, long-range, 120$^\circ$ antiferromagnetic order condenses at $T_N = 38$~K. We show here through neutron scattering measurements under in-plane uniaxial stress and in-plane magnetic field that this occurs through a spontaneous lifting of the three-fold rotational symmetry of the nonmagnetic lattice, which relieves the interlayer frustration. We also show through resistivity measurements that uniaxial stress can suppress thermal magnetic disorder within the antiferromagnetic phase.

Optimized design of RC deep beams based on performance metrics applied to strut and tie model and in-plane stress conditions

Abstract Deep beams do not behave according to classical beam theory. The nonlinearity of strain distribution within these elements requires application of strut and tie models (STM) or other alternatives to evaluate the complex stress field. Although the design of these elements is a common task for structural engineers, limited research is found on assessing effectiveness of the results. The purpose of this work is to compare, in a systematic approach, different design solutions for a deep beam using selected performance metrics which are strain energy, reinforcement ratio, maximum load, structural efficiency, safety factor and cracking behavior. A deep beam (2.85 m of height, 4.20 m of length and 0.2 m of thickness) with a square opening (0.7 m x 0.7 m) close to one of the supports was subjected to a uniform loading at the top surface while resting on supports at both ends. A simplified finite element model (FEM) of this beam was developed simulating concrete with elastic linear stress-strain behavior and disregarding steel reinforcements. This model allowed determination of elastic stress fields necessary to subsequent analyses. Four STM were then developed, supposing the total load respectively represented by one (STM-1), two (STM-2), four (STM-4) or eight (STM-8) concentrated loads equally spaced along the top of the beam. Additionally, an in-plane stress field method (SFM) was applied to the design of the same beam subjected to uniform loading on the top surface. After design and detailing of the reinforcement for each situation, nonlinear FEMs were used to predict the ultimate conditions. The strain energy reduced significantly comparing results from STM-1 to STM-2 and subsequently to STM-4 and remained at a low level in STM-8 and SFM. The reinforcement ratio reduced systematically from STM-1 to STM-8, was minimum with the SFM and the same behavior was followed by maximum load. The structural efficiency (maximum load/reinforcement ratio) increased from STM-1 to STM-8, with maximum efficiency at STM-8 and was slightly below with SFM. The safety factor reduced systematically from STM-1 to STM-8 and was slightly lower with SFM, but in all cases was above acceptable limits found in design codes.

Bistable metallic materials produced by nanocrystallization process

Through a localized nanocrystallization process using surface mechanical attrition treatment (SMAT), a new method is proposed to build bistable and multistable metallic shells. The impacts from randomly fast-moving balls during the nanocrystallization process accumulate plastic deformations, which induce nanotwins and mesh grains into nanoscales significantly increasing elastic behavior range of the treated shells. The in-plane self-equilibrium residual stress field, which is induced from the stretching plastic deformations in the treated region under the constraint of the untreated region, renders nanostructured shells bistable characteristics. An effective numerical modelling is carried out to analyze the bistable behavior and predict their stable configurations. A flat plate with multiple nanostructured regions is manufactured and numerically studied, which is capable of holding multiple stable configurations. In addition, the developed bistable and multistable shells can be further mechanically processed to modify their stable configurations.

An improved definition for mode I and mode II crack problems

In this research, the common definitions for mode I and mode II are evaluated and improved. For this purpose, the in-plane linear elastic stress field around the crack tip is written as a set of infinite series expansions. Mode I and mode II fields are classically defined as symmetric and anti-symmetric parts of these expansions, respectively. There is also a constant term called “T-stress” in these expansions; parallel to the crack line and independent of the distance from the crack tip. Previous definitions assume that T-stress exists only in pure mode I or combined mode I and mode II conditions. Based on these definitions, T-stress always vanishes in pure mode II. However, the published results of several analytical and experimental researches indicate that the constant stress term can exist in mode II stress field, as well. In this paper, some examples are presented which indicate the presence and importance of T-stress in pure mode II conditions. Then, the classical definition for mode I and mode II is modified to make it consistent with the results presented in the literature.

Wellbore stability analysis using strain hardening and/or softening plasticity models

In this paper, the general shear strain hardening and/or softening Drucker-Prager models based on the common triaxial compression test results have been introduced to model the mechanical behaviour of rock formations. These elastoplastic models are then adopted to develop rigorous analytical solutions for the drained wellbore drilling problem subjected to in-plane isotropic stress field, and to further predict the borehole collapse failure. The illustration numerical examples show the distributions of stress components and the progressive development of the plastic deviatoric strain, in addition to the effective stress trajectory for a rock point at the borehole surface due to the wellbore drilling. The advantage of the desired wellbore drilling curve presented lies in the fact that it can track the deformation responses of the borehole surface from elastic to elastoplastic states and down to the peak and ultimate/residual conditions, thus offering much more flexibility and more appropriate mud pressure design over the conventional elastic-brittle models. The critical (minimum) mud pressures, predicted by the current elastoplastic analysis based on different wellbore instability criteria, are compared with the value corresponding to the elastic theory. The solution procedure proposed can also be utilized for the tunnelling analysis and extended to the design of lining required to stabilize a tunnel.

A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part I: Model formulation

Current consensus suggests a trade-off for predicting accurate 3D stress fields in multilayered plates. Relatively expensive layerwise models facilitate accurate stress predictions from underlying model assumptions, whereas more efficient equivalent single-layer theories often rely on variationally inconsistent transverse stresses from recovery steps. In contrast, we show that variationally consistent 3D stress fields are predicted from an equivalent single-layer model using a contracted form of the Hellinger–Reissner functional. Notably, this procedure facilitates computationally efficient analysis of 3D heterogeneous plates, i.e. laminates comprising layers with material properties that differ by multiple orders of magnitude and that can also vary continuously in-plane. The model includes effects of higher-order transverse shearing and zig-zag deformations by expressing the in-plane stress field as a Taylor series expansion of global and local higher-order stress resultants. Equilibrated transverse stress assumptions are derived from Cauchy’s 3D equilibrium equations, which identically satisfy the interfacial and surface traction equilibrium conditions when applied within a variational statement that enforces the 3D equilibrium equations as constraints, hence the Hellinger–Reissner mixed-variational statement. By using inherently equilibrated stress fields as model assumptions, it is shown that only the classical membrane and bending equilibrium equations have to be enforced explicitly. As a result, a contracted Hellinger–Reissner functional emerges that requires fewer displacement Lagrange multipliers than when independent stress fields are assumed. The governing equations are derived in a generalised framework such that the order of the model, and hence its accuracy, increases automatically when implemented in a computer code. By refining the order of the stress assumptions, the model is adaptable to plate-like structures ranging from thin engineering laminates to highly heterogeneous, thick laminates comprising straight-fibre or tow-steered variable-stiffness laminates, foam, honeycomb or other compliant layers.