Principal Stress Trajectories Research Articles

In Situ Pure Shear Tests on Textured Magnesium AZ31B Sheets

Pure shear tests of textured magnesium AZ31B sheet samples were carried out using a 5 kN Kammrath & Weiss in situ tension-compression stage in a scanning electron microscope in combination with real-time electron backscatter diffraction lattice orientation mapping. The sample design was optimized to produce a pure shear stress in the central gauge zone. Distributions of the deformation twins were correlated with finite element simulations using a linear-elastic constitutive law considering large deformations to show that twins form in areas where the principal compressive stress σ3* is a maximum and that they form normal to the trajectories of that minor principal stress. Mappings of the same area at different load values revealed the formation and growth of individual twins and their relationship to the internal elastic strain of individual grains as indicated by the internal grain disorientation. All twins observed were of the extension type, with an 86.3° disorientation with respect to the parent grains. A more detailed study was conducted using transmission electron microscopy to correlate with the EBSD observations and to further elucidate the twin structures within samples.

Fiber Reinforcement of 3D Printed Concrete by Material Extrusion Toolpaths Aligned to Principal Stress Trajectories

Additive manufacturing of cementitious materials is a rapidly growing branch of manufacturing both in research and industry, particularly the variant of material deposition by extrusion. This process results in a strong anisotropy in mechanical properties, owing largely to the interfaces between adjacent filaments. This anisotropy is even more pronounced when fiber reinforced mortars or continuous entrained reinforcement components such as cables are used. To exploit orientation-dependent performance, the print path can be designed to align with the principal (tensile) stress trajectories. However, obtaining an appropriate print path based on this concept poses several challenges, related to the filling of intermediate spaces between two trajectories. In this paper, an approach for planning such a robot toolpath is presented, elaborated, and illustrated by means of a case study on a well-known reference case. The main features of the tool planning method are the relaxation of the offset width, the avoidance of toolpaths with acute angles by intersecting offset curves, and a continuous toolpath.

Numerical modelling of an orthopedic brace with increased functional characteristics for the treatment of idiopathic scoliosis.

Orthotic braces play a key role in the correction of spinal deformities. The effectiveness of these devices depends on the design and distribution of corrective forces transmitted through the corset shell. The present study aimed to reduce the weight of the orthosis and improve its functionality while maintaining its corrective function. The distribution of corrective forces transmitted by the orthosis was evaluated using the finite element method (FEM). Areas of the orthosis, which had minimal impact on the overall stiffness, were identified and material from these areas was removed. The modified orthosis shell was subjected to minor adjustments to maintain its corrective stiffness. With the modifications made, a 39% reduction in the weight of the orthosis was achieved, while maintaining its corrective stiffness. This indicates that the corrective function was largely preserved. The study provides a novel approach to orthosis design demonstrating that optimizing the structure using the distribution of maximum principal stress trajectories can significantly improve the functionality of the brace. The proposed method offers potential advances in the design of various types of orthoses, contributing to developments in the field.

An Analysis of Trabecular Bone Structure Based on Principal Stress Trajectory.

To understand the mechanism of Wolff's law, a finite element analysis was performed for a human proximal femur, and the principal stress trajectories of the femur were extracted using the principal stress visualization method. The mechanism of Wolff's law was evaluated theoretically based on the distribution of the principal stress trajectories. Due to the dynamics of the loads, there was no one-to-one correspondence between the stress trajectories of the fixed load and the trabeculae in the cancellous architecture of the real bone. The trabeculae in the cancellous bone were influenced by the magnitude of the principal stress trajectory. Equivalent principal stress trajectories suitable for different load changes were proposed through the change in load cycle and compared with the anatomical structure of the femur. In addition, the three-dimensional distribution of the femoral principal stress trajectory was established, and the adaptability potential of each load was discussed. The principal stress visualization method could also be applied to bionic structure design.

CAD-integrated stiffener sizing-topology design via force flow members (FFM)

Stiffener design plays a crucial role in the lightweight design of stiffened panels. We propose a CAD-integrated stiffener design approach for shells. The problem is partitioned into two steps: (1) generating clear force flow members (FFM) based on the direction of principal stress, and (2) introducing a nodal variable-based sizing-topology optimization for further lightweight design enhancement. Specifically, we generate dense principal stress trajectory families using the iterative tracing technique, and extract FFM through hierarchical clustering. Generating high-precision principal stress trajectories on free-form surfaces is a challenging task. Our method differs from previous approaches in that we employ tensor transformations to convert principal stress directions from the 3D physical space into the 2D parameter space. We then create NURBS curve-based FFM in the 2D parameter space. The NURBS-based free-form deformations (NFFD) technique is utilized to map FFM from the 2D parameter space to the 3D physical space, ensuring FFMs are precise and watertight constructions of geometries. We propose a nodal variable-based sizing-topology method for the lightweight design of FFM. The heights of FFM control points are designated as design variables. Steadily eliminating pseudo-heights is achieved by using the Sigmoid function to penalize the design variables. Moreover, we introduce an NFFD-based isogeometric analysis method for stiffened panels modeled by NFFD-based FFM to realize the integration of design and analysis. The effectiveness and robustness of the presented method are demonstrated through various benchmarks and examples with industrial backgrounds. The proposed optimization framework exhibits strong application potential in the lightweight design of stiffened panels.

Principal Stress Trajectories in Plasticity under Plane Strain and Axial Symmetry

The two families of principal stress trajectories can be regarded as an orthogonal curvilinear coordinate system under plane strain and axial symmetry. Under plane strain, the equilibrium equations in conjunction with a yield criterion comprise a statically determinate system. Under axial symmetry, a statically determinate system results from the above equations supplemented with the hypothesis of Haar von Karman. In both cases, the compatibility equations for mapping the principal line coordinate system to a given coordinate system show that the scale factors of the former satisfy a simple algebraic or transcendental equation for many yield criteria. Using this equation, one can develop a method for reducing boundary value problems in plasticity to purely geometric problems. The method is independent of any flow rule that can be chosen to calculate displacement or velocity fields, as well as independent whether elastic strains are included. The present paper summarizes available results related to using principal stress trajectories in plasticity and emphasizes the advantages of the method above.

Nonlinear description of active earth pressure for finite backfill based on principal stress trajectory method subjected to steady unsaturated seepage

To simplify calculations, soil cohesion, horizontal shear stress, and unsaturated properties have usually been neglected in previous earth pressure analytical methods. This negligence has certain theoretical flaws and does not conform to the actual situation. Especially for finite backfill, which often occurs, it is worth continuing to expand research in this field to seek a more rigorous solution. This work presents an analytical framework for estimating the active earth pressure of finite backfill on inclined retaining walls under steady unsaturated seepage conditions. The extended Mohr–Coulomb failure criterion expresses soil shear strength, and a unified effective stress method considering the suction stress is adopted. For a classic failure model, a calculation method using curved differential elements is established based on the principal stress trajectory. The proposed method fully considers soil cohesion, arching effect, and unsaturated properties. Relative to the existing test and theoretical results, the rationality of this method is verified. The sensitivities of relevant parameters are discussed based on different assumed soil types. The results show that seepage significantly affects the lateral earth pressure. The earth pressure distributions vary greatly under different unsaturated types for various soils. The results can provide a reference for the selection of engineering backfill.

3D printing of continuous carbon fibre reinforced polymer composites with optimised structural topology and fibre orientation

This study presents a sequentially coupled optimisation of structural topology and fibre orientation for 3D printing of continuous carbon fibre reinforced polymer composites. Topology optimisation was first performed to obtain the geometry of the structure under a specific load, and then continuous fibres were placed along the identified principal stress trajectories. Composite preforms were 3D printed by a material extrusion-based technique followed by post-processing with vacuum bagging using epoxy for infusion. The case of Messerschmitt-Bolkow-Blohm (MBB) beam under three-point bending test was studied and a path-based model was also built to analyse the effect of customised fibre placement and their lightweight performance. Experimental results showed that 3D printed composites with optimised fibre orientation achieved 305% and 256% higher strength and stiffness than Markforged® printed composites. By comparing with the traditionally-manufactured composites and aerospace-grade aluminium alloy, this study demonstrated the potential of manufacturing ultra-lightweight composite structures via 3D printing and the benefits of fibre orientation optimisation in lightweight design of composites with topology optimisation.

In-situ three-dimensional X-ray investigation on micro ductile fracture mechanism of a high-Mn steel with delayed necking effect

This study investigates the micro ductile fracture mechanism of a high-Mn steel with delayed necking effect (DN-HMn steel) under monotonic tension. A random-oriented micro-grain structure cooperating with intragranular twins in the DN-HMn steel contributes to its high tensile strength of 880 MPa and high uniform elongation of over 55%, accounting for 92.8% of the total elongation. Three-dimensional (3D) visualization of the ductile fracture process is presented, and characteristics of void evolution are identified. It was observed that voids distribute and grow uniformly during deformation, with larger voids forming a double-helix spatial distribution pattern at high strain levels, which fits the principal shear stress trajectory. The number and volume of voids in the DN-HMn steel showed a significant increase, with the porosity increasing from 0.071‰ (initial state) to 1.580‰ (before fracture). Intervoid necking was found to be the mode of void coalescence, which occurred close to the necking strain of the sample. The DN-HMn steel demonstrated a significant volume expansion of 4.58% throughout the elongation process, which breaks the volume conservation law of classical metal plasticity theory. This study opens new perspectives for understanding the ductile fracture mechanism of this steel.

A 3D torus-slice model for limit equilibrium analysis of shield tunnel face under soil arching effect

This paper proposes a torus-slice model to estimate the tunnel face critical support pressure, (σcrit) from the shield machine under the influence of soil arching. The failure envelope of the tunnel face is created by a cluster of discrete points using a spatial discretization technique. The vertical earth pressure loaded on the failure soil mass in front of the face is determined by an arching mechanism based on the rotation trajectory of the major principal stress. The tunnel face support pressure distribution is derived from the stress interaction using a stepwise method between the horizontal slices divided from the torus. The comparisons between the centrifuge tests and analytical solutions demonstrate the accuracy of the proposed model. The pressure distribution is further validated by the results of a FLAC3D model and analytical solutions. The failure envelope is compared to the geometry of the shear band from the numerical model, indicating the rationality of the proposed method. With an improved geometry considered, the proposed torus-slice model demonstrates the advantages over the classical silo-wedge model. Finally, the effects of material and geometrical parameters on a required tunnel face pressure are discussed. The results show that the soil internal friction angle φ’, cohesion c’, the soil-shield interface friction angle δ, and the geometric shape factor Tm all have significant effects on the tunnel face stability.

Trajectories of principal stresses after plastic deformation

Improving research efficiency of mechanical stresses in metals can be achieved by studying and taking into account the fields of the trajectories of the main stresses (isostats), which qualitatively characterize the stress state. The purpose of the work was to determine the possibility of detecting the isostates in a St3 steel sample, following its plastic deformation. The methodology is based on the magnetoelastic method, manifested by the dependence of magnetic permeability of ferromagnetic materials upon mechanical stresses, acting in them. The method is implemented using the IMN-4M monophase magnetoelastic meter of mechanical stress: the base of the device gage is 5 mm, the angle meter error is ± 2°. The sample is a plate with dimensions of 150 x 150 x 4 mm. Coordinate grids with 10 x 10 mm cells were applied to its front and back surfaces. Using the IMN-4M device, we measured the values of tilt angles for tangents to principal stress trajectories in all nodes, then we created isostates by consecutive adjustment of the direction and radius of curvature for a raised curve. The plastic deformation of a plate was performed by shooting small lead pellets with kinetic energy of 1916 J. The maximum plate curvature in the impact zone of main charge equaled 7 mm. After a shot, the measurements were repeated in the same nodes of coordinate grids, and the isostates were plotted again. The experiment has shown that: the magnetoelastic method makes it possible to detect the isostates even after plastic deformation; the trajectories are present both on front and back surfaces of the deformed plate; isotropic points and parallel isostates are available in fields, formed by the shot. Izotropic zones were formed on front and back surfaces, in which isostates can not be plotted due to random orientation of tangents. The "disturbances" of fields with disruption of regular trajectory flows were detected. The results of the work may be useful for researchers and technical engineers, engaged in the study of elastic and plastic deformations.

Production-induced pressure-depletion and stress anisotropy changes near hydraulically fractured wells: Implications for intra-well fracture interference and fracturing treatment efficacy

This study investigates the changes in the principal stress trajectories during development of hydrocarbon (and/or geothermal) reservoirs with hydraulically fractured wells. Our analysis indicates four phases in the well-life with typical stress states, i.e., Pre-fracturing Phase (Stress State 0): the natural stress state prior to the drilling intervention; Fracturing Phase (Stress State 1): stress state prevailing during fracturing treatment; Flowback Phase (Stress State 2): stress state prevailing during flowback; and Production Phase (Stress State 3): stress state prevailing during production. The various stress changes are computed and visualized using the Linear Superposition Method (LSM). Two episodes of stress trajectory alterations occur, a first one during the Fracturing Phase (transition from Stress State 0 to 1), and a second one during the Flowback Phase (transition from Stress States 1 to 2), with respectively positive and negative fracture net pressures. During the Production Phase (Stress State 3), the spatial advance of the pressure depletion around the fractured well system due to production was modeled using recently developed Gaussian pressure transient equations. Our new results show that the early stress reversals (Stress State 1) near the pressured fractures during fracturing treatment are short-lived. In addition, the residual stress change magnitude during flowback (Stress State 2) depends on the final fracture-width aperture. In any case, the local stress reversals due to engineering interventions are a short-term phenomenon and remain limited to the near-fracture regions. The regions with the reversed stress will increase when more stages are fractured, assuming the elevated fracture pressure is not fully released before the next stage is completed. Subsequently, the stress anisotropy decreases during production as a result of pressure depletion. Our improved analysis of the stress reversal phenomenon is important for optimizing drilling plans for infill wells, and for improving fracturing treatment designs.

Experimental investigation and simulation on load-transfer paths in optimally designed RC deep beams

The main failure pattern for reinforced concrete deep beams is shear failure due to the low shear-span ratio. The Topology Optimization for the stress of rebar in this study is explored for aided-design of deep beams, which proves that the optimization can transform the brittle shear failure to ductile bending failure without reducing the bearing capacity of deep beams, compared with the traditional empirical method. In the process of increasing load, the displacement fields on the surface of the deep beams were measured successively by non-contact displacement gauges, and converted to the principal strain distribution. Then the distribution is used to fit correspondingly the principal stress trajectory and load-transfer paths of the specimens. The results show that, for the specimen designed with Topology Optimization, the built-in inclined main reinforcements are in line with the direction of the concrete tensile stress in the web of the specimens, and effectively restrict the development of width of the inclined cracks. Consequently, the integrity and directness of the load-transfer paths from the loading point to the supports are guaranteed. Meanwhile, the stress fields of the specimens correspond to the Michell-type structure, hence the stress of their rebar is almost under a full stress state. Therefore, the optimal design presents a more reasonable reinforcement layout, which can better control the accumulation of damage and the change of the load-transfer paths during the loading process, and finally achieve the desired failure pattern.

An improved method of active earth pressure on rigid retaining wall under movement modes considering arching effects

AbstractBoth wall movement mode and soil arching are important to the active earth pressure on rigid retaining wall, but the method of active earth pressure under wall movement modes considering arching effects is yet to be studied. A series of minor principal stress trajectories described by the power function with the exponent b is assumed to analyze the soil arching effect using the equivalent friction coefficient between adjacent soil layers. The horizontal Winkler elastic foundation beam model induced by wall movement modes is presented. Then based on the trajectories and wall movement modes the horizontal flat‐element method of active earth pressure on rigid retaining wall is improved and verified. The active earth pressures on the retaining wall under translation (T), rotation about top (RT) mode and rotation about bottom (RB) mode calculated by the improved method are in a better agreement with the experimental results than other analytical studies. The application point of the resultant active earth pressure on retaining wall increases from 0.25 H to 0.49 H with varies of wall movement modes considering soil arching effect. The larger equivalent friction coefficient is, the more notable soil arching effect is. The most notable soil arching effect is described by the approximate linear minor principal stress trajectory whose power function with exponent of b approaches zero from positive direction. The active earth pressure on retaining wall under RT mode considering the most notable soil arching effect is simplified and suggested for the engineering practice.

Review of Selected Issues in Anisotropic Plasticity under Axial Symmetry

The present review paper consists of two main parts, which are not connected. The first part is devoted to a general axisymmetric elastic–plastic plane stress solution, assuming polar anisotropy. Strains are infinitesimal. The principal stress trajectories coincide with the principal axes of anisotropy. No restrictions are imposed on the yield criterion other than the conventional restrictions imposed on the yield criteria in plasticity. The plastic portion of the strain rate tensor is determined from the associated flow rule. A simple example illustrates the general solution. The second part is devoted to the stationary ideal flow theory for anisotropic materials under axial symmetry. The elastic portion of the strain tensor is neglected. A piece-wise linear yield criterion is adopted. This criterion generalizes Tresca’s yield criterion. The existence of ideal flow is proven. It is also shown that the available solutions for Tresca’s yield criterion can be used for deriving solutions for the yield criterion under consideration. Miscellaneous topics are shortly discussed in the third part of the paper.

Stress-adapted fiber orientation along the principal stress directions for continuous fiber-reinforced material extrusion

A proven method to enhance the mechanical properties of additively manufactured plastic parts is the embedding of continuous fibers. Due to its great flexibility, continuous fiber-reinforced material extrusion allows fiber strands to be deposited along optimized paths. Nevertheless, the fibers have so far been embedded in the parts contour-based or on the basis of regular patterns. The outstanding strength and stiffness properties of the fibers in the longitudinal direction cannot be optimally utilized. Therefore, a method is proposed which allows to embed fibers along the principal stresses into the parts in a load-oriented manner. A G-code is generated from the calculated principal stress trajectories and the part geometry, which also takes into account the specific restrictions of the manufacturing technology used. A distinction is made between fiber paths and the matrix so that the average fiber volume content can be set in a defined way. To determine the mechanical properties, tensile and flexural tests are carried out on specimens consisting of carbon fiber-reinforced polyamide. In order to increase the influence of the principal stress-based fiber orientation, open-hole plates are used for the tensile tests, as this leads to variable stresses across the cross section. In addition, a digital image correlation system is used to determine the deformations during the mechanical tests. It was found that the peak load of the optimized open-hole plates was greater by a factor of 3 and the optimized flexural specimens by a factor of 1.9 than the comparison specimens with unidirectional fiber alignment.

Passive earth pressures with sloping backfill based on a statically admissible stress field

Passive earth pressure is a fundamental problem in the field of soil-structure interaction. In this study, a fully plastic solution obeying the Mohr-Coulomb criterion was successfully developed to estimate passive earth pressure against rigid retaining walls with various sloping backfills, wall roughness, and wall rear face inclination. The numerical procedures using a shooting technique and the 5th order adaptive Runge-Kutta method were successfully developed for boundary-value and initial-value problems, respectively. The developed procedures successfully coped with the numerical integral in non-smooth domains then statically admissible stress fields in the sloping backfills are explored. The stress distributions together with the trajectories of principal stresses in various sloping backfills are presented to demonstrate the load transfer mechanisms of cohesionless soil in various geometries of the backfill. The effects of the wall roughness, the wall rear face inclination, and backfill inclination on the load transfer mechanism of the cohesionless soil behind the retaining wall are investigated. The obtained results of passive earth pressure coefficients (Kp) are validated by comparing with an exact solution and some well-known experimental works. Thus, highly accurate values of passive earth pressure coefficients on rigid retaining walls with various sloping backfills, wall roughness, and wall rear face inclinations are provided for engineering practice.

Optimization of Continuous Fiber Path Planning for an Additively Manufactured Open-Hole Specimen

This study presents a numerical method for optimizing the quantity and the placement of reinforcements along the principal-stress trajectories. The model representing carbon fiber composite structures consists of solids and embedded one-dimensional beam elements. Based on the Runge-Kutta method, the reinforcing structure is optimized considering the manufacturability of additive manufacturing (AM). For a case study, the optimization method is performed on an open-hole specimen. The Young’s modulus and the tensile strength of the optimized structure show an increase of more than 30 % and ~50 % in the simulation, respectively, compared to the reference specimen from another study. Robotic additive manufacturing is used to fabricate the specimen for experimental validation. The prediction of absolute values of tensile strength are reliable comparing to the experimental test, however, there is a deviation of more than 30 % in the linear-elastic behavior possibly due to the presence of voids in the printed part.

Study on the Calculation Method of Active Earth Pressure and Critical Width for Finite Soil Behind the Retaining Wall

The method to determine the active earth pressure and critical width for finite soil behind the retaining wall in mountainous areas is one of the concerns of geotechnical engineering. In order to study the active earth pressure distribution of the finite soil against the retaining wall and determine the critical width of the boundary between the finite soil and the semi-infinite soil, this study focuses on investigating a retaining wall with finite cohesionless backfill. The shape of the failure surface is assumed to be a cycloid passing through the heel of the wall in the limit equilibrium state. Considering the deflection of soil principal stress induced by wall–soil friction effect, a calculation method of active earth pressure for finite soil is proposed by using an arc-shaped small principal stress trajectory, and the rationality of this method is verified. On this basis, a calculation formula of the critical width for finite soil is proposed. The influence of the internal friction angle and the wall–soil friction angle on the critical width of finite soil is examined. The results indicate that the active earth pressure of finite soil presents a nonlinear drum distribution along the height of the retaining wall under the failure mode of the cycloidal surface. The maximum value of active earth pressure is close to the bottom of the wall. The critical width of finite soil decreases with the increase of the internal friction angle, and its variation rate decreases gradually. The critical width of finite soil increases with the increase of the wall–soil friction angle, and its variation rate also increases gradually. Under different internal friction angles and wall–soil friction angles, the critical width values of finite soil calculated by the assumption of the cycloidal failure surface are smaller than those calculated by the Coulomb earth pressure calculation method.

Effect of tunnelling-induced ground loss on the distribution of earth pressure on a deep underground structure

This paper investigated the stress-transfer mechanisms in soil around a circular tunnel (CT) and a rectangular tunnel (RT) using the finite element method (FEM). Compared with the CT, there was a self-weight stress zone in the local soil above the RT, and the shear stress was concentrated only near the sliding surface, which caused the trajectory of the minor principal stress in this area to take an inverted arch shape. Then, a multi-arch model consisting of the end-bearing arch and the friction arch was proposed for estimating the distribution of the vertical earth pressure on a deep RT in dry sand. Unlike the CT, the friction arch in the RT was based on the assumption that the minor principal stress was a circular arc. The distribution factor of the vertical earth pressure on the RT was obtained. Note that a modified coefficient in the distribution factor was introduced, which was obtained by numerical simulation at different cover depth ratios and internal friction angles. Finally, the theoretical, experimental and numerical results were compared to evaluate the validity of the proposed model. The theoretical results in this paper were in good agreement with the experimental and numerical results.