General Stress State Research Articles

The buckling sphere: A symbiosis of mechanics and geometry

The basis of this work is a novel symbiosis of mechanics of solids and spherical geometry to quantify and illustrate the variation of the “non-membrane” percentage of the strain energy in the prebuckling region of linear elastic beams, arches, plates and shells, and structures assembled of such one-dimensional and two-dimensional members. The zenith angle of an arbitrary point of a specific curve on an octant of the unit sphere, called buckling sphere, is related to this energy percentage. For the limiting case of buckling from a membrane stress state this curve degenerates to a point, characterized by zero values of both spherical coordinates. For all other stress states the azimuth angle increases with the proportionally increasing load. Its magnitude at the stability limit correlates with a global quantity that depends on both the “non-membrane” deformations and the stiffness of the structure at incipient buckling. The azimuth angle is computed with the help of the so-called consistently linearized eigenproblem, which is solved by means of the Finite Element Method. This eigenvalue problem is the basis for a hypothesis for the “non-membrane” percentage of the strain energy. In the theoretical part of the paper, the concept of the buckling sphere is presented. The subsequent numerical investigation consists of four examples, referring to buckling from a membrane stress state, a pure bending stress state, and a general stress state. The practical motive for this research is the intention to investigate the influence of “non-membrane” action just before buckling on the initial postbuckling behavior of elastic structures.

Yield function of the orthotropic material considering the crystallographic texture

On the basis of the energy approach it is reported a development of the yield function and the constitutive equations for the orthotropic material with consideration of the crystal lattice constants and parameters of the crystallographic texture for the general stress state. For practical use in sheet metal forming analysis it is considered different loading scenarios: plane stress and plane strain states. Using the proposed yield function, the influence of single ideal components on the shape of yield surface was analyzed. The six texture components investigated here were cube, Goss, copper, brass, S and rotated cube, as these components are typically observed in rolled sheets from FCC alloys.

Molecular dynamics at constant Cauchy stress.

The Parrinello-Rahman algorithm for imposing a general state of stress in periodic molecular dynamics simulations is widely used in the literature and has been implemented in many readily available molecular dynamics codes. However, what is often overlooked is that this algorithm controls the second Piola-Kirchhoff stress as opposed to the true (Cauchy) stress. This can lead to misinterpretation of simulation results because (1) the true stress that is imposed during the simulation depends on the deformation of the periodic cell, (2) the true stress is potentially very different from the imposed second Piola-Kirchhoff stress, and (3) the true stress can vary significantly during the simulation even if the imposed second Piola-Kirchhoff is constant. We propose a simple modification to the algorithm that allows the true Cauchy stress to be controlled directly. We then demonstrate the efficacy of the new algorithm with the example of martensitic phase transformations under applied stress.

Material Constitutive Models for Creep and Rupture of SiC/SiC Ceramic-Matrix Composites (CMCs) Under Multiaxial Loading

Material constitutive models for creep deformation and creep rupture of the SiC/SiC ceramic-matrix composites (CMCs) under general three-dimensional stress states have been developed and parameterized using one set of available experimental data for the effect of stress magnitude and temperature on the time-dependent creep deformation and rupture. To validate the models developed, another set of available experimental data was utilized for each model. The models were subsequently implemented in a user-material subroutine and coupled with a commercial finite element package in order to enable computational analysis of the performance and durability of CMC components used in high-temperature high-stress applications, such as those encountered in gas-turbine engines. In the last portion of the work, the problem of creep-controlled contact of a gas-turbine engine blade with the shroud is investigated computationally. It is assumed that the blade is made of the SiC/SiC CMC, and that the creep behavior of this material can be accounted for using the material constitutive models developed in the present work. The results clearly show that the blade-tip/shroud clearance decreases and ultimately becomes zero (the condition which must be avoided) as a function of time. In addition, the analysis revealed that if the blade is trimmed at its tip to enable additional creep deformation before blade-tip/shroud contact, creep-rupture conditions can develop in the region of the blade adjacent to its attachment to the high-rotational-speed hub.

The Mohr–Coulomb criterion for intact rock strength and friction – a re-evaluation and consideration of failure under polyaxial stresses

Abstract. Darley Dale and Pennant sandstones were tested under conditions of both axisymmetric shortening and extension normal to bedding. These are the two extremes of loading under polyaxial stress conditions. Failure under generalized stress conditions can be predicted from the Mohr–Coulomb failure criterion under axisymmetric shortening conditions, provided the best form of polyaxial failure criterion is known. The sandstone data are best reconciled using the Mogi (1967) empirical criterion. Fault plane orientations produced vary greatly with respect to the maximum compressive stress direction in the two loading configurations. The normals to the Mohr–Coulomb failure envelopes do not predict the orientations of the fault planes eventually produced. Frictional sliding on variously inclined saw cuts and failure surfaces produced in intact rock samples was also investigated. Friction coefficient is not affected by fault plane orientation in a given loading configuration, but friction coefficients in extension were systematically lower than in compression for both rock types. Friction data for these and other porous sandstones accord well with the Byerlee (1978) generalization about rock friction being largely independent of rock type. For engineering and geodynamic modelling purposes, the stress-state-dependent friction coefficient should be used for sandstones, but it is not known to what extent this might apply to other rock types.

The Stability of Inclined and Fractured Wellbores

SummaryThe theory of linear-elastic fracture mechanics is used to develop an analytical solution for a wellbore in an isotropic elastic medium when drilled inclined to a general state of 3D far-field stress, and attached to which are an arbitrary number of N straight and axially aligned fractures. Axial alignment refers to the special case where the borehole axis is the existing common interface of all planes defined by the fracture faces. The solution uses a familiar load decomposition and coordinate-transform scheme. Within the fracture-mechanics context of the analysis, this scheme translates into a set of three fundamental subproblems comprising a uniaxial stress problem, together with an in-plane mixed mode (Modes I and II), and a Mode III antiplane-crack/cavity-interaction problem.The overall solution is obtained by superposing the solutions to these subproblems. A numerical example is presented to demonstrate its usefulness in the stability analysis of inclined and fractured wellbores. Attention has been directed toward the underlying significance that the results would bring in fundamental understanding of lost-circulation events. For this purpose, the criteria for a possible extended margin of the mud weight that secures stable states of a fractured wellbore are recognized and quantified. These criteria include the wellbore-wall refracturing and the existing fractures propagation.

Critical state behaviors of a coarse granular soil under generalized stress conditions

Critical state line (CSL) is the central concept in soil mechanics. A series of true triaxial compression tests under the constant- $${p}'$$ and constant-b loading condition were carried out to investigate the CSL of a coarse granular soil. It was observed that the intermediate principal stress ratio (i.e., the b-value) greatly influenced the CSLs in both $$q{-}{p}'$$ and $$e{-}{p}'$$ spaces. The CSL slope in the $$q{-}{p}'$$ space decreased with an increase in b-value. The intercept and gradient of the CSL in the $$e{-}{p}'$$ space decreased with increasing b-value. CSLs incorporating the effects of the b-value in $$q{-}{p}'$$ and $$e{-}{p}'$$ spaces were extended to three-dimensional critical state surfaces (TCSSs) in $$q{-}{p}'{-}b$$ and $$e{-}{p}'{-}b$$ spaces. Two empirical equations were proposed for the two TCSSs in $$q{-}{p}'{-}b$$ and $$e{-}{p}'{-}b$$ spaces, respectively. The predictions by the two equations were in good agreement with the corresponding experimental data. The relationship between the excess friction angle (the difference between the peak state and critical state friction angles) and initial state parameter was influenced by the b-value. However, the relationship between the maximum dilatancy and initial state parameter was independent of the b-value.

Decomposed theorem of a transversely isotropic elastic plate for extensional deformation

Without ad hoc assumptions, a decomposed theorem of a transversely isotropic plate for extensional deformation are derived and studied based on transversely isotropic elastic theory. Firstly, from the Elliott–Lodge solution and Lur’e method, the displacement and stress components are obtained in terms of mid-plane displacements and transverse normal strain. Secondly, the exact equations of the plate are obtained under homogeneous boundary conditions. The general stress state of the plate consists of three parts: the generalized plane-stress state, the shear state, and the Papkovich–Fadle state. At last, the decomposed form of a transversely isotropic elasticity plate for extensional deformation is obtained, and the decomposed theorem is proven strictly for the first time.

1H NMR-based metabolomics of Daphnia magna responses after sub-lethal exposure to triclosan, carbamazepine and ibuprofen

Pharmaceuticals and personal care products are a class of emerging contaminants that are present in wastewater effluents, surface water, and groundwater around the world. There is a need to determine rapid and reliable bioindicators of exposure and the toxic mode of action of these contaminants to aquatic organisms. 1H nuclear magnetic resonance (NMR)-based metabolomics in combination with multivariate statistical analysis was used to determine the metabolic profile of Daphnia magna after exposure to a range of sub-lethal concentrations of triclosan (6.25–100μg/L), carbamazepine (1.75–14mg/L) and ibuprofen (1.75–14mg/L) for 48h. Sub-lethal triclosan exposure suggested a general oxidative stress condition and the branched-chain amino acids, glutamine, glutamate, and methionine emerged as potential bioindicators. The aromatic amino acids, serine, glycine and alanine are potential bioindicators for sub-lethal carbamazepine exposure that may have altered energy metabolism. The potential bioindicators for sub-lethal ibuprofen exposure are serine, methionine, lysine, arginine and leucine, which showed a concentration-dependent response. The differences in the metabolic changes were related to the dissimilar modes of toxicity of triclosan, carbamazepine and ibuprofen. 1H NMR-based metabolomics gave an improved understanding of how these emerging contaminants impact the keystone species D. magna.

Derivation, parameterization and validation of a creep deformation/rupture material constitutive model for SiC/SiC ceramic-matrix composites (CMCs)

The present work deals with the development of material constitutive models for creep-deformation and creep-rupture of SiC/SiC ceramic-matrix composites (CMCs) under general three-dimensional stress states. The models derived are aimed for use in finite element analyses of the performance, durability and reliability of CMC turbine blades used in gas-turbine engines. Towards that end, one set of available experimental data pertaining to the effect of stress magnitude and temperature on the time-dependent creep deformation and rupture, available in the open literature, is used to derive and parameterize material constitutive models for creep-deformation and creep-rupture. The two models derived are validated by using additional experimental data, also available in the open literature. To enable the use of the newly-developed CMC creep-deformation and creep-rupture models within a structural finite-element framework, the models are implemented in a user-material subroutine which can be readily linked with a finite-element program/solver. In this way, the performance and reliability of CMC components used in high-temperature high-stress applications, such as those encountered in gas-turbine engines can be investigated computationally. Results of a preliminary finite-element analysis concerning the creep-deformation-induced contact between a gas-turbine engine blade and the shroud are presented and briefly discussed in the last portion of the paper. In this analysis, it is assumed that: (a) the blade is made of the SiC/SiC CMC; and (b) the creep-deformation behavior of the SiC/SiC CMC can be represented by the creep-deformation model developed in the present work.

A closed-form criterion for dislocation emission in nano-porous materials under arbitrary thermomechanical loading

Our traditional view of void nucleation is associated with interface debonding at second-phase particles. However, under extreme dynamic loading conditions second-phase particles may not necessarily be the dominant source of void nucleation sites. A few key experimental observations of laser spall surfaces support this assertion. Here, we describe an alternative mechanism to the traditional view, namely shock-induced vacancy generation and clustering followed by nanovoid growth mediated by dislocation emission. This mechanism only becomes active at very large stresses. It is therefore desirable to establish a closed-form criterion for the macroscopic stress required to activate dislocation emission in porous materials. Following an approach similar to Lubarda and co-workers, we derive the desired criterion by making use of stability arguments applied to the analytic solutions for the elastic interactions of dislocations and voids. Our analysis significantly extends that of Lubarda and co-workers by accounting for a more general stress state, finite porosity, surface tension, as well as temperature and pressure dependence. The resulting simple stress-based criterion is validated against a number of molecular dynamics simulations with favorable agreement.

A Multiscale Model for the Quasi-Static Thermo-Plastic Behavior of Highly Cross-Linked Glassy Polymers

We present experimentally validated molecular dynamics predictions of the quasi-static yield and postyield behavior for a highly cross-linked epoxy polymer under general stress states and for different temperatures. In addition, a hierarchical multiscale model is presented where the nanoscale simulations obtained from molecular dynamics were homogenized to a continuum thermoplastic constitutive model for the epoxy that can be used to describe the macroscopic behavior of the material. Three major conclusions were achieved: (1) the yield surfaces generated from the nanoscale model for different temperatures agree well with the paraboloid yield criterion, supporting previous macroscopic experimental observations; (2) rescaling of the entire yield surfaces to the quasi-static case is possible by considering Argon’s theoretical predictions for pure compression of the polymer at absolute zero temperature; (3) nanoscale simulations can be used for an experimentally free calibration of macroscopic continuum models, opening new avenues for the design of materials and structures through multiscale simulations that provide structure–property–performance relationships.

Micromechanical investigation of the fracture behavior of powder materials

Fracture of compacted powder has been studied experimentally and numerically using a micromechanical approach. In the experimental investigation, the compacts are crushed in two different directions to account for general stress states and a microscopy study shows that fracture of the powder granules plays a significant role in the fracture process. The numerical analysis is based on the Discrete Element Method (DEM) and a novel approach is presented to account for the fracture of the particles in the numerical model. The force–displacement relations for two particles in contact, which are needed in DEM, are derived using micomechanical experiments together with finite element analyses of the contact problem. The contact model accounts for plastic compression, elastic unloading and adhesive bonding together with friction and tangential bonding. The model shows a very good agreement with the experimental data both for the elastic behavior during unloading and, if failure of the particles is accounted for, the fracture of the compacts.

On the energetics of tensile and shear void coalescences

This paper investigates the mechanisms of tensile and shear void coalescences in ductile materials from energetics perspective. By examining and comparing the elastic and plastic energies of a voided cell throughout its deformation history, the onset of and final coalescence can be distinctly established. This energy-based approach offers a single unified method and criterion for determining the occurrences of both mechanisms. This paper also reports a novel micromechanics model that considers general stress states described by three macroscopic normal stresses and one macroscopic shear stress. Detailed formulation of the model is presented that includes the homogenization-based derivation and implementation of a 4×4 orthogonal transformation matrix, which relates the macroscopic deformation rate of the cell to displacement rates of non-physical degrees-of-freedom (DOFs), and the polar decomposition of the macroscopic deformation gradient tensor which admits the explicit determination of the logarithmic strain measures and rotation angle.In terms of stress ratios, ρ1(=Σ11/Σ22),ρ2(=Σ12/Σ22),ρ3(=Σ33/Σ22), it is analytically shown that multiple macroscopic stress states {ρ1,ρ2,ρ3} can exist that result in the same stress triaxiality T and Lode parameter L. Specifically, it is shown that for a prescribed pair of T and L and in the absence of shear stress, at most six stress states {ρ1,0,ρ3} are possible. On the other extreme in the presence of shear stress, an infinite number of stress states is possible, due to the existence of Mohr's circle for this stress state. This model, together with the proposed energy-based criteria, is used to examine void coalescence under multiple stress-state conditions for any given T and L. Numerical results have shown that the presence of shear stress has a significant effect of reducing the effective strains for the onset of and final void coalescences. In addition, a relationship has also been established between shear angle and effective strain at the onset of shear void coalescence.

Mechanical Behavior of Granular Materials under Continuously Varying b Values Using DEM

AbstractThe influence of the intermediate stress ratio on the mechanical behavior of granular materials under generalized stress conditions is not well understood, particularly in regard to relationships between macroscale behavior and microscale response. The objective of this paper is to study the influence of the intermediate stress ratio, specified by b[=(σ2−σ3)/(σ1−σ3)], on the mechanical behavior of granular materials under generalized stress conditions, using the discrete element method (DEM). Selected b values were used to illustrate the influence of the intermediate principal stress σ2 on the major and minor principal stresses, σ1 and σ3, respectively. More-generalized stress conditions were identified by continuously varying b values. Specifically, 11 stress paths under truly triaxial conditions, following the stress-controlled method for a three-dimensional sphere, were simulated with continuously varying b values as well as constant b values. Differences in the stress–strain relationship resul...

Cyclic Behaviors of Granular Materials under Generalized Stress Condition Using DEM

AbstractMost behaviors of granular materials have been studied under the limit conditions of stress paths such as cyclic triaxial compression and extension tests. Furthermore, the macrobehavior and micromechanical response of granular materials under more comprehensive cyclic loading tests in the generalized stress condition are not well understood. The objectives of this paper are to ensure the ability of the Discrete Element Method (DEM), to simulate the macrobehavior of granular materials under various cyclic stress paths of the generalized stress condition, and to explore the micromechanical response. To this end, four different cyclic stress paths under generalized stress simulations, following the stress-controlled method on a sample of spheres, were conducted. To check the validity of DEM results, the experimental results of the real sand under cyclic stress programs were used. The macrobehaviors, indicated by the stress–strain–dilation relationship and the directions of principal strain increment ...

Forming limit prediction of anisotropic material subjected to normal and through thickness shear stresses using a modified M–K model

In some recent sheet metal forming processes such as hydroforming and incremental forming processes, normal and through thickness stresses are induced over the metal sheet, and therefore, prediction of forming limits requires considering effect of these stresses. In this investigation, a modified M–K model has been utilized to consider general stress state including normal and through thickness shear stresses on anisotropic metal sheet. This is achieved by assuming additional force equilibrium and geometrical compatibility between groove and matrix, and the numerical model has been solved by Newton–Raphson method to calculate the limiting strains. Forming limit diagrams resulting from this model have been evaluated with some published experimental data, and a good accordance between the results has been observed. It has also been concluded that exertion of normal and through thickness stresses enhances forming limits significantly.

Engineering properties of lightweight cellular cemented clay−fly ash material

Lightweight Cellular Cemented (LCC) material has wide applications in infrastructure rehabilitation and in the construction of new facilities. The roles of water content, cement content, air content, and fly ash (FA) replacement on the engineering properties, including unit weight, flow and strength of LCC clay−FA material, are investigated, analyzed and presented in this article. The engineering properties are strongly controlled by the generalized stress state, w/wL, where w is the water content and wL is the liquid limit. The FA replacement reduces wL, resulting in a change in w/wL. The workable state, recommended to produce the LCC clay−FA material, is w>1.5wL. The flowability is independent of cement content and approximated in terms of w/wL and air content in logarithmic function. The void/cement ratio (V/C), defined as the ratio of the void volume to the cement volume in the mix, is found to be the dominant parameter governing the strength development in LCC clay−FA material. The fabric (arrangement of clay particles, clusters and pore spaces) reflected from both air foam content and water content is taken into consideration by the void volume while the inter-particle forces (levels of cementation bond) are governed by the input of cement (cement volume). A strength equation in terms of V/C at a particular curing time is introduced using Abrams’ law as a basis. From the critical analysis of test results, a mix design method to attain the target unit weight, flowability and strength is suggested. This method is beneficial from both engineering and economic viewpoints.

A 3D mechanistic model for brittle materials containing evolving flaw distributions under dynamic multiaxial loading

We present a validated fully 3D mechanism-based micromechanical constitutive model for brittle solids under dynamic multiaxial loading conditions. Flaw statistics are explicitly incorporated through a defect density, and evolving flaw distributions in both orientation and size. Interactions among cracks are modeled by means of a crack-matrix-effective-medium approach. A tensorial damage parameter is defined based upon the crack length and orientation development under local effective stress fields. At low confining stresses, the wing-cracking mechanism dominates, leading to the degradation of the modulus and peak strength of the material, whereas at high enough confining stresses, the cracking mechanism is completely shut-down and dislocation mechanisms become dominant. The model handles general multiaxial stress states, accounts for evolving internal variables in the form of evolving flaw size and orientation distributions, includes evolving anisotropic damage and irreversible damage strains in a thermodynamically consistent fashion, incorporates rate-dependence through the micromechanics, and includes dynamic bulking based on independent experimental data. Simulation results are discussed and compared with experimental results on one specific structural ceramic, aluminum nitride. We demonstrate that this 3D constitutive model is capable of capturing the general constitutive response of structural ceramics.

Evaluation of leaf features in forest trees: Methods, techniques, obtainable information and limits

Tree leaves are interfaces between the whole organism and the environment. Leaves display a series of attributes that are linked to specific functions (functional leaf traits—FLT) and/or show responses to biotic and abiotic stress factors (stress response traits, SRT), which can be subdivided into: (a) morphological traits; (b) chemical traits; (c) physiological traits; (d) symptoms. The analysis of FLT is a useful tool for tree species and provenance phenotyping, due to the adaptation of trees to environmental stress. Additionally, FLT can be used as response factor in long term and large spatial scales surveys of forest conditions. Despite these potential benefits of leaf traits in the assessment of ecosystem health and functioning, leaf sampling in forests is time-consuming and costly, especially in forests with a complex vertical and horizontal structure and in remote forest areas. Once a foliar sample has been collected, many different analyses can be carried out; however, analyses should be technically simple and able to be performed within one day following the leaf collection (i.e., on fresh samples), or after air-drying the leaves themselves (analysis of dried specimens). This paper reports the results of leaf sampling and foliar analyses carried out in previous research projects and revises the current state-of-the-art. The leaf traits that are easily obtainable from leaf sampling are listed, together with the operational procedures necessary for their measurement, described in a standardized protocol. Their ecological and functional relevance is discussed in relation to their potential information (as indicators of climatic stress, drought, air and soil pollution, tree light-use and competition, plant nutritional status, health and general plant stress conditions). Finally, this review provides suggestions for the elaboration and reporting of data, and proposes some guidelines to improve the effectiveness of foliar analysis in the assessment of forest ecosystem health, properties and functioning.