Inelastic Volume Research Articles

Nanoindentation creep dependent deformation process of silica and soda lime silicate glass

In this study, the time dependent deformation of soda lime silicate and silica glass surfaces is investigated with nanoindentation creep tests. The results reveal that the nanomechanical properties of both glasses decrease with holding time. In addition, a quick recovery in the residual depth of nanoindentation imprint of both glasses has also been found after the indenter tip is retracted and before atomic force microscopy. Furthermore, the inelastic volume of glass surfaces within nanoindentation tests is quantified into densified and shear flow with annealing, and it is found that both the densified and shear flow volume of both glasses increase with holding time, but the recovery ratio of both glasses decreases with holding time, which is due to a faster increase in indentation volume with holding time compared with densified volume. Finally, a model of creep-induced deformation of glass surfaces is proposed to complement the creep-induced deformation process of silicate glasses.

Mechanical characteristics and permeability evolution of salt rock under thermal-hydro-mechanical (THM) coupling condition

The mechanical characteristics and gas tightness of salt rock play a decisive role in the safety evaluation of salt caverns. Triaxial compression coupled permeability tests were performed on Chinese impure salt rocks under thermal-hydro-mechanical (THM) coupling condition. The results revealed that the peak strength decreased with the decrease of confining pressure and the increase of temperature. The permeability of salt rock decreased with the increase of confining pressure, while the increase of temperature led to an increase in permeability under lower effective confining pressure, and a decrease in permeability under higher effective confining pressure. Moreover, there is a critical effective confining pressure, which controls the evolution trend of salt rock permeability with temperature. The dilatancy boundary and failure strength envelope of impure salt rock at different temperatures were given, which can be expressed by exponential and hyperbolic functions respectively. An empirical formula for the evolution of the final permeability of the salt rock with the effective confining pressure at different temperatures was established. Considering the effects of effective confining pressure and inelastic volume strain, a new model was proposed to predict the permeability evolution of salt rock under THM coupling condition at different temperatures. The results of this paper can help understand the mechanical characteristics and permeability evolution of salt rock under THM coupling condition, and provide the necessary basis for the safety evaluation of salt caverns.

Experimental study on the effective measurement of the total homogenization temperature of high internal pressure CO<sub>2</sub>-rich fluid inclusions in orogenic gold deposits: A case study of the Yufeng gold deposit, Eastern Tianshan Orogen

形成于中-深成条件下的造山型金矿的成矿流体属于H₂O-NaCl-CO₂体系，金矿化阶段的石英中发育大量的富CO₂流体包裹体。长期以来，用常规冷热台对其进行显微测温热力学研究时，较高的内压力会造成绝大多数CO₂充填度>0.4的包裹体在尚未达到完全均一状态前就发生爆裂或泄露，导致所测的完全均一温度值明显偏离真实值，甚至无法测得完全均一温度，这制约了对造山型金矿床成矿条件的认知。针对该问题，本文以东天山玉峰金矿床富金矿石中的H₂O-NaCl包裹体（A型）和H₂O-NaCl-CO₂包裹体（AC型）为研究对象，介绍了利用最新型热液金刚石压腔（HDAC-VT型号）测试富CO₂高内压的AC型包裹体完全均一温度的实验流程与结果，并将实验数据与Linkam THMSG600冷热台测得的A型、AC型包裹体的完全均一温度进行了对比分析。实验结果表明，热液金刚石压腔能够有效阻止富CO₂流体包裹体在升温过程中发生爆裂、泄露等非弹性体积改变现象的发生，从而获得有效的完全均一温度。同时，本文还提出了一个新的压力-温度拟合线两线相交法，对热液金刚石压腔所测富CO₂流体包裹体的完全均一温度数据进行校正，以最大程度上减少外压力造成的影响，获得更为接近真实成矿流体的完全均一温度。基于此，获得玉峰金矿床的成矿温度和成矿压力分别为312~343℃和181~268 MPa。本文的实验研究工作显示了热液金刚石压腔在中-深成造山型金矿富CO₂成矿流体的研究领域具有重要应用前景。;The ore-forming fluids of orogenic gold deposits normally belong to a H₂O-NaCl-CO₂ system, and there are usually lots of CO₂-rich fluid inclusions hosted in the quartz at the gold mineralization stage. The trapped fluid inclusions in this type of gold deposit usually have high internal pressures since they are formed at a mid-deep depth in the earth's crust. For a long time, almost all researchers carry out the homogeneous thermomechanical properties study on CO₂-rich inclusions (CO₂ phase filling ratio>0.4) under a 1atm pressure condition. However, when the internal pressure is greater than the external confining pressure, the fluid inclusion will be deformed, such as stretching, leakage and decrepitating, which eventually lead to the measured total homogenization temperature (THT) deviate significantly from the actual value, or even cannot be measured. In order to solve these problems, we choose H₂O-NaCl (A-type) and H₂O-NaCl-CO₂ (AC-type) inclusions in quartz minerals from Yufeng orogenic gold deposit in Eastern Tianshan Orogen as the research object to measure their THTs with high internal pressures by using the latest hydrothermal diamond-anvil cell (HDAC; HDAC-VT type). At the same time, this manuscript compares the THTs obtained by a HDAC-VT with those by a Linkam THMSG600 heating-freezing stage. Our experimental results show that the HDAC can effectively prevent the inelastic volume change of CO₂-rich fluid inclusions during heating process, and consequently obtain an effective THT. In order to minimize the influence of external pressure on the measured THTs and obtain more reliable and accurate metallogenic temperatures, we propose a new two pressure-temperature fitting lines intersection method to calibrate the THTs of CO₂-rich fluid inclusions measured by a HDAC. Based on these improvements, the ore-forming temperatures and pressures of the Yufeng gold deposit are estimated to be 312~343℃ and 181~268MPa, respectively. The experimental work in this paper suggests that the HDAC has an important application prospect in the research field of CO₂-rich fluid inclusions from orogenic gold deposits.

Decoupling of indentation modulus and hardness in silicate glasses: Evidence of a shear- to densification-dominated transition

Prediction of the composition dependence of Calcium aluminosilicate (CAS) glasses mechanical properties is essential to aid in the development of new glasses with tuned properties for specific applications. In this investigation, we use microindentation to measure the hardness (H) and modulus (M) and the inelastic volume that dissipates the inelastic energy is elucidated using annealing and AFM to decompose it into contributions from densification and shear flow. We chose a family of CAS glasses with a constant CaO/Al2O3 ratio but varying SiO2 content, as well as a commercial >99% SiO2 glass. We observe a decoupling between M & H, M systematically decreases upon increasing SiO2 content, however, H exhibits a non-monotonic behavior with a minimum around 79% SiO2. The inelastically deformed volumes show a high contribution from shear flow for SiO2-poor glasses and a high contribution of densification for SiO2-rich glasses, demonstrating a shear to densification dominated transition between these two extremes.

Is crystal plasticity non-conservative? Lessons from large deformation continuum dislocation theory

Crystal plasticity for large plastic deformations is usually modeled as volume conserving. However, there is experimental evidence that huge amounts of vacancies are produced during plastic deformation. The only regularly considered mechanism of non-conservative dislocation motion is climb of edge dislocations, but this is supposed to be enabled by high vacancy concentrations rather than producing them. In the current contribution we show that large deformation kinematics applied to evolving fields of geometrically necessary dislocations accounts for kinking and jogging of dislocations when cutting through one another. Therefore, dislocation fields in multiple slip situations will not be confined to slip planes and their motion will necessarily involve non-conservative motion of jogs. Because inelastic volume changes in crystals are only possible in the presence of point defects, we find that large deformation continuum dislocation theory must be coupled to an evolving vacancy concentration. We suggest to treat vacancy diffusion as an independent inelastic deformation mechanism yielding a three-way multiplicative decomposition of the deformation gradient. Small numerical examples are used to illustrate the kinematics of emerging and moving kinks and jogs, and the ensuing production of vacancy concentration. Continuum thermodynamics is used to derive coupled evolution equations for geometrically necessary dislocation density vectors and vacancy concentration. We conclude that large deformation crystal plasticity as an averaged theory of dislocations may not be conservative and that the usual kinematic assumption Lp=∑sγ˙sms⊗ns does not hold in multiple slip deformation.

A phenomenological model for the inelastic stress–strain response of a potato tuber

In this article, we synthesise concepts from the theories of plasticity, volumetric growth, and pseudo-hyperelasticity to develop a phenomenological constitutive model for potato tuber tissue. Here, particular focus is laid on the inhomogeneity of the constitutive behaviour across different anatomical zones of a potato tuber. The internal stresses are expressed in terms of an elastic contribution to the local strains and are controlled by a kinetic memory variable that traces the past strain history. Inelastic strains, by contrast, are considered as a combination of a plastic component and a pure volumetric component mimicking the eviction of fluid. Considering tubers of cultivar Belana, we identify the relevant constitutive parameters based on stress–stretch curves obtained from single-cycle uniaxial compression experiments on samples extracted from different anatomical tuber parts. The stress–strain response of the tissue is characterised by a pronounced and well-reproducible hysteresis which displays slight stiffening along the initial loading branch. Considering the parenchyma microstructure, the significant reduction in tissue turgidity on unloading may be caused by the eviction of intracellular fluid from the parenchyma cells and an inelastic cell wall response. The experiments show that the inelastic volume change varies approximately linearly with the axial stretch. Additionally, histological analyses are presented in an attempt to quantify microstructural inhomogeneities.

New insights into the indentation size effect in silicate glasses

When glasses resist permanent deformations, for applications such as a windshield or the cover of a touch-screen device, it is important to understand how they respond to inelastic energy dissipation processes. Many glasses exhibit the so-called Indentation Size Effect (ISE), where the indentation hardness is dependent on the maximum force exerted on the probe. In this study, we perform microindentation on silica and soda lime silicate glasses over a wide range of maximum forces and extract the Vickers hardness by the Oliver and Pharr method. The inelastic volume responsible for dissipating the inelastic energy is decomposed into densification and plastic flow, using surface topography and annealing. We show that the ISE is intimately linked to these mechanisms. Finally, we hypothesize the cause of the ISE is an increase in plasticity in a zone of material under the indenter probe experiencing reduced viscosity due to high strain rates and shear thinning.

Anisotropic finite hyper-elastoplasticity of geomaterials with Drucker–Prager/Cap type constitutive model formulation

The formulation of large strain anisotropic hyper-elastoplasticity of geomaterials is examined. Attention is given to the role of structure tensors (also called fabric tensors), especially in context of the Eshelby–Mandel stress and large inelastic volume changes attributable to porosity. Both (hyper-)elastic and inelastic orthotropic symmetry, reducing to the particular case of transverse isotropy, are considered. Specific material assumptions and constitutive choices are identified for the development of a novel Anisotropic Drucker–Prager/Cap (ADPC) model formulated within the intermediate configuration consistent with multiplicative split of the deformation gradient. The model is calibrated to existing experimental measurements, including high pressure large strain triaxial compression of lithographic (Solnhofen) limestone and triaxial compression measurements on Tournemire shale assessing elastoplastic anisotropy. Manifest implications of constitutive theory are investigated, including consequences of recognizing (or not) the Eshelby–Mandel stress as energy conjugate to the plastic velocity gradient and including (or not) contribution from the skew-symmetric parts of the Mandel stress to the plastic anisotropy. Numerical simple shear experiments and large deformation simulated indentation experiments are provided in order to investigate model predictions and demonstrate the overall robustness in finite element modeling.

Nonlinear thermomechanics for analysis of weak shock profile data in ductile polycrystals

Modeling of shock compression of isotropic, polycrystalline elastic-plastic solids that deform by dislocation glide is undertaken. Internal energy density of the material depends on a thermoelastic Eulerian strain tensor, entropy, and an internal state variable representative of dislocation density. Dislocation glide is incompressible, but inelastic volume changes arising from residual local strain fields and core effects of dislocations are captured. A semi-analytical method is advanced for extracting inelastic constitutive response information from particle velocity histories of polycrystalline samples under planar shock loading. The only parameters entering the procedure are fundamental thermoelastic properties and assumed bounds on the fraction of plastic work corresponding to energy storage of generated dislocations in the lattice. Densities of statistically stored and geometrically necessary dislocations, in addition to shear stress, plastic strain, plastic strain rate, and temperature, are outcomes of the analysis. The model is implemented for polycrystalline aluminum and copper. Certain results are compared with others in the literature obtained under different kinematic and thermodynamic assumptions. Effects of choices of nonlinear thermoelastic formulations and higher-order elastic constants on extracted deviatoric stress profiles can be substantial. Stored energy of dislocations and commensurate residual volume changes together can notably affect extracted shear stress. Stored energy significantly affects predicted entropy production and its contribution to temperature rise. The calculated density of geometrically necessary dislocations required to maintain compatibility of total strain across the structured steady plastic wave front is small relative to the line density of total dislocations.

Dilatancy as a measure of fracturing development in the process of rock damage

Abstract Dilatancy understood as inelastic volume changes of rock under conditions of differential stress not only foreshadows brittle rock failure [1, 2, 3], preceding such phenomena as earthquakes or mining-induced rockbursts, but also defines the kinematics of rock deformation [4, 5]. This article shows that there is a quantitative relationship between the dilatancy occurring at different levels of rock load and the development of fracturing that accompanies the process of rock degradation. Based on the mechanical laboratory test results of Wustenzelleer fine-grained sandstone, the article presents analyses of plastic volume changes in rock samples and a certain measure of fracture development accompanying the process of their degradation. Thanks to the characteristics obtained through mechanical tests with unloading, the inelastic volumetric strain of the samples was established, and the scalar damage variable (CDM - continuum damage mechanics approach) was determined at various load levels. These results demonstrated the relationship between dilatancy and rock stiffness changes describing the development of fracturing. Three load intervals were determined, in which the nature of the analysed phenomena, i.e. dilatancy and damage describing the development of cracks, undergoes significant changes. The highest intensity of both phenomena was obtained in the post-critical load range, between the ultimate strength limit and the residual strength of the samples, in which rock samples undergo the process of degradation the most. Obtained dependencies allow the correlation to be taken into account in damage mechanics models describing the variable stiffness of rocks in the loading process.

An experimentally constrained constitutive model for geomaterials with simple friction–dilatancy relation in brittle to ductile domains

Complexity of the mechanical behavior of geomaterials makes it very difficult to formulate constitutive models (both at micro- and macro-scales) valid for different loading conditions and deformation regimes. To make progress in understanding this complexity, we take advantage of the large set of data for the synthetic rock analog GRAM1, a granular, frictional, dilatant, and cohesive material formed of bonded rigid particles. We use also data from literature for two real rocks. All data are from conventional triaxial tests conducted for a wide range of confining pressures covering the material behavior from brittle fracturing to ductile flow. The data processing allowed to define both the yield function and the inelastic volume strain as functions of the mean stress σm and the accumulated inelastic strain γ¯p. The internal friction coefficient and dilatancy factor calculated from these functions were shown to be different but evolving very similarly with σm and γ¯p for all the three materials. This allowed to relate the yield and plastic potential functions and thereby to complete the constitutive formulation within the framework of the classical elastoplasticity theory. The obtained results are also used to elucidate the relation between the yield surface and failure envelope as well as the meaning of the internal friction coefficient derived from the failure envelope which is routinely used in geomechanical applications and which is very different from the internal friction coefficient derived from the yield function.

Phase Field Theory and Analysis of Pressure-Shear Induced Amorphization and Failure in Boron Carbide Ceramic

A nonlinear continuum phase field theory is developed to describe amorphization of crystalline elastic solids under shear and/or pressure loading. An order parameter describes the local degree of crystallinity. Elastic coefficients can depend on the order parameter, inelastic volume change may accompany the transition from crystal to amorphous phase, and transitional regions parallel to bands of amorphous material are penalized by interfacial surface energy. Analytical and simple numerical solutions are obtained for an idealized isotropic version of the general theory, for an element of material subjected to compressive and/or shear loading. Solutions compare favorably with experimental evidence and atomic simulations of amorphization in boron carbide, demonstrating the tendency for structural collapse and strength loss with increasing shear deformation and superposed pressure.

Creep and Dilatancy Behavior of Rock Salt around Underground Caverns

In the present paper, the time-dependent behavior of a salt cavern was analyzed using finite element method. A viscoplastic model considering inelastic volume changes was utilized to describe the time-dependent mechanical behavior of rock salt. The model was then implemented in a finite element procedure to analyze the stress and deformation of rock salt around an underground cavern using an axisymmetric representation. Finally, finite element analyses results were compared with those obtained by a viscoplastic model neglecting inelastic volume changes.

Finite element modeling of concrete structures reinforced with internal and external fibre-reinforced polymers

Externally bonded fibre-reinforced-polymer (FRP) laminates and fabrics have been successfully used for strengthening damaged or deficient concrete members, whereas internal FRP reinforcements are becoming an efficient alternative to steel reinforcement, particularly in corrosive environments. Despite the enormous progress that has been observed in the last decade, further research is still required to consolidate recent developments and expand the scope of application of FRPs for structural uses. Nonlinear finite element analysis combined with laboratory testing constitutes an efficient approach for pursuing this objective. The scope of this paper is to illustrate, through a selection of a wide variety of typical applications, the contribution of a refined three-dimensional (3-D) constitutive model for investigating the nonlinear response of concrete structures reinforced with internal and external FRPs. The analyses are carried out using a general and portable constitutive concrete model implemented as a user-defined subroutine at Gauss integration point level in commercial finite element software. The constitutive law follows a 3-D hypoelastic approach that models the nonlinear behaviour of concrete using a scalar damage parameter that accounts for the anisotropic behaviour of partially confined concrete and the inelastic volume expansion upon reaching the peak strength. In tension, the model adopts a macroscopic approach that is directly integrated into the concrete law. It simulates implicitly the reinforcing bar – concrete interaction using tension-stiffening factors modified according to the nature of reinforcement that vary as a function of the member strain. The applications include results of well-known test series published in the literature on beams with external and internal FRP reinforcement, slabs with internal reinforcements, bond failure analysis of external FRP, and the effect of confinement on the behaviour in compression of circular and square elements. The paper demonstrates the ability of the concrete model to correctly simulate the behaviour of structural elements reinforced with FRPs at service load level and reproduce failure mechanisms and loads that are consistent with the experimental observations.Key words: constitutive model, nonlinear analysis, finite element, reinforced concrete, glass-fibre-reinforced polymer (GFRP), carbon-fibre-reinforced polymer (CFRP), strengthening, steel.

Shear and compaction band formation on an elliptic yield cap

Conditions for shear and compaction localization are examined for stress states on an elliptic yield cap in the space of Mises equivalent shear stress and mean normal compressive stress σ. Localization is predicted to occur when the slope of the hydrostatic stress versus inelastic volume strain curve falls to a critical value kcrit. When applied to the standard triaxial test, the analysis reveals a transition from compaction localization to shear localization as lateral confining stress σc decreases below a particular value. This value also corresponds to the largest value of kcrit, which is zero if normality is satisfied and slightly positive if not, for either shear or compaction bands. For σc larger than the transition value, compaction bands are the only mode of localized deformation predicted, and kcrit becomes increasingly negative with increasing σc. For σc smaller than the transition value, both shear bands and compaction bands are possible, but the value of kcrit is larger for shear bands. These results are consistent with experimental observations of compaction bands on relatively flat portions of the stress versus strain curve, corresponding to kcrit ≈ 0, and with their occurrence in a limited range of σc. As σc is decreased from the transition value, the predicted angle between the normal to the shear band and the maximum compression direction increases rapidly from 0° (for a compaction band) to 20°–30°. This rapid increase provides a possible explanation for the infrequent observation of very low angle shear bands.

Effects of inelastic volume increase on fractured rock behaviour

This study investigated the correlation between inelastic volumetric deformation and the bearing capacity of fractured rock. Triaxial compression tests on marble specimens have been performed under constant and controlled deformation rates using a servocontrolled loading machine supported by an electro-hydraulic volume-measuring unit connected to a data acquisition system. The triaxial compression tests were carried out on two different marble samples under constant confining pressures ranging from 1–12.5 MPa. The results indicate that stress–inelastic volumetric strain curves are generally linear through the initial part of the post-failure curve where fractured rock behaviour is dominant. It was found that brittle rock loses its strength in the post-failure region depending on volumetric deformation; the slopes of post-failure curves change with an increase in confining pressure. As a result it is concluded that there is a relationship between strength and volumetric strain of fissured rock.

Inelastic constitutive properties and shear localization in Tennessee marble

AbstractThe inelastic response of Tennessee marble is modelled by an elastic plastic constitutive relation that includes pressure dependence of yield, strain‐softening and inelastic volume strain (dilatancy). Data from 12 axisymmetric compression tests at confining pressures from 0 to 100 MPa are used to determine the dependence of the yield function and plastic potential, which are different, on the first and second stress invariants and the accumulated inelastic shear strain. Because the data requires that the strain at peak stress depends on the mean stress, the locus of peak stresses is neither a yield surface nor a failure envelope, as is often assumed. Based on the constitutive model and Rudnicki and Rice criterion, localization is not predicted to occur in axisymmetric compression although faulting is observed in the tests. The discrepancy is likely due to the overly stiff response of a smooth yield surface model to abrupt changes in the pattern of straining. The constitutive model determined from the axisymmetric compression data describes well the variation of the in‐plane stress observed in a plane strain experiment. The out‐of‐plane stress is not modelled well, apparently because the inelastic normal strain in this direction is overpredicted. In plane strain, localization is predicted to occur close to peak stress, in good agreement with the experiment. Observation of localization on the rising portion of the stress–strain curve in plane strain does not, however, indicate prepeak localization. Because of the rapid increase of mean stress in plane strain, the stress–strain curve can be rising while the shear stress versus shear strain curve at constant mean stress is falling (negative hardening modulus). Copyright © 2001 John Wiley & Sons, Ltd.

Theory of compaction bands in porous rock

Compaction bands are narrow planar zones of localized purely compressive (without shear) deformation that form perpendicular to the most compressive principal stress. Such bands have been observed in high porosity rocks in the laboratory and in the field. Because compaction presumably decreases permeability, these bands can act as barriers to flow within reservoirs. Reexamination of the results of Rudnicki and Rice (J. Mech. Phys. Solids, 1975) for shear localization, with corrections by Perrin and Leblond (J. Appl. Mech., 1993), reveals that they admit solutions for compaction bands in a range of parameters that is representative of porous rock. Solutions for compaction bands are possible when the inelastic volume deformation is compactive and is associated with a “cap” on the yield surface. The expression for the critical hardening modulus (related to the slope of the shear stress vs. shear strain curve at constant mean stress) at which compaction bands are predicted to form differs from that for shear localization. For parameters representative of porous rock, axisymmetric compression is the most favorable deviatoric stress state for formation of compaction bands. Comparison of conditions for shear localization and compaction band formation suggests that either may occur depending on the stress path and magnitude of the confining stress.

Conditions for compaction bands in porous rock

Reexamination of the results of Rudnicki and Rice for shear localization reveals that solutions for compaction bands are possible in a range of parameters typical of porous rock. Compaction bands are narrow planar zones of localized compressive deformation perpendicular to the maximum compressive stress, which have been observed in high‐porosity rocks in the laboratory and field. Solutions for compaction bands, as an alternative to homogenous deformation, are possible when the inelastic volume deformation is compactive and is associated with stress states on a yield surface “cap.” The cap implies that the shear stress required for further inelastic deformation decreases with increasing compressive mean stress. While the expressions for the critical hardening modulus for compaction and shear bands differ, in both cases, deviations from normality promote band formation. Inelastic compaction deformation associated with mean stress (suggested by Aydin and Johnson) promotes localization by decreasing the magnitude of the critical hardening modulus. Axisymmetric compression is the most favorable deviatoric stress state for formation of compaction bands. Predictions for compaction bands suggest that they could form on the “shelf” typically observed in axisymmetric compression stress strain curves of porous rock at high confining stress. Either shear or compaction bands may occur depending on the stress path and confining stress. If the increase in local density and decrease in grain size associated with compaction band formation result in strengthening rather than weakening of the band material, formation of a compaction band may not preclude later formation of a shear band.

Failure analysis of R/C columns using a triaxial concrete model

Inelastic failure analysis of concrete structures has been one of the central issues in concrete mechanics. Especially, the effect of confinement has been of great importance to capture the transition from brittle to ductile fracture of concrete under triaxial loading scenarios. Moreover, it has been a challenge to implement numerically material descriptions, which are susceptible to loss of stability and localization. In this article, a novel triaxial concrete model is presented, which captures the full spectrum of triaxial stress and strain histories in reinforced concrete structures. Thereby, inelastic dilatation is controlled by a non-associated flow rule to attain realistic predictions of inelastic volume change at various confinement levels. Different features of distributed and localized failure of the concrete model are examined under confined compression, uniaxial tension, pure shear, and simple shear. The performance at the structural level is illustrated with the example of a reinforced concrete column subjected to combined axial and transverse loading.