Uniaxial Curves Research Articles

Numerical modeling and design sensitivity of structural and seismic behavior of UHPC bridge piers

Advanced behavior of Ultra-high performance concrete (UHPC) is attracting a growing interest among the construction industry worldwide. Currently, UHPC is commonly used in precast bridge deck joints and connections. As the UHPC market grows, the material will have a great potential to extend to larger structural applications. However, there is no guidance yet on best practices or optimum structural designs that fully utilize the UHPC superior mechanical properties for larger components. The objective of this study is to better understand the overall behavior, failure mechanism, and effect of reinforcement and design details of UHPC bridge columns using detailed finite element modeling. The pushover response of a two-column bridge pier with the typical geometry and gravity loads obtained from a representative California bridge is investigated when UHPC is used instead of conventional concrete for the columns. A detailed sensitivity/parametric analysis is conducted to assess the effect of different steel fiber ratio, longitudinal reinforcement ratio, and steel grades of reinforcement bars on the overall structural behavior of the columns. The total strain crack model, as readily implemented in DIANA FEA, is utilized with user-defined input to model the UHPC constitutive material behavior. The UHPC stress-strain relationships in tension and compression are independently defined using uniaxial curves from the literature. For comparison, the two-column bent of conventional concrete is modeled and used as the reference case to evaluate the relative increase in load capacity of the UHPC bridge columns.

Stress distribution in front of the crack – analytical solutions vs. numerical. Can the differences be minimized?

It is shown that it is possible to obtain such parameters as α and Q, which, when used in the analytical formulae proposed by O’Dowd and Shih, can lead to stress distributions similar to those obtained numerically. The numerical solution obtained after calibration of the stress-strain uniaxial curve and assuming large strains is expected to be close to the stress distribution. Thus, the analytical solution, after correction is also close to the “real” stress distribution. These new values of α and Q can now be used in fracture criteria proposed within the scope of classical nonlinear fracture mechanics.

Stress Distribution in Front of the Crack - Analytical Solutions vs. Numerical. Can the Differences be Minimized?

It is shown that it is possible to obtain such material parameters as α and Q, which, when used in the analytical formulae proposed by Hutchinson, Rice and Rosengren and O’Dowd and Shih, can lead to stress distributions similar to those obtained numerically (except for the region at the immediate crack front). The numerical solution obtained after calibration of the stress-strain uniaxial curve and assuming large strains is expected to be close to the “"real” stress distribution. Thus, the analytical solution is also close to the “real” stress distribution. These new values of α and Q can now be used in fracture criteria proposed within the scope of classical nonlinear fracture mechanics.

Multiaxial fatigue analysis of complex welded joints in notch stress approach

The notch stress method is often recommended for fatigue assessment of complex welded joints as it can consider the influence of weld toe geometric parameters on local stress. However, the application of the notch stress method is still in the exploratory stage in multiaxial fatigue compared with the traditional nominal stress method and structural stress method. Considering the geometric effect at the weld toe, the notch stress method describes the local stress with an accurate geometry model and the complex stress states at the damage point under multiaxial loading are determined in the local coordinate system. Using the notch stress method, six classical multiaxial fatigue criteria are re-evaluated for a series of typical welded joints in the literatures under both proportional and non-proportional loads. In order to evaluate the different criteria quantitatively, a statistical analysis is performed and the probability of non-conservative prediction is determined for each multiaxial fatigue criterion. The results show that the S-N curve for notch stress under multiaxial loading has a lower inclination and higher fatigue class compared with the uniaxial curve recommended by IIW.

Are the mechanical field parameters sufficient to predict uniquely the failure due to the ductile or cleavage mechanisms?

The most often observed failure mechanisms in metallic alloys are divided in two groups: brittle and ductile. Brittle failure mechanism may proceed along grain boundaries (failure due to the creep process or aggressive environment) or along the cleavage planes within the grain. Ductile fracture mechanism is most often the result of voids nucleation–growth–coalescence process or by dislocation slip along slip planes. Tests were performed at three different temperatures: +20°C, –20°C, –50°C, on five different specimen geometries, designed to provide different stress triaxialities, Lode factors as well as critical strains and stresses at the moment of final failure. Numerical analyses were performed after careful calibration of the real stress – logarithmic strains uniaxial curves. Calibration followed modified Bai–Wierzbicki procedure.

Determination of mechanical properties using sharp macro-indentation method and genetic algorithm

To determine mechanical properties, instrumented indentation is a non-destructive one where traditional methods (such as tensile test) are not accessible. However, there are several serious challenges when trying to correctly use this technique.Here, a method to predict the steel stress-strain uniaxial curve is presented accompanied by an experimental verification. A wide range of Finite Element (FE) analyses using Vickers indentation were performed in order to generate an accurate equation. The experimental validation was followed by a general new procedure to achieve material properties by maximum precision and without any insight observation. Five dimensionless parameters obtained from the load-penetration (P-h) curve were used. Error function was constituted using a linear combination by these five. Genetic Algorithm (GA) was finally employed to estimate yield stress and strain-hardening exponent. A maximum of 4 percent error was observed between the stress-strain curves obtained by the macro indentation test and those determined by the conventional tensile test.

The relevance of transverse deformation effects in modeling soft biological tissues

Hyperelastic constitutive models for anisotropic biological materials are frequently based on orthotropic incompressible stored energy functions. The material parameters of these models are then obtained through an optimization procedure as to fit some stress-strain experimental data. For example, in arterial wall mechanics the material data usually employed for the Holzapfel-Gasser-Ogden and the Gasser-Ogden-Holzapfel models are two uniaxial tension curves from circumferential and axial specimens. The transverse strains from these specimens are frequently not taken into consideration. In this paper we analyze the evolution of those strains, showing that an unrealistic behaviour may be predicted. We then show how transverse strains may be prescribed using our What-You-Prescribe-Is-What-You-Get (WYPIWYG) model in a very intuitive way, still capturing the longitudinal stress-strain behavior in an exact manner without employing any constitutive parameter. This is possible because, in contrast to what it is usually done, we exactly solve the equilibrium and compatibility equations without imposing the shape of the stored energy function. Furthermore, we show that the small strains formulation is naturally recovered and that the physical insight from the infinitesimal theory is preserved. In fact, for incompressible materials, the present approach can be considered as a natural extension of the infinitesimal continuum elastic framework to large strains. This new physical insight clearly shows that if some subclasses of orthotropic incompressible material models are determined with just two uniaxial curves, then the transverse behavior should be contrasted with additional experimental observations.

Endochronic theory of inelastisity modeling microfracture and hardening of material

This article offers a tensor-parametric variants of endochronic elasticity theory, considering mi- crotension, microfracture and hardening of material. There are given the solutions for defining rela- tions under uniaxial loading. The relationship is set between some variants of the theory. The au- thors analyzed the influence of the Endochronic parameter and the parameter of hardening material in the form of uniaxial curves of “strain~strain”. The article contains results of some numerical ex-periments, including a scheme with strain of a softening material.

Identification of the hardening behavior of solids described by three-parameter Voce law using spherical indentation

Abstract

Brain tissue deforms similarly to filled elastomers and follows consolidation theory

Slow, large deformations of human brain tissue—accompanying cranial vault deformation induced by positional plagiocephaly, occurring during hydrocephalus, and in the convolutional development—has surprisingly received scarce mechanical investigation. Since the effects of these deformations may be important, we performed a systematic series of in vitro experiments on human brain tissue, revealing the following features. (i) Under uniaxial (quasi-static), cyclic loading, brain tissue exhibits a peculiar nonlinear mechanical behaviour, exhibiting hysteresis, Mullins effect and residual strain, qualitatively similar to that observed in filled elastomers. As a consequence, the loading and unloading uniaxial curves have been found to follow the Ogden nonlinear elastic theory of rubber (and its variants to include Mullins effect and permanent strain). (ii) Loaded up to failure, the “shape” of the stress/strain curve qualitatively changes, evidencing softening related to local failure. (iii) Uniaxial (quasi-static) strain experiments under controlled drainage conditions provide the first direct evidence that the tissue obeys consolidation theory involving fluid migration, with properties similar to fine soils, but having much smaller volumetric compressibility. (iv) Our experimental findings also support the existence of a viscous component of the solid phase deformation. Brain tissue should, therefore, be modelled as a porous, fluid-saturated, nonlinear solid with very small volumetric (drained) compressibility.

Determining Representative Stress and Representative Strain in Deriving Indentation Flow Curves Based on Finite Element Analysis

A new method [1] to evaluate indentation flow curves using an instrumented indentation test has been applied to many materials for several years. Though the method produces relatively good results compared to uniaxial tensile tests, a few parameters had not been verified by theoretical or numerical methods. In this study, proportional constants of representative strain and representative stress were verified using finite element analysis and proven to be unaffected by the elastic property and strain level. The constants were generally dependent on the plastic property; however, one combination of the constants is independent of all properties. The values of this combination are consistent with early research and produced overlapping indentation flow curves with uniaxial curves.

구속응력을 받는 콘크리트 구조물 해석을 위한 콘크리트 구성모델

This paper presents a hypoplastic model for three-dimensional analysis of concrete structures under monotonic, cyclic, proportional and non-proportional loading. The constitutive model is based on the concept of equivalent uniaxial strains that allows the assumed orthotropic model to be described via three equivalent uniaxial stress-strain curves. The characteristics of these curves are obtained from the ultimate strength surface in the principal stress space based on the Willam-Warnke curve. A cap model is added to consider loading along or near the hydrostatic axis. The equivalent uniaxial curve is based on the Popovics and Saenz models. The post-peak behavior is adjusted to account for the effects of confinement and to describe the change in response from brittle to ductile as the lateral confinement increases. Correlation studies with available experimental tests are presented to demonstrate the model performance. Tests with monotonic loading on specimens under constant lateral confinement are considered first, followed by biaxial and triaxial tests with cyclic loads. The triaxial test example considers non-proportional loading.

Evaluation of anisotropic yield functions for aluminum sheets

The anisotropic behaviour of some rolled aluminum alloys is investigated using the phenomenological approach via some of the recently proposed 3-D yield functions. The directional variations of the yield stress and the so-called R-values in the plane of the sheet are predicted using the 3-D yield functions and the results are compared to previously reported experimental measurements. The hardening properties of the alloys examined are extracted from the experimental uniaxial curves along the rolling direction (RD). The forming limit curves are produced using the Marciniak and Kuczynski (Marciniak, Z., Kuczynski, K., 1967 Limit strains in the process of stret-forming sheet metals. Int. J. Mech. Sci. 9, 609–620) approach and the predictions of different yield functions are compared to the experimental curves. In general, although the refinements incorporated to the yield function proposals in chronological order produced better agreement with the experimental observations (at the expense of some complexity in the formulation), the overall performance of each criteria examined could nevertheless vary with the type of the application, as can be anticipated from the phenomenological nature of the approach.

Cyclic stress–strain behaviour of circumferentially notched cylindrical bars at high temperature

Cyclic stress–strain tests were undertaken at temperatures between 550°C and 820°C on plain and doubly-notched specimens of three alloy steels: 20Cr/25Ni/Nb, 316 and 1CrMoV respectively. Integrated axial ‘strains’ were measured between the minimum sections of the (semi-circular) notches using a longitudinal extensometer, while surface hoop strains were measured on one notch by means of a diametral extensometer. Cyclic hardening occurred in plain specimens of 316 steel, cyclic softening occurred in the 1CrMoV steel and the 20Cr/25Ni/Nb alloy showed stable behaviour. These effects were also demonstrated in notched specimens, although to a lesser degree. From hysteresis loops determined during multiple step tests, the cyclic deformation response of the notched regions was expressed in terms of an ‘equivalent gauge length’. Comparison was also made between the equivalent (von Mises) stress–strain curves deduced at a representative (skeletal) point in the minimum section with stress–strain data obtained from uniaxial (plain) specimens. The equivalent curves calculated from standard relations were found to be lower than the uniaxial curves and possible reasons for this are suggested.

A 3D hypoplastic model for cyclic analysis of concrete structures

This paper presents a hypoplastic model for three-dimensional analysis of concrete structures under monotonic, cyclic, proportional and non-proportional loading. The constitutive model is based on the concept of equivalent uniaxial strains that allow the assumed orthotropic model to be described via three equivalent uniaxial stress–strain curves. The characteristics of these curves are obtained from the ultimate strength surface in the principal stress space based on the Willam–Warnke curve. A cap model is added to consider loading along or near the hydrostatic axis. The equivalent uniaxial curve is based on the Popovics and Saenz models. The post-peak behavior is adjusted to account for the effects of confinement and to describe the change in response from brittle to ductile as the lateral confinement increases. Correlation studies with available experimental tests are presented to demonstrate the model performance. Tests with monotonic loading on specimens under constant lateral confinement are considered first, followed by biaxial and triaxial tests with cyclic loads. The triaxial test example considers non-proportional loading.

Anisotropic yield functions with plastic-strain-induced anisotropy

Abstract In most anisotropic yield functions, the stress exponent, M , associated with the shape of the yield surface is usually independent of plastic-strain accumulation. This does not allow for different work-hardening characteristics under various strain states, as has been observed in aluminum alloys. Assuming coefficients characterizing anisotropy do not change with plastic deformation, the M value should vary with plastic strain, relaxing the isotropic hardening assumption. To verify this, plane-strain tests along with numerical analysis were carried out with 2008-T4 aluminium and 70/30 brass. The effective stress and effective plastic-strain curve assuming plane strain and plane stress was fit to the corresponding stress-strain data obtained in uniaxial tension. This was done by allowing M value to vary with effective plastic-strain. Hill's 1979 (case iv),Hosford's 1979 and Barlat's 1991 (6 component) yield functions were evaluated. Results showed that, with all the yield functions tested, the aluminum exhibited substantial variation of M value especially at larger strains while the brass showed minor change. Relevant numerical analysis indicated that, even though all the yield functions showed noticeable changes of M as strain increases in order for the plane-strain curve to match with the uniaxial curve, this variation of M will not affect much to the prediction with Hosford's and Barlat's yield functions, of which the typically valid M is much higher than that of Hill's. FEM simulation of plane-strain sheet forming with 2008-T4 aluminium alloy verified that implementation of varying M -value with Hill's yield function led to better agreement with experimental measurements, while the variation of M with Barlat's yield function exhibited little influence on the strain prediction.

Nonlinear Biaxial Stress-Strain Law for Concrete

A constitutive model for plain concrete is developed to match experimental data on monotonic proportional biaxial loading. The material representation is presented in a form applicable for use with the finite curves to be duplicated from uniaxial curves. The results compare favorably with experimental curves as well as two other analytical representations of concrete under monotonic biaxial loading.

A New Slip Theory of Plasticity

A slip theory is presented for the calculation of the incremental stress-strain relations of a face-centered cubic polycrystal such as aluminum and its alloys, from its experimental uniaxial curves. This theory satisfies the conditions of equilibrium, condition of continuity of displacement as well as the slip characteristics of the component crystals. Numerical calculations based on this theory give results which are in good agreement with known experimental data on thin wall cylinders subject to different ratios of incremental axial and torsional loadings after being compressed beyond the elastic range.

Large Deflection of Axisymmetric Shell with Creep Strain

A numerical method for analyzing large deflections of shells of revolution under axisymmetric loading with arbitrary creep characteristics is given. Creep strain is treated as an equivalent external loading. Two nonlinear differential equations similar to Reissner’s equations are presented and solved by the iterative scheme and the finite difference approximation. The multiaxial stress-strain-time relationship is formulated from the empirical uniaxial curve based on the second deviated stress and strain invariants. Examples are given for a corrugated tube subject to an axial load at an elevated temperature. The results show the nonlinear effects due to finite strain and creep strain. The method shown may also be applied to axisymmetric shells subject to thermal and plastic strains.