Shear band formation analysis in soils by the subloading surface model with tangential stress rate effect

The generalized elastoplastic constitutive equation for soils is proposed based on the subloading surface model extended so as to describe the dependence of both the magnitude and the direction of inelastic stretching on the stress rate tangential to the subloading surface [Int J Plasticity 17 (2001) 117]. It would be applicable to the analysis of deformation of soils in both normal-yield and subyield states for not only lower but also higher stress ratio than that in the critical state. Then, the shear band formation in the rectangular specimen subjected to the biaxial compression under the undrained plane strain condition is analyzed by the generalized equation, and thus the condition for shear band formation and the shear band inclination are discussed in relation to material properties and the state of stress, i.e. the stress-ratio and the normal-yield ratio. These results reveal that the tangential stretching term makes easy to fulfill the necessary condition of shear band formation for not only normal-yield but also subyield states, and further the formation is affected by the material parameter prescribing the approaching degree of the stress to the normal-yield state.     

A constitutive equation for the Mullins effect in stress controlled uniaxial extension experiments

A one-dimensional constitutive equation for the Mullins effect in rubberlike materials, which is motivated by the two phase microstructural material model proposed by Mullins and Tobin [I], is developed in [2]. The constitutive equation is used in [2] to predict the effect of stress softening on the small transverse vibration of a rubber string loaded in uniaxial extension. The two functions which comprise the constitutive equation were assumed to be monotone, but no further analysis of the actual nature of these functions was necessary.In this paper, we examine more closely how the results of a stress controlled uniaxial extension experiment can be used to gain insight into the specific nature of the microstructural strain and the strain amplification functions which comprise the constitutive equation. We examine experimental representations of the two functions which are independent of any special microstructural interpretations. Stress controlled uniaxial extension experiments with buna-n, neoprene, and silicone rubber cords are examined. We demonstrate how the experimental data can be applied to yield representations of the functions of interest to within a multiplicative constant; but no attempt is made to find specific analytical representations of these functions. For buna-n and neoprene samples, we observe behavior consistent with our monotone assumptions, while anomalous behavior is observed with silicone rubber.     

A non-linear uniaxial integral constitutive equation incorporating rate effects,creep and relaxation

A previously proposed first order non-linear differential equation for uniaxial viscoplasticity, which is non-linear in stress and strain but linear in stress and strain rates, is transformed into an equivalent integral equation. The proposed equation employs total strain only and is symmetric with respect to the origin and applies for tension and compression. The limiting behavior for large strains and large times for monotonic, creep and relaxation loading is investigated and appropriate limits are obtained. When the equation is specialized to an overstress model it is qualitatively shown to reproduce key features of viscoplastic behavior. These include: initial linear elastic or linear viscoelastic response: immediate elastic slope for a large instantaneous change in strain rate normal strain rate sensitivity and non-linear spacing of the stress-strain curves obtained at various strain rates; and primary and secondary creep and relaxation such that the creep (relaxation) curves do not cross. Isochronous creep curves are also considered. Other specializations yield wavy stress-strain curves and inverse strain rate sensitivity. For cyclic loading the model must be modified to account for history dependence in the sense of plasticity.     

On the explicit differential operator formulae for the Oldroyd rate type constitutive equation

We have established the conditions under which the implicit Oldroyd type constitutive equation is equivalant to a differential type constitutive equation. Physical interpretations of these conditions are given. Finally some explicit forms for Maxwell type and Oldroyd type constitutive equations are given as illustrations.     

Photoplastic stress analysis considering rate effect

The possibilities of cellulose acetate as a photoplastic model material used in the uniaxial tension test were presented by the authors in Ref. 1. Empirical formulas were proposed for photoplastic stress analysis considering the effect of strain rate and temperature. In this paper, the stress-strain-optic laws derived from the uniaxial tension test were confirmed precisely under the biaxial stress field. Consequently, the biaxial stress-strain-optic laws of cellulose acetate can be represented by simply substituting the difference of the principal stress and the difference of the principal strain into the terms of the stress and the strain in the empirical formulas for the uniaxial stress state, respectively. Therefore, the empirical formulas considering the influence of strain rate were obtained for the photo-viscoelastoplastic plane-stress analysis during the usual static loading. This method was applied to an elastoplastic problem of a finite plate with a circular hole under uniaxial tension. The distribution of stress and strain, the stress-concentration factors in the elastoplastic region and the development of the plastic region were obtained experimentally taking the rate effect into account. As a result, the stress on a minimum section in a model increased as head speed increased. On the other hand, the distribution of strain, the development of the plastic region and the stress-concentration factors were almost never influenced by head speed.     

Elastoplastic stress analysis for cylindrical shell with flat strip imperfection

Based on unequidistant B-spline function, generalized spline subdomain displacement mode of rotational shell is obtained by taking double-direction interpolation of spline. The elastoplastic constitutive equation of shells is established by using the endochronic theory.According to the initial deflection theory of shells, the elastoplastic stress analysis of cylindrical shells with flat strip geometrical imperfection is studied. Numerical results show that the geometrical imperfection has a great effect on the stress distribution of shells.     

Study on the generalized prandtl-reuss constitutive equation and the corotational rates of stress tensor

I.IntroductionTilepl'ogl'ess11as.toifcertainextent,beenmadeintheelastic-plasticconstitutivetheoryatII[litedefbrlllations.Coil'paredwitllotherconstitutiverelations,thegeneralizedPrandtlReuss(P-R)equatiollsareextensivelystudiedandwidelyapplied.IndevelopingthegeneralizedP-Requation.itisusuallyassumedthatthedeformationrate(thesymmetricpartorvelocitygradiellt)isdecolllposedintotheelasticpartandplasticpart.TheplasticLIcf\'l.llliltlollrittcobeystilenormalfi(,xvrilleasillthecaseofinfinitcsilllnld…     

On stress rate and plasticity constitutive equations referred to a body-fixed coordinate system

Investigators have viewed the stress rate in two different ways: the material (body-fixed) point of view and the Eulerian point of view. We discuss the Zaremba–Jaumann rate and Oldroyd’s rate from the material viewpoint and apply them to the material formulation of a theory of plasticity for materials undergoing anisotropic plastic deformation. Significant advantages of the material formulation are that the derivation of equations is straight forward, the distortion of yield surface can be easily accounted for, and the issue of self-consistent elastic equation does not arise.     

An admissible form of an inelastic constitutive equation—II. Examples of elastoplastic constitutive equation

Some examples of elastoplastic constitutive equation are presented using the general theory reported in the preceding paper (Part I). Some examinations of them are given to show that the theory is self-consistent and useful especially for anisotropic materials or materials with anisotropy resulting from plastic deformation. Mises' and Yoshimura's yield functions and a kind of quadratic function are adopted as the yield function. Formulae of r-value after arbitrary pre-straining are given which are of paramount importance in the field of press-forming of sheet metals. Several examples of stress-strain curves for various loading paths are also given.     

Normal and tangential stress fluctuations during jamming

Under the application of high shear rates, concentrated suspensions of colloidal particles jam. The stress necessary to induce their flow abruptly increases and exhibits very large fluctuations. At the transition, the first normal stress difference increases and develops fluctuations. The sign of the first normal stress difference changes with the volume fraction of the suspension, which becomes dilatant at high volume fraction. I review an experimental study of this jamming transition and experimental evidence of the dilatancy of the suspension.     

Differential constitutive equation for entangled polymers with partial strand extension

Beginning with a formal statement of the conservation of probability, we derive a new differential constitutive equation for entangled polymers under flow. The constitutive equation is termed the Partial Strand Extension (PSE) equation because it accounts for partial extension of polymer strands in flow. Partial extensibility is included in the equation by considering the effect of a step strain with amplitude E on the primitive chain contour length. Specifically, by a simple scaling argument we show that the mean primitive chain contour length after retraction is L=L ₀ E ^1/2, not the equilibrium length L ₀ as previously thought. The equilibrium contour length is infact recovered only after a characteristic stretch relaxation time λ _s that is bounded by the reptation time and longest Rouse relaxation time for the primitive chain. The PSE model predictions of polymer rheology in various shear and extensional flows are found to be in good to excellent agreement with experimental results from several groups. Received: 16 July 1997 Accepted: 22 January 1998     

水泥砂浆的一个热粘弹性率型损伤本构模型

利用SHPB实验系统及自行研制的混凝土类材料快速高温加热设备,对水泥砂浆试件进行了不同 温度(20~600℃)和3种冲击速度下的实验,得到了不同温度和冲击速度下水泥砂浆试件的应力应变关系曲 线。基于ZWT粘弹性本构模型,并且考虑高温下水泥砂浆损伤演化规律都服从Weibull分布,提出了一个水 泥砂浆的热粘弹性率型损伤本构模型。通过数据拟合,获得了本构模型的相关参数,结果表明:理论预测和实 验结果吻合良好。     

Bifurcation analysis for sand samples with a non-linear constitutive equation

Summary A thoroughly non-linear constitutive equation connecting stress-rates and strain-rates is shown to yield realistic predictions for the conditions at which bifurcation may appear in the course of axisymmetric and plane homogeneous deformations of sand samples. The features of the non-homogeneous deformation after the bifurcation are also realistically predicted, a fact which is of importance in understanding and describing rupture phenomena (formation of shear bands) in geomechanics.

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Übersicht In dieser Arbeit wird gezeigt, daß eine nichtlineare Beziehung zwischen der Spannungsrate und der Deformationsgeschwindigkeit realistische Voraussagen für Verzweigungen liefert, die sich im Verlauf von achsensymmetrischer und ebener Verformung von Sandproben einstellen können. Die Geometrie der inhomogenen Verformung, die sich nach der Verzweigung einstellt, kann ebenfalls realistisch vorausgesagt werden. Diese Tatsache ist von Bedeutung für das Verständnis und die Beschreibung von Brucherscheinungen (Bildung von Scherfugen u. ä.) in der Geomechanik.

     

A constitutive equation for a non-Newtonian liquid

The K-BKZ constitutive equation for non-Newtonian liquids usually uses as invariants the coefficients of the characteristic equation of a certain finite strain tensor. We show that the use of the roots of this equation, the squares of the principal stretches, is more efficient. We use a product of the linear relaxation spectrum and a simple sum of powers of the principal stretches as the energy function in the K-BKZ theory. The predictions of this theory agree with Meissner's data to a good approximation over many decades.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.     

Extended thermodynamics and the Jeffrey's constitutive equation

Extended irreversible thermodynamics provides an evolution equation for the viscous pressure tensor which reduces to the Jeffrey's constitutive equation in the long-wave limit. in contrast with Jeffrey's equation, the equation obtained in extended irreversible thermodynamics leads to finite speed of propagation for shear pulses. The nonlocal effects are included into the theory by allowing the entropy to depend on higher-order fluxes, instead of spatial gradients. The use of the former ones is clearly advantageous in the high-frequency domain.     

Prediction of primary normal stress difference from shear viscosity data using a single integral constitutive equation

Summary Based on a single integral constitutive equation with a strain-dependent memory function, a relation between the primary normal stress function and the shear viscosity function is proposed. According to this theory, the primary normal stress function can be obtained from viscosity data by simple differentiation of the viscosity function with respect to the shear rate , and multiplication by a factor (–1/n). The material parametern is thereby associated with the strain dependence of the memory function.This relation was compared with the viscosity and primary normal stress data of six polymer melts, three polymer solutions, and an aluminium-soap solution, which were measured by several research groups and are available in the literature. In spite of the vast differences in physical constitution and chemical structure of the melts and solutions considered, agreement between predicted and measured values was encouraging.

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Zusammenfassung Ausgehend von einer Zustandsgleichung vom Integraltyp mit deformationsabhängiger Gedächtnisfunktion wird eine einfache Beziehung zwischen der ersten Normalspannungsfunktion und der Scherviskositätsfunktion vorgeschlagen. Nach dieser Theorie kann man die erste Normalspannungsfunktion aus Viskositätsdaten erhalten, indem man die Viskositätsfunktion nach der Schergeschwindigkeit ableitet und den entstehenden Ausdruck mit einem Faktor (–1/n) multipliziert. Dabei hängt die Materialgrößen mit der Deformationsabhängigkeit der Gedächtnisfunktion zusammen.Diese Beziehung wurde mit den Viskositäts- und Normalspannungsdaten von sechs Polymerschmelzen, drei Polymerlösungen und einer Aluminiumseifenlösung verglichen, die von verschiedenen Forschergruppen gemessen wurden und die in der Literatur verfügbar sind. Trotz der großen Unterschiede im physikalischen Zustand und in der chemischen Struktur der betrachteten Schmelzen und Lösungen wurde eine gute Übereinstimmung zwischen der Theorie und den experimentellen Daten gefunden.

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With 9 figures and 2 tables