Stress-deformation Relation Research Articles

A FINITE AMPLITUDE WAVE PROPAGATION PROBLEM FOR A PSEUDO-ELASTIC SOLID

A constitutive model for an incompressible pseudo-elastic rubberlike solid is proposed. The term pseudo-elastic refers to a solid which has different stress-deformation relations for loading and unloading so that a hysteresis effect is exhibited, and the virgin reference configuration is recovered when the solid is unloaded. The model is based on an isotropic hyperelastic strain energy function W(F), where F is the deformation gradient tensor and det[F] = 1, and the stress-deformation relation for loading from an unstressed state is the same as that for a hyperelastic solid with strain energy function W(F). Loading (unloading) is assumed to occur when dW(F) > 0 (dW(F) < 0) and neutral loading when dW(F) = 0. Finite amplitude wave propagation in a semi-infinite pseudo-elastic string, which is subjected to a cycle of tensile loading and unloading at the free end is considered. Numerical results, obtained from an adaptation of the method of characteristics, are presented for a particular form of the proposed...

Modeling Injection Molding Processes with Deformable Porous Preforms

This paper deduces a new model aimed at simulating injection molding processes under isothermal conditions. These processes can be generally stated as infiltration problems in initially dry porous materials. The spatial domain is then divided by the infiltration front into two time-dependent subdomains, the dry and the wet porous preforms, both being allowed to deform under the action of the liquid pressure. It is shown that the model calls for the definition of the stress-deformation relationship of both the dry and the wet preforms, which are assumed to behave elastically and inelastically, respectively. The coupled flow/deformation problem in the two regions (separated by an interface) is formulated with the corresponding boundary conditions and with the proper evolution equations determining the motion of the boundaries. The mathematical problem is solved numerically, highlighting the importance of inertial terms in the early stage of the infiltration and focusing on the influence of the mechanical properties of the material and on the deformation of the preform during the infiltration process.

Estimation of Debonding Temperature of Joints Made from Two Different Materials by Using Energy Release Elements.

A new method using energy release elements(EREs)for evaluating adhesive strength, which we suggested in our previous paper, is used to estimate debonding temperature of joints made from resin and steel. This estimated temperature agrees well enough for practical use with those measured by Terasaki et al. over all sizes and shapes of test specimens. This agreement shows that the new method can be widely applied for estimating adhesive strength. When joint components are thicker than 2 mm and bonding angle is greater than 90 degrees, detail of the stress-deformation relation of ERE does not affect the estimation of debonding temperature. And different ERE characteristics determined by using one experimental datum give almost the same estimated temperature. These estimated temperature agree with those measured to the same extent as those estimated by Terasaki et al. by using energy release rate g as an evaluating parameter.

Pure volumetric compaction of a prestressed nonlinear hyperelastic solid with reference to poroelastic materials

A homogeneous hyperelastic solid is considered. It is assumed that the pressure of the solid is only a function of volumetric deformation. This requires that the free energy of the material is the sum of two terms. The first term is a function of volumetric deformation and the second is a function of distortion. It is further assumed that at any state of stress, the material can be uniformly and isotropicly compacted by application of boundary tractions. Static equilibrium restricts the functional form of the free energy term that depends on volumetric deformation. It is shown that the boundary traction that induces a uniform compaction in the prestressed material is the sum of a uniform pressure applied to the entire boundary, and a constant proportion of the traction that induced the prestress. The relevance of this to the mechanical behavior of saturated nonlinear poroelastic solids is discussed. The resulting nonlinear stress-deformation relation allows to identify the part of deformation that is induced by pore pressure and the part induced by differential stress.

Simulation of fracture in a quasi-brittle material in direct tension–a lattice model

In this paper, results of simulation of stress-deformation response of a quasi-brittle material, like concrete, in direct tension using lattice model earlier proposed by Bhattacharya [2] (Bhattacharya GS. Simulation of tension softening and size effect in quasi-brittle materials–by lattice and fractal models. PhD thesis submitted to Indian Institute of Science, Bangalore, 1996.) are reported. A total of nine direct tension plate specimens are taken from the literature for which experimental stress–deformation relations are available. Seven of them are double edge notched (DEN) and two are single edge notched (SEN) specimens. The results of simulations are compared with experimental results and it is encouraging to see that the results match closely. An expression to find a parameter α used in this model is proposed. Thus the load–deflection behaviour including the phase of softening of any direct tension plate specimen of given dimensions can be approximately simulated. The model idealizes concrete as a heterogeneous material and heterogeneity is introduced through a technique incorporating singular fractal functions. A remarkable feature of this model is that the simulation can be carried out with a small number of members in the lattice and there is no need to remove fractured members in successive load steps. Each member is made to undergo tension softening rather than brittle fracture

Parametric sensitivity of fracture behaviour of concrete

The fictitious crack method (FCM) is applied to determine the load-deflection diagrams of notched plain concrete beams under three-point bending using various forms of strain softening in the stress-deformation relationship. The results indicate that there is a need to determine a more realistic relationship.

Large strain analysis of soft biological membranes: Formulation and finite element analysis

This paper presents a general formulation of thin incompressible membranes and investigates the behavior of soft biotissues using the finite element method. In particular the underlying hyperelastic model is chosen to examine the highly non-linear constitutive relation of blood vessels which are considered to be perfectly elastic, homogeneous and (nearly) incompressible. First, the stress-deformation relation and the elastic tangent moduli are derived in a very general material setting which is subsequently specified for blood vessels in terms of Green-Lagrangian strains. Based on the principle of virtual work the finite element equations are provided and briefly discussed. Consistent linearization of the weak form of equilibrium and the external pressure term ensures a quadratic convergence rate of the iterative solution procedure. On the computational side of this work an effort was undertaken to show a novel approach on the investigation of soft tissue biomechanics. Representative numerical analyses of problems in vascular mechanics are discussed that show isochoric finite deformations (large rotations and large strains). In particular, a numerical simulation of the interaction between an inflated balloon catheter and a plaque deposit on the wall of a blood vessel is presented.

Modeling of Stress-deformation Relationships in a Collision Belt:Taiwan

Seismic refraction data from onshore and offshore experiments in theeastern-northeastern Taiwan region were used to study the velocity structureby the two-dimensional ray-tracing method.In the velocity model,astructural fault boundary located beneath the Longitudinal Valley was usedto separate the northern Coastal Range(CR)on the eastern side from theeastern flank of the Central Range(EFCR)on the western side.The P-wave velocities from the surface to the depth of 12-15 km varied from 3.9 to5.8 km/s beneath the CR and from 4.8 to 6.1 km/s beneath the EFCR.Comparingthe velocity structures along various latitudes,it was found that theCR extends northward to 24.2°N.The velocity structures of the CR,theHsinchen Ridge(HR)and the Yaeyama Ridge(YR)indicate that the HRand the YR both belong to the same type of tectonic unit as the CR.To thenorth of 24.2°N,the velocity structure of the Ilan Ridge(IR),located betweenthe EFCR and the southwestern end of the Ryukyu arc,is similar tothat of the EFCR;hence,probably indicating it is the northeastern extensionof the EFCR.This suggests that the EFCR bends eastward and belongsto the same tectonic unit as the southwestern Ryukyu arc.From acomparison of the velocity structures of the CR,EFCR and of other typicalcontinental arcs,orogens and oceanic arcs in the literature,it can be concludedthat the northern CR belongs to an oceanic arc and that the EFCRis a continental arc.Further more,from the analysis of the velocity structuresbeneath the CR and EFCR,it is believed that the upper crust of theCR is weaker in strength than the EFCR,which means that the arc-continentcollision is not an appropriate model for the formation of Taiwan island.

International comparison of coupled thermo-hydro-mechanical models of a multiple-fracture bench mark problem: DECOVALEX phase I, bench mark test 2

An international inter-code bench mark comparison has been carried out under the auspices of DECOVALEX, an international cooperative project for the development of coupled (thermo-hydro-mechanical) models and their validation against experiments in nuclear waste isolation. The subject bench mark problem, designated BMT2, involves a system of nine blocks of intact hard rock separated by two pairs of relatively soft fractures. The coupled processes of conductive, convective and dispersive heat transport; nonisothermal fluid flow; and thermo-mechanical stresses and displacements in both the intact rock and the fractures were simulated. Five research teams representing four European and North American countries participated in this bench-marking exercise using two fundamentally different types of numerical models, i.e. finite-element models and distinct-element models. All teams simulated the fractures as discrete features. For the coupled hydro-mechanical behaviour of the fractures the modelling teams have used either the empirical, nonlinear Barton-Bandis-Bakhtar constitutive relation or a simpler Coulomb friction like stress-deformation relation in conjunction with Snow's parallel-plate fracture flow model. For this bench mark problem it was found that (1) heat convection significantly affects the distribution of temperature, thermal stresses and displacements; and (2) the predominant coupled effect is fracture closure caused by thermal expansion of the rock blocks. Comparison of the modelling results obtained by the various teams have indicated very good overall agreement. In particular, the results calculated using finite-element and distinct-element codes are quite similar. Practically all the discrepancies among simulation results can be ascribed to different idealization, constitutive relations, or parameter values adopted by the participating research teams. In-depth discussions among participants have enhanced our understanding of the simulated processes and provided guidance for model improvements.

Direct tension tests on notched and un-notched plain concrete specimens

Direct tension tests were conducted on plain concrete specimens in a stiff loading frame which allowed complete stress-deformation relations to be obtained. The suitability of the specimen shape and the testing apparatus was verified by finite-element analysis. The behaviours of notched and un-notched specimens with different water-cement ratios were examined and compared. Piecewise linear and exponential stress-deformation relations are proposed and interpreted physically. Relationships between fracture parameters fracture energy and brittleness indices) and basic strength properties are also presented.

Fracture mechanics concepts applied to moment capacity and rotational capacity of reinforced concrete beams

The theoretical stress-strain relation in the compression zone of a concrete beam is determined by means of an approximate application of a model with strain softening and localization, i.e. with the descending branch described as a stress-deformation relation instead of a stress-strain relation. This results in a stress-strain relation for the compression zone, which is strongly dependent on the depth of this zone. This is in contradiction with existing design rules, which assume that no such size dependence exists. The theoretical results are confirmed by comparisons with test results regarding rotational capacities. Further test comparisons are recommended.

A two material constant, nonlinear elastic stress constitutive equation including the effect of compressibility

A new constitutive equation for compressible, nonlinear elasticity is derived. The stress-deformation relation takes the final form σ ij = 3k 2J (J 2 3 - 1) δ ij + μ J (B ij - j 2 3 δ ij) where B ij is the left Cauchy-Green deformation tensor. J is the Jacobean of the deformation, and k and μ are the two governing mechanical properties having the same interpretation as in linear, isotropic elasticity theory. In this nonlinear theory, k and μ are separately determinable from large deformation conditions of pure dilatation and simple shear, respectively. Under isochoric conditions, the distortional part of the theory takes the form of the kinetic theory of rubber elasticity. Under conditions of small volume change, the theory coincides with the molecular theory of Flory derived for elastomers, but given explicit from only in the one dimensional case.

AN EVALUATION METHOD OF LONGITUDINAL STIFFNESS OF SHIELD TUNNEL LININGS FOR APPLICATION TO SEISMIC RESPONSE ANALYSES

This paper presents a practical method to evaluate the longitudinal stiffness of shield tunnel linings which is of great importance in the seismic response analyses of shield tunnels. Linings of shield tunnels are constructed assembling a number of segments and joints, and have structural discontinuity in the longitudinal direction. In this paper, relationships between stress and deformation of segments-joints system, both for axial force and for bending moment, are formulated into the form of “equivalent stiffness”, i. e., the stiffness of uniform beam having the same stress-deformation relation as that of segments-joints system.Modeling the lining into a uniform beam with the equivalent stiffness, it makes possible to analyze behavior of shield tunnels by means of seismic deformation method as well as other pipe-like underground structures. Parameters which have influence on the equivalent stiffness are also discussed.

Nonlinear acoustics of micro-inhomogeneous media

Acoustic waves can interact in micro-inhomogeneous media much more intensively than in homogeneous media. This has been repeatedly observed in experiments with ground species, marine sediments, porous materials and metals. This paper considers two models of such media which seem to be applicable to the description of these results. One of them is based on the consideration of nonlinear sound scattering by separate spherical cavities in liquids and solids. The second model is based on the phenomenological stress-deformation relation in solids with microplasticity which often has hysteresis (heritage) properties associated with the micro-inhomogeneities. In metals, for example, it is caused by the movement of dislocations. Different nonlinear effects in such media (harmonic and combination frequency generation, nonlinear, variations of resonance frequency amplitude-dependent losses) are considered. Some results of experiments with metallic resonators supporting the theory developed here are also presented. These mechanisms may determine the nonlinear properties of real soils and rocks summarized in a table given in the paper.

Tensile Tests and Failure Analysis of Concrete

The tensile strength of concrete is receiving an increasing amount of attention since the loading capacity and durability of structures are being studied more thoroughly, and since nonlinear fracture mechanics and numerical methods require a complete stress deformation relation. The theoretical background of the application of the tensile strength of concrete in fracture analysis is given. Test results are presented that were obtained from deformation‐controlled uniaxial tensile tests under static and cyclic loading of lightweight and normalweight concrete. The results have been used to establish material models for finite element calculations. Numerical examples show the usefulness of these models for static and cyclic loading conditions.

Recent Advances in the Phenomenological Theory of Rubber Elasticity

Abstract We have shown that representations for the strain-energy function in terms of the stretches λ1, λ1, λ1, with particular reference to the Valanis-Landel hypothesis, provide a very good correspondence between theoretical stress-deformation relations and existing experimental data for a variety of different deformations of rubberlike materials. The stress-deformation relations themselves have a simple structure which makes for clear interpretation of the experimental results. As was emphasized in Section II.B., however, more data from biaxial tests on rubberlike materials of varying constitution are needed to support those already available. Also, further data from a variety of different experiments in which inhomogeneous deformations occur would be valuable. Finally, with reference to the effects of compressibility, data relating to volume changes—possibly from biaxial tests in the presence of hydrostatic pressure—would enable a more systematic approach to the construction of strain-energy functions for compressible materials to be pursued.

Rock joint compliance tests for compression and shear loads

A series of compression and shear tests were conducted at large scale on joint surfaces in granite and slate specimens, with special equipment being used to isolate the deformations of the joints from those of the rock material. The matrix of compliance components for a rock joint is discussed in relation to these standard tests and it is shown that one pair of compliance components can be determined from a compression test and a second pair from a shear test. Any attempt to base predictions of rock mass behaviour on the combined use of these two pairs of components should take into account the different stress paths to which they relate. The nature of the contacts between the asperities on adjacent joint surfaces is considered as a means of suggesting the form of the stress-deformation relations, both in compression and shear. This approach is shown to be of great assistance in helping to explain the test results and fit the observed compliance characteristics. For the compression tests, the test results indicate a pattern of decreasing compliance with load, an increase in compliance with initial apeture, and high levels of recoverability of deformation. In the shear tests, the curves for the relative displacements indicate there are three zones of different behaviour, elastic, transition and sliding. There appear to be links between the way in which forces are transferred at asperities and the typical patterns of observed compliance.

Local and global bifurcation phenomena in plane-strain finite elasticity

Bifurcation, global non-uniqueness and stability of solutions to the plane-strain problem of an incompressible isotropic elastic material subject to in-plane dead-load tractions are considered. In particular, for loading in equibiaxial tension, bifurcation from a configuration in which the in-plane principal stretches are equal is shown to occur at a certain critical value of the tension (which depends on the form of strain-energy function). Results concerning the global invertibility of the elastic stress-deformation relations are obtained and then used to derive an equation governing the deformation paths branching from this critical value. The stability of each branch is also examined. The analysis is carried through for a general form of strain-energy function and the results are then illustrated for a particular class of strain-energy functions.

Impact of a cone on a deformable string

The motion of a flexible deformable string and the process of propagation of the stress waves through it, following a transverse impact of a cone, are investigated. Equations of motion are derived and an analytic solution of a self-similar problem is obtained for an arbitrary, single-valued stress-deformation relationship. Possible wave patterns of the string motion are studied. It is shown that in the case of complete encirclement the motion of a string with a single kink is impossible and a new wave pattern of motion caused by a transverse impact is obtained for the first time. An established wave system with two kinks is used to derive the conditions at the strong shock waves. A numerical algorithm is given for a scheme with linear strengthening and the deformation versus impact rate grapns are constructed.

Inequalities associated with the inversion of elastic stress-deformation relations and their implications

AbstractFor an elastic solid the constitutive law can be written in terms of the deformation gradient α and its conjugate nominal stress s ≡ s(α), and also in terms of the right stretch u and its conjugate stress τ ≡ τ(u). It is shown that for isotropic elastic solids s(α) is invertible, in the local sense, for all u in the domain of u-space where τ(u) is locally invertible, with the exception of certain configurations which correspond to planes in τ-space.In the global sense a given s corresponds to four distinct τ's, and s is invertible to give four distinct α's when the corresponding τ's are uniquely invertible. That there are four branches of the inversion α(s) is of fundamental importance in that it clarifiesthe extent to which non-uniqueness of solution of boundary-value problems can be expected.The implications of these results in respect of the complementary variational principle are discussed, and the controversy surrounding the use of nominal stress in this principle resolved.Consequences of the required restrictions on τ(u) are examined and discussed in relation to inequalities which may be regarded as entailing physically reasonable response. It is intimated that τ(u) is invertible in the domain of elastic response.