Nonlinear Stress-strain Characteristics Research Articles

Dynamics of Rectangular Footings by Finite Elements

A finite element solution in three dimensions for rectangular footings resting on soil medium has been presented for the vertical mode of vibration. Soil medium has been idealized as an elastic half space. Both linear and nonlinear, as well as depth dependent, soil properties have been considered for analysis. For discretizing into a finite number of elements, the semi-infinite elastic medium is idealized as a bounded region by adopting appropriate boundary conditions. A simple hexadedron isoparametric element with eight nodes (ZIB8) has been taken as basic element. Experimental and analytical studies indicate that the nature of contact pressures at the footing-soil interface changes with the magnitudes of operational frequencies, thus all probable contact pressures have been incorporated in the analysis. Effects of embedment, static surcharge, nonhomogeneity, and nonlinear stress-strain characteristics are shown. Influences of the aforementioned factors on the resonant frequencies have been reported. Results are compared with the existing solutions and presented graphically.

Stress dependent behaviour of a particulate material

This paper considers the non-linear stress-strain characteristics of a particulate material using the classical elasticity concepts of modulus of elasticity and Poisson's ratio and shows that these terms are stress-dependent. In addition to the first loading of the material a number of unloading/reloading cycles are carried out under conditions representative of practical loading cases. Finally, recommendations are made regarding the analysis of problems involving particulate materials subjected to three-dimensional stress and strain.

Significance of cyclic confining stress in repeated-load triaxial testing of granular material : BROWN, SF HYDE, AF 10F, 1T, 8R. TRANSP. RES. REC., N537, 1975, P49–58

A number of research projects have been concerned with determining the nonlinear stress-strain characteristics of granular materials by using repeated-load triaxial tests. Improvements in materials testing equipment have lead to experiments in which the confining stress and the vertical stress were applied cyclically. There is an apparent difference in the resilient Poisson's ration determined under constant and under variable confining stresses. This difference is best explained by considering the behavior of granular materials in terms of volumetric stress-strain and shear stress-strain relationships. By doing this, one can show that Poisson's ratio is a function of the ratio of volumetric strain to shear strain. Resilient strains were investigated separetely from permanent strain because the permanent strain developed depended on the loading history of the sample and because the resilient behavior was not affected by any previous subfailure stress applications. To keep materials testing techniques simple, a constant confining stress equal to the mean value of a cyclic confining stress can be considered to be equal to the cyclic confining stress for determining the resilient modulus and permanent deformation.

Dynamic stiffness and damping of transformer pressboard during axial short circuit vibration

Data on the dynamic stress-strain characteristics of transformer pressboard during transient loading like that occurring in power transformer coils during short circuits are presented for wide ranges of preload stress, peak stress, and temperature. A nine-element dynamic stiffness and damping representation for pressboard is described which is based on these measured results and the physical nature of oil-immersed pressboard. It is shown that the representation accurately duplicates the highly nonlinear dynamic stress-strain characteristics of pressboard over wide ranges of conditions and provides a means for reliably calculating the large transient forces and displacements that can cause failure of transformer coils during short circuits.

Non-linear characteristics of engineering soils

The paper deals with the non-linear stress-strain characteristics of engineering soils and investigates some of the more important factors relating to the elasto-plastic behaviour. A review of previous literature describes some of the work already performed in this area of study, and discusses the applicability of the theories proposed, most of which have been established using results obtained from a standard tri-axial test. A non-linear stress-strain theory is formulated, using results from a “true” tri-axial test, which also takes into account the behaviour of the soil during unloading and subsequent reloading. This behaviour is shown to depend on the stress levels attained in the unloading/reloading hysteresis loop in which the reloading portion of the loop is closely linearly elastic. Finally, the suitability of the proposed equations in typical engineering calculations is discussed and a worked example is included which demonstrates how the proposed incremental stress-strain equations may be applied in analysis.

Experimental investigation of strains in fabric under biaxial and shear forces.

The paper defines the experimental phase of an objective to obtain the mechanical characteristics and coefficients required by the generalized form of Hooke's law for nylon-polyurethane-coated fabric. Test specimens were cylindrical fabric sleeves and were loaded in axial tension by an Instron, in hoop tension by pressurizing, and in shear by a torquing fixture. An extensive amount of strain data is included for a wide combination of the three membrane loads. The tests indicate highly nonlinear stress-strain characteristics of the fabric and a strong dependency on all three membrane loads.

Stress Analysis of Contractive or Dilative Soil

Nonlinear plastic stress-strain characteristics of drained soil, as described by Rowe's stress-dilatancy and by the Cambridge soil models, have been utilized in the solution of simple boundary value problems. Only plane strain conditions have been considered, and basic soil parameters reflecting frictional characteristics and stiffness appropriate to the plane strain state have been employed. The method of computation used has been the finite element (displacement) method. Nonlinearity has been incorporated using initial strain rather than tangent modulus techniques. Analysis of one problem involving contractive soil shows that for one type of stress path, the Cam clay model agrees better with experiment than does the modified Cam clay model. In the case of a similar problem involving dilative soil, the stress-dilatancy theory models the soil behavior well in a case where elasticity solutions predict failure at two-thirds of the true failure load.

Determination of stress in rock with linear or non-linear elastic characteristics

Determination of Stress in Rock with Linear or Non-Linear Elastic Characteristics A non-linear law of elasticity is proposed in which each principal stress is expressed as the summation of two series, one a function of the dilatory (hydrostatic) or octahedral normal strain and the other of the deviatory or octahedral shear strain. The constants in the series can be obtained from a simple uniaxial compression or tension test on the material. These expressions can be used to determine the stress from strain readings in rock having non-linear stress-strain characteristics and the application of these expressions when using the CSIR triaxial strain cell in such rock is demonstrated.

Cyclic Stress-Strain Characteristics of Clay

The decreases in moduli and increases in yield strain occurring during cyclic loading of sensitive clay under simple shear conditions are examined. The loading conditions are representative of those associated with the upward propagation of a seismic shear wave through a large mass of soil; i.e., the strain is completely reversed during each cycle at the rate of 1 cps. The moduli and yield strain reported are those defining the bilinear hysteresis models which best fit the nonlinear stress-strain characteristics obtained from the tests. For San Francisco Bay Mud the moduli are constant for strain levels greater than 2% and for loading cycles after the fiftieth. Yield strain, however, is proportional to cyclic strain level for at least 200 cycles. The strength and moduli measured under static loading conditions are decreased by cyclic loading involving large strains, but only the modulus is decreased by loading restricted to strains below about 1%.

Theory of Folding of Stratified Viscoelastic Media and Its Implications in Tectonics and Orogenesis1

This paper presents an introduction to the theory of folding of stratified viscoelastic media under compression and discusses its significance in the context of tectonics and orogenesis. Simplified derivations are given for results obtained earlier by the writer as particular cases of more elaborate theories. The writer emphasizes the mechanism involved in folding. The paper begins with a discussion of the buckling of an elastic rod that is under axial compression and is restrained laterally by viscous dashpots. The analysis then proceeds to the analogous problem for an elastic and a viscous plate surrounded by a viscous medium. Results of some of the more complex problems previously analyzed by the writer are also applied and discussed. Experimental verification of the theory by model tests is presented in a companion paper (Biot, Ode, and Roever, 1961). A new feature of the present approach is the emphasis on rate phenomena and time histories in tectonic folding. In purely static problems of elastic buckling, a sharply defined wavelength is associated with the instability. By contrast, for viscoelastic media, the present theory leads to the concept of dominant wavelength and band width selectivity in analogy with the theory of electric wave filters. This is well illustrated by the gradual appearance of near-regular folds when a purely viscous layer surrounded by a viscous medium of lower viscosity is subjected to a compression in a direction tangent to the layer. The theory is applied to specific examples of geological interest. These include the case of a single layer or a superposition of layers. Previous theoretical work by the writer is applied to the discussion of folding, under the simultaneous action of gravity forces and a horizontal compression, for a single layer or a superposition of layers lying at the surface of a deep substratum of lower viscosity. The case of a continuously inhomogeneous medium under similar forces is also included. Using accepted values of rock viscosity and elastic moduli, the writer finds that the time required for significant folding to take place agrees very well with the geological time scale. Folding may occur under tectonic stresses that are small in comparison with the crushing strength of the, rock. The time history of folding depends, of; course, on initial irregularities of the layers, but9 after sufficient time the folding becomes fairly! insensitive to the magnitude and the distribution of the initial disturbances. The writer concludes ; that the viscous mechanism tends to predominate in tectonic folding. As a theoretical consequence, the wavelength of the folds will, in general, not be sensitive to the magnitude of the tectonic stresses unless gravity forces become important. The calculated wavelengths are in good agreement with the range of observed values. The point at which plastic or brittle failure occurs is found to depend primarily not on the magnitude of the strain but on deformation rates. The theory can be applied to materials with nonlinear stress-strain characteristics, and the procedure to accomplish this is briefly discussed. Certain nonlinear features resulting from the geometry of deformation are shown o t have a bearing on the regularity of the folds.

Elastic and Plastic Bending of Beams

This paper presents a method of analyzing fiber stresses of beams which includes the full effect of nonlinear stress-strain characteristics of materials throughout the elastic and plastic range of bending. Various shaped cross sections commonly encountered in aircraft practices are considered. A new term is proposed for the measurement of remote fiber stresses of cross sections of beams under pure bending. Eqs. (11) and (12) or Eqs. (14) and (15) are the final equations derived for calculating the norstress on the rectangular beams. Eqs. (17) and (18) are for • , £, I, Z, J~~L, L, and T beams; Eqs. (21) and (22) for circular beams; and Eqs. (23) and (25) for tubular beams. A stress-strain formula is also recommended for use of calculations of integrals involved in the stress and moment equations, as well as for the purpose of drawing stress-strain curves in the absence of test data.