Finite Strain Solutions Research Articles

A New Numerical Finite Strain Procedure for a Circular Tunnel Excavated in Strain-Softening Rock Masses and Its Engineering Application

The small strain theory underestimates the self-bearing capacity of rock masses, especially for a soft rock tunnel under high geostress. To perform an efficient and accurate calculation and provide a reference for the stiffness design of a tunnel, the finite strain solution for a circular tunnel in Mohr–Coulomb strain-softening rock masses with a non-associated flow rule was derived as three sets of differential equations under the Lagrangian coordinate, which are in the residue region, the softening region, and the elastic region, respectively. Based on the bisection method, an iteration procedure for solving the finite strain solution was proposed to approximate the boundary condition at infinity, the values of two adjacent boundaries, and the initial values on the excavation boundary. This numerical procedure was verified by comparing with self-similar solutions, recursive solutions, and FLAC simulation results. In the calculation example, the relative error on boundaries can be decreased to less than 10−8 after only 10 times iteration and the time for each calculation is less than 15 s. Applying this procedure on the sensibility analysis and stiffness reliability design for the Zhongyi tunnel, a support stiffness of 4.3 MPa/m is recommended to guarantee a tunnel displacement lower than 0.5 m.

A finite strain solution for strain-softening rock mass around circular roadways

This paper presents a semi-analytical finite strain solution for circular roadways excavated in a strain-softening rock mass with a non-associated flow rule. The evolutions of both strength parameters and Young’s modulus are considered, and the finite strains are applied to describe both the elastic and plastic behaviors with the logarithmic strain in Lagrangian coordinates. By transforming the derivative of displacement with a new variable that is correlated with displacement, the stress and displacement in the elastic region are obtained in the form of integral equations. Considering the assumption of constant material parameters within a very small region, the plastic region is divided into a number of concentric annuli, whose closed-form solutions of displacement and stress can be recursively obtained. The solution is validated by similarity solutions and numerical simulation. Both the roadway wall displacement and thickness of the plastic (residual) region appear to be smaller than those of the small strain solution. For the elasto-perfectly-plastic rock mass, the finite strain solution is approximately the same as that of small strain solution. However, for the strain-softening rock mass, with the decrease of Young’s modulus, the thickness of plastic and residual regions shows the first increase and then decrease trend, finally approaching nil; the difference between the thickness of softening and residual region gradually decreases. The decreasing Young’s modulus deterioration coefficient and increasing dilation angle significantly produce the displacement. Both the displacement and the thickness of plastic and residual regions decrease with the increase of Poisson’s ratio. When Young’s modulus is large, the tangential stress gradually decreases with the increasing radius. When Young’s modulus is small, the tangential stress firstly increases to a maximum value, then decreases gradually. Different from the small strain solutions, the roadway wall displacement of finite strain solutions cannot be larger than the excavation radius, which is in compliance with the practical underground engineering condition. The predicted displacement and thickness in both plastic and residual regions are in accordance with the field test results.

A simple numerical procedure for the elasto-plastic coupling finite strain analysis of circular tunnels in strain-softening rock masses

This paper presents a numerical finite strain solution for a circular tunnel in Mohr-Coulomb and generalized Hoek-Brown strain-softening rock masses. By approximating the differential equations with a difference method in the deformed coordinates, an efficient procedure is presented to calculate the displacements and stresses. The proposed solution is verified by another numerical method for both elasto-brittle-plastic and strain-softening finite strain examples. The parametric analysis indicates that (1) with the decrease in Young’s modulus of the residual region, the displacement increases, and the thickness of the softening region decreases. The displacement and the thickness of the softening and residual regions remain relatively stable when the deterioration coefficient of Young’s modulus is less than 0.01. (2) The displacement and the thickness of the softening and residual regions decrease with the increasing Poisson’s ratio, and increase with the increasing parameter α for the generalized Hoek-Brown rock mass. (3) With the increase in the in-situ stress, the dimensionless displacement of the tunnel wall increases and tends to 1.0, and the thicknesses of plastic and residual region first increase to their maximum values and then decrease. The finite strain solution is necessary for the displacement prediction and supporting design, rather than the small strain solution.

Elasto-plastic solutions for expanding cavities in strain-hardening and/or softening soils

This paper presents an analytical finite strain solution for cavity expansion in strain-hardening and/or softening Mohr-Coulomb soil subjected to a non-associated flow rule. The evolutions of material parameters, i.e. deformation and strength parameters, are totally considered. The finite strain is applied to describe the plastic behavior with the logarithmic strain. By assuming constant material properties in a very small region, the plastic region is divided into a series of concentric annuli. The solution can be recursively obtained according to the closed-form solution of each annulus, and is suitable to soil with any complex material evolution types. For the cavity with the same cavity expansion a/a0, the required cavity pressure of strain-hardening and/or softening soils is larger and/or smaller than that of elasto-perfectly plastic soils for both spherical and cylindrical cavity expansions. With respect to the same cavity pressure, the displacement and stress around cavity of strain-hardening and/or softening models are larger and/or smaller than those of perfectly plastic model. With respect to the same cavity expansion displacement, the increases in the strength parameters including cohesive strength and friction enlarge the cavity pressure, and slightly decrease the plastic and residual radii.

On Phenomenological Failure Loci of Metals under Constant Stress States of Combined Tension and Shear: Issues of Coaxiality and Non-Uniqueness

The present study investigates how the choice of characterization test and the composition of the stress state in terms of tension and shear can produce a non-unique failure locus in terms of stress triaxiality under plane stress conditions. Stress states that are composed of tensile and simple shear loadings result in a loss of proportionality between the cumulative strain and stress such that the principal frames become non-coaxial despite a constant stress triaxiality. Consequently, it is shown that the conventional interpretation of a failure locus in plane stress is based upon an implicit assumption of proportional coaxial loading. The use of simple shear tests along with traditional in-plane tensile tests for fracture characterization is only one “path” that can be taken in terms of the stress triaxiality, which may produce a bifurcation at uniaxial tension while the tension–torsion path does not. In general, the failure locus in terms of the equivalent strain is a failure surface and must consider the composition of the stress state that produces a given triaxiality. A comprehensive review of phenomenological fracture loci within a modified Mohr-Coulomb (MMC) framework is performed to highlight how the choice of stress states obtained using different characterization tests can change the apparent fracture locus of a material. The finite strain solutions for the work conjugate equivalent strain are derived for various loading paths that produce the same stress triaxiality. It is then shown that accounting for non-coaxiality leads to equivalent failure strains that are even higher than previously reported in tension–torsion tests within the literature. The equivalent plastic strains integrated from finite-element simulations are work-conjugate by definition. The equivalent strains estimated from the cumulative principal strains using DIC strain measurement depend upon a coaxial or non-coaxial assumption. Finally, an analytical solution for the onset of diffuse necking that accounts for the stabilizing influence of shear loading against a tensile instability is considered. Even under plane stress conditions, a failure surface arises in terms of the equivalent strain at necking, the stress triaxiality, and the severity of shear loading.

Interfacial stresses within boundary between martensitic variants: Analytical and numerical finite strain solutions for three phase field models

The origin of a large elastic stress within an interface between martensitic variants (twins) within a finite strain phase field approach has been determined. Notably, for a sharp interface this stress is absent. Three different constitutive relations for the transformation stretch tensor versus order parameters have been considered: a linear combination of the Bain tensors (kinematic model-I, KM-I), an exponential-logarithmic combination (KM-II) of the Bain tensors, and a stretch tensor corresponding to simple shear (KM-III). An analytical finite-strain solution has been found for an infinite sample for tetragonal martensite under plane stress condition. In particular, explicit expression for the stresses have been obtained. The maximum interfacial stress for KM-II is more than twice that which corresponds to KM-I. Stresses are absent for KM-III, but it is unclear how to generalize this model for multivariant martensitic transformation. An approximate analytical solution for a finite sample has been found as well. It shows good correspondence with numerical results obtained using the finite element method. The obtain results are important for developing phase field approaches for multivariant martensitic transformations coupled to mechanics, especially at the nanoscale.

A finite strain solution for the elastoplastic ground response curve in tunnelling: rocks with non‐linear failure envelopes

SummaryThis paper generalizes the finite strain Coulomb solution of Vrakas and Anagnostou (Int J Numer Anal Meth Geomech 2014; 38(11): 1131–1148) for the classic tunnel mechanics problem of the ground response curve to elastoplastic grounds satisfying a non‐linear Mohr's failure criterion. A linear (Coulomb‐type) plastic potential function is used, leading to a non‐associated flow law, and edge plastic flow is considered in the plastic zone. The solution for a general non‐linear Mohr's failure criterion is semi‐analytical in that it requires the evaluation of definite integrals. In the special case of the Hoek–Brown criterion, however, these integrals are calculated analytically, resulting in a rigorous closed‐form series solution. The applicability of the derived solution is illustrated through the example of the Yacambú‐Quibor tunnel, where very large deformations were observed when crossing of weak graphitic phyllites. Copyright © 2016 John Wiley & Sons, Ltd.

A new approach for elasto-plastic finite strain analysis of cantilever beams subjected to uniform bending moment

The reliability and limits of solutions for static structural analysis depend on the accuracy of the curvature and deflection calculations. Even if the material model is close to the actual material behavior, physically unrealistic deflections or divergence problems are unavoidable in the analysis if an appropriate fundamental kinematic theory is not chosen. Moreover, accurate deflection calculation plays an important role in ultimate strength analysis where in-plane stresses are considered. Therefore, a more powerful method is needed to achieve reliable deflection calculation and modeling. For this purpose, a new advanced step was developed by coupling the elasto-plastic material behavior with precise general planar kinematic analysis. The deflection is generated precisely without making geometric assumptions or using differential equations of the deflection curve. An analytical finite strain solution was derived for an elasto-plastic prismatic/non-prismatic rectangular cross-sectioned beam under a uniform moment distribution. A comparison of the analytical results with those from the Abaqus FEM software package reveals a coherent correlation.

A simple equation for obtaining finite strain solutions from small strain analyses of tunnels with very large convergences

This paper presents a novel, very simple, accurate, theoretically well-founded and widely applicable relationship expressing the tunnel convergences obtained from large strain elasto-plastic analyses as a hyperbolic function solely of the corresponding small strain convergences. It can thus be used for ‘self-correcting’ small strain solutions, removing the need for large strain elasto-plastic analyses at least at the preliminary design stage and quantifying a hitherto unknown error caused by disregarding the geometric non-linearity. The proposed relationship can be proved rigorously for the plane strain rotationally symmetric ground response problem with a general elasto-plastic constitutive law with or without dilatancy and hardening. Numerical analyses of characteristic two- and three-dimensional excavation problems show that this relationship is generally applicable, irrespective of the in situ stress state and the tunnel geometry. It is therefore very useful from an engineering point of view for the design of tunnels crossing heavily squeezing ground, where the convergences may be so large (sometimes well in excess of 10% of the tunnel radius) that the usual small strain elasto-plastic analyses are deficient.

A simple equation for obtaining finite strain solutions from small strain analyses of tunnels with very large convergences

A simple equation for obtaining finite strain solutions from small strain analyses of tunnels with very large convergences

Numerical and analytical solutions with finite strains for circular inflated membranes considering pressure–volume coupling

Abstract The classical problem of the circular inflated membrane fixed at its rim and expanded under gas pressure is studied in this paper. Numerical and analytical solutions considering pressure–volume coupling are investigated. It should be pointed out that the application of the analytical solutions described in this section to closed chambers considering pressure volume coupling requires the adoption of recursive solutions scheme. The deformed configuration and volume are updated followed by the pressure update through the ideal gas law. The analytical solutions presented in the literature for this problem consider only the presence of small strains. In the present study the solutions given by Hencky, Fichter and Campbell are studied. A novel finite strain solution is derived, as an extension of Fichter׳s solution. The numerical solution is based on the finite element method and also considers finite strain kinematics. The results of numerical and analytical models for an enclosed circular membrane are compared, where the effects of small and finite strains with or without the consideration of the pressure–volume coupling are highlighted.

Assessment of Consolidation Behavior of Golden Horn Marine Dredged Material

ABSTRACT Polluted sea bottom sediments from Golden Horn in Istanbul have been dredged and stored on land at a disposal site. Reclamation of the disposal site was highly dependent on prediction of self-weight consolidation behavior of the dredged material, which is analyzed numerically using software which employs a nonlinear finite strain solution algorithm (Fox and Berles 1997). The material parameters needed for the numerical model are determined using a seepage-induced consolidation testing system and prediction of the numerical model is tested against experimental observations in a slurry consolidation model tank. The numerical modeling of the field behavior at the disposal site could be successfully accomplished using sediment property data from the seepage-induced consolidation test.

Analytical solutions of linear finite- and small-strain one-dimensional consolidation

Analytical solutions are presented for linear finite-strain one-dimensional consolidation of initially unconsolidated soil layers with surcharge loading for both one- and two-way drainage. These solutions complement earlier solutions for initially unconsolidated soil layers without surcharge and initially normally consolidated soil layers with surcharge. Small-strain solutions for the consolidation of initially unconsolidated soil layers with surcharge loading are also presented, and the relationship between the earlier solutions for initially unconsolidated soil without surcharge and the corresponding small-strain solutions, which was not addressed in the earlier work, is clarified. The new solutions for initially unconsolidated soil with surcharge loading can be applied to the analysis of low stress consolidation tests and to the partial validation of numerical solutions of non-linear finite-strain consolidation. They also clarify a formerly perplexing aspect of finite-strain solution charts first noted in numerical solutions. Copyright (C) 2004 John Wiley Sons, Ltd.

Finite strain solutions for a compressible elastic solid

Several closed form finite strain equilibrium solutions are presented for a special compressible isotropic elastic material which was proposed as a model for foam rubber by Blatz and Ko. These solutions include bending of a cylindrical sector into another sector or a rectangular block, bending of a block into a sector, expansion, compaction or eversion of cylinders or spheres, and torsion and extension of circular cylinders or tubes.

Finite strain solutions in compressible isotropic elasticity

Three classes of compressible isotropic elastic solids are introduced, for each of which the strain energy, expressed as a function of the three principal invariants of the stretch tensors, is linear in two of its arguments and nonlinear in the third argument. One of these is the class of harmonic materials. Several deformation fields are examined, for which the governing equations, for general compressible isotropic elastic response, reduce to a nonlinear ordinary differential equation. For the three special classes of materials, this differential equation may be solved in closed form, giving a deformation field which is independent of the material response function, or its solution may be reduced to a single quadrature, involving the nonlinear material response function.

On instabilities in large deformation simple shear loading

We obtain finite strain solutions for an elastic-perfectly-plastic isotropic material under simple shear loading. The effects of using the Jaumann, Green-Naghdi, and Truesdell stress rates are examined. An analytic solution is obtained using the Jaumann rate, and numerical solutions are obtained for the Green-Naghdi and Truesdell rates. The shear stress, as a function of the shear strain is nonmonotonic, and consequently ‘unstable’, for all three stress rates, which is contrary to previously published results.