An investigation of hydraulic fracturing. Part I: one-phase flow model

The prediction of the growth of a hydraulic fracture in an oil bearing formation based on the injection rate of fluid is valuable in applications of the waterflood technique in secondary oil recovery. In this paper, the problem of hydraulic fracture growth is studied under the assumption of uniform distribution of pressure in the fracture and unidirectional permeating flow in an infinitely large isothermal linearly elastic porous medium saturated with a one-phase incompressible fluid. The condition of plane strain is imposed in the study. A comparison of the constant fracture toughness criterion based on the asymptotic value for large crack growth with the crack tip ductility criterion for an ideally plastic solid under plane strain and small-scale yielding conditions indicates that the effect of ductility of rock on the crack growth is so small that the steady state value of the energy release rate can be reached within a short period of crack growth. Thus we can employ the constant fracture toughness criterion in our study. The analysis includes the effects of both fracture volume increase and leak-off of fluid from the surface of the fracture. A nonlinear singular integro-differential equation can be formulated for the quasi-static hydraulic fracture growth under a prescribed injection rate. It is solved numerically by a modified fourth order Runge-Kutta method.     

Symmetric prismatic tensegrity structures: Part I. Configuration and stability

This paper presents a simple and efficient method to determine the self-equilibrated configurations of prismatic tensegrity structures, nodes and members of which have dihedral symmetry. It is demonstrated that stability of this class of structures is not only directly related to the connectivity of members, but is also sensitive to their geometry (height/radius ratio), and is also dependent on the level of self-stress and stiffness of members. A catalogue of the structures with relatively small number of members is presented based on the stability investigations.     

A Bending-Gradient model for thick plates. Part I: Theory

This is the first part of a two-part paper dedicated to a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called the Bending-Gradient plate theory is described in the present paper. It is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case of the Bending-Gradient plate theory when the plate is homogeneous. However, we demonstrate also that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In part two (Lebée and Sab, 2011), the Bending-Gradient theory is applied to multilayered plates and its predictions are compared to those of the Reissner–Mindlin theory and to full 3D Pagano’s exact solutions. The main conclusion of the second part is that the Bending-Gradient gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

A model for rolling deformation with grain subdivision. Part I: The initial stage

A model is presented for plastic deformation with grain subdivision into parallel bands. The experimental reference is the subdivision into “cell blocks” observed in rolled aluminium. The model maintains intragranular strain continuity between the bands with relaxed constraints. One version of the model maintains intergranular strain continuity by imposing identical strains in all grains. Another version does not provide formal fulfilment of intergranular strain continuity, but it tries to minimize strain discontinuity by selection of the appropriate physical solutions. Part I deals with the initial stage of grain subdivision at low strain. Part II (Leffers, T. 2001. A model for rolling deformation with grain subdivision. Part II: the subsequent stage. Int. J. Plasticity 17, 491–511.) deals with the subsequent stages at higher strains and the resulting rolling texture.     

An elastoplastic constitutive relation of whisker-reinforced composite for meso damage: Part I – formulation

An elastoplastic constitutive relation is developed for meso damage of whisker-reinforced composites. A model is constructed that includes orientation distribution of whiskers and slip systems as well as interface and crystal sliding. Evolution of damage will be addressed. Given in Part I is the formulation while examples will be illustrated in Part II.     

An explicit algebraic Reynolds stress and heat flux model for incompressible turbulence: Part I Non-isothermal flow

Tensor representation theory is used to derive an explicit algebraic model that consists of an explicit algebraic stress model (EASM) and an explicit algebraic heat flux model (EAHFM) for two-dimensional (2-D) incompressible non-isothermal turbulent flows. The representation methodology used for the heat flux vector is adapted from that used for the polynomial representation of the Reynolds stress anisotropy tensor. Since the methodology is based on the formation of invariants from either vector or tensor basis sets, it is possible to derive explicit polynomial vector expansions for the heat flux vector. The resulting EAHFM is necessarily coupled with the turbulent velocity field through an EASM for the Reynolds stress anisotropy. An EASM has previously been derived by Jongen and Gatski [10]. Therefore, it is used in conjunction with the derived EAHFM to form the explicit algebraic model for incompressible 2-D flows. This explicit algebraic model is analyzed and compared with previous formulations including its ability to approximate the commonly accepted value for the turbulent Prandtl number. The effect of pressure-scrambling vector model calibration on predictive performance is also assessed. Finally, the explicit algebraic model is validated against a 2-D homogeneous shear flow with a variety of thermal gradients. Dedicated to the memory of the late Professor Charles G. Speziale of Boston University     

A hybrid phenomenological model for ferroelectroelastic ceramics. Part I: Single phased materials

In this part I of a two part series, a rate-independent hybrid phenomenological constitutive model applicable for single phased polycrystalline ferroelectroelastic ceramics is presented. The term “hybrid” refers to the fact that features from macroscopic phenomenological models and micro-electromechanical phenomenological models are combined. In particular, functional forms for a switching function and the Helmholtz free energy are assumed as in many macroscopic phenomenological models; and the volume fractions of domain variants are used to describe the internal material state, which is a key feature of micro-electromechanical phenomenological models. The approach described in this paper is an attempt to combine the advantages of macroscopic and micro-electromechanical material models. Its potential is demonstrated by comparison with experimental data for barium titanate. Finally, it is shown that the model for single phased materials cannot reproduce the material behavior of morphotropic PZT ceramics based on a realistic choice for the material parameters. This serves as a motivation for part II of the series, which deals with the modeling of morphotropic PZT ceramics taking into account the micro-structural specifics of these materials.     

Relaxational interactions and viscoelasticity of polymer melts. Part I: model development

Rheological equations of state for the concentrated solutions and melts of high polymers are derived by applying a structural approach. The dynamics of a macromolecule are considered on the basis of the fundamental model of the polymer chain, e.g., the bead-spring model. The drag forces describing correlations of motion macromolecules are determined by means of the relaxation equations. The oscillatory shearing flow of the melts is studied on the basis of the equations derived. Expressions for the dynamic modulus and relaxation times are determined. As can be judged from the form of the dependence of the dynamic modulus on frequency, the relaxation time distribution is the same as in real materials. The relaxation spectrum of high polymers has a terminal zone with abnormally long relaxation times.     

Buoyancy-driven instability in a vertical cylinder: Binary fluids with Soret effect. Part I: General theory and stationary stability results

Buoyancy-driven instability of a monocomponent or binary fluid which is completely contained in a vertical circular cylinder is investigated, including the influence of the Soret effect for the binary mixture. The Boussinesq approximation is used, and the resulting linear stability problem is solved using Galerkin's technique. The analysis considers various types of fluid mixtures, ranging from gases to liquid metals, in cylinders with a variety of radius-to-height ratios. The flow structure is found to depend strongly on both the cylinder aspect ratio and the magnitude of the Soret effect. Comparisons are made with experiments and other theories, and the predicted stability limits are shown to agree closely with observations.     

Stability of travelling fronts in a model for flame propagation Part II: Nonlinear stability

This paper is concerned with the nonlinear stability of the travelling-wave solutions of the multidimensional thermodiffusive model for flame propagation, with unit Lewis number. The model consists in a semilinear parabolic equation in an infinite cylinder, with Neumann boundary conditions. We prove that any solution which is initially close to a travelling wave will converge to a translate of that wave.     

Slip-line modeling of machining with a rounded-edge tool—Part I: new model and theory

The effect of tool edge roundness attracts growing attention from the international machining research community due to ever accelerating applications of precision, super-precision, micro-, and nano-machining technologies in a wide variety of modern industries. A new slip-line model for machining with a rounded-edge tool and its associated hodograph are proposed in this paper. The model consists of 27 slip-line sub-regions, each sub-region having its own physical meaning. It is demonstrated that the model simultaneously takes into account nine effects, such as the shear-zone effect and the size effect, which commonly occur in machining. Eight groups of machining parameters, such as the ploughing (parasitic or non-cutting) force and the chip up-curl radius, can be simultaneously predicted from the model. Furthermore, the model incorporates eight slip-line models previously developed for machining during the last six decades as special cases. An additional special case that involves a parallel-sided shear zone can also be derived from the new model. A mathematical formulation of the model is established based on Dewhurst and Collins's (1973) matrix technique for numerically solving slip-line problems. A purely analytical equation is proposed to predict the thickness of the primary shear zone. This equation is also employed to predict the shear strain-rate in the primary shear zone.     

An Incompressible Fluid in a Weakly Deformable Shell. Part I: Analysis of the Model

We consider the flow of a viscous incompressible fluid in a pipe when the boundary is a deformable shell of Naghdi type. We prove that the corresponding system of partial differential equations has a solution when the deformation of the shell is smooth and small enough. We propose an algorithm that uncouples the unknowns and prove its convergence.     

Inelastic behavior of an AS4/PEEK composite under combined transverse compression and shear. Part I: experiments

The nonlinear behavior in shear and transverse compression of unidirectional AS4/PEEK and their interaction are investigated experimentally. The composite is rate dependent even at room temperature and its rate exponent is similar to that of neat PEEK. The material is tested under pure shear, pure compression and under biaxial loading histories. The biaxial tests are performed in a custom facility on thin strips of the material. The facility allows freedom to choose the loading path in the biaxial stress and strain spaces of interest. Tests are performed for three biaxial loading paths. In the first, the specimen is sheared then compressed while the shear stress is held constant; in the second, the specimen is compressed then sheared while the compressive stress is held constant; and in the third, the specimen is loaded simultaneously by proportional amounts of compression and shear. It was found that the induced deformation is influenced significantly by the loading history followed. Also, initial loading in shear or compression has only a modest effect on subsequent loading of the other type. An unorthodox yielding behavior for the composite results from this lack of interaction. Finally, the stresses at failure are found to trace an elliptical path in the shear–transverse compression plane, but the failure stress state is not significantly affected by the loading path followed.     

A mathematical model of sulphite chemical aggression of limestones with high permeability. Part I. Modeling and qualitative analysis

We introduce a degenerate nonlinear parabolic–elliptic system, which describes the chemical aggression of limestones under the attack of SO₂, in high permeability regime. By means of a dimensional scaling, the qualitative behavior of the solutions in the fast reaction limit is investigated. Explicit asymptotic conditions for the front formation are derived.     

Necessary and sufficient fracture criteria for a composite with a brittle matrix. Part 1. Weak reinforcement

Fracture of a composite medium with a brittle matrix is studied. The brittle or plastic material of the reinforcing elements is highly deformable. For normalrupture macrocracks, necessary criteria of brittle strength and sufficient criteria of quasibrittle strength are proposed. Simple analytical dependences of the macrocrack length on the loading parameter, structural, rigidity, and strength parameters of the medium, and damage parameters of the material of the components are obtained. The critical loads for these criteria may differ substantially even if the reinforcement coefficient is small and the material of reinforcing elements is highly deformable. If the necessary criterion is satisfied, crack extension occurs and microcracks are formed in the bonds of the structure located ahead of the macrocrack tip. The number of damaged bonds depends on the macrocrack opening and characteristics of postcritical deformation of the damaged bonds.     

A model for rolling deformation with grain subdivision. Part II: The subsequent stage

Part I of the present work dealt with the initial stage of plastic deformation with grain subdivision into two band families. It covered the situation with different average strains in the individual grains (the non-Taylor case) and the situation with identical average strains in the individual grains (the Taylor case). For the latter situation part I included solutions with 5 plus 3 and with 4 plus 4 active slip systems in the two band families, respectively. In part II we deal with the subsequent stage of plastic deformation (finite strains) for the Taylor case with four active slip systems in each band family. In the subsequent stage the cooperation between the two band families leads to an energetic advantage for grain subdivision, and it leads to a simulated texture with decreased sharpness in accordance with experimental observations. Finally, the physics behind grain subdivision are discussed on the basis of the results in Parts I and II and various general considerations.     

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I: Theoretical formulation and model validation

This paper is first of the two papers dealing with analytical investigation of resonant multi-modal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables. Part I deals with theoretical formulation and validation of the general cable model. Approximate nonlinear partial differential equations of 3-D coupled motion of small sagged cables – which account for both spatio-temporal variation of nonlinear dynamic tension and system asymmetry due to inclined sagged configurations – are presented. A multi-dimensional Galerkin expansion of the solution of nonplanar/planar motion is performed, yielding a complete set of system quadratic/cubic coefficients. With the aim of parametrically studying the behavior of horizontal/inclined cables in Part II [25], a second-order asymptotic analysis under planar 2:1 resonance is accomplished by the method of multiple scales. On accounting for higher-order effects of quadratic/cubic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations of resonant nonlinear normal modes reveal the dependence of cable response on resonant/nonresonant modal contributions. Depending on simplifying kinematic modeling and assigned system parameters, approximate horizontal/inclined cable models are thoroughly validated by numerically evaluating statics and non-planar/planar linear/non-linear dynamics against those of the exact model. Moreover, the modal coupling role and contribution of system longitudinal dynamics are discussed for horizontal cables, showing some meaningful effects due to kinematic condensation.     

A viscoelastic damage model for polycrystalline ice,inspired by Weibull-distributed fiber bundle models. Part I: Constitutive models

We consider a constitutive model for polycrystalline ice, which contains delayed-elastic and viscous deformations, and a damage variable. The damage variable is coupled to the delayed-elastic deformation by a fiber bundle ansatz. We construct an isotropic theory, which can be calibrated with experimental data. Furthermore, we generalize the theory to a damage model in terms of rank-four tensors. This general model allows the evolution of anisotropic damage.