A two-strain-gage technique for determining mode I stress-intensity factor

A finite element analysis and a experimental test were performed to show that the terms with r^−1/2 and ^1/2 in the eigenfunction expansion of the strains can describe the crack tip strain distribution with sufficient accuracy. A set of two linear equations can be obtained to determine the stress intensity factor K_I using only two strain-gages. Errors within 5% can be achieved provided that the two strain-gages are placed at the appropriate locations. The technique can be developed to treat crack bodies with irregular geometry and complex loading.     

A finite deformation theory of higher-order gradient crystal plasticity

For higher-order gradient crystal plasticity, a finite deformation formulation is presented. The theory does not deviate much from the conventional crystal plasticity theory. Only a back stress effect and additional differential equations for evolution of the geometrically necessary dislocation (GND) densities supplement the conventional theory within a non-work-conjugate framework in which there is no need to introduce higher-order microscopic stresses that would be work-conjugate to slip rate gradients. We discuss its connection to a work-conjugate type of finite deformation gradient crystal plasticity that is based on an assumption of the existence of higher-order stresses. Furthermore, a boundary-value problem for simple shear of a constrained thin strip is studied numerically, and some characteristic features of finite deformation are demonstrated through a comparison to a solution for the small deformation theory. As in a previous formulation for small deformation, the present formulation applies to the context of multiple and three-dimensional slip deformations.     

An efficient parallel and fully implicit algorithm for the simulation of transient free-surface flows of multimode viscoelastic liquids

The parallelization of a fully implicit and stable finite element algorithm with relative low memory requirements for the accurate simulation of time-dependent, free-surface flows of multimode viscoelastic liquids is presented. It is an extension of our multi-stage sequential solution procedure which is based on the mixed finite element method for the velocity and pressure fields, an elliptic grid generator for the deformation of the mesh, and the discontinuous Galerkin method for the viscoelastic stresses [Dimakopoulos and Tsamopoulos [12], [14]]. Each one of the above subproblems is solved with the Newton–Rapshon technique according to its particular characteristics, while their coupling is achieved through Picard cycles. The physical domain is graphically partitioned into overlapping subdomains. In the process, two different kinds of parallel solvers are used for the solution of the distributed set of flow and mesh equations: a multifrontal, massively parallel direct one (MUMPS) and a hierarchical iterative parallel one (HIPS), while viscoelastic stress components are independently calculated within each finite element. The parallel algorithm retains all the advantages of its sequential predecessor, related with the robustness and the numerical stability for a wide range of levels of viscoelasticity. Moreover, irrespective of the deformation of the physical domain, the mesh partitioning remains invariant throughout the simulation. The solution of the constitutive equations, which constitutes the largest portion of the system of the governing, non-linear equations, is performed in a way that does not need any data exchange among the cluster's nodes. Finally, indicative results from the simulation of an extensionally thinning polymeric solution, demonstrating the efficiency of the algorithm are presented.     

High-speed fiber spinning process with spinline flow-induced crystallization and neck-like deformation

The dynamics and stability of the high-speed fiber spinning process with spinline flow-induced crystallization and neck-like deformation have been studied using a simulation model equipped with governing equations of continuity, motion, energy, and crystallinity, along with the Phan-Thien–Tanner constitutive equation. Despite the fact that a simple one-phase model was incorporated into the governing equations to describe the spinline crystallinity, as opposed to the best-known two-phase model [Doufas et al. J Non-Newton Fluid Mech, 92:27–66, 2000a]; [Kohler et al. J Macromol Sci Phys, 44:185–202, 2005] that treats amorphous and crystalline phases separately in computing the spinline stress, the simulation has successfully portrayed the typical nonlinear characteristic of the high-speed spinning process called neck-like spinline deformation. It has been found that the criterion for the neck-like deformation to occur on the spinline is for the extensional viscosity to decrease on the spinline, so that the spinning is stabilized by the formation of the spinline neck-like deformation. The accompanying linear stability analysis explains this stabilizing effect of the spinline neck-like deformation, corroborating a recent experimental finding [Takarada et al. Int Polym Process, 19:380–387, 2004].This paper was presented at the 2nd Annual European Rheology Conference 2005 on April 21–23, 2005, in Grenoble, France.     

The finite rotation parameter of a deformable body and gyromagnetic effect

The derivation of Synge for the finite rotation formula in terms of Euler parameter is put into tensor form.Hence a further clarification of the geometric meaning of the orthogonal transformation obtained in the author’s paper[7] is made. The tensor properties of rotation axis vector are also discussed. By means of the method of co-moving coordinate system established in[8],[9], we explain the gyromagnetic effect,and derive a simple formula for calculating body couple which is induced by magnetization.     

Rate equations for viscoplastic materials subjected to finite strains

The constitutive equations for plasticity proposed by Voyiadjis [1984] and Voyiadjis & Kiousis [1987] are modified here in order to introduce rate sensitivity in the plastic region. Some of the basic concepts of the theory of viscoplasticity outlined by Naghdi & Murch [1963], Perzyna & Wojno [1966], and Eisenberg & Yen [1981] are used in this work in order to obtain the proposed viscoplastic constitutive model for finite strain deformation analysis.Uniaxial loading-reverse loading tests are conducted so as to check the validity of the proposed constitutive model as well as to determine its material parameters. The model is effectively used in simulating numerically the obtained experimental results at finite strains.     

Constitutive relations for isotropic or kinematic hardening at finite elastic–plastic deformations

The rate-type constitutive relations of rate-independent metals with isotropic or kinematic hardening at finite elastic–plastic deformations were presented through a phenomenological approach. This approach includes the decomposition of finite deformation into elastic and plastic parts, which is different from both the elastic–plastic additive decomposition of deformation rate and Lee’s elastic–plastic multiplicative decomposition of deformation gradient. The objectivity of the constitutive relations was dealt with in integrating the constitutive equations. A new objective derivative of back stress was proposed for kinematic hardening. In addition, the loading criteria were discussed. Finally, the stress for simple shear elastic–plastic deformation was worked out.     

Inelastic constitutive equations of polycrystalline metals subjected to finite deformation part I: Effect of grain rotation

In this article, a set of inelastic constitutive equations of polycrystalline metals is derived by combining a finite deformation kinematics of single crystal component, and a shear stress-shear strain relation of slip system based on a thermoactivated motion of dislocation. Interactions among grains are incorporated by “constant deformation gradient assumption.” The forms of these equations are rather simple internal variable theory types. By using these equations, some fundamental effects of grain rotations on inelastic behaviors of polycrystalline metals in a finite deformation range under complex loading and elevated temperature conditions are demonstrated. Some comments are given on a problem of plastic spin tensor.     

Non-self-similar crack growth in elastic-plastic finite thickness plate

This work is concerned with non-self-similar crack growth in medium strength metal plates while the loading step, plate thickness and material properties are altered. The three-dimensional elastic-plastic finite element stress analysis is combined with the strain energy density criterion for modeling the material damage process from crack initiation to final global instability including the intervening stage of slow crack growth. Both inelastic deformation and crack growth are accounted for each increment of loading such that the redistribution of stresses and strains are made for each new crack profile. Numerical results are obtained for the center cracked plate configuration under uniform extension with twenty-seven (27) different combinations of specimen thickness, loading step and material type. The fracture toughness S_c being related to K_1c for three different materials are predicted analytically from the corresponding uniaxial tensile test data. Effective strain energy density factor and half crack length are defined so that the results can be compared with their two-dimensional counterparts. Crack growth resistance curves (R-curves) are constructed by plotting as a function of . The condition is found to prevail during slow crack growth. Translation and/or rotation of the lines can yield results other than those calculated and serve a useful purpose for scaling component size and test time. The minimum thickness requirement for the ASTM valid K_1c test is also discussed in connection with predictions based on the strain energy density criterion. The corresponding K_1c for smaller specimens that exhibit moderate ductility and nonlinearity can also be obtained analytically. In such cases, the influence of loading step can be significant and should not be neglected. Notwithstanding the shortcomings of the theory of plasticity, the qualitative features of non-self-similar crack growth are predicted by the strain energy density criterion. Any refinements on the analytical modeling of the material damage process would only affect the results qualitatively, a subject that is left for future investigation.     

Buckling and postbuckling of moderately thick plates

This paper gives the basic differential equations for finite deflections of elastic plates according to Reissner’s approximate stress distributions. The buckling and postbuckling problems of elastic rectangular plates, including the effect of transverse shear deformation, are solved and discussed, by using perturbation method suggested in ref. [8]. The postbuckling equilibrium paths of perfect and imperfect moderately thick rectangular plates are presented and compared with the results based on thin plate theory.     

Bending analysis of functionally graded CNT reinforced doubly curved singly ruled truncated rhombic cone

The bending analysis of functionally graded carbon nanotube (CNT) reinforced doubly curved singly ruled truncated rhombic cone is investigated. In this study, a simple C⁰ isoparametric finite element formulation based on third order shear deformation theory is presented. To characterize the membrane-flexure behavior observed in a CNT reinforced truncated rhombic cone, a displacement field involving higher-order terms in in-plane fields is considered. The proposed kinematics field incorporates for transverse shear deformation and nonlinear variation of the in-plane displacement field through the thickness to predict the overall response of the CNT reinforced truncated rhombic cone in an accurate sense. The material properties of the CNT reinforced truncated rhombic cone are estimated according to the rule of mixture. The present model eliminates the need of shear correction factor and imposed zero-transverse shear strain at upper and lower surface of the truncated rhombic cone. The new feature in present model is simultaneous inclusion of twist curvature in strain field as well as curvature in displacement field that makes it suitable for moderately thick and deep truncated rhombic cone. The proposed new mathematical model is implemented in finite element code written in FORTRAN. The proposed model has been validated with analytical, experimental, and finite element results from the literature. This is first attempt to study bending response of CNT reinforced doubly curved singly ruled truncated rhombic cone. The effect of CNT distribution, boundary condition, loading pattern, and other geometric parameters are also examined.     

A new look at the laminar flow of power law fluids through granular beds

Summary The paper deals with laminar flow of power law fluids through granular beds. A critical review of the assumptions concerning the capillary model of the bed, applied by various authors, led us to the conclusion that the derivation of the correlation eq. [13] given byChristopher andMiddleman was based on a too simplified model of the granular bed. Taking advantage of the approach presented in the classical works ofKozeny andCarman (which seems to be partly overlooked by some authors, including our own previous works) a modified correlation equation for power law fluids [21], a corrected formula for shear rate in the bed [29] and for Deborah number [32], as well as corrected correlation equation for fluids exhibiting memory effects [34] were presented.

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Zusammenfassung Diese Arbeit betrifft laminare Strömungen von Potenzgesetzflüssigkeiten durch Kornschüttungen. Eine kritische Prüfung der Annahmen, die von verschiedenen Autoren für das Kapillar-Modell der Schüttung gemacht worden sind, führt uns zu der Folgerung, daß die Herleitung der Korrelationsgleichung [13] nachChristopher undMiddleman auf einem übervereinfachten Modell der Kornschüttung basiert. Unter Nutzbarmachung der Annahmen, die in den klassischen Arbeiten vonKozeny undCarman dargestellt worden sind (sie wurden sowohl von manchen anderen Autoren als auch in unseren früheren Arbeiten nicht beachtet), werden nun eine modifizierte Korrelationsgleichung für die Potenzgesetzflüssigkeiten [21], eine korrigierte Formel für die Schergeschwindigkeit in der Schüttung [29], eine korrigierte Formel für die Deborah-Zahl [32] und eine korrigierte Korrelationsgleichung für Flüssigkeiten, die Gedächtnis-Effekte zeigen [34], angegeben.

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Notation A constant in eq. [9] - d _p effective particle diameterd _p = 6/a (wherea is the specific surface of the bed), m - f _BK modified friction factor, defined by eq. [1] - k power law parameter, N s ⁿ /m² - K Kozeny constant, defined by eq. [8] - K ₀ constant depending on the shape of the channel cross-section - K ₁ constant, defined by eq. [5] - l bed height, m - l _e channel length, m - n power law parameter - p pressure drop due to friction, N/m² - r _h hydraulic radius, defined by eq. [6], m - s bed permeability, defined by eq. [16], m² - v ₀ mean linear velocity related to an empty crosssection of the column, m/s - v _e mean linear velocity in the channel, m/s - shear rate at the wall of the channel, s^–1 - shear rate at the wall of the channel calculated according to the formula [29], s^–1 - bed porosity - characteristic time of the fluid, s - friction factor, defined by eq. [25] - µ dynamic viscosity of the fluid, N s/m² - parameter, defined by eq. [15], N s ⁿ /m¹⁺ⁿ - De Deborah number, defined by eq. [33] - De ^* Deborah number, defined by eq. [32] - Re _BK modified Reynolds number, defined by eq. [2] - Re _BK modified Reynolds number, defined by eq. [26] - Re _BK ^* modified Reynolds number, defined by eq. [23] - Re _CM modified Reynolds number byChristopher andMiddleman, defined by eq. [14] - Re _CM modified Reynolds number, defined by eq. [17] With 3 figures and 1 table     

Formulation of the problem of the motion of a small body in a perturbed flow

The classical problem of the hydrodynamic reactions on a body of arbitrary shape moving in a fluid at rest [1] was generalized by Sedov to the case of an accelerated translational flow [2]. In the expressions for the hydrodynamic reactions, the shape of the body is represented only by the coefficients λ_ij of the added masses and the volume Ω of the body. In the general case of motion of a body in a nontranslational flow the shape of the body cannot be represented by a finite set of coefficients in the determination of the hydrodynamic reactions. An important simplification occurs in the small-body formulation, which again leads to expressions for the force and torque similar to the classical expressions. The problem of the motion of a small body in a perturbed nontranslational flow was posed by Grigoryan and Yakimov [3], and with allowance for deformation of the body by Yakimov [4]. Later studies containing this formulation have been reviewed by Vil'khovchenko and Yakimov [5]. The aim of the present paper is to formulate the small-body problem more precisely. The order of smallness of the terms in the earlier studies was estimated solely as a function of the power of a small parameter — the size of the body. In the present paper it is shown that if it is additionally required that the final expression for the reactions should contain only principal terms containing the components ν_i and ω_i of the translational and angular velocities, and also terms describing the flow structure, then the expression found by Grigoryan and Yakimov [3] for the hydrodynamic reaction is valid. The terms are estimated on the basis of dimensional analysis. Such arguments have already been used by the author for special examples [6, 7].     

Unsteady three‐dimensional boundary layer flow due to a stretching surface in a micropolar fluid

In this paper, the unsteady three‐dimensional boundary layer flow due to a stretching surface in a viscous and incompressible micropolar fluid is considered. The partial differential equations governing the unsteady laminar boundary layer flow are solved numerically using an implicit finite‐difference scheme. The numerical solutions are obtained which are uniformly valid for all dimensionless time from initial unsteady‐state flow to final steady‐state flow in the whole spatial region. The equations for the initial unsteady‐state flow are also solved analytically. It is found that there is a smooth transition from the small‐time solution to the large‐time solution. The features of the flow for different values of the governing parameters are analyzed and discussed. The solutions of interest for the skin friction coefficient with various values of the stretching parameter c and material parameter K are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Two-dimensional stress wave analysis in incompressible elastic solids

Two-dimensional stress waves in a general incompressible elastic solid are investigated. First, basic equations for simple waves and shock waves are presented for a general strain energy function. Then the characteristic wave speeds and the associated characteristic vectors are deduced. It is shown that there usually exist two simple waves and two shock waves. Finally, two examples are given for the case of plane strain deformation and antiplane strain deformation, respectively. It is proved that, in the case of plane strain deformation the oblique reflection problem of a plane shock is not solvable in general.     

A dynamic model of bridging fiber pull-out of composite materials

An elastic analysis of an internal central crack with bridging fibers parallel to the free surface in an infinite orthotropic anisotropic elastic plane is carried out. In this paper a dynamic model of bridging fiber pull-out is presented for analyzing the distributions stress and displacement of composite materials with the internal central crack under the loading conditions of an applied non-uniform stress and the traction forces on crack faces yielded by the fiber pull-out model. Thus the fiber failure is determined by maximum tensile stress, the fiber breaks and hence the crack propagation should occur in self-similar fashion. By reducing the dynamic model to the Keldysh–Sedov mixed boundary value problem, a straightforward and easy analytical solution can be attained. When the crack extends, its fibers continue to break. Analytical study on the crack extension under the action of an inhomogeneous point force Px/t, Pt is obtained for orthotropic anisotropic body, respectively; and it can be utilized to attain the concrete solutions of the model by the ways of superposition.     

The creep deformation of vibrating structures

In a recent paper [1] it was shown that the evaluation of certain bounding solutions for a structure subjected to cyclic loading was equivalent to assuming that the cycle time Δt was short compared with a stress redistribution time. Comparisons between values which are likely to occur in creep design situations indicated that Δt may often be assumed to be small and the bounding solution may be expected to closely approximate the actual stress history. In this paper the solution for the limiting case when Δt → 0 is evaluated for a class of constitutive relationships which may be expressed in terms of a finite number of state variables. Strain-hardening viscous, visco-elastic and Bailey-Orowan equations are discussed and particular solutions for which the residual stresses remain constant in time are derived. The solution for a non-linear visco-elastic model indicates that, for the stationary cyclic state, the constitutive equation need only predict the creep strain over a discrete number of cycles and need not predict the strains during a cycle. This observation should considerably simplify creep analysis.The solution of a simple example demonstrates the similarity between the predicting of the various constitutive relationships for isothermal problems. In fact they provide virtually identical solutions when expressed in terms of reference stress histories. The finite element solution of a plate containing a hole and subjected to variable edge loading is also presented for a viscous material. The solutions show behaviour which is similar to that of the two bar structure.     

General hyperbolic difference formulas for linear and quasilinear hyperbolic equations

The method of non-standard finite elements was used to develop multilevel difference schemes for linear and quasilinear hyperbolic equations with Dirichlet boundary conditions. A closed form equation of kth-order accuracy in space and time (O(Δt^k, Δx^k)) was developed for one-dimensional systems of linear hyperbolic equations with Dirichlet boundary conditions. This same equation is also applied to quasilinear systems. For the quasilinear systems a simple iteration technique was used to maintain the kth-order accuracy. Numerical results are presented for the linear and non-linear inviscid Burger's equation and a system of shallow water equations with Dirichlet boundary conditions.