A class of exact solutions of the dynamical equations for rigid-viscoplastic bodies

Summary We here discuss the class of plane, irrotational flows of a rigid-perfectly plastic or of a rigid-viscoplastic body. Such flows turn out to be of a limited variety: they are either familiar hydrodynamical flows or are geometrically similar to a model flow whose velocity field can be represented in parametric form in terms of the complex error function. Qualitative features of such flows are discussed as well as the motivation for considering them.

---

Zusammenfassung Betrachtet wird die Klasse der ebenen, wirbelfreien Strömungen in einem starr-idealplastischen oder starrviskoplastischen Körper. Solche Strömungen sind entweder von bekanntem Typ oder sind einer Modellströmung ähnlich, deren Geschwindigkeitsfeld eine parametrische Darstellung besitzt, die durch die komplexe Fehlerfunktion beschreibbar ist. Sowohl die qualitativen Hauptmerkmale als auch die Bedeutung solcher Strömungen werden besprochen.

---

---

This paper summarizes research described in more detail in (1).     

On the dynamics of swimming linked bodies

In this paper we study the motion of three linked ellipses moving through a viscous fluid in two dimensions. The angles between the ellipses change with time in a specified manner (the gait) and the resulting time varying configuration is similar to the appearance of a swimming leech. We simulate the motion using the particle method Smoothed Particle Hydrodynamics (SPH) which we test by convergence studies and by comparison with the inviscid results of Kanso et al. (2005) [2] and the viscous results of Eldredge (2006, 2007, 2008) [4], [5], [6]. We determine how the average speed and power output depends on the amplitude and oscillation frequency of the gait. We find that the results fit simple scaling rules which can be related to the analytical results of G.I. Taylor for the swimming of long narrow animals (1952). We apply our results to estimate the speed of a swimming leech with reasonable accuracy, and we determine the minimum power required to propel the bodies at a specified average speed.     

Minimum principles in the dynamics of isotropic rigid-plastic and rigid-viscoplastic continuous media

Summary The paper discusses the incremental boundary value problem for rigid-viscoplastic isotropic continua subjected to dynamic actions. A pair of dual extremum theorems reduces the problem to the minimization of some functionals in a class of functions defined by appropriate constraints. The first theorem takes as variables stresses and accelerations, the latter only the accelerations. Some conclusions end the paper.

---

Sommario Viene discussa la soluzione incrementale dei problemi al contorno per i mezzi continui rigido-viscoplastici isotropi soggetti ad azioni dinamiche. Si dimostrano due teoremi duali che riportano il problema alla minimizzazione di opportuni funzionali in una classe di funzioni definita attraverso vincoli appropriati. Il primo teorema concerne le tensioni e le accelerazioni, il secondo le sole accelerazioni. Alcune considerazioni conclusive chiudono il lavoro.

---

---

The results presented in the paper form part of a Research supported by the National Research Council (C.N.R.).     

On the solution of three-dimensional boundary-value problems for layered bodies with nonplanar interfaces

The interfaces play an important role in various buildup bodies, and also in the composite materials and structural elements. Special monographs [7, 8] have been devoted to this question, presenting the results of scientific studies of the physical and chemical phenomena on the interfaces, the mechanical behavior, and the role of the interfaces in the damage processes, and also their influence on the basic mechanical properties of the composites. In many cases the interfaces deviate from the ideal geometric shapes: planar (in the layered composites), circular cylindrical (in the fibrous composites), and spherical (in the granular composites). Numerous theoretical and experimental studies confirm this. Thus, in the explosive welding of metals (and nonmetals) there form wavy surfaces, the sections of which may be close to sinusoids, for example in the welding of niobium and copper [9]. If the densities of the materials differ significantly, then the sinusoidal nature of the interface distorts as illustrated in [12] for the example of the welding of lead and steel. In addition, in view of the nature of the technological processes [10] the interfaces may become curved in the layered composite materials and deviate locally or periodically from the ideal coordinate planes. Theoretical and experimental studies have shown that the shape of the interface has a significant influence on the physical and mechanical processes and phenomena (bond strength, stress concentration, wave diffraction, thermal conduction, and so on). Numerous publications that are cited in the survey works [1, 3, 11] confirm this. A second variant of the boundary shape perturbation method was developed in [4, 5] for the solution of the three-dimensional boundary-value problems for nonorthogonal surfaces that are close to the coordinate planes. It was assumed that the equations of the interfaces are linear relative to the small parameter characterizing the degree of deviation from the coordinate planes. This narrowed significantly the class of the examined boundary-value problems and their practical importance. In the present work we examine the three-dimensional boundary-value problems of the mechanics of layered bodies with interfaces that are described by nonlinear equations relative to a small parameter. We construct in general form the recurrence relations and the differential operators of the boundary conditions, making it possible to solve the three-dimensional boundary-value problems with the accuracy that is required for applications. We examine particular cases and present one of the possible criteria for evaluating the accuracy of the approximate solutions that are obtained with the aid of the described variant of the boundary shape perturbation method.S. P. Timoshenko Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 23–32, February, 1994.     

On the nonlinear dynamics of elastically interacting asymmetric rigid bodies

A symmetric mathematical model is developed to describe the spatial motion of a system of space vehicles whose structure is represented by regular geometrical figures (Platonic bodies). The model is symmetrized by using the Euler-Lagrange equations of motion, the Rodrigues-Hamilton parameters, and quaternion matrix mathematics. The results obtained enable us to model a wide range of dynamic, control, stabilization, and orientation problems for complex systems and to solve various problems of dynamic design for such systems, including estimation of dynamic loading on the basic structure during maneuvers in space __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 126–132, January 2006.     

Non-overlapping conditions for crack face displacement of anisotropic bodies

Crack tip stress and displacement fields are useful for studying the fracture behavior of cracks in both isotropic and anisotropic materials. Under certain boundary conditions, crack surfaces could overlap, a condition that could be more prevalent for the anisotropic case as compared with isotropic materials. Conditions can be derived for different loading conditions and material properties such that overlap of the crack faces would not occur.     

虚位移原理与变形体能量法的教学衔接

论述了理论力学教学中如何恰当处理质点系的虚位移原理与变形体能量原理的主要内在关联,由质点系虚功方程简便地导出虚功方程应用于变形体的一般形式,并由此导出卡氏第一、第二定理和莫尔定理的普遍形式,并给出了相应例题,为理论力学教学中讲授虚位移原理增添了重要新内容,同时为后续课程打下理论基础,实现了相关内容的自然衔接．     

On an analytic solution in the theory of large elastoviscoplastic strains

Analytic solutions for motions of viscoplastic media under pressure drop [1, 2] have become classical. They have served for the progress in the theory and as a necessary tool for testing numerical methods for problems with unknown boundaries (the boundaries of rigid cores and stagnation regions). In the present paper, we essentially generalize these solutions to the case in which one cannot neglect elastic properties. It turns out that the assumption that the medium is incompressible permits obtaining the analytic solution with large strains taken into account, while without this assumption the analytic solution cannot be obtained even for small strains in elastic strain regions.     

On the displacement potential solution of plane problems of structural mechanics with mixed boundary conditions

The present paper describes the advancement of displacement potential approach in relation to solution of plane problems of structural mechanics with mixed mode of boundary conditions. Both the conditions of the plane stress and the plane strain are considered for analyzing the displacement and stress fields of the structural problem. Using the finite difference technique based on the present displacement potential approach for the case of the plane stress and the plane strain conditions, firstly an elastic cantilever beam subjected to a pure shear at its tip is solved and these two solutions (plane stress and plane strain) are compared with Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951); secondly the above-mentioned displacement potential approach for the case of the plane stress and the plane strain conditions are applied to solve a one-end fixed square plate subjected to a combined loading at its tip. Effects of plane stress and plane strain on the elastic field of the plate are discussed in a comparative fashion. Limitations of Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951) over the displacement potential approach for the case of the plane stress and the plane strain conditions are not only discussed but also the superiority of the present displacement potential approach for the case of the plane stress and the plane strain conditions are reflected in the present research work.     

Analysis of a benchmark solution for non-Newtonian radial displacement in porous media

We present an analytical formulation useful to interpret the key phenomena involved in non-Newtonian displacement in porous media and an analysis of the results obtained by considering the uncertainty associated with relevant problem parameters. To derive a benchmark solution, we consider the radial dynamics of a moving stable interface in a porous domain saturated by two fluids, displacing and displaced, both non-Newtonian of shear-thinning power-law behavior, assuming the pressure and velocity to be continuous at the interface, and constant initial pressure. The flow law for both fluids is a modified Darcy’s law. Coupling the nonlinear flow law with the continuity equation, and taking into account compressibility effects, yields a set of nonlinear second-order partial differential equations. Considering two fluids with the same flow behavior index n allows transformation of the latter equations via a self-similar variable; further transformation of the equations incorporating the conditions at the interface shows for n<1 the existence of a compression front ahead of the moving interface. Solving the resulting set of nonlinear equations yields the positions of the moving interface and compression front, and the pressure distributions; the latter are derived in closed form for any value of n. A sensitivity analysis of the model responses is conducted both in a deterministic and a stochastic framework. In the latter case, Global Sensitivity Analysis (GSA) of the benchmark analytical model is adopted to study how the effects of uncertainty affecting selected parameters: (a) the fluids flow behavior index, (b) the relative total compressibility and mobility in the displaced and displacing fluid domains, and (c) the domain permeability and porosity, propagate to state variables. The relative influence of input parameters on model outputs is evaluated by means of associated Sobol indices, calculated via the Polynomial Chaos Expansion (PCE) technique. The goodness of the results obtained by the PCE is assessed by comparison against a traditional Monte Carlo (MC) approach.