A perturbation-incremental method for strongly non-linear non-autonomous oscillators

A perturbation-incremental method is extended for the analysis of strongly non-linear non-autonomous oscillators of the form , where g(x) and are arbitrary non-linear functions of their arguments, and ε can take arbitrary values. Limit cycles of the oscillators can be calculated to any desired degree of accuracy and their stabilities are determined by the Floquet theory. Branch switching at period-doubling bifurcation along a frequency-response curve is made simple by the present method. Subsequent continuation of an emanating branch is also discussed.     

Size-dependent sharp indentation-II: a reverse algorithm to identify plastic properties of metallic materials

In part I of this paper (Cao and Lu, J. Mech. Phys. Solids, in press), a closed-form expression of the size-dependent sharp indentation loading curve has been proposed. In this second part, which concerns the direct application of the analytical model, a reverse algorithm has first been established to extract the plastic properties of metallic materials on a small scale where the size effect caused by geometrically necessary dislocations is significant. Second, from the viewpoint of the mathematical theory of inverse problems, the properties of the present inverse problem i.e. the existence, uniqueness and stability of the solution, have been investigated systematically. The results have identified the extent to which the plastic properties of ductile materials can be determined effectively using the present method. Third, experimental verifications of the reverse algorithm using a standard Berkovich indenter have been carried out for 316 stainless steel and pure titanium, respectively. The results show that, by taking a maximum indentation depth of 1.5 and , respectively for the two materials, a good engineering estimation of the representative stresses, σ_0.033, in the absence of a strain gradient can be made using the present method, which can be used in conjunction with the representative stress corresponding to another indenter with a different tip apex angle to determine the plastic properties of metallic materials, i.e. the yield strength σ_y and the strain hardening exponent n. The material length scale can also be identified by using the present algorithm. Experimental results show that it has the correct order of magnitude, but is more sensitive to data errors than the identified representative stress.     

Geometrical properties of the vorticity vector derived using large-eddy simulation

In this paper, the geometrical properties of the resolved vorticity vector derived from large-eddy simulation are investigated using a statistical method. Numerical tests have been performed based on a turbulent Couette channel flow using three different dynamic linear and nonlinear subgrid-scale stress models. The geometrical properties of have a significant impact on various physical quantities and processes of the flow. To demonstrate, we examined helicity and helical structure, the attitude of with respect to the eigenframes of the resolved strain rate tensor and negative subgrid-scale stress tensor -τ_ij, enstrophy generation, and local vortex stretching and compression. It is observed that the presence of the wall has a strong anisotropic influence on the alignment patterns between and the eigenvectors of , and between and the resolved vortex stretching vector. Some interesting wall-limiting geometrical alignment patterns and probability density distributions in the form of Dirac delta functions associated with these alignment patterns are reported. To quantify the subgrid-scale modelling effects, the attitude of with respect to the eigenframe of -τ_ij is studied, and the geometrical alignment between and the Euler axis is also investigated. The Euler axis and angle for describing the relative rotation between the eigenframes of -τ_ij and are natural invariants of the rotation matrix, and are found to be effective for characterizing a subgrid-scale stress model and for quantifying the associated subgrid-scale modelling effects on the geometrical properties of .     

Discrete dislocation plasticity analysis of the wedge indentation of films

The plane strain indentation of single crystal films on a rigid substrate by a rigid wedge indenter is analyzed using discrete dislocation plasticity. The crystals have three slip systems at ±35.3^° and 90^° with respect to the indentation direction. The analyses are carried out for three values of the film thickness, 2, 10 and , and with the dislocations all of edge character modeled as line singularities in a linear elastic material. The lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation are incorporated through a set of constitutive rules. Over the range of indentation depths considered, the indentation pressure for the 10 and thick films decreases with increasing contact size and attains a contact size-independent value for contact lengths . On the other hand, for the films, the indentation pressure first decreases with increasing contact size and subsequently increases as the plastic zone reaches the rigid substrate. For the 10 and thick films sink-in occurs around the indenter, while pile-up occurs in the film when the plastic zone reaches the substrate. Comparisons are made with predictions obtained from other formulations: (i) the contact size-independent indentation pressure is compared with that given by continuum crystal plasticity; (ii) the scaling of the indentation pressure with indentation depth is compared with the relation proposed by Nix and Gao [1998. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 43, 411-423]; and (iii) the computed contact area is compared with that obtained from the estimation procedure of Oliver and Pharr [1992. An improved technique for determining hardness and elastic-modulus using load and displacement sensing indentation experiments, J. Mater. Res. 7, 1564-1583].     

Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector

This paper discusses boundary conditions appropriate to a theory of single-crystal plasticity (Gurtin, J. Mech. Phys. Solids 50 (2002) 5) that includes an accounting for the Burgers vector through energetic and dissipative dependences on the tensor G=curlH^p, with H^p the plastic part in the additive decomposition of the displacement gradient into elastic and plastic parts. This theory results in a flow rule in the form of N coupled second-order partial differential equations for the slip-rates , and, consequently, requires higher-order boundary conditions. Motivated by the virtual-power principle in which the external power contains a boundary-integral linear in the slip-rates, hard-slip conditions in which

(A)

on a subsurface S_hard of the boundary

for all slip systems α are proposed. In this paper we develop a theory that is consistent with that of (Gurtin, 2002), but that leads to an external power containing a boundary-integral linear in the tensor , a result that motivates replacing (A) with the microhard condition

(B)

on the subsurface S_hard.

We show that, interestingly, (B) may be interpreted as the requirement that there be no flow of the Burgers vector across S_hard.What is most important, we establish uniqueness for the underlying initial/boundary-value problem associated with (B); since the conditions (A) are generally stronger than the conditions (B), this result indicates lack of existence for problems based on (A). For that reason, the hard-slip conditions (A) would seem inappropriate as boundary conditions.Finally, we discuss conditions at a grain boundary based on the flow of the Burgers vector at and across the boundary surface.     

Limit analysis of multi-layered plates. Part I: The homogenized Love-Kirchhoff model

The purpose of this paper is to determine , the overall homogenized Love-Kirchhoff strength domain of a rigid perfectly plastic multi-layered plate, and to study the relationship between the 3D and the homogenized Love-Kirchhoff plate limit analysis problems. In the Love-Kirchhoff model, the generalized stresses are the in-plane (membrane) and the out-of-plane (flexural) stress field resultants. The homogenization method proposed by Bourgeois [1997. Modélisation numérique des panneaux structuraux légers. Ph.D. Thesis, University Aix-Marseille] and Sab [2003. Yield design of thin periodic plates by a homogenization technique and an application to masonry wall. C. R. Méc. 331, 641-646] for in-plane periodic rigid perfectly plastic plates is justified using the asymptotic expansion method. For laminated plates, an explicit parametric representation of the yield surface is given thanks to the π-function (the plastic dissipation power density function) that describes the local strength domain at each point of the plate. This representation also provides a localization method for the determination of the 3D stress components corresponding to every generalized stress belonging to . For a laminated plate described with a yield function of the form , where σ^u is a positive even function of the out-of-plane coordinate x₃ and is a convex function of the local stress σ, two effective constants and a normalization procedure are introduced. A symmetric sandwich plate consisting of two Von-Mises materials ( in the skins and in the core) is studied. It is found that, for small enough contrast ratios (), the normalized strength domain is close to the one corresponding to a homogeneous Von-Mises plate [Ilyushin, A.-A., 1956. Plasticité. Eyrolles, Paris].     

Non-Newtonian pseudoplastic fluids: Analytical results and exact solutions

A one layer model of laminar non-Newtonian fluids (Ostwald-de Waele model) past a semi-infinite flat plate is revisited. The stretching and the suction/injection velocities are assumed to be proportional to x^1/(1−2n) and x⁻¹, respectively, where n is the power-law index which is taken in the interval . It is shown that the boundary-layer equations display both similarity and pseudosimilarity reductions according to a parameter γ, which can be identified as suction/injection velocity. Interestingly, it is found that there is a unique similarity solution, which is given in a closed form, if and only if γ=0 (impermeable surface). For γ≠0 (permeable surface) we obtain a unique pseudosimilarity solution for any 0≠γ≥−((n+1)/3n(1−2n))^n/(n+1). Moreover, we explicitly show that any pseudosimilarity solution exhibits similarity behavior and it is, in fact, similarity solution to a modified boundary-layer problem for an impermeable surface. In addition, the exact similarity solution of the original boundary-layer problem is used, via suitable transverse translations, to construct new explicit solutions describing boundary-layer flows induced by permeable surfaces.     

Incompressible laminar 2D steady thermal boundary layers with temperature-dependent kinematic viscosity and thermal diffusivity

We prove that the incompressible 2D steady thermal boundary layer equations with temperature-dependent kinematic viscosity ν and thermal diffusivity α is maximally symmetric provided the Prantl number Pr=ν/α is constant and or ν=K₂(AT+B)^K₁ if we neglect energy dissipation and if we take into account dissipation. This result corroborates assumptions often made in applications. When we disregard dissipation, the symmetry Lie algebra assumes the forms L^r⊕L^∞, where L^∞ is an infinite-dimensional Lie algebra and L^r is an r-dimensional Lie algebra with r∈{3,4,5,6}. If we include dissipation, r∈{2,3}. We notice that dissipation has a symmetry breaking effect.We also show how the symmetries can be employed for the calculation of invariant solutions.     

Hall effects on the magnetohydrodynamic shear flow past an infinite porous flat plate subjected to uniform suction or blowing

An analysis is made of Hall effects on the steady shear flow of a viscous incompressible electrically conducting fluid past an infinite porous plate in the presence of a uniform transverse magnetic field. It is shown that for suction at the plate, steady shear flow solution exists only when S²<Q, where S and Q are the suction and magnetic parameters, respectively. The primary flow velocity decreases with increase in Hall parameter m. But the cross-flow velocity first increases and then decreases with increase in m. Similar results are obtained for variation of the induced magnetic field with m. It is further found that for blowing at the plate, steady shear flow solution exists only when , where S₁ is the blowing parameter.     

Dynamic fracture of a kane-mindlin plate

We study dynamic crack problems for an elastic plate by using Kane-Mindlin's kinematic assumptions. The general solutions of the Laplace transformed displacements and stresses are first derived. Path independent integrals for stationary cracks subjected to transient loads and steadily growing cracks are deduced. For a stationary crack in a very thin plate subjected to impact loads, the crack tip dynamic stress intensity factor (DSIF), K₁(t), is related to the far field plane stress one, K₁⁰(t), by where ν is Poisson's ratio. For a crack steadily growing with speed V, the crack tip DSIF, K₁(V), is given by where K₁⁰(V) is the plane stress DSIF and A(V) and B(V) are known functions of V. These results are applied to compute the DSIF for a semi-infinite stationary crack in an unbounded plate subjected to impact pressure on the crack faces. The results of DSIF for a finite crack in an infinite plate under uniform impact pressure on the crack surfaces show that for each plate thickness, the maximum DSIF is higher than that for the plane stress case.     

A Semilinear Schrödinger Equation in the Presence of a Magnetic Field

We consider the semilinear stationary Schrödinger equation in a magnetic field: (–i+A)² u+V(x)u=g(x,|u|)u in ^N , where V is the scalar (or electric) potential and A is the vector (or magnetic) potential. We study the existence of nontrivial solutions both in the critical and in the subcritical case (respectively g(x,|u|)=|u|^{2 * –2} and |g(x,|u|)|c(1+|u| ^{p –2}), where 2<p<2^*). The results are obtained by variational methods. For g critical we use constrained minimization and for subcritical g we employ a minimax-type argument. In the latter case we also study the existence of infinitely many geometrically distinct solutions.     

Irreducible structure, symmetry and average of Eshelby's tensor fields in isotropic elasticity

The strain field ?(x) in an infinitely large, homogenous, and isotropic elastic medium induced by a uniform eigenstrain ?⁰ in a domain ω depends linearly upon . It has been a long-standing conjecture that the Eshelby's tensor field S^ω(x) is uniform inside ω if and only if ω is ellipsoidally shaped. Because of the minor index symmetry , S^ω might have a maximum of 36 or nine independent components in three or two dimensions, respectively. In this paper, using the irreducible decomposition of S^ω, we show that the isotropic part S of S^ω vanishes outside ω and is uniform inside ω with the same value as the Eshelby's tensor S⁰ for 3D spherical or 2D circular domains. We further show that the anisotropic part A^ω=S^ω-S of S^ω is characterized by a second- and a fourth-order deviatoric tensors and therefore have at maximum 14 or four independent components as characteristics of ω's geometry. Remarkably, the above irreducible structure of S^ω is independent of ω's geometry (e.g., shape, orientation, connectedness, convexity, boundary smoothness, etc.). Interesting consequences have implication for a number of recently findings that, for example, both the values of S^ω at the center of a 2D C_n(n?3,n≠4)-symmetric or 3D icosahedral ω and the average value of S^ω over such a ω are equal to S⁰.