Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law

Starting from Eshelby’s solution of the equivalent inclusion problem, an approximate solution is proposed in order to model interface debonding of a spherical inhomogeneity isolated in a uniform matrix. Both phases are linear elastic but the interface traction-separation law is non-linear. A semi-analytical incremental model is developed which is suitable for any type of loading. For computational efficiency, the model relies on two simplifying assumptions: (i) the eigenstrain is uniform inside the inhomogeneity and (ii) the interface compliance is averaged over inhomogeneity’s surface when computing the average strain within the inhomogeneity. An extensive parametric study is conducted for three loading modes and 144 combinations of non-dimensional parameters. The predictions are assessed against full-field finite element solutions based on two error measures of the mean stress field inside the inhomogeneity. The results show that the mean error value is acceptable in all cases and indicate the parameter ranges for which the model is most accurate.     

Dynamically Equilibrium Shape of Intrusive Vortex Formations in the Ocean

The problem of finding the dynamically equilibrium shape of a rotating mass of liquid with homogeneous density (lens) submerged in a stratified ocean at rest on the rotating Earth is formulated. An equation for the shape of the interface between water masses is derived. An exact solution of the problem for an anticyclonically rotating lens in a linearly stratified ocean in the neighborhood of the lens depth shows that the dynamically equilibrium shape of the interface is a triaxial ellipsoid inclined with respect to the horizon and similar to an ellipsoid of revolution for real parameters of the phenomenon. The limiting values of the latitudes at which these formations can exist are determined. Degeneration of the shape with decrease in the intrinsic lens angular rate is investigated.     

Bifurcation phenomena in the rigid inclusion power law matrix composite sphere

Bifurcation of interface separation related to cavity nucleation is analyzed for a radially loaded composite sphere consisting of a rigid inclusion separated from a power law matrix by a uniform, non-linear cohesive zone. Equations for the spherically symmetric and non-symmetric problems are obtained from a hyperelastic finite strain theory by a limiting process that preserves non-linear matrix and interface response at infinitesimal strain. A complete solution to the symmetric problem is presented including bifurcation load, stresses, and evolution of elasto-plastic boundary and interface separation. An analysis of non-symmetric bifurcation, under symmetric conditions of geometry and loading, yields the bifurcation load and first non-symmetric mode shape associated with rigid inclusion displacement. An energy analysis is carried out for both symmetric and non-symmetric problems in order to assess stability of spherically symmetric states to spherically symmetric and non-symmetric “rigid body mode” perturbations.Results are provided for an interface force law that captures interface failure in normal mode and linear response in shear mode. For the symmetric problem, (i) there are threshold parameter values above which bifurcation will generally not occur, (ii) threshold values below which there do not exist equilibria in the post bifurcation regime, (iii) bifurcation occurs after attainment of the maximum interface strength. For the non-symmetric problem, (i) bifurcation always occurs, although it can be delayed by interfacial shear, (ii) for the smooth interface, non-symmetric bifurcation occurs after attainment of the maximum interface strength and always precedes symmetric bifurcation.     

Problems of Hydrodynamics for a Triaxial Ellipsoid

Problems of motion of a triaxial ellipsoid in an ideal liquid and in a viscous liquid in the Stokes approximation and also equilibrium shapes of the rotating gravitating liquid mass are considered. Solutions of these problems expressed via four quadratures depending on four parameters are significantly simplified because they are expressed via the only function of two arguments. The efficiency of the proposed approach is demonstrated by means of analyzing the velocity and pressure fields in an ideal liquid, calculating the added mass of the ellipsoid, determining the viscous friction, and studying the equilibrium shapes and stability of the rotating gravitating capillary liquid. The pressure on the triaxial ellipsoid surface is expressed via the projection of the normal to the impinging flow velocity. The shape of an ellipsoid that ensures the minimum viscous drag at a constant volume is determined analytically. A simple equation in elementary functions is derived for determining the boundary of the domains of the secular stability of the Maclaurin ellipsoids. An approximate solution of the problem of equilibrium and stability of a rotating droplet is presented in elementary functions. A bifurcation point with non-axisymmetric equilibrium shapes branching from this point is found.     

Generalization of the Prandtl problem for a creep model

An approximate solution to the problem of compression of an infinite layer of material between rough parallel plates is constructed with the creep equations being fulfilled. Constitutive relations in accordance with which the equivalent stress tends to a finite value as the equivalent strain rate tends to infinity are used. The behavior of the solution in the neighborhood of the maximum friction surface is studied. It is shown that the existence of the solution depends on one of the parameters included in the constitutive equations. If the solution exists, the equivalent strain rate tends to infinity in the neighborhood of the maximum friction surface, and the asymptotic behavior of the solution depends on the same parameter. It is established that there is a range of this parameter in which the nature of the change in the equivalent strain rate near the maximum friction surface is the same as in the solutions for rigid plastic materials.     

An extension of the time–temperature superposition principle to non-linear viscoelastic solids

An experimental data treatment is introduced to manage with the tensile test responses of highly non-linear viscoelastic solids such as solid propellants. This treatment allows the representation of a set of strain–stress curves by a single intrinsic non-linear response which is found independent of the experimental conditions of rate and temperature. To obtain this result, two independent normalization factors are applied both on the stress and strain axis. The requirement of a normalization factor applied to the strain measure produces a pseudo-strain which is found to be viscoelastic in nature. It is believed that the existence of such a viscoelastic strain measure is the characteristic feature of non-linear viscoelasticity. To confer some generality to the principle, the validity is also checked for volumetric and multiaxial stress response of solid propellant. To illustrate the potential application of the principle, it is applied to build up an idealized material database upon which numerical identification of constitutive non-linear models can be easily performed. Finally, generalization is extended to other filled elastomers such as a carbon black filled SBR.     

Folding of fiber composites with a hyperelastic matrix

This paper presents an experimental and numerical study of the folding behavior of thin composite materials consisting of carbon fibers embedded in a silicone matrix. The soft matrix allows the fibers to microbuckle without breaking and this acts as a stress relief mechanism during folding, which allows the material to reach very high curvatures. The experiments show a highly non-linear moment vs. curvature relationship, as well as strain softening under cyclic loading. A finite element model has been created to study the micromechanics of the problem. The fibers are modeled as linear-elastic solid elements distributed in a hyperelastic matrix according to a random arrangement based on experimental observations. The simulations obtained from this model capture the detailed micromechanics of the problem and the experimentally observed non-linear response. The proposed model is in good quantitative agreement with the experimental results for the case of lower fiber volume fractions but in the case of higher volume fractions the predicted response is overly stiff.     

Upper yield points of non-linear viscoplastic materials

An approximate solution to the non-linear differential equation governing the behavior of a non-linear, viscoplastic material in tension is given. This solution is used to generate an accurate estimate for the upper yield point for the material. The results are compared to those previously obtained by other means.     

Some results for approximate strain and rotation tensor formulations in geometrically non-linear Reissner-Mindlin plate theory

Finite element deflection and stress results are presented for four flat plate configurations and are computed using kinematically approximate (rotation tensor, strain tensor or both) non-linear Reissner-Mindlin plate models. The finite element model is based on a mixed variational principle and has both displacement and force field variables. High order interpolation of the field variables is possible through p-type discretization. Results for some of the higher order approximate models are given for what appears to be the first time. It is found that for the class of example problems examined, exact strain tensor but approximate rotation tensor theories can significantly improve the solution over approximate strain tensor models such as the von Kármán and moderate rotation models when moderate deflections/rotations are present. However, for each of the problems examined (with the exception of a postbuckling problem) the von Kármán and moderate rotation model results compared favorably with the higher order models for deflection magnitudes which could be reasonably expected in typical aeroelastic configurations.     

Failure prediction in anisotropic sheet metals under forming operations with consideration of rotating principal stretch directions

An approximate macroscopic yield criterion for anisotropic porous sheet metals is adopted to develop a failure prediction methodology that can be used to investigate the failure of sheet metals under forming operations. Hill's quadratic anisotropic yield criterion is used to describe the matrix normal anisotropy and planar isotropy. The approximate macroscopic anisotropic yield criterion is a function of the anisotropy parameter R, defined as the ratio of the transverse plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial loading conditions. The Marciniak–Kuczynski approach is employed here to predict failure/plastic localization by assuming a slightly higher void volume fraction inside randomly oriented imperfection bands in a material element of interest. The effects of the anisotropy parameter R, the material/geometric inhomogeneities, and the potential surface curvature on failure/plastic localization are first investigated. Then, a non-proportional deformation history including relative rotation of principal stretch directions is identified in a critical element of a mild steel sheet under a fender forming operation given as a benchmark problem in the 1993 NUMISHEET conference. Based on the failure prediction methodology, the failure of the critical sheet element is investigated under the non-proportional deformation history. The results show that the gradual rotation of principal stretch directions lowers the failure strains of the critical element under the given non-proportional deformation history.     

Asymptotic analysis of the non-linear behavior of long anisotropic tubes

An asymptotically correct beam model is obtained for a long, thin-walled, circular tube with circumferentially uniform stiffness (CUS) and made of generally anisotropic materials. By virtue of its special geometry certain small parameters cause unusual non-linear phenomena, such as the Brazier effect, to be exhibited. The model is constructed without ad hoc approximations from 3D elasticity by deriving its strain energy functional in terms of generalized 1D strains corresponding to extension, bending, and torsion. Large displacement and rotation are allowed but strain is assumed to be small. Closed-form expressions are provided for the 3D non-linear warping and stress fields, the 1D non-linear stiffness matrix and the bending moment–curvature relationship. In bending, failure could be caused by limit-moment instability, local buckling or material failure of a ply. A procedure to determine the failure load is provided based on the non-linear response, neglecting micro-mechanical failure modes, post-failure behavior, and hygrothermal effects. Asymptotic considerations lead to the neglect of local shell interlaminar and transverse shear stresses for the thin-walled configuration. Results of the theory are illustrated for a few symmetric, antisymmetric angle-ply and unsymmetric layups and show that some previously published theories are not asymptotically correct.     

Dynamics and Stability of a Liquid Triaxial Ellipsoid

The problem of the dynamics of a liquid triaxial ellipsoid confined inside a uniformly rotating rigid shell is formulated and solved within the framework of the Poincare-Joukovski problem. The system of three non-linear first-order differential equations has kinetic energy and circulation integrals representing two ellipsoids with displaced centers in parameter space. All the steady-state and time-dependent solutions are studied; time-dependent solutions exist in four zones, where they are represented by elliptic Jacobi functions, and on three zone boundary interval (represented in terms of elementary functions). On the three other boundary intervals there exist liquid steady-state ellipsoids for which the vortex vector either coincides with one of the principal axes of the ellipsoid or lies in one of its principal planes of symmetry.     

Variational approach to non-linear boundary value problems for elasto-plastic incompressible bending plate

Non-linear boundary value problems for inelastic isotropic homogeneous incompressible bending plate, within the range of J₂-deformation theory, are considered. An existence of the weak solution of the non-linear problem with clamped boundary condition is obtained in H²(Ω) by using monotone operator theory and Browder-Minty theorem. For linearization of the non-linear problem a monotone iteration scheme is constructed. It is shown that the sequence of potentials obtained from the sequence of approximate solutions (i.e. iterations), is a monotone decreasing one. Convergence of the iteration process in H²-norm is proved by using the convexity argument. Numerical solutions, based on finite-difference scheme, are given for linear bending problems with rigid clamped as well as simply supported boundary conditions. Further numerical examples are presented to illustrate the convergence of approximate solutions and monotonicity of the potentials as applied to the non-linear problems.     

On the brachystochrone with dry friction

The problem of the brachystochrone with dry friction is governed by an autonomous set of highly non-linear ordinary differential equations, which contain a small parameter μ (the coefficient of friction). Using perturbation technique an approximate solution is constructed.     

An investigation into non-linear vibrations of thin plates

The variational and modified forms of the von Kármán-type non-linear plate equations are considered in the context of the Rayleigh-Ritz and Galerkin methods. An approximate analysis of the non-linear vibrations of thin elastic plates including inplane inertia is presented. The quantitative study confirms that the inplane inertia effects are negligible for thin plates provided the non-linearity is not too large. It is observed that the non-linear inertia terms in the transverse equation of motion should be retained in any such study. The analysis is simplified by neglecting the inplane inertia and applied to constrained and unconstrained plates. A different type of inplane boundary condition termed ‘the partially constrained’ is studied, and the inadequacy of replacing the unconstrained condition by means of an average-zero stress condition is clearly demonstrated. It is observed that in most of the cases considered the Galerkin method yields lower bounds for the non-linear coefficient of the modal equation. In all cases the Galerkin results yield less stiff models than the Rayleigh-Ritz method. The general significance of the convergence of the two methods beyond the scope of the title problem is highlighted.     

Boundary Layer Solutions in Elastic Solids

Circumferential shear deformation in an annular domain is studied for a large class of incompressible isotropic elastic materials. It is demonstrated that large strains are confined in a region adjacent to a boundary, in analogy to the boundary layer phenomenon in fluid mechanics. The size of this region is quantified. An approximate solution technique for the deformation of nonlinear elastic solids, proposed by Rajagopal [7], is further studied. In this solution, akin to the boundary layer approximation in classical fluid mechanics, the full nonlinear problem is solved in a relatively small region of large strain, while the linearized problem is solved in the remaining region. Error estimates for the approximate solution are obtained.     

A quasi-secant continuous model for the analysis of long-span cable-stayed bridges

In this paper a non-linear continuous model for the statical analysis of long-span cable-stayed bridges with fan scheme and H-shaped towers is proposed. This model is based on a quasi-secant large-displacement formulation of the stays-deck interaction and involves an Euler-Bernoulli/De Saint Venant model for the girder. An approximate closed-form solution of the statical problem, in which flexural and torsional terms are coupled, is obtained by means of a perturbative technique and according to the prevailing “truss-like” structural behavior. Results for some study-cases are compared with those relevant to both classical models and numerical solutions, proving effectiveness and applicability of the proposed model.     

一种利用岩块特征长度估算二维渗吸量的新方法

 对于存在大量亲水疏油岩块的天然裂缝性油藏,自发渗吸是基质岩块和裂缝系统油水交换的重要驱动机制．对于一维渗吸问题,前期存在自相似解析解,后期可以用近似解析解描述,从而可以很方便地估算出岩块剩余油饱和度和岩块出油速率．本文研究了二维矩形岩块在不同边界条件下渗吸过程的近似解析解,通过引入特征长度,将二维岩块渗吸问题等效为一维,从而可以有效利用一维近似解析解来估算出二维岩块渗吸过程中剩余油饱和度和岩块出油速率随时间的变化．本文提出了一个新的混合特征长度,在渗吸前期取岩块面积与岩块开放边界长度之比;在渗吸后期,特征长度根据前期特征长度和Ma 等提出的特征长度按饱和度插值确定．通过数值检验比较了几种常见的特征长度取法和本文提出的混合特征长度取法,结果表明新的混合特征长度总体性能优于其他已有的特征长度取法,并且适用于本文列举的所有边界条件．     

On the Torsion of a Class of Nonlinear Fiber Composite Cylinders

This paper presents an analysis of the torsion of a solid or annular circular cylinder consisting of nonlinear material in the form of an elastic matrix with embedded unidirectional elastic fibers parallel to the cylinder axis. The specific class of composite considered is one for which nonlinear fiber-matrix interface slip is captured by uniform cohesive zones of vanishing thickness. Previous work on the effective antiplane shear response of this material leads to a stress–strain relation depending on the interface slip together with an integral equation governing its evolution. Here, we obtain an approximate single mode solution to the integral equation and utilize it to solve the torsion problem. Equations governing the radial distributions of shear stress and interface slip are obtained and formulae for torque–twist rate are presented. The existence of singular surfaces, i.e., surfaces across which the slip and the shear stress experience jump discontinuities are analyzed in detail. Specific results are presented for an interface force law that allows for interface failure in shear.     

Non-linear response of a beam to periodic loading

The steady state response of a non-linear beam under periodic excitation is investigated. The non-linearity is attributed to the membrane tension effect which is induced in the beam when the deflection is not small in comparison to its thickness. The effects of multimode participation are investigated for simply supported and clamped boundary conditions. The finite element technique is used to formulate the non-linear differential equations of the straight beam and the method of averaging is used to obtain an approximate solution to the non-linear equations under harmonic loading. An analog computer was used to simulate the non-linear beam equation which was subjected to harmonic excitation. The agreement between theoretical and experimental values is reasonably good.