Direct localized boundary-domain integro-differential formulations for physically nonlinear elasticity of inhomogeneous body

A static mixed boundary value problem (BVP) of physically nonlinear elasticity for a continuously inhomogeneous body is considered. Using the two-operator Green-Betti formula and the fundamental solution of an auxiliary linear operator, a non-standard boundary-domain integro-differential formulation of the problem is presented, with respect to the displacements and their gradients. Using a cut-off function approach, the corresponding localized parametrix is constructed to reduce the nonlinear BVP to a nonlinear localized boundary-domain integro-differential equation. Algorithms of mesh-based and mesh-less discretizations are presented resulting in sparsely populated systems of nonlinear algebraic equations.     

Radial integration boundary integral and integro-differential equation methods for two-dimensional heat conduction problems with variable coefficients

This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional heat conduction problems with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the boundary-value problem (BVP) to a boundary-domain integral equation (BDIE) or boundary-domain integro-differential equation (BDIDE). The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Localized direct boundary-domain integro-differential formulations for scalar nonlinear boundary-value problems with variable coefficients

Mixed boundary-value Problems (BVPs) for a second-order quasi-linear elliptic partial differential equation with variable coefficients dependent on the unknown solution and its gradient are considered. Localized parametrices of auxiliary linear partial differential equations along with different combinations of the Green identities for the original and auxiliary equations are used to reduce the BVPs to direct or two-operator direct quasi-linear localized boundary-domain integro-differential equations (LBDIDEs). Different parametrix localizations are discussed, and the corresponding nonlinear LBDIDEs are presented. Mesh-based and mesh-less algorithms for the LBDIDE discretization are described that reduce the LBDIDEs to sparse systems of quasi-linear algebraic equations.     

Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D

A numerical implementation of the direct boundary-domain integral and integro-differential equations, BDIDEs for treatment of the Dirichlet problem for a scalar elliptic PDE with variable coefficient in a three-dimensional domain is discussed. The mesh-based discretisation of the BDIEs with tetrahedron domain elements in conjunction with collocation method leads to a system of linear algebraic equations (discretised BDIE). The involved fully populated matrices are approximated by means of the H-Matrix/adaptive cross approximation technique. Convergence of the method is investigated.     

Nonlinear static/dynamic analysis for elasto-plastic laminated plates with interfacial damage evolution

A new analysis model, which includes the effects of interfacial damage, geometrical nonlinearity and material nonlinearity, is presented for elasto-plastic laminated plates. Based on the model, the nonlinear equilibrium differential equations for elasto-plastic laminated plates with interfacial damage are established. The finite difference method and iteration method are adopted to solve these equations. The nonlinear static and dynamic behaviors for the elasto-plastic laminated plates under the action of transverse loads are analyzed. Effects of interfacial damage on the stress and displacement distribution and nonlinear dynamic response are discussed in the numerical examples together with the comparison of nonlinear mechanical behaviors between the elastic and elasto-plastic laminated plates. Numerical results show that both the interfacial damage and plastic deformation put obvious influence on the mechanical properties of structures.     

Wave propagation in elasto-plastic granular systems

Due to the nonlinear nature of the inter-particle contact, granular chains made of elastic spheres are known to transmit solitary waves under impulse loading. However, the localized contact between spherical granules leads to stress concentration, resulting in plastic behavior even for small forces. In this work, we investigate the effects of plasticity in wave propagation in elasto-plastic granular systems. In the first part of this work, a force–displacement law between contacting elastic-perfectly plastic spheres is developed using a nonlinear finite element analysis. In the second part, this force–displacement law is used to simulate wave propagation in one-dimensional granular chains. In elasto-plastic chains, energy dissipation leads to the formation and merging of wave trains, which have characteristics very different from those of elastic chains. Scaling laws for peak force at each contact point along the chain, velocity of the leading wave, local contact and total dissipation are developed.     

Numerical solution of two-dimensional mixed problems with variable coefficients by the boundary-domain integral and integro-differential equation methods

This paper presents a formulation of the boundary-domain integral equation (BDIE) and the boundary-domain integro-differential equation (BDIDE) methods for the numerical solution of two-dimensional mixed boundary-value problems (BVP) for a second-order linear elliptic partial differential equation (PDE) with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the BVP to a BDIE or BDIDE. The numerical formulation of the BDIDE employs an approximation for the boundary fluxes in terms of the potential function within the domain cells; therefore, the solution is fully described in terms of the variation of the potential function along the boundary and domain. Linear basis functions localised on triangular elements and standard quadrature rules are used for the calculation of boundary integrals. For the domain integrals, we have implemented Gaussian quadrature rules for two dimensions with Duffy transformation, by mapping the triangles into squares and eliminating the weak singularity in the process. Numerical examples are presented for several simple problems with square and circular domains, for which exact solutions are available. It is shown that the present method produces accurate results even with coarse meshes. The numerical results also show that high rates of convergence are obtained with mesh refinement.     

Mesh-based numerical implementation of the localized boundary-domain integral-equation method to a variable-coefficient Neumann problem

An implementation of the localized boundary-domain integral-equation (LBDIE) method for the numerical solution of the Neumann boundary-value problem for a second-order linear elliptic PDE with variable coefficient is discussed. The LBDIE method uses a specially constructed localized parametrix (Levi function) to reduce the BVP to a LBDIE. After employing a mesh-based discretization, the integral equation is reduced to a sparse system of linear algebraic equations that is solved numerically. Since the Neumann BVP is not unconditionally and uniquely solvable, neither is the LBDIE. Numerical implementation of the finite-dimensional perturbation approach that reduces the integral equation to an unconditionally and uniquely solvable equation, is also discussed.     

Elasto-plastic analysis of anisotropic plates and shells by the Semiloof element

An elasto-plastic analysis of thin plates and shells by means of the finite element displacement method is considered for anisotropic materials, using the Semiloof curved shell element. The elasto-plastic analysis is based on the Huber-Mises yield criterion extended by Hill for anisotropic materials. The yield function is generalized by introducing anisotropic parameters of plasticity which are updated during the material strain hardening history. Numerical examples are presented and compared with available solutions. The effects of anisotropy on these solutions are also discussed. The practical problem of an aerofoil blade is also considered and some results presented.     

A numerical solution of nonlinear parabolic-type Volterra partial integro-differential equations using radial basis functions

In this paper, an effective numerical method for solving nonlinear Volterra partial integro-differential equations is proposed. These equations include the partial differentiations of an unknown function and the integral term containing the unknown function which is the “memory” of problem. This method is based on radial basis functions (RBFs) and finite difference method (FDM) which provide the approximate solution. These techniques play the important role to reduce a nonlinear partial integro-differential equation to a linear system of equations. Some illustrative examples are shown to describe the method. Numerical examples confirm the validity and efficiency of the presented method.     

钢框架极限承载力的有限变形理论分析和试验研究

本文为钢框架极限承载力的全过程分析提供了一种有效的方法。该法基于非线性连续介质力学理论,导出了结构的几何刚度矩阵,它不仅包含了轴力对结构刚度的影响,而且还包含了剪力和弯矩的影响。根据内力屈服面塑性流动理论,把经典的塑性铰概念加以推广,既考虑了塑铰性处内力之间的相互影响,也考虑了弹性卸载的影响。进行了四榀钢框架极限承载力的试验,试验与理论分析结果符合良好。     

非线性分数阶积分微分方程边值问题正解的存在性

研究一类具有两个分数阶导数项的非线性分数阶积分微分方程积分边值问题。首先将原问题转化为只有一个导数项的等价形式,通过定义等价问题的上下解,再利用单调迭代技术建立了原问题正解的存在性与唯一性定理,给出了求其唯一正解的迭代格式和误差估计。最后给出实例说明所得结论的有效性和适用性。     

A mixed finite element formulation for non-linear analysis of plane problems

Based on the incremental non-linear theory of solid bodies and the Hellinger-Reissncr principle, a mixed updated Lagrangian formulation of the large displacement motion of solid bodies is derived, and an associated mixed finite element model is developed. The model contains the displacements and stresses as the nodal degrees of freedom. The model is used for the large deformation elasto-plastic analysis of plane problems. In solving non-linear problems, the Newton-Raphson method with arc-length control is adopted to trace the post-buckling response. The computational steps to calculate the elasto-plastic stress increments at Gauss points in the elasto-plastic analysis by the present mixed model are described in detail. Numerical results are presented and compared with those of the displacement model and existing solutions to show the accuracy of the present mixed model in the large deformation elasto-plastic analysis of plane problems.     

Banach空间非线性混合型微分-积分方程解的存在唯一性

在较宽的条件下研究了Banach空间中非线性混合型微分—积分初值问题解的存在唯一性及解的迭代逼近和误差估计，改进并推广了最近的一些结果。     

A method to extract the mechanical properties of particles in collision based on a new elasto-plastic normal force–displacement model

We present a general method to extract the elasto-plastic mechanical properties such as the coefficient of restitution, Young’s modulus, Poisson’s ratio, and the yield stress of granular materials from simple experimental measurements. These mechanical properties are important in the simulation of granular flows, but are not readily available and/or cannot be measured through direct experiments. The method developed here is based on the new elasto-plastic normal force–displacement (NFD) model [presented in Proceedings of the Royal Society of London, Series A 455 (1999b) 4013]. Using the Vu-Quoc and Zhang NFD model to simulate both the dropping test and the compression test of a particle, the mechanical properties of the particle can be extracted by minimizing, in the least-square sense, the difference between the experimental and simulated force–displacement relation. An application of the proposed method to soybeans is presented.     

Modeling of surface and subsurface crack behavior under contact load in the presence of lubricant

A new mathematical model for lubricated elastic solids weakened by cracks is proposed. Surface and subsurface cracks are taken into account, and the interaction of lubricant with elastic solids within cavities of surface cracks is regarded as the most interesting aspect of the problem. The boundary conditions characterizing the behavior of lubricant within crack cavities such as pressure rise in crack cavities fully filled with lubricant as well as other boundary and additional conditions are derived. The problem is reduced to a system of integro-differential equations with nonlinear boundary conditions in the form of alternating equations and inequalities. A new iterative numerical method is developed for solution of the proposed problem. The method guarantees conservation of lubricant volumes trapped within closed crack cavities and allows for all three functions (normal and tangential displacement jumps and normal stress applied to crack faces) characterizing the problem solution to be determined simultaneously. Examples of numerical results for surface and subsurface cracks are presented and numerical and asymptotic results for small subsurface cracks are compared to each other. The numerical analysis indicates that depending on a surface crack orientation its normal stress intensity factor may be two or more orders of magnitude higher than the one for a similar subsurface one.     

索穹顶结构非线性分析的杆索有限元法

索穹顶结构是大变形柔性组合结构，其受力分析属于几何非线性问题，求解较复杂。本文应用有限元法，结合索穹顶结构特征，提出三节点等参数索单元与两节点杆单元相结合的混合有限元模式。基于Lagrangian坐标描述法和虚功原理推导了空间拉索和杆单元的大位移刚度矩阵，并建立了索穹顶结构非线性分析的增量平衡方程，利用NewtonRaphson法进行了实例计算。结果表明，本文方法精度很高，适合于索穹顶结构的空间非线性分析。     

A mathematical model of a biosensor

This paper is concerned with the modelling of the evolution of a chemical reaction within a small cell. Mathematically the problem consists of a heat equation with nonlinear boundary conditions. Through an integro-differential equation reformulation, an asymptotic result is derived, a perturbation solution is developed, and a modified product integration method is discussed. Finally, an alternative integral formulation is presented which acts as a check on the previous results and permits high accuracy numerical solutions.     

Adaptive reproducing kernel particle method using gradient indicator for elasto-plastic deformation

An adaptive meshless method based on the multi-scale Reproducing Kernel Particle Method (RKPM) for analysis of nonlinear elasto-plastic deformation is proposed in this research. In the proposed method, the equivalent strain, stress, and the second invariant of the Cauchy–Green deformation tensor are decomposed into two scale components, viz., high- and low-scale components by deriving them from the multi-scale decomposed displacement. Through combining the high-scale components of strain and the stress update algorithm, the equivalent stress is decomposed into two scale components. An adaptive algorithm is proposed to locate the high gradient region and enrich the nodes in the region to improve the computational accuracy of RKPM. Using the algorithm, the high-scale components of strain and stress and the second invariant of the Cauchy–Green deformation tensor are normalized and used as criteria to implement the adaptive analysis. To verify the validity of the proposed adaptive meshless method in nonlinear elasto-plastic deformation, four case studies are calculated by the multi-scale RKPM. The patch test results show that the used multi-scale RKPM is reliable in analysis of the regular and irregular nodal distribution. The results of other three cases show that the proposed adaptive algorithm can not only locate the high gradient region well, but also improve the computational accuracy in analysis of the nonlinear elasto-plastic deformation.     

An analysis for the elasto-plastic problem of the moderately thick plate using the meshless local Petrov–Galerkin method

A meshless local Petrov–Galerkin method for the analysis of the elasto-plastic problem of the moderately thick plate is presented. The discretized system equations of the moderately thick plate are obtained using a locally weighted residual method. It uses a radial basis function (RBF) coupled with a polynomial basis function as a trial function, and uses the quartic spline function as a test function of the weighted residual method. The shape functions have the Kronecker delta function properties, and no additional treatment to impose essential boundary conditions. The present method is a true meshless method as it does not need any grids, and all integrals can be easily evaluated over regularly shaped domains and their boundaries. An incremental Newton–Raphson iterative algorithm is employed to solve the nonlinear discretized system equation. Numerical results show that the present method possesses not only feasibility and validity but also rapid convergence for the elasto-plastic problem of the moderately thick plate.