A comparative study between two smoothing strategies for the simulation of contact with large sliding

The numerical simulation of contact problems is still a delicate matter especially when large transformations are involved. In that case, relative large slidings can occur between contact surfaces and the discretization error induced by usual finite elements may not be satisfactory. In particular, usual elements lead to a facetization of the contact surface, meaning an unavoidable discontinuity of the normal vector to this surface. Uncertainty over the precision of the results, irregularity of the displacement of the contact nodes and even numerical oscillations of contact reaction force may result of such discontinuity. Among the existing methods for tackling such issue, one may consider mortar elements (Fischer and Wriggers, Comput Methods Appl Mech Eng 195:5020–5036, 2006; McDevitt and Laursen, Int J Numer Methods Eng 48:1525–1547, 2000; Puso and Laursen, Comput Methods Appl Mech Eng 93:601–629, 2004), smoothing of the contact surfaces with additional geometrical entity (B-splines or NURBS) (Belytschko et al., Int J Numer Methods Eng 55:101–125, 2002; Kikuchi, Penalty/finite element approximations of a class of unilateral contact problems. Penalty method and finite element method, ASME, New York, 1982; Legrand, Modèles de prediction de l’interaction rotor/stator dans un moteur d’avion Thèse de doctorat. PhD thesis, École Centrale de Nantes, Nantes, 2005; Muñoz, Comput Methods Appl Mech Eng 197:979–993, 2008; Wriggers and Krstulovic-Opara, J Appl Math Mech (ZAMM) 80:77–80, 2000) and, the use of isogeometric analysis (Temizer et al., Comput Methods Appl Mech Eng 200:1100–1112, 2011; Hughes et al., Comput Methods Appl Mech Eng 194:4135–4195, 2005; de Lorenzis et al., Int J Numer Meth Eng, in press, 2011). In the present paper, we focus on these last two methods which are combined with a finite element code using the bi-potential method for contact management (Feng et al., Comput Mech 36:375–383, 2005). A comparative study focusing on the pros and cons of each method regarding geometrical precision and numerical stability for contact solution is proposed. The scope of this study is limited to 2D contact problems for which we consider several types of finite elements. Test cases are given in order to illustrate this comparative study.     

An RVE-based micromechanical analysis of fiber-reinforced composites considering fiber size dependency

An RVE-based micromechanical elastic damage model considering fiber size dependency is presented to predict the effective elastic moduli and interfacial damage evolution in fiber-reinforced composites. To assess the validity of the present model, the predictions based on the proposed micromechanical elastic model are compared with Hashin’s theoretical bounds [Hashin Z. Analysis of properties of fiber composites with anisotropic constituents. J Appl Mech: Trans ASME 1979;46:543–50]. The proposed micromechanical elastic damage model is then exercised under uniaxial loading conditions to show the overall elastic damage behavior of the proposed micromechanical framework and to illustrate fiber size effect on the behavior of the composites. Moreover, comparisons between the present prediction and experimental data are made to further illustrate the capability of the proposed micromechanical framework for predicting the elastic damage behavior of fiber-reinforced composites.     

Uncertain dynamic response of a deterministic elastic–plastic beam

The counterintuitive phenomenon of elastic–plastic beam dynamics was demonstrated by Symonds and Yu (ASME J Appl Mech 1985;52:517). An analytical model has been developed to explain this phenomenon from a deterministic viewpoint. However, experimental evidence in (Int J Impact Eng 1991;11(3):341; Int J Impact Eng 1991;11(4):445) showed that the response of this deterministic system is uncertain, which is studied qualitatively in the present paper based on parametric sensitive characteristics of the deterministic system and parametric uncertainty of the studied system. FEM and Monte Carlo method are applied to study this phenomenon.     

Inherent relations between the Bueckner integral and the J k -integral or the M-integral in piezoelectric materials containing multiple defects

This paper deals with the inherent relations between the Bueckner work conjugate integral and the J _k -integral (k = 1, 2) or the M-integral in piezoelectric materials with a number of arbitrarily oriented and distributed defects such as cracks, voids and inclusions. The explicit expression of the Bueckner integral is derived in piezoelectric materials by using its path-independent property and asymptotic feature for each pair of the complex potential functions in the series expansion forms. It is concluded that the J _k -integral or the M-integral are only three different special cases of the Bueckner integral when three different complementary fields are introduced, respectively, and when the closed integral contour encloses all the defects. In other words, there are universal relations between the Bueckner integral and the J _k -integral or M-integral whatever the detailed configurations of the multiple defects are. It is also concluded that both components of the J _k -integral vanish when the contour selected to calculate the invariant integral encloses all defects, providing that no resultant force acting on each defect exists. This leads to the independence of the M-integral from the global coordinate shifts, and its value is significantly influenced by the mechanical–electrical feature of the damaged piezoelectric materials, e.g., the material properties, the remote loading conditions, the defect configurations, etc.     

Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions

We discuss the modeling of the coefficient of restitution as a function of the incoming velocity in elasto-plastic collisions with normal frictionless impact, and compare the results from nonlinear finite-element analysis to those of two recent normal force displacement models: One by Thornton (ASME J. Appl. Mech. 64 (1997) 383) and one by Vu-Quoc and Zhang (Proc. R. Soc. London, Ser. A 455 (1999) 4013) which is the displacement-driven counterpart of the force-driven model proposed by Vu-Quoc, Zhang, and Lesburg (ASME. J. Appl. Mech. 67 (2000) 363). The resulting values of the coefficient of restitution are also compared to those from the model proposed in Stronge (in: R.C. Batra, A.K. Mal, G.P. MacSithigh (Eds.), Impact Waves and Fractures, ASME AMD 205 (1995) 351). The relationships among the coefficient of restitution, the incoming velocity, the collision time, the contact force/displacement, the normal pressure distribution are presented and discussed. These results establish the better accuracy provided by the model proposed by Vu-Quoc, Zhang, and Lesburg, when compared to previously proposed models.     

A damage model for crack prediction in brittle and quasi-brittle materials solved by the FFT method

In this paper, we present a damage model and its numerical solution by means of Fast Fourier Transforms (FFT). The FFT-based formulation initially proposed for linear and non-linear composite homogenization (Moulinec and Suquet in CR Acad Sci Paris Ser II 318:1417–1423 1994; Comput Methods Appl Mech Eng 157:69–94 1998) was adapted to evaluate damage growth in brittle materials. A non-local damage model based on the maximal principal stress criterion was proposed for brittle materials. This non-local model was then connected to the Griffith criterion with the aim of predicting crack growth. By using the proposed model, we carried out several numerical simulations on different specimens in order to assess the fracture process in brittle materials. From these studies, we can conclude that the present FFT-based analysis is capable of dealing with crack initiation and crack growth in brittle materials with high accuracy and efficiency.     

On the approximation in the smoothed finite element method (SFEM)

This letter aims at resolving the issues raised in the recent short communication (Int. J. Numer. Meth. Engng 2008; 76 (8):1285–1295. DOI: 10.1002/nme.2460 ) and answered by (Int. J. Numer. Meth. Engng 2009; DOI: 10.1002/nme.2587 ) by proposing a systematic approximation scheme based on non‐mapped shape functions, which both allows to fully exploit the unique advantages of the smoothed finite element method (SFEM) (Comput. Mech. 2007; 39 (6):859–877. DOI: 10.1007/s00466‐006‐0075‐4 ; Commun. Numer. Meth. Engng 2009; 25 (1):19–34. DOI: 10.1002/cnm.1098 ; Int. J. Numer. Meth. Engng 2007; 71 (8):902–930; Comput. Meth. Appl. Mech. Engng 2008; 198 (2):165–177. DOI: 10.1016/j.cma.2008.05.029 ; Comput. Meth. Appl. Mech. Engng 2007; submitted; Int. J. Numer. Meth. Engng 2008; 74 (2):175–208. DOI: 10.1002/nme.2146 ; Comput. Meth. Appl. Mech. Engng 2008; 197 (13–16):1184–1203. DOI: 10.1016/j.cma.2007.10.008 ) and resolve the existence, linearity and positivity deficiencies pointed out in (Int. J. Numer. Meth. Engng 2008; 76 (8):1285–1295). We show that Wachspress interpolants (A Rational Basis for Function Approximation. Academic Press, Inc.: New York, 1975) computed in the physical coordinate system are very well suited to the SFEM, especially when elements are heavily distorted (obtuse interior angles). The proposed approximation leads to results that are almost identical to those of the SFEM initially proposed in (Comput. Mech. 2007; 39 (6):859–877. DOI: 10.1007/s00466‐006‐0075‐4 ). These results suggest that the proposed approximation scheme forms a strong and rigorous basis for the construction of SFEMs. Copyright © 2009 John Wiley & Sons, Ltd.     

Micromechanics-based viscoelastic damage model for particle-reinforced polymeric composites

The objective of this study is to develop a micromechanics-based viscoelastic damage model that can predict the overall viscoelastic behavior of particle-reinforced polymeric composites undergoing damage. The emphasis here is that the present model successfully combines a rate-dependent viscoelastic constitutive model and a damage model. The Laplace transform based on the Boltzmann superposition principle and the ensemble-volume averaged method suggested by Ju and Chen (Acta Mech 103:103–121, 1994a; Acta Mech 103:123–144, 1994b) are extended toward effective viscoelastic properties. Further, the probability of the distribution function of Weibull (J Appl Mech 18:293–297, 1951) is adopted to describe a damage model that is dependent on damage parameters. A series of numerical simulations including parametric studies, and experimental comparisons are carried out to give insight into the potential capacity of the present micromechanics-based viscoelastic damage framework.     

Boundary element solution of the two-dimensional sine-Gordon equation using continuous linear elements

This article studies the boundary element solution of two-dimensional sine-Gordon (SG) equation using continuous linear elements approximation. Non-linear and in-homogenous terms are converted to the boundary by the dual reciprocity method and a predictor–corrector scheme is employed to eliminate the non-linearity. The procedure developed in this paper, is applied to various problems involving line and ring solitons where considered in references [Argyris J, Haase M, Heinrich JC. Finite element approximation to two-dimensional sine-Gordon solitons. Comput Methods Appl Mech Eng 1991;86:1–26; Bratsos AG. An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions. Appl Numer Anal Comput Math 2005;2(2):189–211, Bratsos AG. A modified predictor–corrector scheme for the two-dimensional sine-Gordon equation. Numer Algorithms 2006;43:295–308; Bratsos AG. The solution of the two-dimensional sine-Gordon equation using the method of lines. J Comput Appl Math 2007;206:251–77; Bratsos AG. A third order numerical scheme for the two-dimensional sine-Gordon equation. Math Comput Simul 2007;76:271–8; Christiansen PL, Lomdahl PS. Numerical solutions of 2+1 dimensional sine-Gordon solitons. Physica D: Nonlinear Phenom 1981;2(3):482–94; Djidjeli K, Price WG, Twizell EH. Numerical solutions of a damped sine-Gordon equation in two space variables. J Eng Math 1995;29:347–69; Dehghan M, Mirzaei D. The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation. Comput Methods Appl Mech Eng 2008;197:476–86]. Using continuous linear elements approximation produces more accurate results than constant ones. By using this approach all cases associated to SG equation, which exist in literature, are investigated.     

A new yield surface distortion model based on Baltov and Sawczuk’s model

In the present study, a new distortion yield surface model is proposed to represent compatible results with experimental observations. The proposed yield surface model is determined numerically during tension–torsion loadings by considering a kinematic hardening model and monotonic loading paths. The experimental results of yield surface determination (Khan et al. in Int J Plast 26:1432–1441, 2010; Naghdi et al. in ASME J. Appl Mech 25:201–209, 1957) represent the nosed and flattened regions in the loading and reverse loading directions, respectively. But, the Baltov and Sawczuk’s yield surface model can only predict nosed or flattened shape in both loading and reversed loading directions, depending on the sign of their model constant. Thus, the elliptic Baltov and Sawczuk’s yield surface is modified by changing the sign of this parameter continuously from loading to reverse loading direction. Relations and convexity of the new model are obtained and discussed. The new model is able to predict properly the shape of the yield surface. The experimental results are in a satisfactory agreement with the new yield surface distortion model predictions.     

Elasto‐plastic finite element analysis of shells with damage due to microvoids

This paper presents a non‐linear finite element analysis for the elasto‐plastic behaviour of thick/thin shells and plates with large rotations and damage effects. The refined shell theory given by Voyiadjis and Woelke (Int. J. Solids Struct. 2004; 41 :3747–3769) provides a set of shell constitutive equations. Numerical implementation of the shell theory leading to the development of the C⁰ quadrilateral shell element (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted) is used here as an effective tool for a linear elastic analysis of shells. The large rotation elasto‐plastic model for shells presented by Voyiadjis and Woelke (General non‐linear finite element analysis of thick plates and shells. 2006, submitted) is enhanced here to account for the damage effects due to microvoids, formulated within the framework of a micromechanical damage model. The evolution equation of the scalar porosity parameter as given by Duszek‐Perzyna and Perzyna (Material Instabilities: Theory and Applications, ASME Congress, Chicago, AMD‐Vol. 183/MD‐50, 9–11 November 1994; 59–85) is reduced here to describe the most relevant damage effects for isotropic plates and shells, i.e. the growth of voids as a function of the plastic flow. The anisotropic damage effects, the influence of the microcracks and elastic damage are not considered in this paper. The damage modelled through the evolution of porosity is incorporated directly into the yield function, giving a generalized and convenient loading surface expressed in terms of stress resultants and stress couples. A plastic node method (Comput. Methods Appl. Mech. Eng. 1982; 34 :1089–1104) is used to derive the large rotation, elasto‐plastic‐damage tangent stiffness matrix. Some of the important features of this paper are that the elastic stiffness matrix is derived explicitly, with all the integrals calculated analytically (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted). In addition, a non‐layered model is adopted in which integration through the thickness is not necessary. Consequently, the elasto‐plastic‐damage stiffness matrix is also given explicitly and numerical integration is not performed. This makes this model consistent mathematically, accurate for a variety of applications and very inexpensive from the point of view of computer power and time. Copyright © 2006 John Wiley & Sons, Ltd.     

Introduction of damage evolution in a scale transition approach for highly-filled particulate composites

The present paper deals with a non-conventional scale transition for modelling the behaviour of highly-filled particulate composites, starting from a methodology initially proposed by Christoffersen [Christoffersen J. Bonded granulates. J Mech Phys Solids 1983;31:55–83] and recently extended by Nadot et al. [Nadot C, Dragon A, Trumel H, Fanget A. Damage modelling framework for viscoelastic particulate composites via a scale transition approach. J Theor Appl Mech 2006;44(3):553–83] in presence of damage. The model thus obtained is here completed with several ingredients allowing to describe damage evolution and in particular a defect nucleation criterion as well as a closure criterion. These criteria are formulated in terms of displacement, and so as to ensure continuity in terms of macroscopic stress. They are finally introduced in an iterative numerical solving procedure which allows to follow damage evolution as a discrete sequence of interfacial debonding including also eventual closure of defects.     

Progressive damage modeling in fiber-reinforced materials

This paper presents an anisotropic damage model suitable for predicting failure and post-failure behavior in fiber-reinforced materials. In the model the plane stress formulation is used and the response of the undamaged material is assumed to be linearly elastic. The model is intended to predict behavior of elastic-brittle materials that show no significant plastic deformation before failure. Four different failure modes – fiber tension, fiber compression, matrix tension, and matrix compression – are considered and modeled separately. The onset of damage is predicted using Hashin’s initiation criteria [Hashin Z, Rotem A. A fatigue failure criterion for fiber-reinforced materials. J Compos Mater 1973;7:448; Hashin Z. Failure criteria for unidirectional fiber composites. J Appl Mech 1980;47:329–34] and the progression of damage is controlled by a new damage evolution law, which is easy to implement in a finite element code. The evolution law is based on fracture energy dissipation during the damage process and the increase in damage is controlled by equivalent displacements. The issues related to numerical implementation, such as mesh sensitivity and convergence in the softening regime, are also addressed.     

Asymptotic fields at the tip of a cohesive crack

Recently, the authors (Xiao and Karihaloo, J Mech Mater Struct 1:881–910, 2006) obtained universal asymptotic expansions at a cohesive crack tip, analogous to the Williams (ASME J Appl Mech 24:109–114, 1957) expansions at a traction-free crack tip for any normal cohesion-separation law (i.e. softening law) that can be expressed in a special polynomial. This special form ensures that the radial and angular variations of the asymptotic fields are separable as in the Williams expansions. The coefficients of the expansions of course depend nonlinearly on the softening law and the boundary conditions. They demonstrated that many commonly-used cohesion-separation laws, e.g., rectangular, linear, bilinear and exponential, can indeed be expressed very accurately in this special form. They also obtained universal asymptotic expansions when the cohesive crack faces are subjected to Coulomb friction. The special polynomial involves fractional powers which seem rather contrived. In this paper, we will show that the asymptotic expansions can be obtained in a separable form even when the cohesion-separation law is in a special polynomial form involving only integer powers.     

A simplified implementation of a gradient-enhanced damage model with transient length scale effects

Gradient-enhanced damage models with constant gradient activity suffer from spurious damage growth at high deformation levels. This issue was resolved by Geers et al. (Comput Methods Appl Mech Eng 160(1–2):133–153, 1998) by expressing the gradient activity parameter as a function of the local equivalent strain at the expense of adding one set of degrees of freedom to those of the standard model. In this contribution, a new formulation of the gradient-enhanced damage model with variable length scale is presented which eliminates the need for the extra set of degrees of freedom. The merits of the proposed formulation are demonstrated, and the choice of the damage evolution law and its impact on the model performance are discussed.     

A fourth‐order compact scheme for the Helmholtz equation: Alpha‐interpolation of FEM and FDM stencils

We propose a fourth‐order compact scheme on structured meshes for the Helmholtz equation given by R(φ):=f( x )+Δφ+ξ²φ=0. The scheme consists of taking the alpha‐interpolation of the Galerkin finite element method and the classical central finite difference method. In 1D, this scheme is identical to the alpha‐interpolation method (J. Comput. Appl. Math. 1982; 8 (1):15–19) and in 2D making the choice α=0.5 we recover the generalized fourth‐order compact Padé approximation (J. Comput. Phys. 1995; 119 :252–270; Comput. Meth. Appl. Mech. Engrg 1998; 163 :343–358) (therein using the parameter γ=2). We follow (SIAM Rev. 2000; 42 (3):451–484; Comput. Meth. Appl. Mech. Engrg 1995; 128 :325–359) for the analysis of this scheme and its performance on square meshes is compared with that of the quasi‐stabilized FEM (Comput. Meth. Appl. Mech. Engrg 1995; 128 :325–359). In particular, we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate O((ξ?)⁴), where ξ, ? represent the wavenumber and the mesh size, respectively. An expression for the parameter α is given that minimizes the maximum relative phase error in a sense that will be explained in Section 4.5. Convergence studies of the error in the L² norm, the H¹ semi‐norm and the l^∞ Euclidean norm are done and the pollution effect is found to be small. Copyright © 2010 John Wiley & Sons, Ltd.