Investigations of interlaminar fracture toughness of laminated polymeric composites

The behavior of interlaminar fracture of fiber reinforced laminated polymeric composites has been investigated in modes I, II, and different mixed mode I/II ratios. The experimental investigations were carried out by using conventional beam specimens and the compound version of the CTS (compact tension shear) specimen. In this study, a compound version of the CTS specimen is used for the first time to determine the interlaminar fracture toughness of composites. In order to verify the results obtained by the CTS tests, conventional beam tests were also carried out. In the beam tests, specimens of double cantilever beam (DCB) and end notched flexure (ENF) were used to obtain the critical rates of the energy release for failure modes I and II. The CTS specimen is used to obtain different mixed mode ratios, from pure mode I to pure mode II, by varying the loading conditions. The highest mixed mode ratio obtained in the experiment was G _I /G _II =60. The data obtained from these tests were analyzed by the finite element method. The separated critical rates G _I and G _II of the energy release were calculated by using the modified virtual crack closure integral (MVCCI) method. The experimental investigations were performed on a unidirectional glass/epoxy composite. The results obtained by the beam and CTS tests were compared. It was found that the interlaminar fracture toughness G _IC ^init of mode I at crack initiation and the corresponding value G _II ^Cinit of mode II obtained by the conventional beam and the CTS tests were in rather good agreement. The experimental results of interlaminar fracture of mixed mode were used to obtain the parameters required for the failure criterion. The two different failure criteria were compared. The best correlation with the experimental data was obtained by using the failure criterion proposed by Wu in 1967 containing linear and quadratic terms of the rates of the energy release.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 307–322, May–June, 1998.     

Mode I,mode II,and mixed-mode I/II interlaminar fracture toughness of GFRP influenced by fiber surface treatment

The interlaminar fracture behavior of unidirectional glass fiber reinforced composites with fiber surface treatment has been investigated in modes I and II and for fixed mode I to mode II ratio of 1.33. The data obtained from these tests have been analyzed by using different analytical approaches. The present investigation is focused on the influence of the glass fiber surface treatment on the interlaminar fracture toughness of unidirectional laminates. Glass fibers with two different fiber surface treatments have been investigated. fiber surface treatment was carried out by using a polyethylene or silane coupling agent in combination with modifying agents. The glass fibers were embedded in the brittle epoxy matrix. Mode I, mode II, and mixed-mode I/II tests were performed in order to determine critical strain energy release rates. Double cantilever beam (DCB), end-notched flexure (ENF), and mixed-mode flexure (MMF) specimens were used. For both types of fiber surface treatment about the same values of mode I initiation fracture toughness G_IC ^init were obtained. It was observed that in mode I interlaminar crack growth in the DCB test for the composite sized by polyethylene, the crack propagation is accompanied by extensive fiber bridging. For both fiber surface treatments interlaminar fracture toughness increases considerably with increasing of crack length. For the fiber surface treatment with the silane coupling agent, the value of mode II initiation fracture toughness G_IIC ^init was about 2.5-times higher in comparison with that of a composite sized by polyethylene. For both types of fiber surface treatments the mixed-mode I/II test has shown a similar behavior to the mode I DCB test.     

On the single fibre pullout and pushout problem: Effect of fibre anisotropy

Based on our recent theoretical model of the interfacial debonding and fibre pullout/pushout stresses, an improved analysis including the effect of fibre anisotropy has been performed for single fibre pullout and pushout. During fibre-matrix debonding the debonded region is considered as an interfacial crack and its extension is dependent on a fracture energy criterion being satisfied. The elastic properties of the fibre are taken to be transversely isotropic about the fibre axis and the matrix is assumed to be isotropic. The results show that the maximum debond stress for fibre pullout and the initial frictional pullout stress increase as the ratio of the Young's moduli of the fibre between the axial and radial directionsE _f /E _f ^r increases, and this is particularly so at large embedded fibre lengths. However, in the case of fibre pushout these debond and pushout stresses decrease asE _f /E _f ^r increases.     

Conditions for Cracking and Debond Development at Partially Bonded Bimaterial Half-Planes

Fracture phenomena at the debond tip of partially bonded bimaterial half-planes subjected to concentrated normal forces, couples, and uniform tension are considered. The crack initiation conditions are described by the stress distribution before the initiation and the energy release rate of the crack immediately after the initiation. The debond development conditions are described by the stress distribution and the energy release rate of the debond before its initiation. When both the crack and the debond have chances to occur, or when cracks can arise in both the materials, the fracture phenomena are predicted by comparing the ratio of energy release rates and the ratio of fracture toughnesses.     

Mixed Micromechanics and Continuum Damage Mechanics Approach to Transverse Cracking in [S, 90n]s Laminates

The stiffness reduction in [S, 90 _n ] _s laminates due to transverse cracking in 90-layers is analyzed using the synergistic continuum damage mechanics (SCDM) and a micromechanics approach. The material constants involved in the SCDM model are determined using the stiffness reduction data for a reference cross-ply laminate. The constraint efficiency factor, which depends on the stiffness and geometry of neighboring layers, is assumed to be proportional to the average crack opening displacement (COD). The COD as a function of the constraint effect of adjacent layers and crack spacing is described by a simple power law. The crack closure technique and Monte Carlo simulations are used to model the damage evolution: the 90-layer is divided into a large number of elements and the critical strain energy rate G _c having the Weibull distribution is randomly assigned to each element. The crack density data for a [0₂/90₄] _s cross-ply laminate are used to determine the Weibull parameters. The simulated crack density curves are combined with the CDM stiffness reduction predictions to obtain the stiffness versus strain. The methodology developed is successfully used to predict the stiffness reduction as a function of crack density in [±/90₄] _s laminates.     

<Emphasis Type="Italic">p</Emphasis>-Geodesic transformations and their groups in second-order tangent bundles induced by concircular transformations of bases

We investigate the flattening properties of the Lie group G _r ^II of transformations of a second-order tangent bundle T ²(M) equipped with the lift ∇ ^II of an affine connection ∇ and the lift g ^II of a metric g on the base of M induced by the Lie group G _r of concircular transformations of the base of M. The obtained results reveal certain geometric features of the induced group G _r ^II within the framework of the theory of p-geodesic mappings.     

PROCRACK-A Software Tool for Finite Element Simulation of Three-Dimensional Fatigue Crack Growth

The lifetime of structural components is limited by fatigue cracks. After initiation from an existing defect the crack grows subcritically until it reaches a critical size. At this point it becomes unstable and the structural component fails. PROCRACK is a powerful tool that enables the commercial finite element software Abaqus to calculate the crack propagation in pre-cracked components. The complete capability of Abaqus can be used to simulate nearly arbitrary geometries. Abaqus/CAE is used for the three-dimensional modeling of the initial crack at a geometrical level by means of points, lines and triangles. The numerical analysis is performed by Abaqus/Standard. PROCRACK automatically generates a tube-shaped submodel around the crack front to calculate the stress intensity factors with high accuracy. The Paris law or the NASGRO equation can be used to calculate the incremental crack growth. The shape of the crack and the finite element mesh are updated in every crack growth step. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fatigue crack growth simulations of homogeneous and bi-material interfacial cracks using element free Galerkin method

In this paper, quasi-static fatigue crack growth simulations of homogeneous and bi-material interfacial cracks have been performed using element free Galerkin method (EFGM) under mechanical as well as thermo-elastic load. The thermo-elastic fracture problem is decoupled into thermal and elastic problems. The temperature distribution obtained by solving heat conduction equation is used as input in the elastic problem to get the displacement and stress fields. Discontinuities in the temperature and displacement fields are captured by extrinsic partition of unity enrichment technique. The values of stress intensity factors have been extracted from the EFGM solution by domain based interaction integral approach. The standard Paris fatigue crack growth law has been implemented for the life estimation of various model problems. The results obtained by EFGM under mechanical and thermo-elastic loads were compared with those obtained by FEM using remeshing approach.     

Interlaminar fracture of multidirectional glass/epoxy laminates under mixed-mode I + II loading

The mixed-mode I + II interlaminar fracture of multidirectional glass/epoxy laminates is investigated. Mixed-mode bending (MMB) tests were performed on specimens with delaminations in 0/θ-type interfaces, with θ varying from 0 to 90°. Preliminary three-dimensional finite-element analyses validated the beam theory model (BTM) used for analysing experimental data. The compliances measured are in a good agreement with BTM predictions. The total critical energy release rate G_c varies linearly with the mode II ratio G_II/G, although some discrepancies are observed in the high-mode II results for the 0/45 and 0/90 specimens. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 3, pp. 349–366, May–June, 2007.     

Shape sensitivity of curvilinear cracks on interface to non-linear perturbations

The 2D-model of an anisotropic, non-homogeneous, bonded elastic solid with a crack on the interface is considered. We state the linear problem with the stress-free boundary condition at the crack faces in addition to the transmission condition at the connected part of the interface. The sensitivity of the model to non-linear perturbations of the curvilinear crack along the interface is investigated. We obtain the asymptotic expansion and the corresponding derivatives of the potential energy functional with respect to the crack length via the material derivatives of the solution. This allows us to describe the growth or stationarity, and the local optimality conditions by the Griffith rupture criterion. The integral expression of the energy release rate for the considered problems is obtained, and the Cherepanov-Rice integral is discussed.     

An analogue of the Duistermaat-van der Kallen theorem for group algebras

Let G be a group, R an integral domain, and V _G the R-subspace of the group algebra R[G] consisting of all the elements of R[G] whose coefficient of the identity element 1 _G of G is equal to zero. Motivated by the Mathieu conjecture [Mathieu O., Some conjectures about invariant theory and their applications, In: Algèbre non Commutative, Groupes Quantiques et Invariants, Reims, June 26–30, 1995, Sémin. Congr., 2, Société Mathématique de France, Paris, 1997, 263–279], the Duistermaat-van der Kallen theorem [Duistermaat J.J., van der Kallen W., Constant terms in powers of a Laurent polynomial, Indag. Math., 1998, 9(2), 221–231], and also by recent studies on the notion of Mathieu subspaces, we show that for finite groups G, V _G also forms a Mathieu subspace of the group algebra R[G] when certain conditions on the base ring R are met. We also show that for the free abelian groups G = ℤ ⁿ , n ≥ 1, and any integral domain R of positive characteristic, V _G fails to be a Mathieu subspace of R[G], which is equivalent to saying that the Duistermaat-van der Kallen theorem cannot be generalized to any field or integral domain of positive characteristic.     

An inner amenable group whose von Neumann algebra does not have property Gamma

We construct inner amenable groups G with infinite conjugacy classes and such that the associated II₁ factor has no non-trivial asymptotically central elements, i.e. does not have property Gamma of Murray and von Neumann. This solves a problem posed by Effros in 1975.     

层状弹性材料界面J积分的产生和特征

层状弹性材料的裂纹方向垂直于界面时,沿围绕裂尖的任意一条封闭路径Γ的J积分（J_Г）由两部分组成,J_Г=J_tip+J_int,这里J_tip表示裂尖产生的J积分,J_int表示Γ所包围的界面产生的J积分.裂尖产生的J积分不随Γ变化,物理含义是裂纹扩展能量释放率;界面产生的J积分随Γ变化,物理含义与裂纹扩展能量释放率无关.界面J积分的产生使J_Г失去了路径无关特性,也失去了实际物理意义.为了有助于理解非均匀材料J积分的含义和局限性,分析了层状弹性材料界面J积分的产生原因和特点.由不同均匀弹性材料组成的层状材料中,应变能密度的跳跃是界面J积分产生的原因,而弹性模量和残余应力在界面处的跳跃可使应变能密度在界面处产生跳跃.层状弹性材料的界面J积分之间具有相互抵消的作用.     

混合型裂纹的疲劳扩展

本文对受单向拉伸疲劳载荷的中心斜裂纹L₃铝试板进行了研究。根据Erdogan和Sih的最大拉应力理论,推导出以△K作为参变量,以裂纹角β₀进行修正的Paris形式的扩展速率表达式。并且进一步论证以更简单的用裂纹长度在x轴上投影的Paris方程来表示。初始裂纹角β₀有20°、30°、45°、60°、80°、90°等各种角度,裂纹尖端有经预制疲劳裂纹尖端与未经预制疲劳裂纹尖端两种情况,比较了这两种情况下疲劳扩展轨迹及疲劳扩展速率。     

Einstein homogeneous bisymmetric fibrations

We consider a homogeneous fibration G/L → G/K, with symmetric fiber and base, where G is a compact connected semisimple Lie group and L has maximal rank in G. We suppose the base space G/K is isotropy irreducible and the fiber K/L is simply connected. We investigate the existence of G-invariant Einstein metrics on G/L such that the natural projection onto G/K is a Riemannian submersion with totally geodesic fibers. These spaces are divided in two types: the fiber K/L is isotropy irreducible or is the product of two irreducible symmetric spaces. We classify all the G-invariant Einstein metrics with totally geodesic fibers for the first type. For the second type, we classify all these metrics when G is an exceptional Lie group. If G is a classical Lie group we classify all such metrics which are the orthogonal sum of the normal metrics on the fiber and on the base or such that the restriction to the fiber is also Einstein.     

Dynamic crack propagation in a 2D elastic body. The out-of plane case

We discuss the propagation of a running crack under shear waves in a rigorous mathematical way for a simplified model. This model is described by two coupled equations in the actual configuration: a two-dimensional scalar wave equation in a cracked bounded domain and an ordinary differential equation derived from an energy balance law. The unknowns are the displacement fields u = u (y, t) and the one-dimensional crack tip trajectory h = h (t). We handle both equations separately, assuming at first that the crack position is known. Existence and uniqueness of strong solutions of the wave equation are studied and the crack-tip singularities are derived under the assumption that the crack is straight and moves tangentially. Using an energy balance law and the crack tip behaviour of the displacement fields we finally arrive at an ordinary differential equation for h (t), called equation of motion for the crack tip. We demonstrate the crack-tip motion with corresponding nonuniformly crack speed by numerical simulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decompositions of complete graphs and complete bipartite graphs into isomorphic supersubdivision graphs

A graph H is called a supersubdivison of a graph G if H is obtained from G by replacing every edge uv of G by a complete bipartite graph K_2,m (m may vary for each edge) by identifying u and v with the two vertices in K_2,m that form one of the two partite sets. We denote the set of all such supersubdivision graphs by SS(G). Then, we prove the following results.

1. Each non-trivial connected graph G and each supersubdivision graph HSS(G) admits an α-valuation. Consequently, due to the results of Rosa (in: Theory of Graphs, International Symposium, Rome, July 1966, Gordon and Breach, New York, Dunod, Paris, 1967, p. 349) and El-Zanati and Vanden Eynden (J. Combin. Designs 4 (1996) 51), it follows that complete graphs K_2cq+1 and complete bipartite graphs K_mq,nq can be decomposed into edge disjoined copies of HSS(G), for all positive integers m,n and c, where q=|E(H)|.

2. Each connected graph G and each supersubdivision graph in SS(G) is strongly n-elegant, where n=|V(G)| and felicitous.

3. Each supersubdivision graph in EASS(G), the set of all even arbitrary supersubdivision graphs of any graph G, is cordial.

Further, we discuss a related open problem.     

On the second Laplacian spectral moment of a graph

Kragujevac (M. L. Kragujevac: On the Laplacian energy of a graph, Czech. Math. J. 56(131) (2006), 1207–1213) gave the definition of Laplacian energy of a graph G and proved LE(G) ⩾ 6n-8; equality holds if and only if G = P _n . In this paper we consider the relation between the Laplacian energy and the chromatic number of a graph G and give an upper bound for the Laplacian energy on a connected graph.     

AlgebraicK-theory of von Neumann algebras

To every von Neumann algebra, one can associate a (multiplicative) determinant defined on the invertible elements of the algebra with range a subgroup of the Abelian group of the invertible elements of the center of the von Neumann algebra. This determinant is a normalization of the usual determinant for finite von Neumann algebras of type I, for the type II₁-case it is the Fuglede-Kadison determinant, and for properly infinite von Neumann algebras the determinant is constant equal to 1. It is proved that every invertible element of determinant 1 is a product of a finite number of commutators. This extends a result of T. Fack and P. de la Harpe for II₁-factors. As a corollary, it follows that the determinant induces an injection from the algebraicK ₁-group of the von Neumann algebra into the Abelian group of the invertible elements of the center. Its image is described. Another group,K ₁ ^w (A), which is generated by elements in matrix algebras overA that induce injective right multiplication maps, is also computed. We use the Fuglede-Kadison determinant to detect elements in the Whitehead group Wh(G).Partially supported by NSF Grant DMS-9103327.     

Subfactors associated to compact Kac algebras

We construct inclusions of the form (B ₀P) ^G (B ₁P) ^G , whereG is a compact quantum group of Kac type acting on an inclusion of finite dimensional C^*-algebrasB ₀B ₁ and on aII ₁ factorP. Under suitable assumptions on the actions ofG, this is a subfactor, whose Jones tower and standard invariant can be computed by using techniques of A. Wassermann. The subfactors associated to subgroups of compact groups, to projective representations of compact groups, to finite quantum groups, to finitely generated discrete groups, to vertex models and to spin models are of this form.