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Stress concentration analysis of thick-walled laminate composites with a loaded circular cut-out by using a first-order shear deformation theory W. Hufenbach, R. Gottwald, B. Grüber, M. Lepper, B. Zhou To cite this version: W. Hufenbach, R. Gottwald, B. Grüber, M. Lepper, B. Zhou. Stress concentration analysis of thick-walled laminate composites with a loaded circular cut-out by using a first-order shear deformation theory. Composites Science and Technology, Elsevier, 2009, 68 (10-11), pp.2238. . HAL Id: hal-00594922 https://hal.archives-ouvertes.fr/hal-00594922 Submitted on 22 May 2011 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Accepted Manuscript Stress concentration analysis of thick-walled laminate composites with a loaded circular cut-out by using a first-order shear deformation theory W. Hufenbach, R. Gottwald, B. Grüber, M. Lepper, B. Zhou PII: DOI: Reference: 10.1016/j.compscitech.2008.04.005 CSTE 4030 To appear in: Composites Science and Technology Received Date: Revised Date: Accepted Date: 13 December 2007 2 April 2008 5 April 2008 S0266-3538(08)00128-0 Please cite this article as: Hufenbach, W., Gottwald, R., Grüber, B., Lepper, M., Zhou, B., Stress concentration analysis of thick-walled laminate composites with a loaded circular cut-out by using a first-order shear deformation theory, Composites Science and Technology (2008), doi: 10.1016/j.compscitech.2008.04.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT Stress concentration analysis of thick-walled laminate composites with a loaded circular cut-out by using a first-order shear deformation theory W. Hufenbach, R. Gottwald, B. Grüber ∗, M. Lepper, B. Zhou Institut für Leichtbau und Kunststoﬀtechnik (ILK), TU Dresden, 01062 Dresden, Germany Abstract Stress concentrations in the vicinity of cut-outs can often be regarded as design drivers for the whole structure. Especially in thick-walled laminate composites not only the extension-bending coupling eﬀects but as well the inﬂuence of the sheardeformation has to be taken into account. For analyzing stress concentration eﬀects in the vicinity of circular cut-outs in thick-walled anisotropic composites, sophisticated semi-analytical solution methods based on a ﬁrst-order shear-deformation theory have been developed at the Institut für Leichtbau und Kunststoﬀtechnik. The comparison of the strains in the vicinity of cut-outs in textile-reinforced composites obtained by the new semi-analytical methods with experimentally measured and numerically determined strain values show a good agreement. Key words: Textile composites, Stress concentrations, Notch, Shear deformation theory Preprint submitted to Elsevier Science 11 April 2008 ACCEPTED MANUSCRIPT 1 Introduction Nowadays multilayered composites are used more and more often even for loadcarrying and safety-relevant structures in all kind of applications for aviation and space technology, for vehicles, for mechanical engineering as well as for the sporting and leisure goods industry. While up to now the reinforcing structure mostly is composed of uni- or bidirectional ﬁbre-reinforced layers, currently textile semi-ﬁnished products in form of multi-axial knitted, woven or braided preforms are getting more and more into the focus of research and application. In the past composites used in applications could mostly be considered as rather thin, but now more and more often thick-walled laminates are used. In the case of such thick composites besides the extension-bending coupling effects, as known from thin-walled laminates, the inﬂuence of the shear-deformations has to be taken into consideration. So it is of great importance to provide adapted calculation methods for critical areas like cut-outs to utilize the large lightweight design potential of thick-walled multilayered composites. The method presented in this study for the linear stress/strain analysis of thickwalled laminate composites with a circular cut-out can be regarded as one module of such a design method and enables the user to compare the inﬂuence or diﬀerent composite designs on the stress/strain distribution. In the literature (for instance in [1–5]), analytical solutions for the analysis of the stress concentration ﬁelds of ﬁbre-reinforced single- and multilayered composite plates with cut-outs or inclusions can be found for various cases of geometry and loads. These solutions are mainly based on the classical laminate ∗ Corresponding author. Tel.: +49-351-463-38146; Fax: +49-351-463-38143 Email address: [email protected] (B. Grüber). 2 ACCEPTED MANUSCRIPT theory and the fundamental works of Lekhnitskii [6]. Solutions for thickwalled multilayered composites based on a higher order laminate theory, for instance a ﬁrst-order shear deformation theory, are not known to the authors. The precise analysis of the stress and strain ﬁelds in the vicinity of cut-outs is often of great importance for the design of thick-walled composites in multimaterial design, since stress concentrations can often be regarded as design drivers for the whole structure. At the Institut für Leichtbau und Kunststoﬀtechnik (ILK) sophisticated analytical solutions for basic problems concerning thin-walled notched single- and multilayered composites have already been developed during recent years [7–11]. Based on the experience gathered during these developments, it was reasoned that for a realistic stress concentration analysis of thick-walled generally structured multilayered composites (MLC), it is important to provide methods which enable a layer-by-layer precalculation of the entire stress and distortion ﬁeld. Hereby it is necessary to pursue superordinate approaches in the expanded stress-deformation analysis, which not only take into consideration the extension-bending coupling eﬀects occurring in non-symmetrical composites but which also account for the inﬂuence of the shear deformation and thus have less restrictions with regard to the lamina materials, the composite lay-up and the thickness. 2 Analytical calculation methods for multilayered composites using a first-order shear deformation theory 2.1 Generalized plate equation for thick-walled laminates 3 ACCEPTED MANUSCRIPT Starting point for the derivation of a generalized plate equation for thickwalled laminates is the deformation law for the single lamina. In this study, Hooke’s law for orthotropic materials σij = Cijkl εkl (1) is used to describe the stress-strain-relationship, with the stress and strain tensors σij and εkl and the stiﬀness tensor Cijkl (i, j, k, l = 1 . . . 3). The use of this deformation law implicates, that the single lamina shall be deemed to be made up of a homogeneous orthotropic material by using homogenization techniques, while in reality the single lamina is inhomogeneous as it is composed of ﬁbres and matrixmaterial. Since in this study neither temperature nor moisture eﬀects are taken into consideration, the terms describing these eﬀects have been omitted in (1). For the incorporation of the shear deformation eﬀects, the following kinematic assumption, which are well known from the Mindlin-Reissner plate theory, are made 1 u(x, y, z) = u0 (x, y) + zψx (x, y), v(x, y, z) = v0 (x, y) + zψy (x, y), w(x, y, z) = w0 (x, y). (2) The displacement functions u0 , v0 and w0 describe the displacements of the midplane of the plate and ψx and ψy the rotations of cross-sections, that were normal to the midplane before deformation. This formulation of the displacement functions was ﬁrst given by Reisner [12,13] and Mindlin [14] 1 x, y, z: global coordinates with x-,y-axis as in-plane axis and z-axis as out of plane axis 4 ACCEPTED MANUSCRIPT in their plate theories. Some examples for theories of even higher order – including not only the shear deformation eﬀects but also, e. g., warping of crossections – can be found in [15]. With the kinematic assumption (2) the strain-displacement relationships for small displacements can be written in the form ⎡ ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ∂ψx ∂u0 0 ⎢ ⎥ ⎢ εx ⎥ ⎢ ⎥ ⎢ εx ⎥ ⎢ κx ⎥ ∂x ∂x ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∂ψy ⎢ ⎥ ∂v 0 0 ⎢ε ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ε κ ⎢ y⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ y y ∂y ⎢ ⎥ ∂y ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ∂ψ ∂ψ ∂u ∂v y ⎢γ ⎥ = ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 x 0 0 ⎢ ⎥+z⎢ =: + z + γ κ + ⎢ xy ⎥ ⎥ ⎢ ⎥ ⎢ xy ⎥ . xy ⎢ ∂y ⎥ ∂x ∂y ∂x ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∂w ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 ⎢ ⎥ γ 0 ⎢γxz ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + ψ 0 x⎥ xz ⎥ ⎢ ∂x ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ∂w ⎦ 0 0 +ψ γyz γ 0 0 y yz ∂y (3) Based on the strain-displacement-relations (3) and Hooke’s deformation law (1), ﬁnally the structural law for thick-walled multilayered laminates according to the ﬁrst-order shear deformation theory is derived ⎡ ⎤ ⎢ Nx ⎥ ⎢ ⎥ ⎢ ⎥ ⎢N ⎥ ⎢ y ⎥ ⎢ ⎥ ⎢ ⎥ ⎢N ⎥ ⎢ xy ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Mx ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ My ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ = ⎢A11 A12 A16 B11 B12 ⎢ ⎢ ⎢ A22 A26 B22 ⎢ ⎢ ⎢ ⎢ A66 symm. ⎢ ⎢ ⎢ ⎢ D11 D12 ⎢ ⎢ ⎢ ⎢ ⎢ symm. D22 ⎢ ⎣ Mxy ⎡ Qy B16 ⎥ ⎡ = ⎤ 2 ⎢−k2 A55 ⎢ ⎣ symm. −k1 k2 A45 ⎥ −k12 A44 ⎥ ⎦ ε0x ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ε0 ⎥ ⎢ y⎥ ⎢ ⎥ ⎢ ⎥ ⎢γ 0 ⎥ ⎢ xy ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ κx ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ κy ⎥ ⎢ ⎥ ⎣ ⎦ ⎥ ⎥ B26 ⎥ ⎥ ⎥ ⎥ B66 ⎥ ⎥ ⎥ ⎥ D16 ⎥ ⎥ ⎥ ⎥ ⎥ D26 ⎥ ⎥ ⎦ D66 ⎤ ⎢Qx ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ , (4) κxy ⎤ ⎡ 0 ⎢−γxz ⎥ ⎢ ⎣ 0 −γyz ⎥ ⎦ , with the shear correction factors 2 k1 , k2 , the stress resultants and moment 2 For details on the shear correction factors see e. g. [16] 5 ACCEPTED MANUSCRIPT resultants ⎡ ⎤ ⎢ Nx ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Ny ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ Nxy ⎡ ⎤ ⎢ σx ⎥ ⎥ h/2 ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ σy ⎥ dz = ⎢ ⎥ ⎥ −h/2 ⎢ ⎣ ⎦ τxy ⎡ ⎡ ⎤ ⎢ ⎢ ⎢ ⎢ ,⎢ ⎢ ⎢ ⎣ Mx ⎥ Mxy Qy ⎡ = , τxy (5) ⎤ h/2 ⎢ τxz ⎥ ⎢ ⎥ ⎢ ⎥ dz ⎣ ⎦ −h/2 ⎤ ⎢ σx ⎥ ⎥ h/2 ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ σy ⎥ z dz = ⎢ ⎥ ⎥ −h/2 ⎢ ⎣ ⎦ ⎥ ⎥ ⎥ My ⎥ ⎥ ⎥ ⎦ ⎤ ⎢Qx ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ , τyz the extensional, extension-bending coupling and bending stiﬀnesses, as known from the classical laminate theory ⎡ ⎡ ⎤ ⎢ Aij ⎥ ⎢ ⎥ ⎢ ⎥ ⎢B ⎥ ⎢ ij ⎥ ⎢ ⎥ ⎣ ⎦ h 2 = Dij − h2 Ci3 Cij − C3j C33 ⎤ ⎢1⎥ ⎢ ⎥ ⎢ ⎥ ⎢ z ⎥ dz ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (i, j = 1, 2, 6) , (6) z2 and the additional shear stiﬀnesses due to the incorporation of the shear deformation eﬀects h 2 Aij = Cij dz (i, j = 4, 5) . (7) − h2 In multilayered textile- or ﬁbre-reinforced composites, a separation of the plate-bending and membrane problems often is not possible due to coupling of force resultants and curvatures as well as coupling of moment resultants and strains occurring in non-symmetrical composites. Therefore, the MindlinReisner plate theory is expanded in such a way, that a uniﬁed approach for the solution of both, the plate bending and the membrane problem is found. Here the formulations of the equilibrium of force and moment resultants at the diﬀerential plate element is supplemented by the membrane force resultants 6 ACCEPTED MANUSCRIPT and a so called generalized plate equation is derived. This system of coupled partial diﬀerential equations (PDES) can be written down by means of a differential operator matrix in compact and clear form as a matrix equation analogously to generalized plate equation for thin-walled laminate composites found in [8] ⎡ ⎡ ⎤ ⎢ u0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0⎥ ⎢ v0 ⎥ ⎥ ⎢ ⎥ ⎥ T⎢ ⎥ ⎥ ⎢ 0 ⎥ Δ ⎢ w0 ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎦ ⎢ ⎥ K ⎢ ψx ⎥ ⎢ ⎥ ⎣ ⎦ ⎤ ⎢A ⎢ ⎢ Δ⎢ ⎢B ⎢ ⎣ B D 0 0 = −P (8) ψy with ⎡ Δ= ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ∂ ∂x 0 ∂ ∂y 0 0 0 0 0 ∂ ∂ ∂y ∂x 0 0 0 0 0 0 0 0 0 0 0 0 0 ∂ ∂x 0 0 0 0 0 ∂ ∂y ∂ ∂ ∂y ∂x 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ∂ − ∂x ∂ − ∂y −1 0 0 −1 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , (9) A, B, D: matrices of extensional, extension-bending coupling and bending stiﬀnesses, q. v. (6), K: matrix of shear correction factors and shear stiﬀness, q. v. (7), P : vector of external loads. 2.2 Boundary conditions and solution method Due to the structure of the PDES (8) it is not possible to transform it into a single partial diﬀerential equation in one function, where all derivations 7 ACCEPTED MANUSCRIPT of the function are of the same order. So the solution method of complex valued displacement functions, as used for the solution of the generalized plate equation in context of the classical laminate theory (for example in [10]), is not applicable here. In this study, an inﬁnite plate with a ﬁnite circular cut-out is selected as a mathematical equivalent model for the notched plate and the following restrictions as for the loading conditions are introduced. The notched plate should be loaded at the inﬁnite outer boundary by constant edge forces per unit length Nx , Ny , Nxy (Fig. 1(a)) and constant edge moments per unit length Mx , My , Mxy (Fig. 1(b)) in such a way, that the static equilibrium conditions are fulﬁlled automatically when looking on the overall plate. At the edge of the notch, the plate can be loaded by a constant normal edge force per unit length Ni and a constant edge bending moment per unit length Mi (Fig. 1(c)). (a) force resultants (b) moment resultants (c) force and moment resultant Fig. 1. Boundary conditions on cut-out and outer boundary In order to take into consideration the eﬀect of the loads on the outer boundary on the stress concentration problem of an inﬁnite plate, the actual state of stress is decomposed using the superposition principle into three states of stress (Fig. 2) as follows I: a ﬁnite, unnotched plate with loads at the outer edge, II: an inﬁnite notched plate with loads at the edge of the notch, adapted in such a way that, with superposition of I and II, an overall unloaded 8 ACCEPTED MANUSCRIPT Fig. 2. Decomposition of the coupled membrane-plate problem by means of superposition, here shown for in-plane loading. notch edge results, III: an inﬁnite notched plate with the corresponding loads at the notch edge. Looking at this decomposition from a diﬀerent point of view, the solution for the stress concentration problem of a notched plate with a loaded cut-out is derived by superpositioning of the solutions for the problem of a notched plate with an unloaded hole and a loaded outer boundary (subproblem I and II) and the problem of an notched plate with a loaded cut-out and an unloaded outer boundary (subproblem III). Of course, if dealing with the problem of an unloaded cut-out, subproblem III can be omitted. Subproblem I, the ﬁnite plate loaded on the outer boundary by constant edge forces and moments, can be solved elementarily and the solutions provide the dynamic boundary conditions for subproblem II, i. e. the force and moment resultants that have to be applied to the boundary of the notch in subproblem II so, that by superposition of I and II an overall unloaded notch edge results. Since the method of complex-valued displacement functions is not applicable here, the Ritz-method is called upon for the solution of each of the subproblems II and III, respectively. By superposing the solutions of all three 9 ACCEPTED MANUSCRIPT subproblems ﬁnally the solution for the stress concentration problem of an inﬁnite plate is derived. To apply the Ritz-method, all equations are transformed into a polar coordinate system with its origin in the centre of the notch ﬁrst. Afterwards the PDES is rewritten into a so-called weak formulation. In the case of the chosen mathematical model of an inﬁnite plate, the shell stress resultants and moments resultants have to disappear for r → ∞. Due to this, the displacements u0 , v0 , w0 and the shear deformations ψx and ψy have to disappear as well for r → ∞. For the Ritz method, the ansatz functions have to be chosen in such a way, that they fulﬁll the kinematic boundary conditions, i. e. they have to tend to zero for r → ∞. Since on the edge of the cut-out only dynamic boundary conditions, i. e. force and moment resultants, are applied, there are no restrictions for the ansatz functions on the notch edge. The ansatz functions are chosen in the form of a series in such a way, that in ϕ-direction a 2π-periodical Fourier series is used while in r-direction a power series is applied u(r, ϕ) = M1 c Um0 m=1 M1 N m a r + a r c (Umn a r c s (Vmn cos(nϕ) + Vmn sin(nϕ)) , m cos(nϕ) + n=1 m=1 m M2 a c Vm0 + v(r, ϕ) = r m=1 M2 m N n=1 m=1 M3 c Wm0 w(r, ϕ) = m=1 sin(nϕ)) , m a r M3 m N a n=1 m=1 M4 Ψcm0 ψr (r, ϕ) = m=1 s Umn r + c s (Wmn cos(nϕ) + Wmn sin(nϕ)) , m a r + 10 (10) ACCEPTED MANUSCRIPT M4 m N a r n=1 m=1 M5 a m c Φm0 ψϕ (r, ϕ) = r m=1 m M5 N n=1 m=1 a r (Ψcmn cos(nϕ) + Ψsmn cos(nϕ) + Φsmn sin(nϕ)) , + (Φcmn sin(nϕ)) . where a is the diameter of the circular notch. By application of the Ritz method the looked-after coeﬃcients for the displacement functions can be computed. Hereby the summation limits are normally chosen in the following form M1 = M2 = M3 , M4 = M5 = M1 + 1, N = free. (11) The values for M1 and N should be chosen with regard to the given problem. In this study M1 = 16 and N = 20 was suﬃcient for all problems. The (semi)analytical calculation method developed here has ﬁnally been implemented into a calculation tool and can now be used on standard oﬃce PCs. 3 Experimental and numerical verification For the veriﬁcation of the developed semi-analytical solution method for thickwalled multilayered composites, extensive experimental and numerical ﬁnite element (FE) investigations have been carried out. Within the scope of the Collaborative Research Centre SFB 639, these studies have been conducted on multilayered composites built from bidirectionally reinforced knitted layers made from hybrid yarn containing glass ﬁbres and polypropylene matrix ﬁbres (GF/PP). The material parameters (12) of the single lamina were determined 11 ACCEPTED MANUSCRIPT experimentally and by the means of X-FEM [17,18] E11 = 15.4 GPa, E22 = 15.4 GPa, E33 = 4.6 GPa, G12 = 1.6 GPa, G13 = 0.9 GPa, G23 = 0.9 GPa, ν12 = 0.134, ν13 = 0.417, (12) ν23 = 0.417. 3.1 Comparison to experimental results In Fig. 3 the experimental set-up for the stress concentration analysis is shown. In this test a textile reinforced GF/PP plate with an un-loaded cut-out is loaded in a unidirectional tension test. For the determination of the displacement and strain ﬁelds the grey-scale correlation method is used as a contactless optical 3D-ﬁeld measurement method. From the experimental investigations the εx and εy strains along the 0◦ , 90◦ and the ±45◦ radians (Fig. 3(b)) are determined and are compared to semi-analytically calculated results. (a) experimental setup (b) directions Fig. 3. Experimental set-up and directions of result evaluation In Fig. 4, some of the results for the calculated and experimentally determined strain decay behaviours along the 0◦ , 90◦ and the ±45◦ radians are shown. From the ﬁgures, an overall good correlation of the semi-analytical and experimental results can be observed. Never the less, the correlation seems to 12 ACCEPTED MANUSCRIPT be not so good at the edge of the notch. This is mainly caused by the used 3D gray-scale measurement equipment used. Due to the limited resolution of the cameras used, the precipitous gradient of the strains near the edge of the cut-out can not be detected with adequate accuracy. (a) 0◦ - and 90◦ -radian (b) ±45◦ -radian Fig. 4. Experimentally determined and analytically calculated strain decay behaviour 3.2 Comparison of analytical and numerical results To reduce the number of expensive and time consuming experimental investigations, an extensive number of comparative ﬁnite-element calculations are performed. Therefore diﬀerent multilayered GF/PP composites (properties of the single layer according to (12)) with diﬀerent lay-ups and diﬀerent external loadings are investigated. Representative results are presented here from the large number of performed verifying calculations. Fig. 6 shows a comparison of the numerically and analytically obtained strains on the laminate top surface for a [±4510 ]s GF/PP laminate with a circular cut-out (diameter ∅ = 30 mm) loaded with a bending moment resultant Mx = 1000 Nm/m according to Fig. 5. 13 ACCEPTED MANUSCRIPT Fig. 5. FE model for a notched GF/PP plate with with a practically far-ﬁeld plate-bending moment In Fig. 6 the strains directly on the edge of the notch (r = 15 mm) and the strains along a concentric circle with radius r = 16 mm are given. From the subﬁgures a very good correlation of the analytical and numerical results can be observed. (a) notch boundary (r = 15 mm) (b) concentric circle (r = 16 mm) Fig. 6. Strains on laminate top surface Overall, it turns out that in case of all examined combinations of composite and load, the numerically determined results and the results calculated by use of the developed analytical methods show a high degree of agreement. Thus, the developed fundamentals for calculations of stress concentrations at notches in thick-walled multilayered composites represent a quick, dependable and easily manageable alternative to a time-consuming FE analysis. 14 ACCEPTED MANUSCRIPT 4 Parameter studies on thick-walled multilayered composite plates with cut-outs 4.1 Comparison with classical laminate theory In the ﬁrst parameter study presented here the eﬀect of a ﬁrst-order shear deformation theory (SDT1) in comparison to the classical laminate theory (CLT) is studied. As exemplary results, the strain distribution on the laminate top surface at the edge of the notch for a [±4510 ]s GF/PP multilayered composite plate with a circular cut-out (diameter ∅ = 30 mm) is shown. The plate was loaded under unidirectional tension (membrane load, Fig. 7(a)) and under plate-bending load (Fig. 7(b)) (b) bending load Mx = 100 Nm/m (a) membrane load Nx = 10 N/m Fig. 7. Strains on laminate top surface In the case of pure membrane load and the symmetric multilayered composite, no diﬀerences between the results using the CLT and the SDT1 can be observed, as expected (see Fig. 7(a)). Naturally, the eﬀects of the shear deformation (Fig. 7(b)) are only relevant in cases where shear force initiated bending eﬀects occur, either caused by the material (non-symmetrical laminates) or caused by the loads(bending load). In these cases a great diﬀerence between the results calculated using the CLT or SDT1, respectively, can be observed. 15 ACCEPTED MANUSCRIPT This phenomena is further studied in the next parameter study concentrating on the inﬂuence of the laminate thickness. 4.2 Inﬂuence of the laminate thickness In a second parameter study, the inﬂuence of the laminate thickness on the strain distribution on the surface of the composite is examined. The platebending moment Mx was chosen according to Mx = h h0 2 · 100 Nm m (13) with h as laminate thickness and h0 = 1 mm as reference thickness. The investigated parameter combinations are shown in Table 1, the material parameter are chosen according to (12). With this choice of Mx in the case of classical laminate theory the strain ﬁelds on the laminate top and bottom respectively are independent of the thicknesses and therefore should be comparable. Some of the results of this parameter study are presented in Fig. 8. Table 1 Laminates and loading parameters for studying the inﬂuence of the laminate thickness thickness h plate bending moment [mm] Mx [Nm/m] [±452 ]s 1 100 [±453 ]s 1.5 225 [±454 ]s 2 400 [±458 ]s 4 1 600 [±4516 ]s 8 6 400 [±4532 ]s 16 25 600 [±4564 ]s 32 102 400 [±45128 ]s 64 409 600 [±45256 ]s 128 1 638 400 lay-up 16 ACCEPTED MANUSCRIPT (a) radial strain (b) radial/tangential shear strain Fig. 8. Inﬂuence of laminate thickness As expected, the strain ﬁelds on the laminate top and bottom surface are independent of the laminate thickness in case of classical laminate theory and the chosen bending load. In the case of the SDT1, the situation is diﬀerent. In this case a great inﬂuence of the thickness on the strain distribution is observed. Hereby the thickness not only quantitatively inﬂuences the curves, especially the extreme values, but also can change the qualitative form of the curves (see Fig. 8). This change in the curve characteristic can be seen exceptionally well for the shear-stress curves in Fig. 8(b). Here the characteristic clearly changes from the curves for the laminate thickness of 1 mm over the curve for the thickness 2 mm to the ﬁnal curve characteristic for a thickness of 8 mm. As a further result of the investigation, it is also found that even for a laminate thickness of 1 mm the inﬂuence of the shear deformation can clearly be observed in comparison to the CLT (q. v. Fig. 8(b)). Simular results were observed for other lay-ups and diﬀerent materials. 4.3 Eﬀects of FE mesh reﬁnements One of the greatest advantages of the developed semi-analytical calculation method compared with the FE method is, that no discretization – as for exam17 ACCEPTED MANUSCRIPT ple the creation of a FE mesh – is needed. The developed methods are mesh independent. In the scope of this study, this advantage is proven by a vast number of comparative FE calculations with diﬀerent mesh reﬁnements. The FE calculation were carried out using the programm system I-Deas Master Series 11 with linear and quadratic 4-node and 8-node shell elements, respectively. Some selected results of this study are presented in Fig. 9 for a [±4510 ]s GF/PP multilayered composite plate with a circular cut-out (diameter ∅ = 30 mm) under a plate-bending load Mx = 1000 Nm/m. Fig. 9. Shear distortion on laminate top surface for diﬀerent FE mesh reﬁnements (linear 4-node shell elements) It is obvious that there is a great inﬂuence of the number of elements at the edge of the notch, i. e. the mesh density (q. v. Fig. 9). Generally the parameter studies on the inﬂuence of the FE mesh density lead to the conclusion, that in all cases there is a great inﬂuence on the stress and strain ﬁelds in the vicinity of the notch. In the case presented here, more than 360 elements on the notch edge are needed to at least qualitatively follow the curve of the analytically calculated strains, and even 1440 elements are not enough to quantitatively reach the analytical results. To overcome this disadvantage of the FE analysis, it is planned to combine the developed semi-analytical mesh-independent calculation methods (for the local stress concentration analysis) with the FE method (for the global structural analysis) by the means of analytical super 18 ACCEPTED MANUSCRIPT elements. 5 Conclusion During recent years the application of thick-walled textile- and ﬁbre-reinforced multilayered composites has expanded considerably. For the application of these materials, local stress concentrations, as they occur in the vicinity of notches, can often be regarded as design drivers for the whole structure. In this study, the mechanically equivalent model of an inﬁnite plate with a circular cut-out, which is subject to combined membrane and plate loads, is dealt with by means of a expanded plate theory based on the Mindlin-Reisner plate theory and semi-analytical methods for a layer-by-layer stress/strain analysis of thick-walled multilayered composites. The presented methods not only calculate stresses, strains and displacements directly at the edge of the notch, but also their distribution throughout the entire area of the plate. For the veriﬁcation of the developed calculation method experimental and a vast number of numerical FE analyses were performed with symmetrical and non-symmetrical composite structures. The comparison of the experimental and numerical results, respectively, with the results obtained by means of the developed solutions show a good agreement and thus indorse the developed theory in an impressive way. The performed parameter studies demonstrate that very complex mechanisms are acting in thick-walled multilayered anisotropic composite plates in the areas of notches. In the case of pure membrane loads and symmetric laminates, the shear deformation eﬀects don’t have to be taken into account. In this case, the more easy-to-handle classical laminate theory should be used for the 19 ACCEPTED MANUSCRIPT analysis of stress concentration eﬀects in multilayered composites, for example according to [8,5]. From the studies carried out it must be concluded, that for dimensioning thick-walled multilayered composites in line with material and component characteristics, a precise layer-by-layer analysis of the distribution of stresses and distortions is absolutely indispensable and especially in the case of the occurrence of bending eﬀects the inﬂuence of shear deformation has also to be taken into account. Starting from this layer-by-layer approach, the development of new failure criteria for notched multilayered laminates is currently one of the focal points of ongoing research work at the ILK. These failure criteria take into consideration so called physically based failure criteria (e. g. according to Puck [19] or Cuntze [20]) and additionally consider the so-called microsupport eﬀect according to Neuber [21]. Acknowledgements We would like to express our gratitude towards the Deutsche Forschungsgemeinschaft (DFG), who funds the subproject B2 within the scope of the Collaborative Research Centre SFB 639 ”Textile-Reinforced Composite Components in Function-Integrating Multi-Material Design for Complex Lightweight Applications”. References [1] Konisch HJ, Whitney JM. Approximate stresses in an orthotropic plate containing a circular hole. Journal of Composite Materials 1975;9:157–167. 20 ACCEPTED MANUSCRIPT [2] Hwu C, Ting TCT. Two-dimensional problems of the anisotropic elastic solid with an elliptic inclusion. Q. Jl. Mech. Appl. Math. 1989;42:553–572. [3] Ukadgaonker VG, Rao DKN. A general solution for stress resultants and moments around holes in unsymmetric laminates. 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