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Journal ArticleDOI

Partition of mixed-mode fractures in 2D elastic orthotropic laminated beams under general loading

01 Aug 2016-Composite Structures (Elsevier)-Vol. 149, pp 239-246
TL;DR: In this article, an analytical method for partitioning mixed-mode fractures on rigid interfaces in orthotropic laminated double cantilever beams (DCBs) under through-thickness shear forces, in addition to bending moments and axial forces, is developed by extending recent work by the authors.
About: This article is published in Composite Structures.The article was published on 2016-08-01 and is currently open access. It has received 20 citations till now. The article focuses on the topics: Pure bending & Bending moment.

Summary (3 min read)

1. Introduction

  • An analytical method for partitioning mixed-mode fractures in orthotropic laminated double cantilever beams (DCBs) with rigid interfaces has been developed in the authors’ recent work [1] by taking 2D elasticity into consideration in a novel way.
  • The present study extends Ref. [1] to include crack tip throughthickness shear forces which commonly occur in practice.
  • The total ERR was determined using the J-integral and related to the individual stress intensity factors by introducing an unknown parameter which, as the solution to an integral equation, must be determined from tabulated numerical data from a limited range of geometries and material configurations.
  • Then, in Section 2.2, a mixed-mode partition theory is established for the DCB under general loading conditions which include crack tip bending moments, axial forces and shear forces.
  • Comparisons for the mixed-fracture mode partition theory are made against results from 2D FEM simulations for a combination of loading conditions in Section 3.

2.1. Mixed-mode partitions with through-thickness shear forces alone at the crack tip

  • Fig. 1b shows the internal loads at the crack tip and the sign convention of the interface normal stress n and shear stress s .
  • The subscript T-P denotes that the pure modes are for the through-thickness shear forces at the crack tip, BP1 and BP2 , and are based on Timoshenko beam theory.
  • Similarly, when BPB PP 1T-2 the pure mode II mode occurs.
  • Obviously, this will not be the case within the context of 2D elasticity.
  • Inside this range the accuracy is much higher; however, outside it the accuracy decreases rapidly and Eq. (9) should be used instead.

2.2. Mixed-mode partitions under general loads

  • The pure-mode-I 2D-1 mode and the pure-mode-II 2D-1 mode have previously been determined in Ref. [1] by introducing correction factors into the beam-theory-based pure-mode-I and pure-mode-II mechanical conditions.
  • Then the pure-mode-I D22 mode and the pure-mode-II D22 mode were obtained by using the orthogonality condition that exists between pure modes.
  • Their explicit expressions are presented here for the easy use of readers with a clearer mechanical interpretation.
  • Again, it is worth noting that in the context of 2D elasticity, the influence of the material properties on the ERR is collectively shown by one effective property E and that the pure modes are affected only by the geometry.
  • This is in agreement with Hutchinson and Suo [24].

3. Numerical verification

  • To verify the present analytical partition theory, a program of 2D FEM simulations was carried out on the DCB shown in Fig. 1a using MSC/NASTRAN.
  • For convenience, therefore, isotropic material constants were used, as follows, and apply for all loading conditions and thickness ratios considered in this section:.
  • In the second loading condition there was a combination of through-thickness shear forces and bending moments at the crack tip and BP1 was varied in the range 000,10000,10 1 BP with 10001 BM .
  • A mesh size of 001.0p was found to provide mesh independence and was used for all simulations in this section.
  • Bending moments, 1M and 2M , were applied as equal and opposite axial forces to the top and bottom corners of each of the upper and lower beam tips respectively.

3.1. Through-thickness shear forces alone at the crack tip

  • This section considers the first loading condition in which there were only through-thickness shear forces at the crack tip and BP2 was varied in the range 000,10000,10 2 BP with 10001 BP .
  • Fig. 4b shows the differences between the ERR partitions GGI from the present analytical theory and from the 2D FEM.
  • Note that in Fig. 4, the subscript ‘th’ notation denotes quantities from the analytical theory whereas the subscript notation ‘FEM’ denotes quantities from the 2D FEM.
  • The areas of slightly increased error on both Fig. 4a and Fig. 4b are a result of the small values of the total ERR G in these regions, therefore magnifying the apparent error between the present analytical theory and 2D FEM.
  • To examine this further, Fig. 5 compares the absolute values of G and GGI from the present analytical theory and the FEM for the cross-sections through Figs. 4a and 4b where 7.01log10 .

3.2. General loading conditions

  • In the second loading condition, there was a combination of through-thickness shear forces and bending moments at the crack tip and BP1 was varied in the range 000,10000,10 1 BP with 10001 BM .
  • To achieve this loading, a DCB tip through-thickness shear force and a DCB tip bending moment were applied as BPP 11 and aPM 11 1000 respectively.
  • There is excellent agreement between the present analytical theory and 2D FEM results for both the total ERR G and the ERR partition GGI for the majority of and BB MP 11 values considered.
  • It is worth noting that the maximum error in Fig. 4c has been capped to 0.15 in order to make clearer comparisons between the present analytical theory and the 2D FEM.
  • Fig. 6 compares the absolute values of G and GGI from the present analytical theory and the FEM for the crosssections through Figs. 4c and 4d where 8.01log10 .

4. Conclusions

  • The authors’ existing analytical partition theory for mixed-mode fractures in 2D elastic orthotropic laminated beams with rigid interfaces [1] has been successfully extended to account for through-thickness shear forces.
  • Results for the present analytical partition theory have been compared to those from the 2D FEM and excellent agreement has been observed, particularly when the total ERR G is not close to zero.
  • This work now offers a means of calculating the 2D elasticitybased ERR partition for an orthotropic laminated DCB under any combination of bending moments, axial forces and shear forces.
  • The correction factors were determined by using the 2D FEM.
  • It is interesting that the correction factors closely follow elegant normal distributions around a symmetric DCB geometry.

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Citations
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Journal ArticleDOI
TL;DR: It is shown that interlayer shearing and sliding near the blister crack tip, caused by the transition from membrane stretching to combined bending, stretching and through-thickness shearing, decreases fracture mode mixity GII/GI, leading to lower adhesion toughness.
Abstract: Interface adhesion toughness between multilayer graphene films and substrates is a major concern for their integration into functional devices. Results from the circular blister test, however, display seemingly anomalous behaviour as adhesion toughness depends on number of graphene layers. Here we show that interlayer shearing and sliding near the blister crack tip, caused by the transition from membrane stretching to combined bending, stretching and through-thickness shearing, decreases fracture mode mixity G II/G I, leading to lower adhesion toughness. For silicon oxide substrate and pressure loading, mode mixity decreases from 232% for monolayer films to 130% for multilayer films, causing the adhesion toughness G c to decrease from 0.424 J m−2 to 0.365 J m−2. The mode I and II adhesion toughnesses are found to be G Ic = 0.230 J m−2 and G IIc = 0.666 J m−2, respectively. With point loading, mode mixity decreases from 741% for monolayer films to 262% for multilayer films, while the adhesion toughness G c decreases from 0.543 J m−2 to 0.438 J m−2. The reason why the surface adhesion of a graphene monolayer is much greater than that of graphene multilayers remains unclear. Here, the authors build a model to show interlayer sliding and fracture mode mixity cause the decrease in adhesion toughness of multilayer graphene.

26 citations


Additional excerpts

  • ...(19), (21) and GII(1 + 1/ρ) =GJ(1 + η), then GII 1⁄4 0:6986GJ ð22Þ...

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TL;DR: In this paper, three analytical models are developed to predict several aspects of the spallation failure of elastic brittle thin films including nucleation, stable and unstable growth, size of spallations and final kinking off.

19 citations

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TL;DR: In this paper, the authors presented robust and efficient predictions of buckling-driven delamination propagation, enabled by a novel analytical modelling approach, using an energy formalism to determine the post-buckling deformation and a crack-tip element analysis employing force and moment resultants acting on the delamination boundary.

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TL;DR: In this paper, a theoretical frame work was developed for strain gage based determination of mixed mode (KI/KII) stress intensity factors (SIFs) in slant edge cracked plate (SECP) made of orthotropic materials.

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Cites background from "Partition of mixed-mode fractures i..."

  • ...Researchers have numerically verified the partition of fracture modes on rigid interfaces in orthotropic laminated double cantilever beams (DCB) under general loading conditions that include crack tip bending moments, axial and shear forces [26,27]....

    [...]

References
More filters
Book ChapterDOI
TL;DR: In this article, the authors describe the mixed mode cracking in layered materials and elaborates some of the basic results on the characterization of crack tip fields and on the specification of interface toughness, showing that cracks in brittle, isotropic, homogeneous materials propagate such that pure mode I conditions are maintained at the crack tip.
Abstract: Publisher Summary This chapter describes the mixed mode cracking in layered materials. There is ample experimental evidence that cracks in brittle, isotropic, homogeneous materials propagate such that pure mode I conditions are maintained at the crack tip. An unloaded crack subsequently subject to a combination of modes I and II will initiate growth by kinking in such a direction that the advancing tip is in mode I. The chapter also elaborates some of the basic results on the characterization of crack tip fields and on the specification of interface toughness. The competition between crack advance within the interface and kinking out of the interface depends on the relative toughness of the interface to that of the adjoining material. The interface stress intensity factors play precisely the same role as their counterparts in elastic fracture mechanics for homogeneous, isotropic solids. When an interface between a bimaterial system is actually a very thin layer of a third phase, the details of the cracking morphology in the thin interface layer can also play a role in determining the mixed mode toughness. The elasticity solutions for cracks in multilayers are also elaborated.

3,828 citations


"Partition of mixed-mode fractures i..." refers background or result in this paper

  • ...Hutchinson and Suo [24] showed that the ERR components of an orthotropic material are essentially the same as their isotropic counterparts, except using the longitudinal tensile modulus....

    [...]

  • ...Li et al. [5], also using Suo and Hutchinson’s work [4], considered the effects of transverse shear loading on a crack between a layered, isotropic, linear elastic material....

    [...]

  • ...This is in agreement with Hutchinson and Suo [24]....

    [...]

  • ...Note that although Suo and Hutchinson’s theory [4] is regarded as the most accurate for bending moments and axial forces, the method developed in Ref....

    [...]

  • ...They combined their results with Suo and Hutchinson’s theory [4] for bending moments and axial forces to provide the ERR and mode partition for layered materials under general loading conditions....

    [...]

Journal ArticleDOI
TL;DR: In this article, a semi-infinite interface crack between two infinite isotropic elastic layers under general edge loading conditions is considered and the problem can be solved analytically except for a single real scalar independent of loading, which is then extracted from the numerical solution for one particular loading combination.
Abstract: A semi-infinite interface crack between two infinite isotropic elastic layers under general edge loading conditions is considered. The problem can be solved analytically except for a single real scalar independent of loading, which is then extracted from the numerical solution for one particular loading combination. Two applications of the basic solution are made which illustrate its utility: interface cracking driven by residual stress in a thin film on a substrate, and an analysis of a test specimen proposed recently for measuring interface toughness.

905 citations


"Partition of mixed-mode fractures i..." refers methods or result in this paper

  • ...[5], also using Suo and Hutchinson’s work [4], considered the effects of transverse shear loading on a crack between a layered, isotropic, linear elastic material....

    [...]

  • ...Hutchinson and Suo [24] showed that the ERR components of an orthotropic material are essentially the same as their isotropic counterparts, except using the longitudinal tensile modulus....

    [...]

  • ...Li et al. [5], also using Suo and Hutchinson’s work [4], considered the effects of transverse shear loading on a crack between a layered, isotropic, linear elastic material....

    [...]

  • ...This is in agreement with Hutchinson and Suo [24]....

    [...]

  • ...Note that although Suo and Hutchinson’s theory [4] is regarded as the most accurate for bending moments and axial forces, the method developed in Ref....

    [...]

Journal ArticleDOI
TL;DR: In this article, a detailed study of the methods of analysing the experimental data obtained from fracture mechanics tests using double-cantilever beam, end loaded split and end notched flexure specimens.
Abstract: One of the most important mechanical properties of a fibre-polymer composite is its resistance to delamination. The presence of delaminations may lead not only to complete fracture but even partial delaminations will lead to a loss of stiffness, which can be a very important design consideration. Because delamination may be regarded as crack propa­gation then an obvious scheme for characterizing this phenomenon has been via a fracture mechanics approach. There is, therefore, an extensive literature on the use of fracture mechanics to ascertain the interlaminar fracture energies, G c , for various fibre-polymer composites using different test geometries to yield mode I, mode II and mixed mode I/II values of G c . Nevertheless, problems of consistency and discussions on the accuracy of such results abound. This paper describes a detailed study of the methods of analysing the experimental data obtained from fracture mechanics tests using double-cantilever beam, end loaded split and end notched flexure specimens. It is shown that to get consistent and accurate values of G c it is necessary to consider aspects of the tests such as the end rotation and deflection of the crack tip, the effective shortening of the beam due to large displacements of the arms, and the stiffening of the beam due to the presence of the end blocks bonded to the specimens. Analytical methods for ascertaining the various correction constants and factors are described and are successfully applied to the results obtained from three different fibre-polymer composites. These composites exhibit different types of fracture behaviour and illustrate the wide range of effects that must be considered when values of the interlaminar fracture energies, free from artefacts from the test method and the analysis method, are required.

373 citations

Journal ArticleDOI
TL;DR: In this paper, the mixed-mode bending (MMB) delamination test has been studied in detail and the results from this test method compared to those obtained from the fixed-ratio mixedmode (FRMM) test method.

205 citations

Journal ArticleDOI
TL;DR: In this paper, the effect of transverse shear on delamination in layered, isotropic, linear-elastic materials has been determined, and expressions for the shear component of the energy-release rate presented in this work have been obtained using finite-element approaches.
Abstract: The effect of transverse shear on delamination in layered, isotropic, linear-elastic materials has been determined. In contrast to the effects of an axial load or a bending moment on the energy-release rate for delamination, the effects of shear depend on the details of the deformation in the crack-tip region. It therefore does not appear to be possible to deduce rigorous expressions for the shear component of the energy-release rate based on steady-state energy arguments or on any type of modified beam theory. The expressions for the shear component of the energy-release rate presented in this work have been obtained using finite-element approaches. By combining these results with earlier expressions for the bending-moment and axial-force components of the energy-release rates, the framework for analyzing delamination in this type of geometry has been extended to the completely general case of any arbitrary loading. The relationship between the effects of shear and other fracture phenomena such as crack-tip rotations, elastic foundations and cohesive zones are discussed in the final sections of this paper.

181 citations


"Partition of mixed-mode fractures i..." refers methods in this paper

  • ...[5], also using Suo and Hutchinson’s work [4], considered the effects of transverse shear loading on a crack between a layered, isotropic, linear elastic material....

    [...]

Frequently Asked Questions (1)
Q1. What are the contributions in "Partition of mixed-mode fractures in 2d elastic orthotropic laminated beams under general loading" ?

An analytical method for partitioning mixed-mode fractures on rigid interfaces in orthotropic laminated double cantilever beams ( DCBs ) under through-thickness shear forces, in addition to bending moments and axial forces, is developed by extending recent work by the authors ( Harvey et al., 2014 ). First, two pure through-thickness-shear-force modes ( one pure mode I and one pure mode II ) are discovered by extending the authors ’ mixed-mode partition theory for Timoshenko beams.