Elastic–plastic analysis of Asymmetric Double Cantilever Beam specimen

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Highlights

  • Non-linear fracture analysis of Asymmetric Double Cantilever Beam is presented.

  • Several closed form analytical solutions of J-integral are obtained.

  • Influence of crack position on J-integral is analyzed.

  • Numerical example included shows influence of material non-linearity on J-integral.

Abstract

The present paper studies theoretically non-linear fracture behavior of Asymmetric Double Cantilever Beam (ADCB) specimen made of unidirectional fiber reinforced polymer composite. The crack arms of ADCB have different heights. Loading configuration consists of two bending moments of the same senses and equal magnitudes applied to the crack arms. The composite material building the beam is assumed to obey stress–strain relation of an elastic-perfectly plastic material. Investigation is performed by means of path independent J-integral. Closed form analytical solutions of J-integral are obtained for every typical level of stresses and strains in the cracked as well as un-cracked beam portion. The influence of crack position on J-integral is also analyzed. A numerical example is presented to show the influence of material non-linearity on J-integral. It is found that the J-integral value increases when material non-linearity is taken into account. This is attributed to the strain energy dissipation due to plastic deformation of the beam.

Introduction

Development of unidirectional fiber reinforced polymer composites requires investigations on their delamination fracture behavior. For this purpose, many beam specimens have been proposed and analyzed theoretically in the specialized literature [1], [2], [3], [4], [5], [6], [7]. Likewise, since the experiments on real specimens are integral part of every research, many important tests have been performed and results proving theory predictions have been reported [8], [9].

One of the specimens for investigation of delamination fracture is Asymmetric Double Cantilever Beam (ADCB) which represents a double cantilever beam with crack arms of equal heights loaded by bending moments of the same senses [10], [11], [12]. Sorensen et al. in [13] have analyzed ADCB specimen where the crack arms are acted upon by two moments of different magnitudes (Fig. 1). They have obtained the following formula for J-integral under plane stress condition:J=21(M12+M22)6M1M24Eb2h3,where b is a beam width.

Formula (1) is found using linear-elastic fracture mechanics which is based on assumption that the relation between stresses and strains obeys Hooke׳s law. However, in the tougher composite systems applicability of linear-elastic fracture mechanics is often limited since the plastic deformation usually begins prior to onset of delamination crack growth. Therefore, the material non-linearity should be taken into account in fracture analysis.

The purpose of the present article is to develop further solution for ADCB by performance of non-linear fracture analysis. Configuration considered represents ADCB-specimen loaded by moments of equal magnitudes and crack arms of different heights: h1>h2 (Fig. 2). It should be noted that the loading conditions to which the beam is subjected contributes to the mixed mode I/II fracture.

In order to perform non-linear fracture analysis, polymer composite forming the beam is assumed to obey stress–strain relation of an elastic-perfectly plastic material in both tension and compression [14], [15] (Fig. 3). It should be noted that the active deformation of the beam is considered in the analysis, i.e. the magnitude of external moments is supposed to increase only [16], [17].

Non-linear fracture analysis is performed by application of path independent J-integral [18], [19]. First, J-integral is solved for linear-elastic regime of material work. The validity of formula obtained is proved by comparison with solution in [13]. Then the magnitude of external moments is supposed to increase which leads to non-linear behavior and redistribution of stresses and strains in the beam cross-sections. Solutions for J-integral are obtained for every characteristic level of stresses in the two crack arms and the un-cracked beam portion.

Section snippets

Solutions of J-integral

When the whole beam works in linear-elastic stage, i.e. the stresses and strains in two crack arms and un-cracked beam portion are smaller than fy and εy, respectively. The magnitude of the moment is M.

Here, the path of integration is chosen to coincide with beam contour (Fig. 4). J-integral is solved using following formula:J=JA+JB+JD,where JA, JB, and JD are the J-integral components in sections А, B, and D of the beam, respectively. J-integral is equal to zero in the other contour segments.

Conclusions

The present article deals with non-linear fracture analysis of ADCB-specimen made of unidirectional fiber reinforced polymer composite. It is assumed that the two crack arms have different heights and are loaded by the moments of same senses and magnitudes. Therefore, at any stage of non-linear analysis the stresses and strains in crack arms and un-cracked beam portion are distributed in different manners.

Non-linear analysis is easily performed using J-integral. Expressions corresponding to the

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