ISSN 0021-8944, Journal of Applied Mechanics and Technical Physics, 2018, Vol. 59, No. 1, pp. 146–152.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
DELAMINATION IN A TWO-DIMENSIONAL
FUNCTIONALLY GRADED BEAM
V. I. Rizov
Abstract: An analytical study of delamination in the crack lap shear beam is performed. It is
assumed that the material is functionally graded along the width and height of the beam. Delami-
nation is studied in terms of the total strain energy release rate by applying methods of linear-elastic
fracture mechanics. An additional analysis of the total strain energy release rate is performed by
considering the strain energies in the beam cross sections ahead of and behind the crack front for
veriﬁcation. The eﬀects of the crack location and material gradient on delamination are evaluated.
Keywords: functionally graded beam, linear-elastic fracture mechanics, analytical approach.
Functionally graded materials are novel inhomogeneous materials whose properties vary continuously along
one or more spatial coordinates . In this kind of materials, interfacial stress concentrations are avoided in
contrast to laminated composites . The fracture behaviour of functionally graded materials has received signiﬁcant
attention from the scientiﬁc community around the world [3, 4]. Although fracture has been extensively studied,
there are some issues that have received relatively less attention. One of these issues is delamination of two-
dimensional functionally graded beams. Thus, the main purpose of the present paper is to perform an analytical
study of delamination in the crack lap shear (CLS) beam that is functionally graded along the width, as well as
along the height of the beam.
1. FORMULATION OF THE PROBLEM
The present paper deals with a delamination fracture analysis of the functionally graded CLS beam con-
ﬁguration shown schematically in Fig. 1. A vertical notch of depth h
is cut in the beam mid-span in order to
generate conditions for delamination fracture. There is a delamination crack located symmetrically with respect
to the beam mid-span (it should be noted that the present study was motivated also by the fact that functionally
graded materials can be built up layer by layer , which is a premise for appearance of delamination cracks between
the layers). The delamination crack is located at a distance h
from the bottom face of the beam (upper crack
arm) and at a distance h
from the top face of the beam (lower crack arm). The beam cross section is a rectangle
of width b and height 2h. The beam length is 2l. The external loading consists of two longitudinal forces F applied
at the centres of the free ends of the beam (see Fig. 1). It is assumed that the beam deforms linear-elastically,
i.e., the material behaviour can be described by Hooke’s law. It is also assumed that the material is functionally
Department of Technical Mechanics, University of Architecture, Civil Engineering and Geodesy, 1046 Soﬁa,
rizov email@example.com. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 59, No. 1,
pp. 171–177, January–February, 2018. Original article submitted December 5, 2016; revision submitted January
2018 by Pleiades Publishing, Ltd.