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the plastically deformed section, an accurate evaluation of the conditions for the failure is
only possible if the both contributions, elastic and plastic, are considered separately.
Therefore, significant errors can appear if plasticity effect is ignored, especially in the
case of ductile materials, typical and recommended for heavy loaded structures.
As strong engineering tool for the evaluation of crack significance, fracture mechanics
offers, if reasonably used, useful solutions with required reliability. To achieve applicable
and reliable results the adequate knowledge and skill are necessary. Inadequate results in
fracture mechanics applications are mainly the consequence of uncertain assumptions and
inaccurate data, and not of the fracture mechanics methods. Fracture mechanics will not
help without the necessary attention and information concerning loading, stress state,
crack data and material characteristics.
2. BASIC APPROACHES
2.1. First developments
Looking back to the history of fracture mechanics development the impression
dominates, that the elastic plastic treatment was from the beginning one of the main
directions. The first theoretical treatment of the crack effect has been performed by
Inglish already in 1913. He defined the stresses at crack tip as reciprocal to the tip radius
size, which therefore for infinite sharp cracks become also infinite. Therefore, every
element with the crack, regardless how small crack is and how small is its loading must
be broken. This could not be the satisfying solution. To solve this dilemma Griffith
(1920) approached to the solution in the other way, connecting the fracture with the
energy necessary to form crack. He assumed that the crack growth require some surface
energy taken from strain energy of the local stresses release when the crack grows. The
fracture occurs when the reduction in strain energy is sufficient to prevail material
resistance. This is essentially a restatement of the first law of thermodynamics.
If the plate in Fig. 1 is under state of constant tension (displacement
Δ
= const.),
prerequisite for crack growth is:
U dW
a
da
Δ
∂⎛ ⎞
−
≥
⎜ ⎟ ∂⎝ ⎠
(1)
where
U
is the
elastic energy
(per thickness unit) contained in the plate in an account of
its tension, and
W
is the work necessary for growth of a crack of initial length
a
. Based on
this solution stress at crack tip becomes final value and no more depend on tip radius only
(assuming that the crack is really sharp).
Irwin and Orowan improved this solution in 1948 considering plastic deformations in
the vicinity of the crack tip. Irwin defined the term on the left of side of expression (1) as
the energy release rate or crack growth force,
G
, necessary to extend the crack and the
term on the right as the material characteristics called fracture toughness,
R
. According to
the definition, the energy release rate (
G
) is the energy quantity per unit length along the
crack contour that is disclosed from the elastic energy of the body and the loading system
when the new crack surface is created. If the crack, under fixed grip conditions (constant
displacement), extends for an increment
da
, the stored energy decreases by
dU
. For this
incremental crack extension to occur,
dU
must be at least as large as
dW
(the work
required to fracture the material and create new crack surface).
The Griffith model, based on energy, with some modifications, is still applied today.