Residual and thermal stresses can influence both brittle fracture and elastic–plastic fracture. Within linear elastic fracture mechanics, the effect of internally self-equilibrating stresses on the crack driving force can be understood in much the same way as the effect of external loading. Under the superposition principle due to Bueckner [
1], the singular component of the stress field at a crack tip caused by the relaxation of stresses during introduction of the crack is identical to that caused by equivalent loading applied to the crack faces. In the case of elastic–plastic fracture however, plastic deformation of material surrounding the crack tip can cause the internally self-equilibrating component of the stress field to change prior to fracture initiation. Consequently, the effects of thermal or residual stresses combine with the action of externally-applied loads in a non-linear manner [
2].
This complicates the prediction of fracture initiation and crack growth. In structural integrity assessment procedures such as R6 Rev. 4 maintained by EDF Energy and others [
3,
4], and the British standard BS 7910:2013 [
5], combinations of residual and applied loading can be accounted for using an interaction factor (denoted
V) to adjust the apparent contribution of residual stress loading to the stress intensity factor at a given crack. During an assessment, this factor is applied to the stress intensity factor calculated for secondary (i.e. self-equilibrating) stresses before it is combined with the corresponding stress intensity factor for primary (i.e. externally-applied) loading. In general, secondary stresses have a greater influence on fracture at low levels of primary loading [
6]. At higher primary load levels, pre-existing residual stresses tend to be partly relaxed by plastic deformation prior to fracture and the formulation of
V within R6 reflects this.
More generally, energy-based criteria are often used to predict elastic–plastic fracture initiation and the presence of residual stress changes the strain energy release rate at a crack. When the
J contour integral is used as a fracture initiation criterion in the presence of thermal and residual stresses, it must be formulated to include terms which would otherwise be equal to zero. For the case of thermal stress [
7]:
$$ J = {\displaystyle \underset{\Gamma}{\int }}\left(W{\delta}_{1i}-{\sigma}_{ij}\frac{\partial {u}_j}{\partial {x}_1}\right){n}_ids + {\displaystyle \underset{A}{\int }}{\sigma}_{ii}\frac{\partial {\varepsilon}_{ij}^{th}}{\partial {x}_1}dA $$
(1)
where Γ is a closed contour surrounding the crack tip for which
n
i
is an outward-facing normal vector and
A is the internal area.
σ
ij
is the stress tensor,
W the strain energy density, and
ε
ij
th
the thermal strain tensor.
x
i
and
u
i
are the position and displacement vectors respectively, and
δ
ij
is the Kronecker delta. As is the case in the absence of thermal/residual stresses, the
J -integral is only equal to the strain energy release rate for ideal non-linear elastic materials; its application to real elastic–plastic materials via models based on incremental plasticity is approximate. Lei proposed a similar expression for
J in the presence of residual stress [
8], closely following a derivation due to Wilson and Yu for the case of thermal stress [
9]:
$$ J = {\displaystyle \underset{\Gamma}{\int }}\left(W{\delta}_{1i}-{\sigma}_{ij}\frac{\partial {u}_j}{\partial {x}_1}\right){n}_ids + {\displaystyle \underset{A}{\int }}\left({\sigma}_{ij}\frac{\partial {\varepsilon}_{ij}}{\partial {x}_1}-\frac{\partial W}{\partial {x}_1}\right)dA $$
(2)
where the total strain
ε
ij
is the sum of all initial, elastic and plastic strains. Using this expression,
J can be estimated so long as the residual stress, total strain, and strain energy density fields in the cracked body are known. To determine these it is normally necessary to simulate the process by which the residual stress field is formed, the introduction of a crack into the residual stress field, and any subsequent loading by externally-applied forces. Models of this nature typically require experimental validation; particularly for residual stresses formed via complex thermo-mechanical processes such as welding [
4,
10]. Alternatively, when residual stresses are measured from a physical specimen it is not normally possible to calculate the
J -integral explicitly. Firstly, due to the difficulty involved in residual stress field characterisation in metals any measured stress field data tends to be insufficiently complete for
J -integral calculation. Secondly, the residual stress field alone insufficient for calculation of the
J -integral; distributions of total strain and strain energy density, including energy dissipated as plastic work for real elastic–plastic materials, are also needed.
In this study we propose a combined experimental/analytical treatment of elastic–plastic fracture in the presence of residual stress. The residual stress and elastic strain fields which exist in a specimen before and after the introduction of a defect are reconstructed from point-wise data measured using neutron diffraction. Using this information along with modelling of the residual stress introduction and fracture loading of the specimen, elastic and elastic–plastic fracture parameters are evaluated explicitly. In this way, the effect of an experimentally-measured residual stress field on elastic–plastic fracture can be analysed without resorting to simplified methods such as the R6 V factor to account for the interaction between residual and applied loading.