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26 KiB
calculation is suitable only for monotonic loading of elastic-plastic materials.
Figure 2.16.1-1 Contour for evaluation of the J-integral.
text_image
x₂ x₁ q Γ n
Following Shih et al. (1986), we rewrite Equation 2.16.1-1 in the form
Equation 2.16.1-2
J = - \oint_ {C + C _ {+} + \Gamma + C _ {-}} \mathbf {m} \cdot \mathbf {H} \cdot \bar {\mathbf {q}} d \Gamma - \int_ {C _ {+} + C _ {-}} \mathbf {t} \cdot \frac {\partial \mathbf {u}}{\partial \mathbf {x}} \cdot \bar {\mathbf {q}} d \Gamma ,
where \bar { \bf q } is a sufficiently smooth weighting function within the region enclosed by the closed contour C + C _ { + } + \Gamma + C . ¡ and has the value \bar { \bf q } = { \bf q } on ¡ and \bar { \mathbf q } = 0 on C ; and m is the outward normal to the domain enclosed by the closed contour, as shown in Figure 2.16.1-2. m = ¡n on \Gamma ; and \mathbf { t } = \mathbf { m } \cdot \pmb { \sigma } is the surface traction on the crack surfaces C _ { + } and C _ { - } .
Figure 2.16.1-2 Closed contour C + C _ { + } + \Gamma + C _ { - } encloses a domain A that includes the crack-tip region as \Gamma 0 :
text_image
C+ C- A m r q m n C
Using the divergence theorem, we convert the closed contour integral into the domain integral
Equation 2.16.1-3
Procedures
J = - \int_ {A} \left(\frac {\partial}{\partial \mathbf {x}}\right) \cdot \left(\mathbf {H} \cdot \bar {\mathbf {q}}\right) d \Gamma - \int_ {C _ {+} + C _ {-}} \mathbf {t} \cdot \frac {\partial \mathbf {u}}{\partial \mathbf {x}} \cdot \bar {\mathbf {q}} d \Gamma ,
where A is the domain enclosed by the closed contour C + C _ { + } + \Gamma + C _ { - } . It is worth noting that the domain A includes the crack-tip region as \Gamma 0 .
If equilibrium is satisfied and W is a function of the mechanical strain--i.e., W = W ( \pmb \varepsilon ^ { m } ) --we have
\left(\frac {\partial}{\partial \mathbf {x}}\right) \cdot \pmb {\sigma} + \mathbf {f} = 0 \quad \mathrm{and} \quad \frac {\partial W}{\partial \mathbf {x}} = \frac {\partial W}{\partial \pmb {\varepsilon} ^ {m}}: \frac {\partial \pmb {\varepsilon} ^ {m}}{\partial \mathbf {x}} = \pmb {\sigma}: \left(\frac {\partial \pmb {\varepsilon}}{\partial \mathbf {x}} - \frac {\partial \pmb {\varepsilon} ^ {t h}}{\partial \mathbf {x}}\right),
where f is the body force per unit volume and \pmb { \varepsilon } ^ { t h } is the thermal strain. Substituting the above two equations into Equation 2.16.1-3 gives
Equation 2.16.1-4
J = - \int_ {A} \left[ \mathbf {H}: \frac {\partial \bar {\mathbf {q}}}{\partial \mathbf {x}} + \left(\mathbf {f} \cdot \frac {\partial \mathbf {u}}{\partial \mathbf {x}} - \boldsymbol {\sigma}: \frac {\partial \boldsymbol {\varepsilon} ^ {t h}}{\partial \mathbf {x}}\right) \cdot \bar {\mathbf {q}} \right] d \Gamma - \int_ {C _ {+} + C _ {-}} \mathbf {t} \cdot \frac {\partial \mathbf {u}}{\partial \mathbf {x}} \cdot \bar {\mathbf {q}} d \Gamma .
To evaluate these integrals, ABAQUS defines the domain in terms of rings of elements surrounding the crack tip. Different "contours" (domains) are created. The first contour consists of those elements directly connected to crack-tip nodes. The next contour consists of the ring of elements that share nodes with the elements in the the first contour as well as the elements in the first contour. Each subsequent contour is defined by adding the next ring of elements that share nodes with the elements in the previous contour. q¹ is chosen to have a magnitude of zero at the nodes on the outside of the contour and to be one (in the crack direction) at all nodes inside the contour except for the midside nodes (if they exist) in the outer ring of elements. These midside nodes are assigned a value between zero and one according to the position of the node on the side of the element.
J-integral in three-dimensions
The J-integral can be extended to three dimensions by considering a crack with a tangentially continuous front, as shown in Figure 2.16.1-3. The local direction of crack extension is again given by q, which is perpendicular to the local crack front and lies in the crack plane. Asymptotically, as r 0 , , the conditions for path independence apply on any contour in the x _ { 1 } { - } x _ { 2 } plane, which is perpendicular to the crack front at s. Hence, the J-integral defined in this plane can be extended to represent the pointwise energy release rate along the crack front as
Equation 2.16.1-5
J (s) = \lim _ {\Gamma \rightarrow 0} \int_ {\Gamma} \mathbf {n} \cdot \mathbf {H} \cdot \mathbf {q} d \Gamma .
Figure 2.16.1-3 Definition of local orthogonal Cartesian coordinates at the point s on the crack front; the crack is in the x _ { 1 } { - } x _ { 3 } plane.
Procedures
text_image
x₂ (perpendicular to plane of crack) r θ x₁ (normal to crack front) x₃ (tangent to crack front) s crack front
For a virtual crack advance \lambda ( s ) in the plane of a three-dimensional crack, the energy release rate is given by
Equation 2.16.1-6
\bar {J} = \int_ {L} J (s) \lambda (s) d s = \lim _ {\Gamma \rightarrow 0} \int_ {A _ {t}} \lambda (s) \mathbf {n} \cdot \mathbf {H} \cdot \mathbf {q} d A,
where L denotes the crack front under consideration; dA is a surface element on a vanishingly small tubular surface enclosing the crack tip ( { \mathrm { i . e . , ~ } } d A = d s d \Gamma ) ; and n is the outward normal to d A . { \bar { J } } can be calculated by the domain integral method similar to that used in two dimensions. To do so, we first convert the surface integral in Equation 2.16.1-6 to a volume integral by introducing a contour surface A _ { o } , , outside surface A _ { t } , external surfaces A _ { e n d s } at the ends of the crack front (the surfaces A _ { e n d s } vanish for the crack whose front forms a closed loop), and the crack faces A _ { c r a c k s } , as shown in Figure 2.16.1-4. It can be seen that A = A _ { t } + A _ { o } + A _ { e n d s } + A _ { c r a c k s } encloses a volume V . A weighting function q¹ is defined such that it has a magnitude of zero on A _ { o } and \bar { \mathbf { q } } = \lambda ( s ) \mathbf { q } on A _ { t } . \bar { \mathbf { q } } is assumed to vary smoothly between these values within A. On the external surfaces A _ { e n d s } where \mathbf { q } is not tangential to the surfaces, it must be made so. This can be done in ABAQUS by using the *NORMAL option to define the surface normals. Then, we can rewrite Equation 2.16.1-6 as
Equation 2.16.1-7
\bar {J} = - \oint_ {A} \mathbf {m} \cdot \mathbf {H} \cdot \bar {\mathbf {q}} d A - \int_ {A _ {e n d s} + A _ {c r a c k s}} \mathbf {t} \cdot \frac {\partial \mathbf {u}}{\partial \mathbf {x}} \cdot \bar {\mathbf {q}} d A,
where m is the outward normal to A ( { \mathrm { a n d } } { \mathbf { m } } = - { \mathbf { n } } { \mathrm { o n } } A _ { t } ) . { \mathbf { t } } = { \mathbf { m } } \cdot { \pmb { \sigma } } is the surface traction on surfaces A _ { e n d s } and the crack surfaces A _ { c r a c k s } .
Figure 2.16.1-4 Surface A = A _ { t } + A _ { o } + A _ { e n d s } + A _ { c r a c k s } encloses a domain volume V that includes the crack-front region as \Gamma 0 :
text_image
Diagram illustrating a mechanical or fluidic system with labeled components including A ends, A_cracks, A_o, and A_i, and directional arrows.
Using the divergence theorem, we obtain
Equation 2.16.1-8
\bar {J} = - \int_ {V} \left[ \mathbf {H}: \frac {\partial \bar {\mathbf {q}}}{\partial \mathbf {x}} + \left(\mathbf {f} \cdot \frac {\partial \mathbf {u}}{\partial \mathbf {x}} - \pmb {\sigma}: \frac {\partial \pmb {\varepsilon} ^ {t h}}{\partial \mathbf {x}}\right) \cdot \bar {\mathbf {q}} \right] d V - \int_ {A _ {e n d s} + A _ {c r a c k s}} \mathbf {t} \cdot \frac {\partial \mathbf {u}}{\partial \mathbf {x}} \cdot \bar {\mathbf {q}} d A.
To obtain J ( s ) at each node set P along the crack front line, \lambda ( s ) is discretized with the same interpolation functions as those used in the finite elements along the crack front:
\lambda (s) = N ^ {Q} (s) \lambda^ {Q},
where \lambda ^ { Q } = 1 at the node set P and all other \lambda ^ { Q } are zero. This expression for \lambda ( s ) is substituted into Equation 2.16.1-8. Finally, the J-integral value at each node set P along the crack front can be calculated as
Equation 2.16.1-9
J ^ {P} = \bar {J} ^ {P} / \int_ {L} N ^ {P} d s.
2.16.2 Stress intensity factor extraction
The stress intensity factors K _ { I } , K _ { I I } , and K _ { I I I } play an important role in linear elastic fracture mechanics. They characterize the influence of load or deformation on the magnitude of the crack-tip stress and strain fields and measure the propensity for crack propagation or the crack driving forces. Furthermore, the stress intensity can be related to the energy release rate (the J -integral) for a linear elastic material through
J = \frac {1}{8 \pi} \mathbf {K} ^ {T} \cdot \mathbf {B} ^ {- 1} \cdot \mathbf {K},
where \mathbf { K } = \lfloor K _ { I } , K _ { I I } , K _ { I I I } \rfloor ^ { T } and B is called the pre-logarithmic energy factor matrix (Shih and
Procedures
Asaro, 1988; Barnett and Asaro, 1972; Gao, Abbudi, and Barnett, 1991; Suo, 1990). For homogeneous, isotropic materials B is diagonal and the above equation simplifies to
J = \frac {1}{\bar {E}} (K _ {I} ^ {2} + K _ {I I} ^ {2}) + \frac {1}{2 G} K _ {I I I} ^ {2},
where \bar { E } = E for plane stress and \bar { E } = E / ( 1 - \nu ^ { 2 } ) for plane strain, axisymmetry, and three dimensions. For an interfacial crack between two dissimilar isotropic materials with Young's moduli E _ { 1 } and E _ { 2 } , Poisson's ratios \nu _ { 1 } and \nu _ { 2 } , and shear moduli G _ { 1 } = E _ { 1 } / 2 ( 1 + \nu _ { 1 } ) and G _ { 2 } = E _ { 2 } / 2 ( 1 + \nu _ { 2 } ) ,
J = \frac {1 - \beta^ {2}}{E ^ {*}} (K _ {I} ^ {2} + K _ {I I} ^ {2}) + \frac {1}{2 G ^ {*}} K _ {I I I} ^ {2},
where
\frac {1}{E ^ {*}} = \frac {1}{2} \Big (\frac {1}{\bar {E} _ {1}} + \frac {1}{\bar {E} _ {2}} \Big), \qquad \frac {1}{G ^ {*}} = \frac {1}{2} \Big (\frac {1}{G _ {1}} + \frac {1}{G _ {2}} \Big)
\beta = \frac {G _ {1} (\kappa_ {2} - 1) - G _ {2} (\kappa_ {1} - 1)}{G _ {1} (\kappa_ {2} + 1) + G _ {2} (\kappa_ {1} + 1)},
and \kappa = 3 - 4 \nu for plane strain, axisymmetry, and three dimensions; and \kappa = ( 3 - \nu ) / ( 1 + \nu ) for plane stress. Unlike their analogues in a homogeneous material, K _ { I } and K _ { I I } are no longer the pure Mode I and Mode II stress intensity factors for an interfacial crack. They are simply the real and imaginary parts of a complex stress intensity factor, whose physical meaning can be understood from the interface traction expressions:
(\sigma_ {2 2} + i \sigma_ {1 2}) _ {\theta = 0} = \frac {(K _ {I} + i K _ {I I}) r ^ {i \varepsilon}}{\sqrt {2 \pi r}}, (\sigma_ {2 3}) _ {\theta = 0} = \frac {K _ {I I I}}{\sqrt {2 \pi r}},
where r and µ are polar coordinates centered at the crack tip. The bimaterial constant " is defined as
\varepsilon = \frac {1}{2 \pi} \ln \frac {1 - \beta}{1 + \beta}.
In this section we describe an interaction integral method ( Shih and Asaro, 1988) to extract the individual stress intensity factors for a crack under mixed-mode loading. The method is applicable to cracks in isotropic and anisotropic linear materials.
Interaction integral method
In general, the J-integral for a given problem can be written as
Procedures
\begin{array}{l} J = \frac {1}{8 \pi} [ K _ {I} B _ {1 1} ^ {- 1} K _ {I} + 2 K _ {I} B _ {1 2} ^ {- 1} K _ {I I} + 2 K _ {I} B _ {1 3} ^ {- 1} K _ {I I I} \\ + (\text { terms not involving } K _ {I}) ]. \\ \end{array}
where I; II; III correspond to 1; 2; 3 when indicating the components of B. We define the J-integral for an auxiliary, pure Mode I, crack-tip field with stress intensity factor k _ { I } , as
J _ {a u x} ^ {I} = \frac {1}{8 \pi} k _ {I} \cdot B _ {1 1} ^ {- 1} \cdot k _ {I}.
Superimposing the auxiliary field onto the actual field yields
\begin{array}{l} J _ {t o t} ^ {I} = \frac {1}{8 \pi} [ (K _ {I} + k _ {I}) B _ {1 1} ^ {- 1} (K _ {I} + k _ {I}) + 2 (K _ {I} + k _ {I}) B _ {1 2} ^ {- 1} K _ {I I} + 2 (K _ {I} + k _ {I}) B _ {1 3} ^ {- 1} K _ {I I I} \\ + (\text { terms not involving } K _ {I} \text { or } k _ {I}) ]. \\ \end{array}
Since the terms not involving K _ { I } or k _ { I } in J _ { t o t } ^ { I } and J are equal, the interaction integral can be defined as
J _ {i n t} ^ {I} = J _ {t o t} ^ {I} - J - J _ {a u x} ^ {I} = \frac {k _ {I}}{4 \pi} (B _ {1 1} ^ {- 1} K _ {I} + B _ {1 2} ^ {- 1} K _ {I I} + B _ {1 3} ^ {- 1} K _ {I I I}).
If the calculations are repeated for Mode II and Mode III, a linear system of equations results:
J _ {i n t} ^ {\alpha} = \frac {k _ {\alpha}}{4 \pi} B _ {\alpha \beta} ^ {- 1} K _ {\beta}, \quad (\mathrm{nosumon} \alpha = I, I I, I I I),
If the k _ { \alpha } are assigned unit values, the solution of the above equations leads to
\mathbf {K} = 4 \pi \mathbf {B} \cdot \mathbf {J} _ {i n t},
where \mathbf { J } _ { i n t } = \lfloor J _ { i n t } ^ { I } , J _ { i n t } ^ { I I } , J _ { i n t } ^ { I I I } \rfloor ^ { T } . The calculation of this integral is discussed next.
Based on the definition of the J -integral, the interaction integrals J _ { i n t } ^ { \alpha } can be expressed as
J _ {i n t} ^ {\alpha} = \lim _ {\Gamma \to 0} \int_ {\Gamma} \mathbf {n} \cdot \mathbf {M} ^ {\alpha} \cdot \mathbf {q} d \Gamma
with M® given as
\mathbf {M} ^ {\alpha} = \pmb {\sigma}: \pmb {\varepsilon} _ {a u x} ^ {\alpha} \mathbf {I} - \pmb {\sigma} \cdot \left(\frac {\partial \mathbf {u}}{\partial \mathbf {x}}\right) _ {a u x} ^ {\alpha} - \pmb {\sigma} _ {a u x} ^ {\alpha} \cdot \frac {\partial \mathbf {u}}{\partial \mathbf {x}}.
The subscript aux represents three auxiliary pure Mode I, Mode II, and Mode III crack-tip fields for ® = I; II; III, respectively. ¡ is a contour that lies in the normal plane at position s along the crack
front, beginning on the bottom crack surface and ending on the top surface (see Figure 2.16.2-1). The limit \Gamma 0 indicates that ¡ shrinks onto the crack tip.
Figure 2.16.2-1 Definition of local orthogonal Cartesian coordinates at the point s on the crack front; the crack is in the x _ { 1 } { - } x _ { 3 } plane.
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x₂ (perpendicular to plane of crack) r θ x₁ (normal to crack front) x₃ (tangent to crack front) s crack front
Following the domain integral procedure used in ABAQUS/Standard for calculating the J-integral, we define an interaction integral for a virtual crack advance \lambda ( s ) :
\bar {J} _ {i n t} ^ {\alpha} = \int_ {L} J _ {i n t} ^ {\alpha} (s) \lambda (s) d s = \int_ {A} \lambda (s) \mathbf {n} \cdot \mathbf {M} ^ {\alpha} \cdot \mathbf {q} d A,
where L denotes the crack front under consideration; dA is a surface element on a vanishingly small tubular surface enclosing the crack tip ( \mathrm { i } . \mathrm { e } . , d A = d s d \Gamma ) ; n is the outward normal to d A ; and q is the local direction of virtual crack propagation. The integral \bar { J } _ { i n t } ^ { \alpha } can be calculated by the same domain integral method as that used for calculating the J -integral.
To obtain J _ { i n t } ^ { \alpha } at each node set P along the crack front line, ¸ is discretized with the same interpolation functions as those used in the finite elements along the crack front:
\lambda (s) = N ^ {Q} (s) \lambda^ {Q},
where \lambda ^ { Q } = 1 at the node set P and all other \lambda ^ { Q } are zero. The result is substituted into the expression for \bar { J } _ { i n t } ^ { \alpha } . Finally, the interaction integral value at each node set P along the crack front can be calculated as
J _ {i n t} ^ {\alpha P} = \bar {J} _ {i n t} ^ {\alpha P} / \int_ {L} N ^ {P} d s.
2.16.3 T -stress extraction
The asymptotic expansion of the stress field near a sharp crack in a linear elastic body with respect to
r, the distance from the crack tip, is
\sigma_ {i j} = \frac {K _ {I}}{\sqrt {2 \pi r}} f _ {i j} ^ {I} (\theta) + \frac {K _ {I I}}{\sqrt {2 \pi r}} f _ {i j} ^ {I I} (\theta) + \frac {K _ {I I I}}{\sqrt {2 \pi r}} f _ {i j} ^ {I I I} (\theta) + T \delta_ {1 i} \delta_ {1 j} + (\nu T + \varepsilon_ {3 3}) \delta_ {3 i} \delta_ {3 j} + O (r ^ {1 / 2})
(Williams, 1957), where r and µ are the in-plane polar coordinates centered at the crack tip. The local axes are defined so that the 1-axis lies in the plane of the crack at the point of interest on the crack front and is perpendicular to the crack front at this point; the 2-axis is normal to the plane of the crack (and thus is perpendicular to the crack front); and the 3-axis lies tangential to the crack front. \varepsilon _ { 3 3 } is the extensional strain along the crack front. In plane strain \varepsilon _ { 3 3 } = 0 ; in plane stress the term ( \nu T + \varepsilon _ { 3 3 } ) \delta _ { 3 i } \delta _ { 3 j } vanishes.
The T -stress represents a stress parallel to the crack faces. It is a useful quantity, not only in linear elastic crack analysis but also in elastic-plastic fracture studies.
The T -stress usually arises in the discussions of crack stability and kinking for linear elastic materials. For small amounts of crack growth under Mode I loading, a straight crack path has been shown to be stable when T < 0 , whereas the path will be unstable and, therefore, will deviate from being straight when T > 0 (Cotterell and Rice, 1980). A similar trend has been found in three-dimensional crack propagation studies by \mathbf { X } \mathbf { u } , Bower, and Ortiz (1994). Hutchinson and Suo (1992) also showed how the advancing crack path is influenced by the T -stress once cracking initiates under mixed-mode loading. (The direction of crack initiation can be otherwise predicted using the criteria discussed in ``Prediction of the direction of crack propagation,'' Section 2.16.4.)
The T -stress also plays an important role in elastic-plastic fracture analysis, even though the T -stress is calculated from the linear elastic material properties of the same solid containing the crack. The early study of Larsson and Carlsson (1973) demonstrated that the T -stress can have a significant effect on the plastic zone size and shape and that the small plastic zones in actual specimens can be predicted adequately by including the T -stress as a second crack-tip parameter. Some recent investigations (Bilby et al., 1986; Al-Ani and Hancock, 1991; Betegón and Hancock, 1991; Du and Hancock, 1991; Parks, 1992; and Wang, 1991) further indicate that the T -stress can correlate well with the tensile stress triaxiality of elastic-plastic crack-tip fields. The important feature observed in these works is that a negative T -stress can reduce the magnitude of the tensile stress triaxiality (also called the hydrostatic tensile stress) ahead of a crack tip; the more negative the T -stress becomes, the greater the reduction of tensile stress triaxiality. In contrast, a positive T -stress results only in modest elevation of the stress triaxiality. It was found that when the tensile stress triaxiality is high, which is indicated by a positive T -stress, the crack-tip field can be described adequately by the HRR solution (Hutchinson, 1968; Rice and Rosengren, 1968), scaled by a single parameter: the J-integral; that is, J-dominance will exist. When the tensile stress triaxiality is reduced (indicated by the T -stress becoming more negative), the crack-tip fields will quickly deviate from the HRR solution, and J -dominance will be lost (the asymptotic fields around the crack tip cannot be well characterized by the HRR fields). Thus, using the T -stress (calculated based on the load level and linear elastic material properties) to characterize the triaxiality of the crack-tip stress state and using the J -integral (calculated based on the actual elastic-plastic deformation field) to measure the scale of the crack-tip deformation provides a two-parameter fracture mechanics theory to describe the Mode I elastic-plastic crack-tip stresses and
Procedures
deformation in plane strain or three dimensions accurately over a wide range of crack configurations and loadings.
To extract the T -stress, we use an auxiliary solution of a line load, with magnitude f , applied in the plane of crack propagation and along the crack line:
\sigma_ {1 1} ^ {L} = \frac {f}{\pi r} \cos^ {3} \theta , \qquad \sigma_ {2 2} ^ {L} = \frac {f}{\pi r} \cos \theta \sin^ {2} \theta , \qquad \sigma_ {1 2} ^ {L} = \frac {f}{\pi r} \sin \theta \cos^ {2} \theta ,
\sigma_ {3 3} ^ {L} = \frac {f}{\pi r} \nu \cos \theta , \quad \sigma_ {1 3} ^ {L} = \sigma_ {2 3} ^ {L} = 0.
The term \sigma _ { 3 3 } ^ { L } = 0 for plane stress.
The interaction integral used is exactly the same as that for extracting the stress intensity factors:
I _ {i n t} = \lim _ {\Gamma \rightarrow 0} \int_ {\Gamma} \mathbf {n} \cdot \mathbf {M} \cdot \mathbf {q} d \Gamma ,
with M as
\mathbf {M} = \pmb {\sigma}: \pmb {\varepsilon} _ {a u x} ^ {L} \mathbf {I} - \pmb {\sigma} \cdot \left(\frac {\partial \mathbf {u}}{\partial \mathbf {x}}\right) _ {a u x} ^ {L} - \pmb {\sigma} _ {a u x} ^ {L} \cdot \frac {\partial \mathbf {u}}{\partial \mathbf {x}}.
In the limit as r 0 _ { : } , using the local asymptotic fields,
T = \bar {E} \Big [ - \frac {I _ {i n t} (s)}{f} - \nu \varepsilon_ {3 3} (s) \Big ],
where \bar { E } = E for plane stress and \bar { E } = E / ( 1 - \nu ^ { 2 } ) for plane strain, axisymmetry, and three dimensions. \varepsilon _ { 3 3 } is zero for plane strain and plane stress.
I _ { i n t } ( s ) can be calculated by means of the same domain integral method used for J-integral calculation and the stress intensity factor extraction, which has been described in ``J -integral evaluation,'' Section 2.16.1, and ``Stress intensity factor extraction,'' Section 2.16.2.
2.16.4 Prediction of the direction of crack propagation
Various criteria have been proposed to predict the angle at which a pre-existing crack will propagate. Among these criteria are the maximum tangential stress criterion (Erdogan and Sih, 1963), the maximum principal stress criterion (Maiti and Smith, 1983), the maximum energy release rate criterion (Palaniswamy and Knauss, 1978, and Hussain, Pu, and Underwood, 1974), the minimum elastic energy density criterion (Sih, 1973), and the T-criterion (Theocaris, 1982). These criteria predict slightly different angles for the initial crack propagation, but they all have the implication that K _ { I I } = 0 at the crack tip as the crack extends (Cotterell and Rice, 1980). In ABAQUS/Standard we provide three criteria for homogeneous, isotropic linear elastic materials: the maximum tangential stress criterion,
the maximum energy release rate criterion, and the K _ { I I } = 0 criterion. K _ { I I I } is not taken into account in what follows, since a generally accepted theory for crack propagation with K _ { I I I } \neq 0 remains to be developed.
Maximum tangential stress criterion
The near-crack-tip stress field for a homogeneous, isotropic linear elastic material is given by
\begin{array}{l} \sigma_ {\theta \theta} = \frac {1}{\sqrt {2 \pi r}} \cos \frac {1}{2} \theta (K _ {I} \cos^ {2} \frac {1}{2} \theta - \frac {3}{2} K _ {I I} \sin \theta), \\ \tau_ {r \theta} = \frac {1}{\sqrt {2 \pi r}} \cos \frac {1}{2} \theta [ K _ {I} \sin \theta + K _ {I I} (3 \cos \theta - 1) ] \\ \end{array}
where r and \theta are polar coordinates centered at the crack tip in a plane orthogonal to the crack front.
The direction of crack propagation can be obtained using either the condition \partial \sigma _ { \theta \theta } / \partial \theta = 0 or \tau _ { r \theta } = 0 ; i.e.,
\hat {\theta} = \cos^ {- 1} \left(\frac {3 K _ {I I} ^ {2} + \sqrt {K _ {I} ^ {4} + 8 K _ {I} ^ {2} K _ {I I} ^ {2}}}{K _ {I} ^ {2} + 9 K _ {I I} ^ {2}}\right),
where the crack propagation angle \hat { \theta } is measured with respect to the crack plane. \hat { \theta } = 0 represents the crack propagation in the "straight-ahead" direction. \hat { \theta } < 0 if K _ { I I } > 0 , while \hat { \theta } > 0 if K _ { I I } < 0 .
Maximum energy release rate criterion
Consider a crack segment of length a kinking out the plane of the crack at an angle { \widehat { \theta } } , as shown in Figure 2.16.4-1. When a is infinitesimally small compared with all other geometric lengths (including the length of the parent crack), the stress intensity factors K _ { I } ^ { k } and K _ { I I } ^ { k } at the tip of the putative crack can be expressed as linear combinations of K _ { I } and K _ { I I } , the stress intensity factors existing prior to kinking for the parent crack:
K _ {I} ^ {k} = c _ {1 1} K _ {I} + c _ {1 2} K _ {I I},
K _ {I I} ^ {k} = c _ {2 1} K _ {I} + c _ {2 2} K _ {I I}.
The { \hat { \theta } } . -dependences of the coefficients c _ { i j } are given by Hayashi and Nemat-Nasser (1981) and by He and Hutchinson (1989).
Figure 2.16.4-1 Contour for evaluation of the J-integral.