Two-dimensional problems in fracture mechanics

The name of problems in the mechanics of deformable solids dealing with the analysis of stresses and displacements in cylinders with tunnel cracks (with crack-like defects), in which the studied pattern is the same in all planes parallel to a cross-section of the body, i.e. when the conditions of plane strain and longitudinal shear are realized either in thin through-cracked plates under the in-plane load (generalized plane stress) or out-of-plane bending loading (thin plates bending). Limit equilibrium conditions of such bodies are also determined.

Two-dimensional problems in fracture mechanics are studied in terms of different theories (rheological models) of elasticity, thermoelasticity, viscoelasticity, plasticity, etc., depending on properties of the body with cracks under consideration (e.g., linear elastic or elastic-plastic materials, viscoelastic medium) and also on the type of external effects (loading, temperature or electromagnetic field). These problems are divided into dynamic, when the geometry of the body and the external loading are time-depended (e.g., propagating crack), and static, when there is no such dependence. Linear static two-dimensional problems of fracture mechanics are studied most widely.

Plane problems of elasticity theory for bodies with cracks.[edit]

Elastic equilibrium of solids is described in elasticity theory by the equations of the planar problem of plane strain of cylindrical bodies when external forces applied to the body are normal to its axis and are the same for all cross-sections, or in the case of a generalized plane stress (i.e. under thin plate deformation by the forces) acting in its plane (cf. also Elasticity theory, planar problem of). The solution is reduced to the determination of the three components $ \sigma _ {x} $, $ \sigma _ {y} $, $ \tau _ {xy} $ of the stress tensor and the two components $ u $, $ v $ of the displacement vector, if a Cartesian coordinate system is chosen so that the $ xy $- plane coincides either with the cross-section of the cylindrical body (plane strain) or with the plane of the middle plate (plane stress state). In the case of a homogeneous isotropic body, when body forces are absent, these values are expressed in terms of two analytic functions $ \Phi ( \zeta ) $, $ \Psi ( \zeta ) $( complex potentials) of the complex variable $ \zeta = x+ iy $, by the Kolosov–Muskhelishvili formulas:

$$ \sigma _ {x} + \sigma _ {y} = 2[ \Phi ( \zeta ) + \overline{ {\Phi ( \zeta ) }}\; ] , $$

$$ \sigma _ {y} - \sigma _ {x} + 2i \tau _ {xy} = 2 [ \overline{z}\; \Phi ^ \prime ( \zeta ) + \Psi ( \zeta ) ] , $$

$$ 2G ( u+ iv) = x \phi ( \zeta ) - \zeta \overline{ {\Phi ( \zeta ) }}\; - \overline{ {\psi ( \zeta ) }}\; , $$

$$ \Phi ( \zeta ) = \phi ^ \prime ( \zeta ) ,\ \Psi ( \zeta ) = \psi ^ \prime ( \zeta ) ; $$

here $ G $ is the shear modulus, $ x = 3- 4 \mu $ for plane strain and $ x = ( 3- \mu )/( 1+ \mu ) $ for generalized plane stress, and $ \mu $ is the Poisson ratio.

In the case of an elastic plane with a crack along a smooth curvilinear contour $ L $, when an arbitrary load is applied to the crack faces

$$ \tag{a1 } N ^ \pm + i T ^ \pm = \ p( t) \pm q( t),\ \ t \in L , $$

and stresses at infinity are absent, the complex potentials may be represented in the form

$$ \tag{a2 } \left . \begin{array}{c} \Phi ( \zeta ) = \frac{1}{2 \pi } \int\limits _ { L } \frac{Q( t) dt }{t - \zeta } , \\ Q( t) = g ^ \prime ( t) - \frac{2iq( t) }{1+ x } , \\ \Psi ( \zeta ) = \frac{1}{2 \pi } \int\limits _ { L } \left [ \frac{Q( \overline{t)}\; - 2i q( \overline{t)}\; }{t- \zeta } \overline{dt}\; - \frac{\overline{t}\; Q( t) }{( t- \zeta ) ^ {2} } dt \right ] , \\ \end{array} \right \} $$

where $ N $ and $ T $ are the normal and the tangential component of the external stress, "+" and "-" indicate limiting values of the respective quantities on the left and on the right from the contour $ L $; the unknown density $ g ^ \prime ( t) $ can be expressed in terms of the displacement jump across $ L $,

$$ g ^ \prime ( t) = \frac{2iG}{1+} x \frac{d}{dt} [ ( u+ iv) ^ {+} - ( u+ iv) ^ {-} ] ,\ \ t \in L . $$

By means of (a2) the boundary value problem (a1) is reduced to a singular integral equation relative to $ g ^ \prime ( t) $:

$$ \tag{a3 } \frac{1} \pi \int\limits _ { L } \left \{ \frac{Q( t)+ iq( t) }{t- t ^ \prime } dt + k _ {1} ( t, t ^ \prime ) [ Q( t) + 2iq( t) ] dt\right . + $$

$$ + \left . k _ {2} ( t, t ^ \prime ) Q( \overline{t)}\; \overline{dt}\; \right \} = p ( t ^ \prime ) ,\ t ^ \prime \in L , $$

where the kernels $ k _ {1} ( t , t ^ \prime ) $ and $ k _ {2} ( t, t ^ \prime ) $ are given by

$$ k _ {1} ( t, t ^ \prime ) = \frac{1}{2} \frac{d}{dt ^ \prime } \mathop{\rm ln} [ ( t- t ^ \prime )( \overline{t}\; - \overline{t}\; {} ^ \prime ) ] ; $$

$$ k _ {2} ( t, t ^ \prime ) = - \frac{1}{2} \frac{d}{dt ^ \prime } \left [ \frac{t- t ^ \prime }{\overline{t}\; - \overline{t}\; {} ^ \prime } \right ] . $$

The solution of (a3) in the class of functions which have integrable singularities at the end points of $ L $ exists and is unique under the additional condition

$$ \tag{a4 } \int\limits _ { L } g ^ \prime ( t) dt = 0 , $$

which provides single-valuedness of displacements when tracing $ L $.

The distribution of stress and displacement near the crack tip is characterized by stress-intensity factors $ K _ {I} $( in the case of symmetry) and $ K _ {II} $( in the case of asymmetry). They are connected with the functions $ g ^ \prime ( t) $ by

$$ K _ {I} ^ \pm - iK _ {II} ^ \pm = \mps \lim\limits _ {t- l ^ \pm } [ \sqrt {2 \pi | t- l ^ \pm | } g ^ \prime ( t) ] , $$

where values with subscript refer to the crack begin $ ( \zeta = l ^ {-} ) $ and values with superscript refer to the crack end $ ( \zeta = l ^ {+} ) $.

In the case of a system of $ N $ curvilinear crack contours $ L _ {n} $( $ n= 1 \dots N $) in an elastic plane, the boundary value problem (a1) can also be reduced to the integral equation (a2), where $ L $ is now the set of all contours $ L _ {n} $, but instead of one condition (a4) $ N $ analogous conditions providing single-valuedness of displacements when tracing each contour $ L _ {n} $ must be satisfied.

If the cracks are situated in a bounded elastic region, then the integral representations of the complex potentials (a2) can be extended in different manners to the case of a multi-connected region with unknown densities on its closed boundary contours. Boundary value problems of elasticity theory for such regions can be reduced to systems of integral equations along open (cracks) and closed (holes and external boundary) contours.

Problems of the bending of thin plates with cracks solved on the basis of the classical Kirchhoff–Love theory are very close to planar problems in mathematical crack theory. These problems are also analyzed on the basis of various more exact theories of the bending of thin plates, which, naturally, leads to a significant complication of their solution.

Longitudinal shear of bodies with cracks.[edit]

Longitudinal shear (or anti-plane deformation) is a stress state in a cylindrical body caused by loads directed along the generatrices of the cylinder and constant along them. The solution can be reduced to the determination of the two components $ \tau _ {xz} $, $ \tau _ {yz} $ of the stress tensor and (one component of) the displacement vector $ w $ in case the deformation axis is directed along the $ z $- axis of a Cartesian coordinate system $ ( x, y, z) $. For a homogeneous isotropic body, when body forces are absent these values can be expressed in terms of an arbitrary analytic function $ F ( \zeta ) $ of the complex variable $ \zeta $ by

$$ 2G w = f ( \zeta ) + \overline{ {f ( \zeta ) }}\; , $$

$$ \tau _ {xz} - i \tau _ {yz} = f ^ { \prime } ( \zeta ) = F( \zeta ) . $$

In the case of an elastic space with tunnel crack along a curvilinear contour $ L $( in a planar cross-section), when an arbitrary shearing load is applied to crack faces

$$ \tag{a5 } \tau _ {nz} ^ \pm = \tau ( t) \pm \mu ( t) ,\ \ t \in L , $$

and stresses at infinity are absent, the function $ F( \zeta ) $ has the form

$$ \tag{a6 } F( \zeta ) = \frac{2}{\pi i } \int\limits _ { L } \frac{H( t) dt }{t- \zeta } ,\ \ H( t) = \gamma ^ \prime ( t) + i \mu ( t) \frac{ds}{dt} , $$

where $ s $ is the arc parameter on $ L $ and the unknown density $ \gamma ^ \prime ( t) $ can be expressed in terms of the displacement jump across $ L $:

$$ \gamma ^ \prime ( t) = ( G / 2) \frac{d}{dt} ( w ^ \pm w ^ {-} ) ,\ \ t \in L . $$

By means of (a6) the boundary value problem (a5) can be reduced to a singular integral equation relative to the function $ \gamma ^ \prime ( t) $:

$$ \mathop{\rm Im} \left [ \frac{1}{\pi i } \frac{dt ^ \prime }{ds ^ \prime } \int\limits _ { L } \frac{H( t) dt }{t - t ^ \prime } \right ] = \tau ( t ^ \prime ) ,\ \ t ^ \prime \in L . $$

The solution of this equation for the class of functions having integrable singularities at the end points of $ L $ exists and is unique if the following additional condition holds:

$$ \int\limits _ { L } \gamma ^ \prime ( t) dt = 0 . $$

This condition provides the single-valuedness of displacements in tracing $ L $.

The distribution of the stresses and displacements near the longitudinal shear crack tip is characterized by a stress-intensity factor $ K _ {III} $, connected with the function $ \gamma ^ \prime ( t) $ by

$$ K _ {III} ^ \pm = \ \mps \lim\limits _ {t \rightarrow l ^ \pm } \left [ \sqrt {2 \pi | t- l ^ \pm | } \gamma ^ \prime ( t) \frac{dt}{ds} \right ] . $$

The stress-intensity factors $ K _ {I} $, $ K _ {II} $, $ K _ {III} $ are the main parameters in linear fracture mechanics. The distribution of stresses and displacements near an arbitrary crack edge in a three-dimensional body may be described by means of these factors. They are often used in various fracture criteria.

Solution methods for the boundary value problem.[edit]

The representation of displacements and stresses in an elastic body by analytic functions is used to obtain explicit solutions for particular problems. The solutions in case of a crack system situated along one and the same straight line or along one and the same circle are obtained by way of reduction to the linear conjugation problem (cf. Riemann–Hilbert problem (analytic functions)). Analytic functions are also used when solving the problem by the method of conformal mapping, which is especially effective in the case of simply-connected regions with edge cracks and also for infinite angles with a sharp opening (crack-type defect).

Some crack problems can be formulated as mixed problems in elasticity theory. Application of integral transformations to these problems gives an opportunity to reduce them to dual integral equations. When using Fourier series in these approaches, one obtains dual equations for series.

The method of singular integral equations is widely used. Exact and approximate solutions for a number of problems have been obtained in terms of this method. In some cases the integral equations can be effectively solved numerically. Singularities of the solution of the problem for crack tips are naturally taken into account when choosing the class of functions in which the solution of the integral equation is sought. Numerical solutions of linear problems have also been found by the methods of edge collocations, body forces, boundary, or finite-elements.

Boundary and finite-elements methods are used for solving two-dimensional problems in fracture mechanics for elastic-plastic body. Analytic solutions are obtained in only some special case, when the plastic deformations are assumed to be localized along lines starting from the crack tip. Especially often the $ \delta _ {c} $- model is used, both in the analytic and the numerical solution, when plastic deformations are modelled by a jump in the normal displacement (in the symmetrical case) of the crack continuation.

For three-dimensional problems see Fracture, mathematical problems of.

References[edit]