 Problems for Chapter 9

Modeling Material Failure

9.3.  Modeling Failure by Crack Growth - Linear Elastic Fracture Mechanics

9.3.1.        Using the equations for the crack tip fields find an expression for the maximum shear stress distribution around the tip of a plane-strain Mode I crack.  Hence, plot approximate contours of successive yield zones

9.3.2.        Briefly describe the way in which the concept of stress intensity factor can be used as a fracture criterion. 9.3.3.        A welded plate with fracture toughness  contains a residual stress distribution  along the line of the weld. A crack with length 2a lies on the weld line.  The solid is subjected to a uniaxial tensile stress  .  Find an expression for the critical value of  that will cause the weld to fracture, in terms of  ,  and a.

9.3.4.        Hard, polycrystalline materials such as ceramics often contain a distribution of inter-granular residual stress.  The objective of this problem is to estimate the influence of this stress distribution on crack propagation through the material.   Assume that

·         The solid has mode I fracture toughness  ·         As a rough estimate, the residual stress distribution can be idealized as  , where L is of the order of the grain size of the solid and  is the magnitude of the stress.

·         A long (semi-infinite) crack propagates through the solid at some time t,the crack tip is located at  ·         The solid is subjected to a remote stress, which induces a mode I stress intensity factor  at the crack tip

9.3.4.1.              If the solid is free of residual stress, what value of  that causes fracture.

9.3.4.2.              Calculate the stress intensity factor induced by the residual stress distribution, as a function of c.

9.3.4.3.              What value of  is necessary to cause crack propagation through the residual stress field?  What is the maximum value of  , and for what crack tip position c does it occur? 9.3.5.        A dislocation, with burgers vector  and line direction  lies a distance d ahead of a semi-infinite crack.  Calculate the crack tip stress intensity factors. 9.3.6.        The figure shows a simple model that is used to estimate the size of the plastic zone at a crack tip.  The crack, with length 2a, together with the plastic zones with length  , are considered together to be a crack with length  .   The solid is loaded by uniform stress  at infinity.  The region with length  near each crack tip is subjected to traction  acting to close the crack.  Using the solutions in Section 9.3.3, calculate an expression for the Mode I crack tip stress intensity factor.   Show that  if   9.3.7.        Suppose that an ASTM compact tension specimen is used to measure the fracture toughness of a steel.  The specimen has dimensions  and  mm.  The crack length was  mm, and the fracture load was 15kN.

9.3.7.1.              Calculate the fracture toughness of the steel.

9.3.7.2.              If the steel has yield stress 800MPa, was this a valid measurement? 9.3.8.        Find expressions for the Mode I and II stress intensity factors for the angled crack shown.  If  , what is the initial direction of crack propagation?  Confirm your prediction experimentally, using a center-cracked specimen of paper. 9.3.9.        A large solid contains a crack with initial length  .  The solid has plane-strain fracture toughness  ,  and under cyclic loading the crack growth rate obeys Paris law  9.3.9.1.              Suppose that the material is subjected to a cyclic uniaxial stress with amplitude  and mean stress  (so the stress varies between 0 and  2  ).  Calculate the critical crack length that will cause fracture, in terms of  and  9.3.9.2.              Calculate an expression for the number of cycles of loading that are necessary to cause a crack to grow from an initial length  to fracture under the loading described in 7.1

9.3.9.3.              Show that the number of cycles to failure can be expressed in the form of Basquin’s law (discussed in Section 9.2.7) as  , where b and D are constants.  Give expressions for b and D in terms of the initial crack length, the fracture toughness, and the material properties in Paris law

(c) A.F. Bower, 2008
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