Problems for Chapter 9

Modeling Material Failure

9.2.  Stress and Strain Based Fracture and Fatigue Criteria

9.2.1.        A flat specimen of glass with fracture strength , Youngâ€™s modulus E and Poissonâ€™s ratio Â is indented by a hard metal sphere with radius R, Youngâ€™s modulus Â and Poissonâ€™s ratio .Â  Using solutions for contact stress fields given in Chapter 5, calculate a formula for the load P that will cause the glass to fracture, in terms of geometric and material parameters.Â  You can assume that the critical stress occurs on the surface of the glass.

9.2.2.        The figure shows a fiber reinforced composite laminate.Â

(i) When loaded in uniaxial tension parallel to the fibers, it fails at a stress of 500MPa.

(ii) When loaded in uniaxial tension transverse to the fibers, it fails at a stress of 250 MPa.Â

(iii) When loaded at 45 degrees to the fibers, it fails at a stress of 223.6 MPa

The laminate is then loaded in uniaxial tension at 30 degrees to the fibers.Â  Calculate the expected failure stress under this loading, assuming that the material can be characterized using the Tsai-Hill failure criterion.

9.2.3.        A number of cylindrical specimens of a brittle material with a 1cm radius and length 4cm are tested in uniaxial tension.Â Â  It is found 60% of the specimens withstand a 150MPa stress without failure; while 30% withstand a 170 MPa stress without failure.

9.2.3.1.              Calculate values for the Weibull parameters Â and m for the specimens

9.2.3.2.              Suppose that a second set of specimens is made from the same material, with length 8cm and radius 1cm.Â Â  Calculate the stress level thatÂ  will cause 50% of these specimens to fail.

9.2.4.        AÂ  beam with length L, and rectangular cross-section Â is made from a brittle material with Youngâ€™s modulus E, Poissonâ€™s ratio , and the failure probability distribution of a volume Â is characterized by Weibull parameters Â and .

9.2.4.1.              Â Suppose that the beam is loaded in uniaxial tension parallel to its length.Â  Calculate the stress level Â corresponding to 63% failure, in terms of geometric and material parameters.

9.2.4.2.              Suppose that the beam is loaded in 3 point bending.Â  Let Â denote the maximum value of stress in the beam (predicted by beam theory).Â  Find an expression for the stress distribution in the beam in terms of

9.2.4.3.              Hence, find an expression for the value of Â that corresponds to 63% probability of failure in the beam.Â Â  Calculate the ratio .

9.2.5.        A glass shelf with length L and rectangular cross-section Â is used to display cakes in a bakery. As a result, it subjected to a daily cycle of load (which may be approximated as a uniform pressure acting on it surface) of the form Â where Â is the time the store has been open, and Â is the total time the store is open each day.Â  As received, the shelf has a tensile strength , and the glass can be characterized by static fatigue parameters Â and .Â Â  Find an expression for the life of the shelf, in terms of relevant parameters.

9.2.6.        A cylindrical concrete column with radius R, cross-sectional radius R, and length L is subjected to a monotonically increasing compressive axial load P.Â  Assume that the material can be idealized using the constitutive law given in Section 9.2.4, with the compressive yield stress-v-plastic strain of the form

where Â and m are material properties.Â  Assume small strains, and a homogeneous state of stress and strain in the column.Â  Neglect elastic deformation, for simplicity.

9.2.6.1.              Calculate the relationship between the axial stress Â and strain , in terms of the plastic properties c, Â and m

9.2.6.2.              Calculate the volume change of the column, in terms of , c, Â and m

9.2.6.3.              Suppose that the sides of the column are subjected to a uniform traction q.Â  Repeat the calculations in parts 9.2.6.1 and 9.2.6.2.

9.2.7.        Suppose that the column described in the previous problem is encased in a steel tube, with (small) wall thickness t.Â  The steel can be idealized as a rigid perfectly plastic material with yield stress .Â  Calculate the relationship between the axial stress Â and strain , in terms of geometric and material properties.Â

9.2.8.        Extend the viscoplastic finite element program described in Section 8.5 to model the behavior of a porous plastic material with constitutive equations given in Section 9.2.5.Â  This will involve the following steps:

9.2.8.1.              Develop a procedure to calculate the stress , the void volume fraction , the effective strain measures Â at the end of the time increment, given their values at the start of the increment and given an increment in plastic strain .Â  You should use a fully implicit update, as discussed in Section 8.5.Â Â  The simplest approach is to set up, and solve, three simultaneous nonlinear equations forÂ  ( Â ) using Newton-Raphson iteration, and subsequently compute the stress distribution.

9.2.8.2.              Calculate the tangent stiffness Â for the material

9.2.8.3.              Implement the new constitutive equations in the viscoplastic finite element program

9.2.8.4.              Test your code by simulating the behavior of a uniaxial tensile specimen subjected to monotonic loading.

9.2.9.        A specimen of steel has a yield stress of 500MPa.Â  Under cyclic loading at a stress amplitude of 200 MPa it is found to fail after Â cycles, while at a stress amplitude of 100MPa it fails after Â cycles.Â  This material is to be used to fabricate a plate, with thickness h, containing circular holes with radius a<<h.Â Â  The plate will be subjected to constant amplitude cyclic uniaxial stress far from the holes, and must have a life of at least Â cycles.Â Â  What is the maximum stress amplitude that the plate can withstand?

9.2.10.    A spherical pressure vessel with internal radius a and external radius b=1.5a is repeatedly pressurized from zero internal pressure to a maximum value .Â  The sphere has yield stress Y, and its fatigue behavior of the vessel (under fully reversed uniaxial tension) can be characterized by Basquinâ€™s law .

9.2.10.1.           Find an expression for the fatigue life of the vessel in terms ofÂ  , and relevant geometric and material properties.Â  Assume that the effects of mean stress can be approximated using Goodmanâ€™s rule.Â  Assume that

9.2.10.2.           Suppose that the vessel is first pressurized to its collapse load and then unloaded, so as to induce a distribution of residual stress in the cylinder.Â Â  It is subsequently subjected to a cyclic pressureÂ  with magnitude .Â  Calculate the mean stress and stress amplitude as a function of position in the vessel wall, and hence deduce an expression for its fatigue life.

9.2.11.    The figure shows a solder joint on a printed circuit board.Â  The printed circuit board can be idealized as a pinned-pinned beam with thickness h, length L, Youngâ€™s modulus E and mass density .Â  The board vibrates in its fundamental mode with a frequency Â and mode shape .Â  The yield stress of solder is so low it can be neglected, and it is firmly bonded to the printed circuit board.Â  As a result, it is subjected to a cyclic plastic strain equal to the strain at the surface of the beam.Â  The fatigue life of solder can be characterized by a Coffin-Manson law .Â Â  Find an expression for the time to failure of the solder joint, in terms of relevant geometric and material parameters.

9.2.12.    A specimen of steel is tested under cyclic loading.Â  It is found to have a fatigue threshold , and fails after Â cycles when tested at a stress amplitude .Â Â Â  Suppose that, in service, the material spends 80% of its life subjected to stress amplitudes , 10% of its life at , and the remainder at .Â Â  Calculate the life of the component during service (assume thatÂ  the mean stress Â during both testing and service).

(c) A.F. Bower, 2008
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