Elasticity/Plate with hole in tension

Plate with hole in a tensile field

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An elastic plate with a circular hole under tension

The BCs are

 

Unperturbed Solution

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The unperturbed part of the Michell solution gives us

 

or,

 

The first term is the axisymmetric term while the second term is the periodic term.

Perturbation

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Similar to previous problem, but we simply choose terms from the Michell solution of the same form (i.e. containing  ) and such that the stresses decay with increasing radius. The relevant terms from the table are:

 

Perturbed Solution

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The perturbed solution is

 

After applying the BCS, we get

 

The stress concentration factor, often referred to as Kt, in this case is   and is the same in both tension and shear.

Example homework problem

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Consider the elastic plate with a hole subject to uniaxial tension.

 
Elastic plate with small circular hole under uniaxial tension
  • Show that the stress function
 

leads to the stresses

 

or, in cartesian coordinates:

 

  • Calculate the stress concentration factors at the hole, both in shear and in tension, and show that they are the same. How far from the hole (in units of hole diameters) does the stress reach 95% of the far field (unperturbed) value?
  • Calculate the displacement field corresponding to this stress field (for plane stress). Plot the deformed shape of the hole.

Solution

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We can use the following Maple code to show the above results.

phi := T*r^2/4*(1 - cos(2*theta)) + A*ln(r) + B*theta + C*cos(2*theta) +   
       D/r^2*cos(2*theta);

srr := 1/r*diff(phi,r) + 1/r^2*diff(phi,theta,theta);
stt := diff(phi,r,r);
srt := -diff((1/r*diff(phi,theta)),r);

srra := collect(simplify(eval(srr, r=a)),{cos});
srta := collect(simplify(eval(srt, r=a)),{cos});

eq1 := coeff(srra, cos(2*theta));
eq2 := coeff(srta, sin(2*theta));
eq3 := 1/2*(T*a^4+2*A*a^2)/a^4;
eq4 := 1/a^2*B;

BB  := solve({eq4=0},{B});
AA  := solve({eq3=0},{A});

sol := solve({eq1=0,eq2=0},{C,D});

phi := subs(BB, phi);
phi := subs(AA, phi);
phi := subs(sol, phi);

srr2 := 1/r*diff(phi,r) + 1/r^2*diff(phi,theta,theta);
stt2 := diff(phi,r,r);
srt2 := -diff((1/r*diff(phi,theta)),r);

srr3 := collect(simplify(srr2),{cos});
stt3 := collect(simplify(stt2),{cos});
srt3 := collect(simplify(srt2),{cos});

The stresses at the hole ( ) are

 

The maximum hoop stress is given at   or  .

At  ,  .

At  ,  .

The maximum shear stress at   is   while that at   is  .

Therefore, the stress concentration factor in tension is  , while that in shear is  .

Both stress concentration factors are equal.


Let us look at the ratio of the hoop stress at   to the far field hoop stress

 

The ratio is

 

This ratio is 0.95 when  , i.e., at a distance of   diameters from the center.


The given stress function is

 

Therefore, the displacement field from the Michell solution is

 

From the stress calculation step, we have

 

After substituting the constants and collecting terms,

 

Replacing   with  , and   with   (for plane stress conditions), we get

 

At  ,

 

The deformed shape is shown below:

 
Deformation of the hole under tension

In cartesian coordinates, the displacement field is given by