# Question on mechanics of material

It has been several years since i last worked on strength of materials, and i can't be certain of what seem to be a simple problem:
Suppose i have a rectangle block. Inside the block is a rectangle cavity with the long side horizontal. Consider the secanarios whereby the interior cavity is rotated 30, 45 , 60 and 90 degrees. In which case is the compressive strength the highest?
Since the max shear is at 45 degrees, the weakest block should be the one with the cavity align at 45. Right? And 30 and 60 should be the same??
philip
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Philip wrote:

materials,
cavity
is
same??
It depends a lot on how you define failure.
A rectangular void is problematic because of unbounded logrithmic stress concentrations at the corners; this leads to local plastic deformation for any load.
Would an elliptically shaped slot be ok? That's a geometry for which there is an analytic answer.
Are you sure it won't fail due to local tension rather than local shear? In the tensile failure case the elliptical void's long axis in the loading direction results in failure at the lowest applied load.
A fracture mechanics approach may not care about details of defect tip geometry but you'll still have to worry about failure mode.
Dave
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Philip wrote:

To add on, The only applied force is acting on the top surface in a downward direction so that there is only compression and no shear.
I was assuming that the failure mode would be shear failure. What I really want to ask if its stupid to put the cavity at a 45 degree angle if tilting it up or down a little will prevent premature shear failure.
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Shear stresses (including concentration effects) will probably be a maximum at 45 degrees compared to +/- a few degrees.
This is covered pretty well in Timeshenko , "Theory of Elasticity."
Things really depend a lot on what you mean by "failure". A slot perpendicular to the loading axis will develop high local shear stresses at its pointy end.
These stresses may cause local plastic flow that'll in effect blunt the slot's tip but not lead to macroscopic failure. Is this ok?
Dave
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snipped-for-privacy@newarts.com wrote:

I suppose it is fine as it's the catastrophic type of failure i'm worried about. Actually i left out some details in the description to keep things easy. The cavity really runs all the way through the block in the x-dir and most of the compression is taken by material in the out-of-plane direction. In this geometry, tilting the cavity 90 degrees of course dosen't make sense. With this new description, will it still be worthwhile to make the cavity at say 30 or 60 degree to lower the chances of shear failure?
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"The cavity really runs all the way through the block in the x-dir and most of the compression is taken by material in the out-of-plane direction. ..."
I'm sorry, I still don't get it. Is the cavity in the x-y plane and the loading in the z direction?
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snipped-for-privacy@newarts.com wrote:

Yes. Considering that the rectangular cavity can rotate about the y-axis, at 0 degree (laying flat), it will be in the xy plane. At 90 degrees, it will be in the yz plane. It extends all the way in the -x and +x direction making two holes in the enclosing block. But it does not extend in the -y and +y direction. And yes, loading is in the z direction.
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Philip wrote:

You really meant a SLOT, didn't you? Not a cavity.
So, you are concerned with the compression weakening aspects of a slot and the orientation of the slot.
This cavity stuff boggles the mind.
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I think I understand.
First let me say that you should *always* avoid sharp corners. A true rectangle will be problematic in any geometry. At least use the biggest fillets you can at the corners; best would be to use a slot with semicircular ends. (The best slot of all would be a cylindrical hole).
Second the bottom line of what follows is that the slot would best be in the x-y plane as you've defined the geometry; ie. the compressive load should be perpendicular to the slot.
Third, a confusing attempt to describe the geometry as I understand it follows:
There is a plate in the y-z plane of thickness x. There is a compressive load in the z direction.
You are going to stab through the plate with a knife.
Shear stresses leading to shear failure of the plate will be a minimum when the knife blade's two axes are perpendicular to the plate's z axis.
Other geometries are hard to describe in a crystal clear fashion as there are six axes involved (3 for the plate & 3 for the knife.)
Perhaps the simplest way to think about it is to imagine the knife blade (or stab wound) like a long envelope lying face up on the desk; the short direction is the width of the blade and the length is the depth of the stab wound (length varies depending on how the blade is tilted with respect to the plate when the stab is made.)
Now put a pencil point in the center of the envelope and hold it in the vertical position with your finger on the eraser; this represents the loading direction. You can tilt the loading direction along the envelope's long axis or along it's short axis.
Tilting the loading direction around the blade's long axis increases the shear stress component which reaches a maximum at 45 degrees. Such tilts don't affect the depth of the stab wound (the length of the envelope.) In a fracture mechanics sense such tilts result in Mode II loading.
Tilting around the blade's short axis also increases the effective shear stress component (but not as rapidly because the depth of the stab wound increases as the angle increases). In a fracture machanics sense such tilts result in Mode III loading.
I'm sorry for the complex description but the geometry is difficult to describe in words and precise mathematical descriptions aren't very useful in visualizing things for me.
I hope the above describes your situation in an understandable way. Are there other constraints on the allowed orientation of the stab?
The ratio of the plate thickness to knife width is a detail that might matter. I've assumed the knife width is big compared to the plate thickness.
Dave
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snipped-for-privacy@newarts.com wrote: <snip>

Dave your description and assumptions are correct. So bottomline is the slot/stab has to be as flat as possible.
jbuch Yeah... a slot, a slot...
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