Solid Mechanics - Uniqueness of Plane Stress State

[goaliematt76]

Homework Statement

My textbook says that the state of plane stress at a point is uniquely represented by two normal stress components and one shear stress component acting on an element that has a specific orientation at the point. Also, the complementary property of shear says that all four shear stresses must have equal magnitude and be directed either toward or away from each other at opposite edges of the element.

Under pure shear, I can prove the complementary property of shear using force and moment balances. When normal and shear stress components are present, I am having difficulty understanding why shear stress and normal stress are unique, and why the complementary property of shear is still valid.

Homework Equations

Force and moment balances

The Attempt at a Solution

I have tried to construct a proof for this (see attached pdf), but I have not been able to complete it. I intentionally set up the directions of the shear stress components to violate the complementary property of shear, since I would like to show what the directions must be mathematically.

I am a third-year mechanical engineering student and I have already taken solid mechanics. This has just always bothered me, and I would like to see a proof for this.

 

[Chestermiller]

τ₂ and τ₄ are the same (this follows from the Cauchy stress relationship), but, in your diagram, they are pointing in the wrong directions. Also, from the Cauchy stress relationship, σ₁ and σ₃, σ₂ and σ₄, and τ₁ and τ₃ are equal. Think of tension in the string, after making a cut and applying compensating forces by hand. The tensions at the two sides of the cut point in opposite directions.

The complementary stress relationship says that τ₂=τ₁.

Chet

 

[goaliematt76]

τ₂ and τ₄ are the same (this follows from the Cauchy stress relationship), but, in your diagram, they are pointing in the wrong directions. Also, from the Cauchy stress relationship, σ₁ and σ₃, σ₂ and σ₄, and τ₁ and τ₃ are equal. Think of tension in the string, after making a cut and applying compensating forces by hand. The tensions at the two sides of the cut point in opposite directions.

The complementary stress relationship says that τ₂=τ₁.

Chet

I know that this is the final result, I would just like to be able to prove it. I drew τ₂ and τ₄ in the wrong directions intentionally because it is not obvious to me why the complementary stress relationship is valid in cases where normal stress is present. I suppose what I am looking for is a proof of the Cauchy stress relationship.