Resolved Shear Stress Compared w/ Shear Stress-Contradictioni

In summary, the conversation discusses the terms "slip plane" and "slip direction" from the textbook "Materials Science and Engineering: An Introduction" by William D. Callister. It also makes assumptions about a material with moderate ductility and a quasi-static uniaxial tensile stress. The nomenclature includes the angles θ, λ, and φ, as well as the areas A and [A][/o]. The question asks for a relationship between shear stress for a uniaxial tensile stress and resolved shear stress geometrically. The attachments show derivations for both shear stresses. However, the two are not directly related and cannot be derived from each other.
  • #1
ltkach2015
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1
TERMS:
Slip Plane: is the plane that has the densest atomic packing—that is, has the greatest planar density.
Slip Direction: corresponds to the direction in this plane that is most closely packed with atoms—that is, has the highest linear density.
TEXTBOOK: Materials Science and Engineering: An Introduction: William D. Callister. Chapter 6 & 7.
ASSUMPTIONS:

-uniaxial tensile stress of a material with moderate ductility
-quasi-static
NOMENCLATURE:
θ = angle of the slip plane as measured from cross section of material
λ = angle of that the applied force makes with the slip direction
φ = it has been said that is the angle between the normal vector and the applied force
θ = angle of the slip plane as measured from the cross section of the material
φ = θ =?; would this be so?
[A][/o] = Area of the materials cross section
A = Area of the slip plane

QUESTION:
Can you show the relationship between shear stress for a typical uniaxial tensile stress (mechanics of materials) to the resolved shear stress geometrically?

It is derived from mechanics of materials principles in Chapter 6 that shear stress is τ = σsinθcosθ. Following that chapter 7 introduces resolved shear stress τ=σcosλcosφ.

I can do both derivations but I cannot relate the two.

ATTACHMENTS: 1(

BEGIN:
-Derivation of resolved shear stress
0) ensure that slip direction lies on slip plane via dot product of the normal vector of the slip plane and the slip direction == 0 (orthogonal)

1) Load in direction of slip direction

τ=σcosλ/A;
[A][/o] = Acosθ
=> τ=σcosλcosφ.-Derivation of shear stress in general:
I can derive via a force balance, sum of forces resulting that resulted to

=> τ = σsinθcosθThank you!
 

Attachments

  • ResolvedShearStress vs Shear Stress.pdf
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Related to Resolved Shear Stress Compared w/ Shear Stress-Contradictioni

1. What is the difference between resolved shear stress and shear stress?

Resolved shear stress is the component of shear stress acting in a specific direction, while shear stress is the force per unit area parallel to a cross-section of a material.

2. How are resolved shear stress and shear stress related?

Resolved shear stress is a component of shear stress, meaning it is a part of the total shear stress acting on a material. The relationship between the two is that the resolved shear stress is the projection of the total shear stress in a specific direction.

3. Can resolved shear stress and shear stress have different values?

Yes, they can have different values. Resolved shear stress is dependent on the orientation and geometry of a material, while shear stress is dependent on the applied force and cross-sectional area.

4. How do resolved shear stress and shear stress affect material strength?

Resolved shear stress and shear stress are important factors in determining the strength of a material. Higher values of these stresses can lead to plastic deformation, resulting in material failure. However, proper design and material selection can help mitigate the effects of these stresses.

5. Are there any contradictions between resolved shear stress and shear stress?

No, there are no contradictions between the two. Resolved shear stress is simply a component of the total shear stress, and they both play important roles in understanding the mechanical behavior of materials.

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