1. Concepts about the science of "Strength of Materials" tasks, consistency, uniformity, priority, brief history

26 -50Principal axes of inertia and principal moments of inertia. Concepts of radius of inertia and ellipse (state, centrifugal, extremity, axis of symmetry, graphic method).
Centrifugal moment of inertia of a plane section relative to a system of X0Y coordinate axes

let it be known.
If we rotate this system of axes by 90 degrees in the positive direction, x ₁ =y and y ₁ =-x .
Centrifugal moment of inertia relative to the new coordinate axes x ₁ Moon _{1 system}

So, when the coordinate axis system is rotated by 90 degrees, the centrifugal moment of inertia changes its sign.
It follows that in this interval there is such a situation of the system of coordinate axes, relative to which the centrifugal moment of inertia of the plane section is equal to zero.
Axes with zero centrifugal moment of inertia are called principal axes, and moments of inertia calculated relative to the principal axes are called principal moments of inertia.
To find the direction of the main axes, we set the equation (6. 33 ) to zero:

From this formula, two values of ₀ differ from each other by 90 degrees. The absolute value of their smallest does not exceed 45 degrees, the main axis passed at this angle is designated as u , and the second as - .
of the main moments of inertia J _u and J _v can be found by putting = _{0 in the formulas} (6. 31 ) and (6. 32 ) , respectively.
In general, the principal moments of inertia (6.34 ) and ( 6.35 ) can be solved using ( 6.36 ) together.
Trigonometric relations
and
is expressed by the equation:


Now the equation (6. 35 ) takes the following form:

By multiplying the last expression with ( 6.33 ), and then subtracting from it, we get the following general relation.

when and will be.
The main moments of inertia have the characteristic of extremity. To believe this, let's differentiate the moment of inertia (6. 31 ) or (6. 32 ) with respect to the variable.

It follows from this . Therefore, moments of inertia with respect to the main axes will have extreme values.
Assuming that the sum of moments of inertia relative to mutually vertical axes is a constant quantity (6. 34 ), the moment of inertia relative to one of the main axes is maximum, and relative to the other - minimum.
The planes passing through the Brus axis and the main axes of its cross section are called main planes.
If a plane section has an axis of symmetry, this axis of symmetry will be the main axis, and the second axis will be perpendicular to it from the center of the section.
Inertia moments and inertia arrows _ some features seeing we go out 1. If mutually vertical ( perpendicular ) axes situation if changed to or any one don't look against on the side direction if changed from the center evasive inertia moment hint will change


27.Plasticity indicators of materials. Comparative work of deformation, hardness, impact viscosity (material, plastic, brittle, relative: residual transfer, viscosity: static, impact; hardness).
Plasticity (from ancient Greek: plastikos - "tendency to work") is the property of changing the size and shape of solid bodies without returning to their original state (plastic deformation) under the influence of external force or tension (tension Plasticity indicators are used to assess the plastic behavior of materials, especially in the context of mechanical deformation, hardness, impact resistance, and viscosity. Here's a discussion on these indicators and their relevance for different material types:

1. 
   Deformation:
   
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     Plastic materials exhibit significant deformation or strain before fracturing.
     
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     Ductility is a measure of a material's ability to undergo plastic deformation without rupture.
     
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     Ductile materials can sustain large deformations and can be elongated or bent without breaking.
     
2. 
   Hardness:
   
   • 
     Hardness is a measure of a material's resistance to localized plastic deformation, such as indentation or scratching.
     
   • 
     Brittle materials tend to have high hardness but low resistance to plastic deformation.
     
   • 
     Plastic materials may have lower hardness values due to their ability to undergo plastic deformation.
     
3. 
   Impact resistance:
   
   • 
     Impact resistance refers to a material's ability to absorb energy when subjected to a sudden impact or shock.
     
   • 
     Ductile materials often exhibit better impact resistance compared to brittle materials.
     
   • 
     The plastic behavior of materials allows them to deform and absorb energy during impact, reducing the risk of fracture.
     
4. 
   Viscosity:
   
   • 
     Viscosity is a measure of a material's resistance to flow or deformation under an applied force.
     
   • 
     In the context of plasticity, viscosity can refer to both static viscosity (resistance to slow deformation) and impact viscosity (resistance to rapid deformation).
     
   • 
     Plastic materials typically have higher viscosity than brittle materials, enabling them to deform without fracture.
     
5. 
   Comparative assessment:
   
   • 
     When comparing plastic and brittle materials, plasticity indicators help distinguish their behavior and suitability for different applications.
     
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     Plastic materials are preferred when deformability, ductility, and impact resistance are crucial, as they can undergo significant deformation and absorb energy.
     
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     Brittle materials, with high hardness and low plasticity, are suitable when rigidity, strength, and resistance to plastic deformation are essential.