Stress, strain, stress-strain relation & Hooke’s law, stress calculation

Stress - One Pascal tension is defined as the ratio of one newton force acting on the surface of one square meter. The stresses in the body depends on the relative position of the body of elementary particles subjected to external forces. In terms of the physical model the behavior of the loaded solid, stress is defined as the quotient of force being response to external loads and the surface on which this force acts:

σ - principal stress

F - force

S - cross-sectional area

The direction along the axis of the cylinder is called a primary direction, and the tension in the principal stress direction. In a plane inclined to the direction of principal stress at any angle to the normal distributed σN and tangential τ.

So we distinguish stress:

- Normal - stress having a direction perpendicular to the section in question, denoted by the Greek letter them σ.

- Tangential - the tension lying in the plane of the section in question, it is denoted with the letter τ.

Strain - The distortion is a temporary or permanent change in the dimensions of the whole body or parts thereof caused the applied load to it.

Deformation: a) the body before the deformation, b) the body after deformation

Deformation structure elements are formed under the action of external forces in different ways to them hooked. It should be here to mark the weight of the body is treated also as according to him, hooking external force.

1. Deformation purely volumetric necessary and sufficient condition that the strain were of a purely volume is strain depending on the fulfillment of

At a purely volumetric deformation relative change in volume is equal to:

1. Deformation purely amorphous necessary and sufficient condition that the deformation of a purely amorphous be met depending on:

Analysis of the stress and strain -In the rod subjected to simple bending occurs uniaxial homogeneous state stress poses only one stress characterized submitting her normal , which depends angry linearly on the coordinates of the point at which we calculate the stress down.

Equation shows small ends of the tension vectors pose lies on a plane which we can name the plane stress down. Edge intersection August plane stress down with the cross-sectional plane to call we will bristle indifferent when It the locus of points at which the value of stress normal conforms

equation:

=0

Substitution to its subsidiaries. Receivables gives the equation of the axis of neutralndopeptidase for the case of simple bending in the plane (X, Z):

z = 0, which shows that that in pondering mode controlled case of stress pose resets the points lie up most of the axis Y, that is the main central axis of inertia of the cross section to the Parallel is a vector of the bending moment. Thus, the neutral axis of the bending angles coincides with the direction vector of the bending moment and the position-up does not depend. Since value of the bending moment.

Hooke's law - the law of mechanics determining the stress-strain relationship. It says that the deformation of the body under the influence of forces acting on them is proportional to this force. Ratio between the force and deflection is often called coefficient (module) elasticity.

This regularity, formulated by Robert Hooke (1635-1703) in the form ut Tensio sic vis (which stress the strength) remains true only for very large deformations, not exceeding called. border Hooke (also called proportional limit), and only for some materials. Hooke's law also assumes that the deformation of the body, in response to forces, followed immediately and completely disappear when the applied forces cease to operate. This simplification is only sufficient for bodies with negligible viscosity.

The simplest example of the application of Hooke's law is static stretching rod. Absolute extension of such rod is proportional to the force applied to the rod to its length and inversely proportional to the cross-sectional area of the rod. The constant of proportionality is the Young's modulus E

where:

F - tensile strength,

S - cross-sectional area,

Δl - the elongation of the rod,

l - length of initial.

In the case of wire or rod of a uniform diameter can be expressed simply: elongation is proportional to the force.

Using the definitions of stress and strain can be said that the relative elongation is proportional to the stress, which can be written:

Hooke's law for a general three-dimensional case, the stresses in the isotropic material can be written as a system of equations:

for linear deflection

for their angular deformation

where:

ε - strain line at the point

σ - stress at the point line,

γ - deformation of amorphous (angle) at the point

τ - the tension in the angled section,

G - modulus of elasticity of shape (transverse) or module Kirchhoff

E - Young's modulus

• - Poisson's ratio.