# Tension equals force

## Tension (mechanics)

The **mechanical tension** (Formula symbol $ \ sigma $) is a term from strength theory, a branch of technical mechanics. It is the force per unit area that acts in an imaginary intersection through a body, a liquid or a gas.

In general, the voltage $ \ sigma $ (engl. *stress*, fr. *contrainte*) the amount of force *F.* (engl. *force*) per area *A.* (engl. *area*):

- $ \ sigma_N = \ frac {| \ vec {F} |} {A} $

The mechanical tension has the same physical dimension as the pressure, namely force per area. Pressure is a special case of mechanical tension.

### Normal and bending stress

With a normal load, the stress is evenly distributed over the surface, with a bending load there is a stress curve, and the *Bending stress* is the maximum value:

The **Normal stress**, d. H. the stress under normal force loading (tension / compression) results from:

- $ \ sigma_N = \ frac F A $,

where $ F = | \ vec {F} _ {\ perp} | $ is the force in the direction of the surface normal and $ A $ is the area.

The bending stress, i.e. H. the stress at moment loading (bending) results from:

- $ \ sigma_M = \ frac M I \ cdot z = \ frac M W $,

where $ M = | \ vec M | $ is the bending moment, $ I $ is the geometrical moment of inertia, $ z $ is the distance from the cross-sectional center of gravity to the edge fiber and $ W $ is the section modulus.

The following sketch illustrates this using a cantilever:

As a vector it has three components and it depends on the orientation of the cutting surface. The sense of direction is only defined when you define which side of the cut surface you are looking at, because tension is the force per unit area that the material that has been cut away in thought exerts on the remaining material. It is therefore directed in opposite directions in the two opposite cutting edges.

### Stress and stress tensor

The stresses acting at a certain point are described in their entirety by the stresses in three intersecting surfaces that intersect at the point, i.e. by three vectors with three components each; these together form the stress tensor.

The stress tensor has the simplest representation if one chooses the three intersecting surfaces each perpendicular to a direction of a Cartesian coordinate system. The three forces in the three sectional areas correspond to the rows of the following matrix:

- $ S = \ begin {bmatrix} \ sigma_ {x} & \ tau_ {xy} & \ tau_ {xz} \ \ tau_ {yx} & \ sigma_ {y} & \ tau_ {yz} \ \ tau_ {zx } & \ tau_ {zy} & \ sigma_ {z} \ end {bmatrix} $

The meaning is shown in the following sketch of a very small volume element cut out:

### Shear, compressive and tensile stress

The diagonal elements $ \ sigma $ represent the *Normal stresses* represent, i.e. the forces that act perpendicular to the surface. You will depending on the direction *Tensile stress* (positive sign) or *Compressive stress* (negative sign) (and their scalar size *print*) called. The off-diagonal elements $ \ tau $ are saved as *Shear stresses* designated. They act tangentially to the surface, so they represent a load on shear.

In the double index, the first index describes the direction in which the outer normal unit vector of the cutting surface points (i.e. on which cutting edge it is knitted) and the second index describes the direction in which the stress acts.

The shear stress curve is used to illustrate the shear stresses that occur in relation to the reference axis within a profile loaded with a transverse force. If the shear force acts outside the center of shear, torsion occurs.

### Principal stress and direction of principal stress

The tensor calculation allows the stress state to be described initially independently of a specific coordinate system and only after the respective calculation method has been derived, the component equations can be adapted to the geometric properties of the body, for example in cylindrical coordinates. In the tensor calculation, the stress tensor is defined as the second level tensor which, scalar multiplied by the outer surface normal of a section surface, results in the force vector per unit area.

Every stress state can be converted into a coordinate system by means of a major axis transformation, in which all shear stresses disappear. If the three normal stresses are combined into a vector in this coordinate system, this can be broken down into two components:

- The component across the room diagonal is a measure of how great the maximum shear stresses can be in other cutting directions, depending on the cutting direction. This proportion alone is relevant when calculating steel structures. He corresponds to the
*Equivalent stress*according to the shape change hypothesis. If he does*Yield stress*exceeds the respective steel grade, the steel deforms plastically. - The component in the direction of the room diagonal describes the pressure; this part is irrelevant for the calculation of steel structures, since it does not lead to shear stresses in any cutting direction and therefore also to no plastic deformation.

The *Principal stresses* can be calculated by solving the equation $ \ det (S- \ sigma E) = 0 $, where *E.* the *3×3*-Unit matrix is. Multiplying the determinant leads to an equation of the third degree, the solutions of which $ \ sigma_ {1} $, $ \ sigma_ {2} $ and $ \ sigma_ {3} $ represent the main stresses we are looking for. They are the eigenvalues of the stress matrix *S.*.

The respective *Main stress direction* results from the equation $ (S- \ sigma E) \ vec e = 0 $, where the calculated principal stress is used for $ \ sigma $. The solutions $ \ vec e_ {1}, \ vec e_ {2}, \ vec e_ {3} $ are eigenvectors of the stress matrix *S.* and indicate the direction of the stresses. In normalized form, they form an orthonormal basis of three-dimensional space, provided none of the principal stresses is equal to zero.

The family of curves of the main stress lines is called *Trajectory*. A geometric representation of the stress components in different coordinate directions in the plane stress state is given by the Mohr stress circle.

Hooke's law establishes the relationship to deformation for elastic deformations. The most important material constants are the modulus of elasticity and Poisson's ratio. The plastic deformation describes the flow condition, the flow law and the hardening law. Newton's approach to viscosity establishes the relationship to the rate of deformation in viscous liquids. The most important material constant is the dynamic viscosity.

The deformation behavior of a body is described by the relationship between stress and deformation tensor. This relationship is called the rheological law and is a material property of the body. Rheology deals with the flow and deformation of bodies.

The tensions that occur without external forces acting on the body are also *Residual stresses* called.

A hydrostatic stress state exists when the three principal stresses are equal.

The voltage in gases is described in the Boyle-Mariotte law and in the Gay-Lussac law.

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