# State of tension

Stress state with stress vectors (red and blue) depending on their effective area (yellow) on a sphere of matter (blue-gray) in a homogeneous stress state

The state of stress is the totality of all conceivable stress vectors in a material point in a loaded body . The stress state defines the stress vector that acts on a surface in a unique way.

Stress vectors are vectors with the dimension force per area and always arise when forces are applied to body surfaces. The tensions in the force introduction points are propagated in the body following the flow of force - apart from disabilities constantly - and form a state of tension in every particle of the body. Around a particle away from imperfections one can imagine an (infinitesimal) small sphere in which the same state of tension prevails everywhere as in the particle, and this state of tension can be visualized as in the picture. The stress vectors are composed of normal stresses (radial in the picture) and shear stresses (tangential in the picture). The stress tensor combines the stress state into a mathematical object.

The body can be rigid , solid , liquid or gaseous . The state of stress is generally a function of time and the location in the body and develops depending on its shape, load ( pressure , shear and, in the case of solids, additional tension , bending , torsion ), material properties and geometric bonds.

## definition

Four area-distributed forces (arrows) on the infinitesimally small tetrahedron (gray) must cancel each other out anytime and anywhere.

Three stress vectors in a point on three different levels, whose surface normals are linearly independent , completely define the stress state in the point.

Because if the voltage vector is searched for on a fourth level through the point, hereinafter referred to as P, then this results as follows in an unambiguous manner from the three known ones. For this purpose, the fourth plane is shifted an infinitesimal piece away from point P in parallel , whereby the four planes cut a tetrahedron out of the body, see picture. Let the tetrahedron be so small that the stress state is uniform in its spatial area. Then the tension vectors multiplied by their tetrahedron area form forces that have to be in equilibrium with the force on the fourth area , because volume effects (acceleration due to gravity, magnetism, ...) tend towards zero in the infinitesimally small tetrahedron as volume-proportional quantities compared to the area-proportional forces of the tension vectors. This clearly defines the force on the fourth side. Purely geometrically, however, the fourth surface is also clearly determined from three tetrahedral surfaces according to content and normal unit vector and consequently the stress vector on the surface. Because the state of stress is uniform in the space of the tetrahedron, this stress vector also acts on the fourth level through P.

A detailed analysis shows that the relationship between the surface normals and the stress vectors must therefore be linear, which is the statement of Cauchy's fundamental theorem with which Augustin-Louis Cauchy introduced the stress tensor as a linear operator between the normal unit vectors and the stress vectors.

## Degree of tension

The degree of the stress state is determined by the number of non-vanishing principal stresses .

Compression test with transverse expansion obstruction

Under uniaxial or uniaxial tension or pressure, a uniaxial stress state can develop with non-zero principal stress in the direction of loading and vanishing principal stresses perpendicular to it. Long, slender, homogeneous rods or ropes that are subjected to tensile loads have, apart from the load application zones, a uniaxial stress state to a good approximation.

In the case of biaxial tension, there is a biaxial or plane stress state (two main stresses in the loading directions, the third main stress perpendicular to the plane is zero). Flat states of tension prevail on unloaded parts of the surface of a body. More on this can be found below.

In irregularly shaped components / specimens (e.g. ISO-V specimen from a notched bar impact test ), in force introduction points, with uneven loading or with transverse strain hindrance as shown in the picture, three-axis, spatial stress states with three non-vanishing principal stresses usually occur .

## Special cases

### Hydrostatic stress state

The hydrostatic stress state is a spatial stress state that develops with all-round tension / pressure, which is ubiquitous on earth when there is no wind. The hydrostatic pressure of the earth's atmosphere creates this state of tension in all otherwise unloaded bodies. The stress vectors are parallel to their normal in each sectional plane, there are no shear stresses in any plane, and all principal stresses are equal. Materials can withstand hydrostatic pressure to a high degree without permanent deformation.

### Level state of tension

Flat states of stress occur with biaxial tension or on unloaded parts of the surface of bodies. In the same way, a plane stress state can also be assumed in thin shells , flight membranes or planar structures far away from force introduction points or other interference points. It can be clearly represented by Mohr's circle of tension .

On unloaded parts of the body surface, the conditions of the plane stress state are exactly met, because the cutting reaction does not occur there in the tangential plane to the surface according to the prerequisite. In the interior of the body, a location-dependent state of stress with a Poisson's ratio not equal to zero can only be approximately level. This is because the elongations in the plane caused by the state of tension also cause location-dependent transverse contractions of the body perpendicular to the plane , in which shearings now occur perpendicular to the plane. These shearings are generally associated with corresponding shear stresses acting perpendicular to the plane. Only when these are negligibly small can we still speak of a level state of stress.

### Homogeneous stress state

The homogeneous stress state is a location-independent stress state. In a homogeneous state of tension, the load-bearing behavior of a material is optimally used.

An equally homogeneous state of distortion arises in an equally homogeneous material. The strain can then be determined using strain gauges , measuring cameras or measuring arms , which are macroscopic measuring devices. If there is a homogeneous state in the area under consideration, the measured elongation provides a directly interpretable value for the elongation of the sample. Accordingly, the homogeneous state of stress is of great importance in material theory and measurement technology.

In this context, the universal deformation is useful, which can be caused in any homogeneous material by stresses introduced exclusively on the surface. A universal deformation with a homogeneous state of tension is created in the case of uniaxial or multi-axial tension, in particular hydrostatic pressure, in the event of shear or torsion.

The homogeneous state of tension is an idealization that can hardly be found in real bodies. Because many bodies have imperfections , cavities , hairline cracks or notches . Material boundaries, force introduction points or areas with internal stresses also have inhomogeneous stress states in their vicinity. These cause inhomogeneous expansions that z. B. can be seen in the determination of residual stress by drilling open. Mathematical methods help to interpret and evaluate these strains. According to the principle of St. Venant , the disturbance subsides with increasing distance and a homogeneous state of tension sets in.

### Stress states in planar beams

Membrane and bending stress state in a dome shell loaded by a single force

The picture shows a dome shell, which is loaded in its middle with a single force. Far away from the introduction of force, the membrane tension state is present (blue in the picture). In the area where the force is introduced, which is a fault, there is a state of bending stress (green).

Under certain conditions, the loads in a shell are primarily diverted to the supports through stresses that are constantly distributed over the wall thickness and parallel to the shell center surface. In such cases, one speaks of a state of tensile stress or membrane stress, which is also present in the state of disk tension in planar structural structures, see disk theory . When the membrane is in tension, the load-bearing behavior of the material is optimally used. The membrane tension condition develops far away from force introduction points and other disturbances.

In the vicinity of imperfections, shells experience a less favorable bending stress state. In the vicinity of the defect, bending stresses and shear stresses that vary over the thickness of the shell arise perpendicular to the shell center surface. According to the St. Venant principle, however, the disturbances quickly subside as the distance to the disturbance point increases. The bending stress state can be compared with the plate stress state of planar surface structures, see plate theory .

## Application in strength theory

The state of stress can be used to characterize deformations in components, with strains then also playing a role. The stress state is particularly suitable for strength considerations in isotropic elastic solids, whereby the knowledge of one or more stresses in the cross section of a component at a certain point or at several certain points is often attempted to draw conclusions elsewhere in the same component. Such strength considerations are the subject of elasticity and plasticity theory . The deformations cause stresses and can often be systematically determined through strength calculations. An approach frequently used is the one that determines the spatial voltage states at a meaningful point in a loaded component, by stretching the component with strain gauges - Measurement measures, this on certain bills in a stress tensor is introduced and then through main axis transform determined extremal voltages.

## Individual evidence

1. ^ H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 , pp. 142 .
2. H. Oertel (ed.): Prandtl guide through fluid mechanics . Fundamentals and phenomena. 13th edition. Springer Vieweg, 2012, ISBN 978-3-8348-1918-5 .
3. C. Truesdell: The non-linear field theories of mechanics . In: S. Flügge (Ed.): Handbuch der Physik . tape III / 3 . Springer, 2013, ISBN 978-3-642-46017-3 , pp. 184 (English).

## literature

• Hans Göldner, Franz Holzweißig : Guide to technical mechanics: statics, strength theory, kinematics, dynamics. 11. verb. Edition. Fachbuchverlag, Leipzig 1989, ISBN 3-343-00497-9
• Eduard Pestel , Jens Wittenburg: Technical Mechanics . Volume 2: Strength of Materials . 2. revised and exp. Edition. Bibliographisches Institut, Mannheim 1992, ISBN 3-411-14822-5