Elasticity theory
The branch of physics that deals with elastic deformations is called the theory of elasticity. It is part of continuum mechanics and characterized in that elastic deformations are reversible after elimination of the external force, the material returns to its original shape. This is no longer the case when it comes to breaks or plastic flow - the latter case is treated by the theory of plasticity.
- 3.1 strain - cross strain coupling
- 3.2 Strain Schiebungs coupling
- 3.3 Schiebungs - Schiebungs coupling
Elasticity tensor
Mechanical stresses are considered to calculate the power approach to a (cut ) surface of a body. A force is used to calculate the normal stress component (), perpendicular to the selected tier, and shear stress () in the plane split. These different voltages are summarized in the stress tensor:
Accordingly, the deformation in the strain tensor are summarized:
A simple pencil eraser can be seen that pulling along the x- axis causes not only a deformation in the x direction, but also the eraser can be made thinner laterally ( transverse contraction ), i.e., depends linearly with the lateral shift and together. If you wanted to Hooke's law, that small deflections cause linear restoring forces (), write in the general case, we see that it is actually a tensor of fourth order ( with 3333 = 81 components) and is Hooke's law as
Or with the Einstein summation convention as
Should be written. In the case of a quadratic nonlinear material results in the context
Between stress tensor and strain tensor. This can be simplified by the Voigt notation for the linear stress-strain relation, however: both are symmetric and therefore only have each six independent components, which can be summed up in one column vector, respectively. This is Hooke's law simplifies to:
The elasticity tensor has been simplified to 36 components; since it is symmetric in addition, he has 21 components in the general case. Depending on the material and the symmetry properties of these components can be further simplified as will be seen below.
Special elasticity laws
Full anisotropy
The full ( triclinic ) anisotropy is the most common form of an elasticity law. It stands out for the engineer by the following properties:
- No planes of symmetry in the material
- 21 independent elastic constants describe the law
- Modulus of elasticity is dependent on the direction
- All pairings available
- Stiffness matrix is fully occupied
Many fiber-plastic composite materials are anisotropic, unidirectional layer outside their principal axes. Engineers try resulting from complete anisotropy effects to use.
Monoclinic anisotropy
The monoclinic anisotropy has little significance for construction materials. The following characteristics distinguish the monoclinic anisotropy:
Rhombic anisotropy ( orthotropy )
Many construction materials are orthotropic, eg technical wood, fabric, many fiber - plastic composites, rolled sheets with texture, etc. The orthotropy should not be confused with the anisotropy. The mere direction-dependent elastic modulus is still no indication of the anisotropy. The orthotropy is a special case of a fully anisotropic elasticity law. The orthotropy is characterized by the following properties:
- 3 planes of symmetry in the material
- 9 independent elastic constants describe the law
- Modulus of elasticity is dependent on the direction
- No strain Schiebungs coupling exists
So Orthotropic materials make no shear distortion when they are stretched. This makes them easy to handle for the designer. Therefore, working in fiber composite technology specifically with orthotropic layers such as the balanced composite angle. Plywood is constructed so that it has orthotropic properties.
Note: The matrix and thus also its inverse are symmetric. Generally, however, are not symmetrical, the constants used in the display for which, and is.
Transverse isotropy
The transverse isotropy is characterized in that the material information can be rotated around an axis without any change. So it is invariant with respect to rotation. An example of a transversely isotropic material is a round timber or a unidirectional layer. The elastic properties of the logs do not change when it rotates about its longitudinal axis. However, the timber has different moduli of longitudinal and transverse to the fiber. The transverse isotropy is characterized by the following properties:
- 3 planes of symmetry in the material
The transverse isotropy is a special case of the general orthotropy.
Isotropy
The isotropic law is the most famous and important material information. With it, almost all metals and unreinforced plastics can be described. And short-fiber- reinforced plastics may be isotropic when randomly distributed reinforcing fibers (see fiber-matrix semi-finished products ). The isotropic elasticity law stands for the designer primarily by the invariance to rotation from. In one construction, it does not matter how the isotropic material is oriented. Rolled metal sheets may have a weak anisotropy.
- An infinite number of planes of symmetry in the material
- 2 independent elastic constants describe the law
- Young's modulus is not dependent on direction, two modules are identical
- No Schiebungs -strain coupling exists
Couplings
The different elasticity laws are distinguished by their couplings. A coupling means the effect that the material reacts with a deformation out of the direction of action of the load.
Strain - cross strain coupling
This is the most famous coupling. It is referred to as Poisson's coupling. The coupling causes the material constricts in train, or widened in pressure. Engineers have learned with the strain coupling handle and apply them selectively, eg during riveting. Virtually all construction materials have this coupling.
- Responsible Terme:
Strain Schiebungs coupling
Especially in anisotropic materials occurs, this coupling. Orthotropic materials do not possess. The strain Schiebungs coupling produces a thrust at an elongation of the material. This is referred to as delay Colloquially. Using the classical laminate theory can be examined whether or not a material has a strain Schiebungs coupling.
- Responsible Terme: and
Schiebungs - Schiebungs coupling
The Schiebungs - Schiebungs coupling occurs only in anisotropic materials. A shift in the plane created out of here also a shift from the plane.
- Responsible Terme: