# Plasticity theory

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The theory of plasticity is the branch of continuum mechanics that deals with irreversible transformations of matter. It describes the state of stress and strain of solid bodies under the influence of a load, but in contrast to the theory of elasticity, it does not deal with reversible deformation.

Beyond the proportionality limit of elasticity theory, different forms of anelastic behavior occur:

• elastic hysteresis : when the load is completely removed, a deformation remains, but this can be reversed by applying a counter-tension.
• Plasticity : an irreversible change in shape that remains after the application of force (example: modeling clay).
• further stretching despite partial relief is called flow .
• A break in the workpiece is also usually associated with an elastic component, i.e. H. some of the elongation (of the fragments) goes back after the break.

Well-known scientists who were involved in the development of the theory of plasticity were, for example, Barré de Saint-Venant and his student Maurice Lévy , as well as Ludwig Prandtl , Richard von Mises , Eugene Cook Bingham , Henri Tresca , Arpad Nadai , Heinrich Hencky , William Prager , Theodore von Kármán , Hilda Geiringer , Rodney Hill , Daniel Drucker , Wadim Sokolowski , Erastus Lee , Horst Lippmann and Lazar Katschanow (LM Kacanov).

## The plastic deformation

In real media, every deformation is only elastic up to a certain limit. If this limit is exceeded, plastic deformation ( plastic flow ) occurs in ductile materials . With plastic deformation, the body does not return to its original shape if the mechanical load responsible for the deformation does not occur. In this case, specifying the positions of points on the solid is no longer sufficient to identify the state of the solid, but the process must also be taken into account, ie in this case it is not a state variable . ${\ displaystyle {\ tilde {\ epsilon}}}$

In the general case the deformation can be given by. The total deformation consists of an elastic part , a plastic part and the temperature-related part: ${\ displaystyle {\ tilde {\ epsilon}}}$${\ displaystyle {\ tilde {\ epsilon}}}$${\ displaystyle {\ tilde {\ epsilon}} ^ {\, {\ rm {E}}}}$${\ displaystyle {\ tilde {\ epsilon}} ^ {\, {\ rm {P}}}}$

${\ displaystyle {\ tilde {\ epsilon}} = {\ tilde {\ epsilon}} ^ {\, {\ rm {E}}} + {\ tilde {\ epsilon}} ^ {\, {\ rm {P }}} + \ alpha \ cdot T \ ,.}$

Elastic-plastic material behavior can be described by a flow condition , a flow law , and a hardening law .

### Yield condition

The yield condition defines all those multiaxial stress states at which the material flows plastically. It is common to specify the yield condition as a convex curved surface in the stress space, which is called the yield locus surface. The material deforms in a purely elastic manner for stress states within the space enclosed by the flow location surface. If the current state of stress is on the yield point surface, plastic flow can occur. Stress states outside the enclosed space are impossible with elasto-plastic material behavior.

Common flow conditions for metallic materials were formulated by Huber, von Mises and Tresca . They are based on the assumption of isotropic behavior.

The yield condition according to R. v. Mises reads:

${\ displaystyle 0 = {\ frac {3} {2}} {\ tilde {s}} ^ {T} \ cdot {\ tilde {s}} - k _ {\ rm {f}} ^ {2}}$,

where denotes the stress deviator and the yield strength . The stress deviator is the stress tensor reduced by the hydrostatic component ${\ displaystyle {\ tilde {s}}}$${\ displaystyle k _ {\ rm {f}}}$

${\ displaystyle {\ tilde {s}} = \ sigma -p \ cdot e \,}$ with and as an identity tensor.${\ displaystyle \ p = {\ frac {\ sigma _ {x} + \ sigma _ {y} + \ sigma _ {z}} {3}}}$${\ displaystyle e}$

According to Tresca, the yield condition is:

${\ displaystyle 0 = {\ frac {\ sigma _ {I} - \ sigma _ {II}} {2}} - k}$,
${\ displaystyle k_ {f} = \ sigma _ {I} - \ sigma _ {II}}$

with and the largest or smallest principal normal stress. The Mohr circles of tension can be used for a graphical interpretation of Tresca’s rule . ${\ displaystyle \ sigma _ {I}}$${\ displaystyle \ sigma _ {II}}$

Both formulations are widely used. The rule of v. Mises is easy to use in the general case. If the position of the main axis system is known, the Tresca rule is often used. For numerical computation, this has the disadvantages that a major axis transformation is necessary and that the flow location area is not continuously differentiable.

### Flow law

The flow law determines the plastic distortion increments. In the case of associated plasticity, this increment is coaxial to the normal vector of the yield point surface at the current stress point. The order of magnitude of the increment is determined by the scalar-valued plastic multiplier.

In the case of non-associated plasticity, one often uses a plastic potential defined for this purpose to determine the plastic distortion direction. The associated case can also be understood as the special case in which the plastic potential and the flow condition project the same area in the stress space.

### Solidification Act

The hardening law defines how the flow condition is modified during flow. Ideally, two different hardening behaviors can be assumed, isotropic and kinematic hardening.

The behavior of the material can be described by isotropic hardening if it is independent of the previous direction of loading or if the direction of loading does not change. Isotropic hardening is expressed by expansion of the flow locus area. This means that the yield strength increases by a certain amount depending on the deformation applied. ${\ displaystyle k_ {f}}$

For example, the Bauschinger effect can be described by kinematic hardening , i.e. H. the elastic limit is significantly lower when loading in the opposite direction than during the previous loading. This phenomenon can be described by shifting the yield locus surface. The yield point remains constant, only the “center of the flow point” changes (back stress) . In the flow rule, the yield stress must then be replaced by the "reduced stress" . ${\ displaystyle k_ {f}}$${\ displaystyle {\ tilde {a}}}$${\ displaystyle {\ hat {\ tilde {s}}} = {\ tilde {\ sigma}} - {\ tilde {a}}}$

### Flow

The deformation does not take place homogeneously in the entire material, but only at energetically preferred crystal structural defects such as dislocations , phase boundaries and amorphous inclusions. Furthermore, the plastic deformation depends on the temperature and strain rate. The flow behavior can be described with a large number of constitutive material laws. There are empirical and metal-physics-based models for this.

## Elementary plasticity theory

The conception of the model first considers a small imaginary cube within the material, on whose opposing surfaces that belong together in pairs, a tension acts in any direction and size. Each of these three stresses can now be broken down in its associated area into one normal stress and two tangential stresses (shear stresses). Mathematically, the stress tensor consisting of a total of nine elements is thus created .

If the position of this cube is changed a little, nothing changes in the applied stresses, but the division into normal and shear stresses will change. It can now be shown that there is a position in which the normal stresses each reach a maximum value and the shear stresses all disappear. This situation can be recognized by the effects of the tensions. Normal stresses cause changes in length and shear stresses cause changes in angle. The directions of the three cube edges in this position can be calculated by a major axis transformation of the stress tensor. If at least the model for a distortion (reshaping) can only be composed of changes in length, it can be assumed that this position is favorable for further mathematical treatment. (A cuboid before reshaping creates a cuboid after reshaping; parallelepipedic reshaping).

This state is also called the "main stress state" and the remaining longitudinal stresses " main stress ". It is then spoken of the elementary plasticity theory.

### application

The elementary theory of plasticity has found wide application in the plastic shaping of metals, particularly in massive forming . There is initially a contradiction in terms, since metals are crystalline , i.e. structured. This anisotropy , however, only exists in the microscopically very small area of ​​the “grains” (around 50 µm in each direction), which in turn, due to the nature of their formation from the liquid (cast) state, are completely random in their orientation. The result is an apparently uniform structure (quasi-isotropy) for a macroscopic body that is practically always present in forming technology.

Another important application of the elementary plasticity theory is the load-bearing method developed in the context of structural engineering .