Work done is the product of force and displacement. By this definition:
Plastic work is done by external force when permanent deformation is formed in a body, as shown in Fig. 6.2.
Work done per unit volume of such body can be expressed as:
In matrix form, the equation above is expressed in:
Permanent deformation caused in plastic state is the summation of elastic and plastic deformations. Therefore, the strain, cle can be divided into two components: elastic strain, dse and plastic strain, dep. Equation above can thus be written as follows:
The work done related to elastic strain is reversible as the body will restore back to its original shape upon unloading. Therefore, the plastic work is solely due to plastic
FIGURE 6.2 Plastic work in a body.
FIGURE 6.3 Definition of plastic work in stress-strain curve.
strain. The area of graph starting from point A to В under the stress-strain curve denotes the plastic work as shown in Fig. 6.3.
Mathematically, plastic work can be defined as:
Associated flow rule
Similar to Hooke's law, plastic behaviour is also governed by its constitutive equations. This rule is known as flow rule. In elastic state, the strains are linearly depending on the state of stress in a body. In plastic state however, this is not the case. The strains are governed by stress and loading history or the body, as the stress-strain relationship of material in plastic zone will be altered from time to time due to the break of bond between particles. To take this condition into account, incremental analysis is required. This method will first set up the timeframe for analysis, and then for each time step, the corresponding stress-strain relationship and resultant deformation will be determined. Through this method, plastic strains for each incremental loading are determined, and the total plastic strain can be derived thereafter.
The strain increment in plastic state is solely depending on the current state of stress, which can also be mathematically defined as:
dX is a non-negative constant for plasticity. Under elastic state, stress is expressed as below:
Express elastic strain in term of state of stress yields the following expression:
The increment of elastic strain can thus be written as follows:
Total strain is the summation of elastic and plastic strains. From Eqs. (6.6) and (6.7), the following elastic-plastic stress-strain relationship yielded:
By expressing the elastic-plastic stress-strain relationship in matrix form yields the following expression:
After simplification we get this: