Consistency of Volume Assumption in Theory of Plasticity

The consistency of volume assumption is an important concept in the theory of plasticity. It states that the volume of a material remains constant during plastic deformation. This assumption allows for the derivation of certain relationships and equations that describe the behavior of materials under plastic deformation.

Relation between True Strains

The consistency of volume assumption leads to a specific relationship between the principle true strains, which are denoted as ϵ1, ϵ2, and ϵ3. These principle true strains represent the strains in three orthogonal directions in a material.

The relationship between the principle true strains can be derived by considering the deformation of an infinitesimally small element in the material. The volume of this element remains constant during plastic deformation, which means that the change in volume due to deformation is zero.

Derivation of the Relationship

Let's consider an infinitesimally small element in a material with initial dimensions dx, dy, and dz in the x, y, and z directions, respectively. After deformation, the dimensions of the element change to (dx + Δdx), (dy + Δdy), and (dz + Δdz).

The change in volume, ΔV, can be calculated as:

ΔV = (dx + Δdx)(dy + Δdy)(dz + Δdz) - dx dy dz

Expanding this equation and neglecting higher-order terms, we get:

ΔV = dx dy Δdz + dx Δdy dz + Δdx dy dz

Since the volume change due to deformation is assumed to be zero, we can equate ΔV to zero:

0 = dx dy Δdz + dx Δdy dz + Δdx dy dz

Dividing through by dx dy dz, we get:

0 = Δdz + Δdy + Δdx

Now, we can substitute the true strains for the changes in dimensions:

0 = ϵ3 + ϵ2 + ϵ1

This equation represents the relation between the principle true strains ϵ1, ϵ2, and ϵ3 under the consistency of volume assumption.

Conclusion

In conclusion, the consistency of volume assumption in the theory of plasticity leads to the relationship ϵ3 + ϵ2 + ϵ1 = 0 between the principle true strains. This relationship is derived by considering the deformation of an infinitesimally small element and assuming that the volume of the material remains constant during plastic deformation.

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