---

Strain-Energy Density

---

Prereqs ▪ strain energy ▪ stress-strain diagram ▪ Normal Strain ▪ load-deformation diagram ▪ normal stress

Quick
The strain-energy density of a material is defined as the strain energy per unit volume. It is equal to the area under the stress-strain diagram of a material, measured from ε_x = 0 to ε_x = ε₁ as described below.


Equation

(Eq1)     u =  U V  =  ∫ ε1   0 σx dεx

Strain-energy density

Nomenclature

F	force
A	area
U	strain energy
V	volume
σx	normal stress in the rod
εx	normal strain

Details

As stated in the lesson Normal Strain, the load-deformation diagram for a rod BC depends upon the length L and the cross-sectional area A of the rod. The strain energy defined by Eq1 from the lesson Strain Energy, therefore, will also depend upon the dimensions of the rod. In order to eliminate the effect of size from the discussion and direct the attention to the properties of the material, the strain energy per unit volume is now considered. Dividing the strain energy U by the volume V (V = AL) of the rod shown in the following figure:



and using Eq1 from the lesson Strain Energy, the following results:

U V

=

∫

x1   0

P A

dx L

Recalling that P/A represents normal stress σ_x in the rod, and x/L the normal strain ε_x:

U V

=

∫

ε1   0

σx dεx

where ε₁ denotes the value of the strain corresponding to the elongation x₁. The strain energy per unit volume, U/V, is referred to as the strain-energy density and is denoted by the letter u. Therefore:

(Eq1)

Strain-energy density = u =  ∫ ε1   0 σx dεx

The strain-energy density u is expressed in units obtained by dividing units of energy by units of volume. Thus, if SI metric units are used, the strain-energy density is expressed in J/m³ or its multiples kJ/m³ and MJ/m³; if U.S. customary units are used, it is expressed in inlb/in³. Note that 1 J/m³ and 1 Pa are both equal to 1 N/m², while 1 inlb/in³ and 1 psi are both equal to 1 lb/in². Thus, strain-energy density and stress are dimensionally equal and could be expressed in the same units.

Referring to the following figure:



it is noted that the strain-energy density u is equal to the area under the stress-strain curve, measured from ε_x = 0 to ε_x = ε₁. If the material is unloaded, the stress returns to zero, but there is a permanent deformation represented by the strain ε_p, and only the portion of the strain energy per unit volume corresponding to the triangular area is recovered. The remainder of the energy spent in deforming the material is dissipated in the form of heat.

---

Related
▪ L - Strain Energy
▪ L - Stress-Strain Diagram
▪ L - Modulus of Toughness
▪ L - Modulus of Resilience