Stress and Strain – Basic Elastic Theory

Applied forces and moments produce internal stresses and deformation within engineering materials. Within the linear-elastic range, these stresses and strains are related through material properties such as Young's modulus, Poisson's ratio and shear modulus.

This page introduces the basic stress-strain relationships used throughout solid mechanics and component stress analysis.

Contents

• Symbols and Units
• Direct Stress
• Direct Strain
• Young's Modulus
• Poisson's Ratio
• Three-Dimensional Normal Stress and Strain
• Shear Stress
• Complementary Shear Stress
• Shear Strain
• Shear Modulus
• Relationships Between Elastic Constants

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Symbols and Units

Stress and elastic moduli are frequently expressed in MPa or GPa rather than Pa. Strain is dimensionless, although engineering strain is often quoted as a percentage, microstrain (µε) or, for shear strain, in radians.

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Direct Stress

For a prismatic bar carrying a concentric axial force F, the average normal stress over a cross-sectional area A is:

$${\sigma }_{\text{avg}}=\frac{F}{A}$$

This is an average stress over the section. The local normal stress at a point is obtained by considering the normal component of force acting on an infinitesimal area:

$$\sigma =\underset{\Delta A\to 0}{\lim }\frac{\Delta F_n}{\Delta A}=\frac{d{F}_n}{dA}$$

where F_n is the force component normal to the area.

A common sign convention takes tensile normal stress as positive and compressive normal stress as negative.

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Direct Strain

For a bar of original length L undergoing an overall change in length δ, the average engineering normal strain is:

$${\epsilon }_{\text{avg}}=\frac{\delta }{L}$$

For a displacement field u(x), the local small normal strain in the x direction is:

$${\epsilon }_x=\frac{du}{dx}$$

Tensile strain is normally taken as positive and compressive strain as negative.

Engineering strain is based on the original dimensions. For small elastic deformations this is normally entirely adequate. At large deformation, distinction between engineering strain and true or logarithmic strain becomes important.

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Young's Modulus

For a linear-elastic material under uniaxial loading, Hooke's law relates normal stress and normal strain:

$$\sigma =E\epsilon$$

Therefore:

$$E=\frac{\sigma }{\epsilon }$$

where E is Young's modulus or modulus of elasticity.

Young's modulus is a measure of elastic stiffness rather than material strength. A material can have a high Young's modulus but relatively low strength, or a high strength without having a correspondingly high modulus.

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Poisson's Ratio

When a material is subjected to uniaxial tensile stress it normally extends in the direction of loading and contracts in the transverse directions. Under compression the opposite occurs.

Within the linear-elastic range, Poisson's ratio ν is defined as the negative ratio of transverse strain to axial strain under uniaxial stress:

$$\nu =-\frac{{\epsilon }_{\text{transverse}}}{{\epsilon }_{\text{axial}}}$$

The minus sign makes Poisson's ratio positive for ordinary materials that contract laterally when stretched.

For a uniaxial stress σ_x, with σ_y = σ_z = 0:

$${\epsilon }_x=\frac{{\sigma }_x}{E}$$

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Three-Dimensional Normal Stress and Strain

For a homogeneous, isotropic, linear-elastic material subjected to normal stresses in three mutually perpendicular directions, the strain in each direction is the sum of the direct strain produced by the stress in that direction and the Poisson strains produced by the other two stresses.

$${\epsilon }_1=\frac{1}{E}\left({\sigma }_1-\nu \left({\sigma }_2+{\sigma }_3\right)\right)$$ $${\epsilon }_2=\frac{1}{E}\left({\sigma }_2-\nu \left({\sigma }_1+{\sigma }_3\right)\right)$$ $${\epsilon }_3=\frac{1}{E}\left({\sigma }_3-\nu \left({\sigma }_1+{\sigma }_2\right)\right)$$

These equations reproduce the relationships represented by the original RoyMech stress-strain diagrams in machine-readable form.

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Shear Stress

Shear stress acts tangentially to the surface on which it is defined. The average shear stress associated with a shear force V acting over an area A is:

$${\tau }_{\text{avg}}=\frac{V}{A}$$

The local shear stress at a point is obtained from the tangential component of force acting over an infinitesimal area:

$$\tau =\underset{\Delta A\to 0}{\lim }\frac{\Delta F_t}{\Delta A}=\frac{d{F}_t}{dA}$$

where F_t is the force component tangential to the area.

Pins, dowels, rivets, bolts and keys are examples of components that may be subjected to significant direct shear loading. Torsion of a circular shaft also produces shear stress, although the stress distribution is not uniform over the cross-section.

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Complementary Shear Stress

Consider a small material element subjected to shear stress. Force equilibrium requires opposing shear stresses on opposite faces, while moment equilibrium requires corresponding shear stresses on the perpendicular faces.

For an element having side dimensions x and y and unit thickness, moment equilibrium gives:

$$\tau xy=\tau \prime yx$$

and therefore:

$$\tau =\tau \prime$$

Thus, in the classical stress element without body couples, shear stresses on mutually perpendicular planes occur as equal complementary pairs. Their directions must be consistent with both force and moment equilibrium.

Shear Stress at a Free Surface

At a traction-free boundary, both the normal and tangential components of traction on the boundary surface are zero. This boundary condition constrains the shear-stress distribution immediately adjacent to the free surface.

For example, the transverse shear stress distribution in a solid circular section under a shear force is non-uniform and falls to zero at the free boundary, as illustrated below.

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Shear Strain

Shear stress produces angular distortion of a material element. Engineering shear strain is conventionally represented by γ.

For the small deformation shown:

$$\gamma =\tan \phi =\frac{{\delta }_x}{y}$$

For small angles, with φ measured in radians:

$$\gamma \approx \phi$$

Engineering shear strain is dimensionless, although it is commonly described in radians because it represents a change in angle.

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Shear Modulus

Within the linear-elastic range, shear stress is proportional to engineering shear strain:

$$\tau =G\gamma$$

Therefore:

$$G=\frac{\tau }{\gamma }$$

where G is the shear modulus, also traditionally called the modulus of rigidity.

Like Young's modulus, the shear modulus is a stiffness property and has units of stress.

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Relationships Between Elastic Constants

For a homogeneous, isotropic, linear-elastic material, Young's modulus E, shear modulus G and Poisson's ratio ν are related by:

$$E=2G\left(1+\nu \right)$$

or:

$$G=\frac{E}{2\left(1+\nu \right)}$$

Including the bulk modulus K:

$$E=3K\left(1-2\nu \right)$$

These relationships apply specifically to isotropic linear-elastic materials. Anisotropic materials require additional independent elastic constants.

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Related RoyMech Mechanics References

• Component Loading – overview of direct, shear, bending, torsional and dynamic loading.
• Failure Modes – yielding, fracture and engineering failure criteria.
• Mohr's Circle – transformation of plane stress and determination of principal stresses.
• Stress-Strain Curve – material behaviour under tensile loading.
• Buckling Notes – stability of compression members.
• Thick Cylinders – stresses in thick-walled pressure cylinders.
• Rotating Discs and Cylinders – stresses produced by rotational loading.
• Torsion – shear stress and deformation due to torque.
• Loaded Flat Plates – plate stress and deflection.
• Beam Design and Analysis – bending stress, shear stress and beam deflection.

Other Mechanics References

• Vectors
• Statics
• Kinematics
• Kinetics and Dynamics
• Mechanical Linkages
• Simple Machines

External References

1. Engineering Strain – eFunda – reference information on engineering strain and material mechanics.
2. Solid Mechanics – Wikibooks – online solid mechanics reference.

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