Stress concentration

A stress concentration is a localized increase of stress caused by geometry or another disturbance of the load path. Holes, notches and abrupt section changes can make the largest local normal stress much greater than a stress calculated from the total force divided by a reference cross-sectional area. A stress concentration factor

K_{t} = σ_{m a x} / σ_{ref}

is meaningful only after both the stress component and its reference have been specified.

The Kirsch equations give an instructive exact result for a circular hole of radius

a

in an infinite, isotropic linear elastic elastic plate under remote tensile stress

σ_{\infty}

in one direction, using a plane stress model. Measure the polar angle

θ

from the loading direction. At the hole boundary,

σ_{rr} = σ_{r θ} = 0, σ_{θθ} (a, θ) = σ_{\infty} (1 - 2 cos 2 θ) .

(1)

θ = π /2, 3 π /2

, the tangential normal stress reaches

3 σ_{\infty}

, giving $K_{t} = 3$ relative to remote stress. At

θ = 0, π

, it is

- σ_{\infty}

. A traction-free hole therefore does not require every component of its stress tensor to vanish: it requires zero traction on the radial boundary.^[ref]

Figure 1.
Tangential stress concentration at a traction-free circular hole
. Original rendering of the exact infinite-plate boundary formula, normalized by positive remote tensile stress. The loading is in the horizontal direction. The highest tangential tension occurs at the top and bottom of the hole; the side points have tangential compression. This is an analytical model, not experimental data. **Details**.

For a finite plate, distinguish gross and net reference areas. A plate of width

W

, thickness

t

and central hole diameter

d

transmits force

P

through gross area

A_{g} = t W

and minimum net area

A_{n} = t (W - d)

. Defining

σ_{g} = P / A_{g}

and

σ_{n} = P / A_{n}

gives the exact conversion

K_{t n} = \frac{σ _{m a x}}{σ _{n}} = K_{t g} \frac{A _{n}}{A _{g}} = K_{t g} (1 - \frac{d}{W}) .

(2)

This is a change of normalization, not a solution for the finite plate's stress field. For an original numerical example, take

W = 40 mm

t = 2 mm

d = 10 mm

and

P = 800 N

. The reference stresses are

10

and

13.333 MPa

. If the local maximum is stipulated as

32 MPa

, the same physical maximum gives

K_{t g} = 3.2

and

K_{t n} = 2.4

. Neither number changes the local stress by choosing a different denominator.

The distinction appears in real historical measurements. In 1939, C. Dumont loaded a perforated aluminum-alloy plate in out-of-plane bending and measured strain and deflection at two loads. The reported peak tangential stress was 1.59 times the computed net-section stress, but 1.85 times the computed stress for an unperforated plate under the same bending forces. These are bending results for that geometry, rather than the infinite-plate tensile factor three.^[ref]

Modern demonstrations can combine a metal plate instrumented with strain gauges and a transparent polymer viewed in a polariscope. The latter uses stress-induced birefringence; its fringes respond to differences of principal stresses. The MIT laboratory compares local and remote strain measurements with qualitative photoelasticity observations around holes and a slot.^[ref]

The elastic stress concentration factor is dimensionless. It differs from the stress intensity factor used to describe a sharp crack's singular field. Local yielding or a crack changes the mechanical problem; multiplying a remote stress by the elastic

K_{t}

does not, by itself, establish a failure load.

Table of contents
- Kirsch equations Stress concentration
- Stress concentration factor Stress concentration

Kirsch equations

 0  0

The Kirsch equations describe the stress tensor around a traction-free circular hole in an infinite linear elastic plate under uniform remote tension. For remote stress

σ_{\infty}

along

x

, hole radius

a

and polar coordinates

(r, θ)

measured from that direction, put

q = a^{2} / r^{2}

. In the classical isotropic plane stress model,

σ_{rr} σ_{θθ} σ_{r θ} = \frac{σ _{\infty}}{2} (1 - q) + \frac{σ _{\infty}}{2} (1 - 4 q + 3 q^{2}) cos 2 θ, = \frac{σ _{\infty}}{2} (1 + q) - \frac{σ _{\infty}}{2} (1 + 3 q^{2}) cos 2 θ, = - \frac{σ _{\infty}}{2} (1 + 2 q - 3 q^{2}) sin 2 θ .

(1)

The fields satisfy equilibrium and the prescribed boundary tractions. They can be obtained using an Airy stress function satisfying the biharmonic compatibility equation.^[ref]

r = a

q = 1

, so

σ_{rr} = σ_{r θ} = 0

while

σ_{θθ} (a, θ) = σ_{\infty} (1 - 2 cos 2 θ) .

(2)

The tangential stress is

3 σ_{\infty}

at the top and bottom of the hole and

- σ_{\infty}

at its sides. A traction-free boundary does not require every stress component to vanish: the outward boundary normal is radial, so radial normal and shear tractions must vanish.

For an original distance comparison along

θ = π /2

, the tangential stress is

σ_{\infty} (1 + q /2 + 3 q^{2} /2)

. At

r = 2 a

it is

1.21875 σ_{\infty}

, already far below the edge value three. At large distance the polar components approach those of the original uniform Cartesian tension.

A finite-width plate, internal hole pressure, plasticity or out-of-plane bending changes the boundary-value problem. The stress concentration factor three is therefore specific to the stated remote-tension model. The MIT laboratory uses local strain measurements and photoelasticity to examine how actual geometries compare with ideal concentration fields.^[ref]

Stress concentration factor ( $K_{t} = σ_{m a x} / σ_{ref}$ )

 0  0

A stress concentration factor compares a specified local peak stress with a defined nominal reference:

K_{t} = \frac{σ _{m a x}}{σ _{ref}} .

(1)

It is dimensionless. Both the stress component and the reference geometry belong in its definition; a number quoted without them can be ambiguous. In the infinite circular-hole Kirsch equations, the maximum tangential tension is three times the remote uniaxial stress, so that particular reference gives

K_{t} = 3

.^[ref]

For an original finite-plate normalization example, let width

W = 50 mm

, thickness

t = 2 mm

, central-hole diameter

d = 10 mm

and tensile force

P = 1000 N

. The gross and minimum-net reference stresses are

σ_{g} = \frac{P}{t W} = 10 MPa, σ_{n} = \frac{P}{t ( W - d )} = 12.5 MPa .

(2)

If the local peak is stipulated as

28 MPa

, the same specimen has

K_{t g} = 2.8

and

K_{t n} = 2.24

. In general

K_{t n} = K_{t g} (1 - d / W)

for these reference definitions. This conversion does not solve the finite plate's stress field or predict its peak.

The reference can also be a nominal bending stress or another stress appropriate to the applied loading. A factor obtained for axial tension cannot automatically be reused for a bending problem with different boundary conditions.

An elastic concentration factor is distinct from the crack-tip stress intensity factor, which has units of stress times square root of length. Nor does

K_{t}

by itself supply fatigue life or a failure load: yielding, defects, residual stress and the loading history require additional analysis. The factor describes amplification within its stated elastic model.

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Stress concentration refers to the occurrence of localized increases in stress within a material, often resulting from geometric discontinuities, such as holes, notches, fillets, or changes in cross-section. When a load is applied to a component, rather than distributing the stress uniformly throughout the material, certain areas may experience significantly higher stress. This can lead to failure at these concentrated locations rather than through the bulk of the material.

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