# Stress concentration

Last updated

A stress concentration (also called a stress raiser or a stress riser) is a location in an object where the stress is significantly greater than the surrounding region. Stress concentrations occur when there are irregularities in the geometry or material of a structural component that cause an interruption to the flow of stress. This arises from such details as holes, grooves, notches and fillets. Stress concentrations may also occur from accidental damage such as nicks and scratches.

## Contents

The degree of concentration of a discontinuity under typically tensile loads can be expressed as a non-dimensional stress concentration factor${\displaystyle K_{t}}$, which is the ratio of the highest stress to the nominal far field stress. For a circular hole in an infinite plate, ${\displaystyle K_{t}=3}$. [1] The stress concentration factor should not be confused with the stress intensity factor, which is used to define the effect of a crack on the stresses in the region around a crack tip. [2]

For ductile materials, large loads can cause localised plastic deformation or yielding that will typically occur first at a stress concentration allowing a redistribution of stress and enabling the component to continue to carry load. Brittle materials will typically fail at the stress concentration. However, repeated low level loading may cause a fatigue crack to initiate and slowly grow at a stress concentration leading to the failure of even ductile materials. Fatigue cracks always start at stress raisers, so removing such defects increases the fatigue strength.

## Description

Stress concentrations occur when there are irregularities in the geometry or material of a structural component that cause an interruption to the flow of stress.

Geometric discontinuities cause an object to experience a localised increase in stress. Examples of shapes that cause stress concentrations are sharp internal corners, holes, and sudden changes in the cross-sectional area of the object as well as unintentional damage such as nicks, scratches and cracks. High local stresses can cause objects to fail more quickly, so engineers typically design the geometry to minimize stress concentrations.

Material discontinuities, such as inclusions in metals, may also concentrate the stress. Inclusions on the surface of a component may be broken from machining during manufacture leading to microcracks that grow in service from cyclic loading. Internally, the failure of the interfaces around inclusions during loading may lead to static failure by microvoid coalescence.

## Stress concentration factor

The stress concentration factor, ${\displaystyle K_{t}}$, is the ratio of the highest stress ${\displaystyle \sigma _{\max }}$ to a nominal stress ${\displaystyle \sigma _{\text{nom}}}$ of the gross cross-section and defined as [3]

${\displaystyle K_{t}={\frac {\sigma _{\max }}{\sigma _{\text{nom}}}}}$

Note that the dimensionless stress concentration factor is a function of the geometry shape and independent of its size. [4] These factors can be found in typical engineering reference materials.

E. Kirsch derived the equations for the elastic stress distribution around a hole. The maximum stress felt near a hole or notch occurs in the area of lowest radius of curvature. In an elliptical hole of length ${\displaystyle 2a}$ and width ${\displaystyle 2b}$, under a nominal or far-field stress ${\displaystyle \sigma _{\text{nom}}}$, the stress at the ends of the major axes is given by Inglis' equation: [5]

${\displaystyle \sigma _{\max }=\sigma _{\text{nom}}\left(1+2{\cfrac {a}{b}}\right)=\sigma \left(1+2{\sqrt {\cfrac {a}{\rho }}}\right)}$

where ${\displaystyle \rho }$ is the radius of curvature of the elliptical hole. For circular holes in an infinite plate where ${\displaystyle a=b}$, the stress concentration factor is ${\displaystyle K_{t}=3}$.

As the radius of curvature approaches zero, such as at the tip of a sharp crack, the maximum stress approaches infinity and a stress concentration factor cannot therefore be used for a crack. Instead, the stress intensity factor which defines the scaling of the stress field around a crack tip, is used. [2]

## Methods for determining factors

There are experimental methods for measuring stress concentration factors including photoelastic stress analysis, thermoelastic stress analysis, [6] brittle coatings or strain gauges.

During the design phase, there are multiple approaches to estimating stress concentration factors. Several catalogs of stress concentration factors have been published. [7] Perhaps most famous is Stress Concentration Design Factors by Peterson, first published in 1953. [8] [9] Finite element methods are commonly used in design today.

## Limiting the effects of stress concentrations

Known as crack tip blunting, a counter-intuitive method of reducing one of the worst types of stress concentrations, a crack, is to drill a large hole at the end of the crack. The drilled hole, with its relatively large size, serves to increase the effective crack tip radius and thus reduce the stress concentration. [4]

Another method used to decrease the stress concentration is by adding a fillet to internal corners. This reduces the stress concentration and results in smoother flow of stress streamlines.

In a threaded component, the force flow line is bent as it passes from shank portion to threaded portion; as a result, stress concentration takes place. To reduce this, a small undercut is made between the shank and threaded portions.

## Examples

• The de Havilland Comet aircraft experienced a number of catastrophic failures that were eventually found to be due to fatigue cracks growing from the high stress concentration caused by the use of punched rivet holes around the automatic direction finder cutouts (sometimes referred to as windows). The square passenger windows were also found to have higher stress concentrations than expected and were redesigned.
• Brittle fractures at the corners of hatches in Liberty ships in cold and stressful conditions in winter storms in the Atlantic Ocean.
• A focus point of stress on an implanted orthosis is very likely to be its point of failure.

## Related Research Articles

In engineering, deformation refers to the change in size or shape of an object. Displacements are the absolute change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strain is the relative internal change in shape of an infinitesimally small cube of material and can be expressed as a non-dimensional change in length or angle of distortion of the cube. Strains are related to the forces acting on the cube, which are known as stress, by a stress-strain curve. The relationship between stress and strain is generally linear and reversible up until the yield point and the deformation is elastic. The linear relationship for a material is known as Young's modulus. Above the yield point, some degree of permanent distortion remains after unloading and is termed plastic deformation. The determination of the stress and strain throughout a solid object is given by the field of strength of materials and for a structure by structural analysis.

Fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If a displacement develops perpendicular to the surface of displacement, it is called a normal tensile crack or simply a crack; if a displacement develops tangentially to the surface of displacement, it is called a shear crack, slip band, or dislocation.

Strength of materials, also called mechanics of materials, deals with the behavior of solid objects subject to stresses and strains. The complete theory began with the consideration of the behavior of one and two dimensional members of structures, whose states of stress can be approximated as two dimensional, and was then generalized to three dimensions to develop a more complete theory of the elastic and plastic behavior of materials. An important founding pioneer in mechanics of materials was Stephen Timoshenko.

In materials science, fatigue is the weakening of a material caused by cyclic loading that results in progressive and localized structural damage and the growth of cracks. Once a fatigue crack has initiated, it will grow a small amount with each loading cycle, typically producing striations on some parts of the fracture surface. The crack will continue to grow until it reaches a critical size, which occurs when the stress intensity factor of the crack exceeds the fracture toughness of the material, producing rapid propagation and typically complete fracture of the structure.

In structural engineering, buckling is the sudden change in shape (deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have buckled. Euler's critical load and Johnson's parabolic formula are used to determine the buckling stress in slender columns.

Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.

The stress intensity factor, , is used in fracture mechanics to predict the stress state near the tip of a crack or notch caused by a remote load or residual stresses. It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials, and is a critical technique in the discipline of damage tolerance. The concept can also be applied to materials that exhibit small-scale yielding at a crack tip.

In materials science, fracture toughness is the critical stress intensity factor of a sharp crack where propagation of the crack suddenly becomes rapid and unlimited. A component's thickness affects the constraint conditions at the tip of a crack with thin components having plane stress conditions and thick components having plane strain conditions. Plane strain conditions give the lowest fracture toughness value which is a material property. The critical value of stress intensity factor in mode I loading measured under plane strain conditions is known as the plane strain fracture toughness, denoted . When a test fails to meet the thickness and other test requirements that are in place to ensure plane strain conditions, the fracture toughness value produced is given the designation . Fracture toughness is a quantitative way of expressing a material's resistance to crack propagation and standard values for a given material are generally available.

The three-point bending flexural test provides values for the modulus of elasticity in bending , flexural stress , flexural strain and the flexural stress–strain response of the material. This test is performed on a universal testing machine with a three-point or four-point bend fixture.The main advantage of a three-point flexural test is the ease of the specimen preparation and testing. However, this method has also some disadvantages: the results of the testing method are sensitive to specimen and loading geometry and strain rate.

The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material. The theoretical concept of J-integral was developed in 1967 by G. P. Cherepanov and independently in 1968 by James R. Rice, who showed that an energetic contour path integral was independent of the path around a crack.

Paris' law is a crack growth equation that gives the rate of growth of a fatigue crack. The stress intensity factor characterises the load around a crack tip and the rate of crack growth is experimentally shown to be a function of the range of stress intensity seen in a loading cycle. The Paris equation is

Material failure theory is the science of predicting the conditions under which solid materials fail under the action of external loads. The failure of a material is usually classified into brittle failure (fracture) or ductile failure (yield). Depending on the conditions most materials can fail in a brittle or ductile manner or both. However, for most practical situations, a material may be classified as either brittle or ductile. Though failure theory has been in development for over 200 years, its level of acceptability is yet to reach that of continuum mechanics.

Thermo-mechanical fatigue is the overlay of a cyclical mechanical loading, that leads to fatigue of a material, with a cyclical thermal loading. Thermo-mechanical fatigue is an important point that needs to be considered, when constructing turbine engines or gas turbines.

According to the classical theories of elastic or plastic structures made from a material with non-random strength (ft), the nominal strength (σN) of a structure is independent of the structure size (D) when geometrically similar structures are considered. Any deviation from this property is called the size effect. For example, conventional strength of materials predicts that a large beam and a tiny beam will fail at the same stress if they are made of the same material. In the real world, because of size effects, a larger beam will fail at a lower stress than a smaller beam.

Polymer fracture is the study of the fracture surface of an already failed material to determine the method of crack formation and extension in polymers both fiber reinforced and otherwise. Failure in polymer components can occur at relatively low stress levels, far below the tensile strength because of four major reasons: long term stress or creep rupture, cyclic stresses or fatigue, the presence of structural flaws and stress-cracking agents. Formations of submicroscopic cracks in polymers under load have been studied by x ray scattering techniques and the main regularities of crack formation under different loading conditions have been analyzed. The low strength of polymers compared to theoretically predicted values are mainly due to the many microscopic imperfections found in the material. These defects namely dislocations, crystalline boundaries, amorphous interlayers and block structure can all lead to the non-uniform distribution of mechanical stress.

The cohesive zone model (CZM) is a model in fracture mechanics in which fracture formation is regarded as a gradual phenomenon in which separation of the surfaces involved in the crack takes place across an extended crack tip, or cohesive zone, and is resisted by cohesive tractions. The origin of this model can be traced back to the early sixties by Barenblatt (1962) and Dugdale (1960) to represent nonlinear processes located at the front of a pre-existent crack.

Concrete is widely used construction material all over the world. It is composed of aggregate, cement and water. Composition of concrete varies to suit for different applications desired. Even size of the aggregate can influence mechanical properties of concrete to a great extent.

Crack tip opening displacement (CTOD) or is the distance between the opposite faces of a crack tip at the 90° intercept position. The position behind the crack tip at which the distance is measured is arbitrary but commonly used is the point where two 45° lines, starting at the crack tip, intersect the crack faces. The parameter is used in fracture mechanics to characterize the loading on a crack and can be related to other crack tip loading parameters such as the stress intensity factor and the elastic-plastic J-integral.

A crack growth equation is used for calculating the size of a fatigue crack growing from cyclic loads. The growth of fatigue cracks can result in catastrophic failure, particularly in the case of aircraft. A crack growth equation can be used to ensure safety, both in the design phase and during operation, by predicting the size of cracks. In critical structure, loads can be recorded and used to predict the size of cracks to ensure maintenance or retirement occurs prior to any of the cracks failing.

Fastran is a computer program for calculating the rate of fatigue crack growth by combining crack growth equations and a simulation of the plasticity at the crack tip.

## References

1. "Stress Concentrations at Holes".
2. Schijve, Jaap (2001). Fatigue of Structures and Materials. Springer. p. 90. ISBN   978-0792370147.
3. Shigley, Joseph Edward (1977). Mechanical Engineering Design (Third ed.). McGraw-Hill.
4. stress at round-tip notches an improved solution
5. "Stresses At Elliptical Holes" . Retrieved 2020-03-13.
6. Rajic, Nik; Street, Neil (2014). "A performance comparison between cooled and uncooled infrared detectors for thermoelastic stress analysis". Quantitative InfraRed Thermography Journal. Taylor & Francis. 11 (2): 207–221. doi:10.1080/17686733.2014.962835.
7. ESDU64001: Guide to stress concentration data. ESDU. ISBN   1-86246-279-8.
8. Peterson, Rudolf Earl (1953). Stress Concentration Design Factors. John Wiley & Sons. ISBN   978-0471683766.
9. Pilkey, Walter D. (1999). Peterson's Stress Concentration Factors (2nd ed.). Wiley. ISBN   0-471-53849-3.