Stress (mechanics)
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Stress | ||
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SI unit pascal | | |
Other units | psi, bar | |
In SI base units | Pa = kg⋅m−1⋅s−2 | |
Dimension |
Part of a series on |
Continuum mechanics |
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In
Stress expresses the internal forces that neighbouring
Strain inside a material may arise by various mechanisms, such as stress as applied by external forces to the bulk material (like
Significant stress may exist even when deformation is negligible or non-existent (a common assumption when modeling the flow of water). Stress may exist in the absence of external forces; such built-in stress is important, for example, in prestressed concrete and tempered glass. Stress may also be imposed on a material without the application of net forces, for example by changes in temperature or chemical composition, or by external electromagnetic fields (as in piezoelectric and magnetostrictive materials).
The relation between mechanical stress, strain, and the strain rate can be quite complicated, although a linear approximation may be adequate in practice if the quantities are sufficiently small. Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow, fracture, cavitation) or even change its crystal structure and chemical composition.
History
Humans have known about stress inside materials since ancient times. Until the 17th century, this understanding was largely intuitive and empirical, though this did not prevent the development of relatively advanced technologies like the
Over several millennia, architects and builders in particular, learned how to put together carefully shaped wood beams and stone blocks to withstand, transmit, and distribute stress in the most effective manner, with ingenious devices such as the capitals, arches, cupolas, trusses and the flying buttresses of Gothic cathedrals.
Ancient and medieval architects did develop some geometrical methods and simple formulas to compute the proper sizes of pillars and beams, but the scientific understanding of stress became possible only after the necessary tools were invented in the 17th and 18th centuries:
Definition
Stress is defined as the force across a small boundary per unit area of that boundary, for all orientations of the boundary.[7] Derived from a fundamental physical quantity (force) and a purely geometrical quantity (area), stress is also a fundamental quantity, like velocity, torque or energy, that can be quantified and analyzed without explicit consideration of the nature of the material or of its physical causes.
Following the basic premises of continuum mechanics, stress is a
Quantitatively, the stress is expressed by the Cauchy traction vector T defined as the traction force F between adjacent parts of the material across an imaginary separating surface S, divided by the area of S.
Normal and shear
In general, the stress T that a particle P applies on another particle Q across a surface S can have any direction relative to S. The vector T may be regarded as the sum of two components: the
If the normal unit vector n of the surface (pointing from Q towards P) is assumed fixed, the normal component can be expressed by a single number, the
Units
The dimension of stress is that of
Causes and effects
Stress in a material body may be due to multiple physical causes, including external influences and internal physical processes. Some of these agents (like gravity, changes in
Conversely, stress is usually correlated with various effects on the material, possibly including changes in physical properties like
The relation between stress and its effects and causes, including deformation and rate of change of deformation, can be quite complicated (although a linear approximation may be adequate in practice if the quantities are small enough). Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow, fracture, cavitation) or even change its crystal structure and chemical composition.
Simple types
In some situations, the stress within a body may adequately be described by a single number, or by a single vector (a number and a direction). Three such simple stress situations, that are often encountered in engineering design, are the uniaxial normal stress, the simple shear stress, and the isotropic normal stress.[13]
Uniaxial normal
A common situation with a simple stress pattern is when a straight rod, with uniform material and cross section, is subjected to tension by opposite forces of magnitude along its axis. If the system is in equilibrium and not changing with time, and the weight of the bar can be neglected, then through each transversal section of the bar the top part must pull on the bottom part with the same force, F with continuity through the full cross-sectional area, A. Therefore, the stress σ throughout the bar, across any horizontal surface, can be expressed simply by the single number σ, calculated simply with the magnitude of those forces, F, and cross sectional area, A.
This analysis assumes the stress is evenly distributed over the entire cross-section. In practice, depending on how the bar is attached at the ends and how it was manufactured, this assumption may not be valid. In that case, the value = F/A will be only the average stress, called engineering stress or nominal stress. If the bar's length L is many times its diameter D, and it has no gross defects or built-in stress, then the stress can be assumed to be uniformly distributed over any cross-section that is more than a few times D from both ends. (This observation is known as the Saint-Venant's principle).
Normal stress occurs in many other situations besides axial tension and compression. If an elastic bar with uniform and symmetric cross-section is bent in one of its planes of symmetry, the resulting bending stress will still be normal (perpendicular to the cross-section), but will vary over the cross section: the outer part will be under tensile stress, while the inner part will be compressed. Another variant of normal stress is the hoop stress that occurs on the walls of a cylindrical pipe or vessel filled with pressurized fluid.
Shear
Another simple type of stress occurs when a uniformly thick layer of elastic material like glue or rubber is firmly attached to two stiff bodies that are pulled in opposite directions by forces parallel to the layer; or a section of a soft metal bar that is being cut by the jaws of a scissors-like tool. Let F be the magnitude of those forces, and M be the midplane of that layer. Just as in the normal stress case, the part of the layer on one side of M must pull the other part with the same force F. Assuming that the direction of the forces is known, the stress across M can be expressed simply by the single number , calculated simply with the magnitude of those forces, F and the cross sectional area, A.
As in the case of an axially loaded bar, in practice the shear stress may not be uniformly distributed over the layer; so, as before, the ratio F/A will only be an average ("nominal", "engineering") stress. That average is often sufficient for practical purposes.[14]: 292 Shear stress is observed also when a cylindrical bar such as a shaft is subjected to opposite torques at its ends. In that case, the shear stress on each cross-section is parallel to the cross-section, but oriented tangentially relative to the axis, and increases with distance from the axis. Significant shear stress occurs in the middle plate (the "web") of I-beams under bending loads, due to the web constraining the end plates ("flanges").
Isotropic
Another simple type of stress occurs when the material body is under equal compression or tension in all directions. This is the case, for example, in a portion of liquid or gas at rest, whether enclosed in some container or as part of a larger mass of fluid; or inside a cube of elastic material that is being pressed or pulled on all six faces by equal perpendicular forces — provided, in both cases, that the material is homogeneous, without built-in stress, and that the effect of gravity and other external forces can be neglected.
In these situations, the stress across any imaginary internal surface turns out to be equal in magnitude and always directed perpendicularly to the surface independently of the surface's orientation. This type of stress may be called isotropic normal or just isotropic; if it is compressive, it is called hydrostatic pressure or just pressure. Gases by definition cannot withstand tensile stresses, but some liquids may withstand very large amounts of isotropic tensile stress under some circumstances. see Z-tube.
Cylinder
Parts with
General types
Often, mechanical bodies experience more than one type of stress at the same time; this is called combined stress. In normal and shear stress, the magnitude of the stress is maximum for surfaces that are perpendicular to a certain direction , and zero across any surfaces that are parallel to . When the shear stress is zero only across surfaces that are perpendicular to one particular direction, the stress is called biaxial, and can be viewed as the sum of two normal or shear stresses. In the most general case, called triaxial stress, the stress is nonzero across every surface element.
Cauchy tensor
Combined stresses cannot be described by a single vector. Even if the material is stressed in the same way throughout the volume of the body, the stress across any imaginary surface will depend on the orientation of that surface, in a non-trivial way.
Cauchy observed that the stress vector across a surface will always be a
Like any linear map between vectors, the stress tensor can be represented in any chosen Cartesian coordinate system by a 3×3 matrix of real numbers. Depending on whether the coordinates are numbered or named , the matrix may be written as
The linear relation between and follows from the fundamental laws of
Change of coordinates
The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr's circle of stress distribution.
As a symmetric 3×3 real matrix, the stress tensor has three mutually orthogonal unit-length eigenvectors and three real eigenvalues , such that . Therefore, in a coordinate system with axes , the stress tensor is a diagonal matrix, and has only the three normal components the
Tensor field
In general, stress is not uniformly distributed over a material body, and may vary with time. Therefore, the stress tensor must be defined for each point and each moment, by considering an infinitesimal particle of the medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point.
Thin plates
Human-made objects are often made from stock plates of various materials by operations that do not change their essentially two-dimensional character, like cutting, drilling, gentle bending and welding along the edges. The description of stress in such bodies can be simplified by modeling those parts as two-dimensional surfaces rather than three-dimensional bodies.
In that view, one redefines a "particle" as being an infinitesimal patch of the plate's surface, so that the boundary between adjacent particles becomes an infinitesimal line element; both are implicitly extended in the third dimension, normal to (straight through) the plate. "Stress" is then redefined as being a measure of the internal forces between two adjacent "particles" across their common line element, divided by the length of that line. Some components of the stress tensor can be ignored, but since particles are not infinitesimal in the third dimension one can no longer ignore the torque that a particle applies on its neighbors. That torque is modeled as a bending stress that tends to change the
Thin beams
The analysis of stress can be considerably simplified also for thin bars, beams or wires of uniform (or smoothly varying) composition and cross-section that are subjected to moderate bending and twisting. For those bodies, one may consider only cross-sections that are perpendicular to the bar's axis, and redefine a "particle" as being a piece of wire with infinitesimal length between two such cross sections. The ordinary stress is then reduced to a scalar (tension or compression of the bar), but one must take into account also a bending stress (that tries to change the bar's curvature, in some direction perpendicular to the axis) and a torsional stress (that tries to twist or un-twist it about its axis).
Analysis
Goals and assumptions
Stress analysis is generally concerned with objects and structures that can be assumed to be in macroscopic
In stress analysis one normally disregards the physical causes of the forces or the precise nature of the materials. Instead, one assumes that the stresses are related to deformation (and, in non-static problems, to the rate of deformation) of the material by known
Methods
Stress analysis may be carried out experimentally, by applying loads to the actual artifact or to scale model, and measuring the resulting stresses, by any of several available methods. This approach is often used for safety certification and monitoring. Most stress is analysed by mathematical methods, especially during design. The basic stress analysis problem can be formulated by
Stress analysis for
Stress analysis is simplified when the physical dimensions and the distribution of loads allow the structure to be treated as one- or two-dimensional. In the analysis of trusses, for example, the stress field may be assumed to be uniform and uniaxial over each member. Then the differential equations reduce to a finite set of equations (usually linear) with finitely many unknowns. In other contexts one may be able to reduce the three-dimensional problem to a two-dimensional one, and/or replace the general stress and strain tensors by simpler models like uniaxial tension/compression, simple shear, etc.
Still, for two- or three-dimensional cases one must solve a partial differential equation problem. Analytical or closed-form solutions to the differential equations can be obtained when the geometry, constitutive relations, and boundary conditions are simple enough. Otherwise one must generally resort to numerical approximations such as the finite element method, the finite difference method, and the boundary element method.
Measures
Other useful stress measures include the first and second
See also
Conjugate variables of thermodynamics | ||||||||
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- Bending
- Compressive strength
- Critical plane analysis
- Kelvin probe force microscope
- Mohr's circle
- Lamé's stress ellipsoid
- Reinforced solid
- Residual stress
- Shear strength
- Shot peening
- Strain
- Strain tensor
- Strain rate tensor
- Stress–energy tensor
- Stress–strain curve
- Stress concentration
- Transient friction loading
- Tensile strength
- Thermal stress
- Virial stress
- Yield (engineering)
- Yield surface
- Virial theorem
References
- ^ a b "12.3 Stress, Strain, and Elastic Modulus - University Physics Volume 1 | OpenStax". openstax.org. Retrieved 2022-11-02.
- ^ "Class Physical-Quantity in theory Physical-Quantities". www-ksl.stanford.edu. Retrieved 2022-11-02.
- ^ a b "What is Shear Stress - Materials - Definition". Material Properties. 2020-07-31. Retrieved 2022-11-02.
- ISBN 0306812835.
- ISBN 0-486-46290-0
- ^ https://archive.org/details/historyofstrengt0000timo_k8r2/page/110/mode/2up, pp.107-110
- ISBN 1-932159-75-4
- ISBN 0-486-40180-4. pages
- ^ ISBN 3-540-43019-9
- ^ (2009) The art of making glass. Lamberts Glashütte (LambertsGlas) product brochure. Accessed on 2013-02-08.
- .
- ^ Sharma, B and Kumar, R "Estimation of bulk viscosity of dilute gases using a nonequilibrium molecular dynamics approach.", Physical Review E,100, 013309 (2019)
- ^ ISBN 9781574447132
- ^ Walter D. Pilkey, Orrin H. Pilkey (1974), "Mechanics of solids" (book)
- ISBN 0-7923-2454-4
- ISBN 3-540-74297-2
- ISBN 978-0-8176-4117-7
Further reading
- Chakrabarty, J. (2006). Theory of plasticity (3 ed.). Butterworth-Heinemann. pp. 17–32. ISBN 0-7506-6638-2.
- Beer, Ferdinand Pierre; Elwood Russell Johnston; John T. DeWolf (1992). Mechanics of Materials. McGraw-Hill Professional. ISBN 0-07-112939-1.
- Brady, B.H.G.; E.T. Brown (1993). Rock Mechanics For Underground Mining (Third ed.). Kluwer Academic Publisher. pp. 17–29. ISBN 0-412-47550-2.
- Chen, Wai-Fah; Baladi, G.Y. (1985). Soil Plasticity, Theory and Implementation. ISBN 0-444-42455-5.
- Chou, Pei Chi; Pagano, N.J. (1992). Elasticity: tensor, dyadic, and engineering approaches. Dover books on engineering. Dover Publications. pp. 1–33. ISBN 0-486-66958-0.
- Davis, R. O.; Selvadurai. A. P. S. (1996). Elasticity and geomechanics. Cambridge University Press. pp. 16–26. ISBN 0-521-49827-9.
- Dieter, G. E. (3 ed.). (1989). Mechanical Metallurgy. New York: McGraw-Hill. ISBN 0-07-100406-8.
- Holtz, Robert D.; Kovacs, William D. (1981). An introduction to geotechnical engineering. Prentice-Hall civil engineering and engineering mechanics series. Prentice-Hall. ISBN 0-13-484394-0.
- Jones, Robert Millard (2008). Deformation Theory of Plasticity. Bull Ridge Corporation. pp. 95–112. ISBN 978-0-9787223-1-9.
- Jumikis, Alfreds R. (1969). Theoretical soil mechanics: with practical applications to soil mechanics and foundation engineering. Van Nostrand Reinhold Co. ISBN 0-442-04199-3.
- Landau, L.D. and E.M.Lifshitz. (1959). Theory of Elasticity.
- Love, A. E. H. (4 ed.). (1944). Treatise on the Mathematical Theory of Elasticity. New York: Dover Publications. ISBN 0-486-60174-9.
- Marsden, J. E.; Hughes, T. J. R. (1994). Mathematical Foundations of Elasticity. Dover Publications. pp. 132–142. ISBN 0-486-67865-2.
- Parry, Richard Hawley Grey (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. 1–30. ISBN 0-415-27297-1.
- Rees, David (2006). Basic Engineering Plasticity – An Introduction with Engineering and Manufacturing Applications. Butterworth-Heinemann. pp. 1–32. ISBN 0-7506-8025-3.
- ISBN 0-07-085805-5.
- Timoshenko, Stephen P. (1983). History of strength of materials: with a brief account of the history of theory of elasticity and theory of structures. Dover Books on Physics. Dover Publications. ISBN 0-486-61187-6.