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60-513: These equations relate the amplitudes of P-waves and S-waves at each side of an interface, between two arbitrary elastic media , as a function of the angle of incidence and are largely used in reflection seismology for determining structure and properties of the subsurface. Zoeppritz was born on 22 October 1881 in Mergelstetten, a small village several miles south of Heidenheim an der Brenz . He studied natural science and geology at

120-412: A deformation mechanism map . Permanent deformation is irreversible; the deformation stays even after removal of the applied forces, while the temporary deformation is recoverable as it disappears after the removal of applied forces. Temporary deformation is also called elastic deformation, while the permanent deformation is called plastic deformation. The study of temporary or elastic deformation in

180-471: A yield surface or a yield criterion . A variety of yield criteria have been developed for different materials. It is often difficult to precisely define yielding due to the wide variety of stress–strain curves exhibited by real materials. In addition, there are several possible ways to define yielding: Yielded structures have a lower stiffness, leading to increased deflections and decreased buckling strength. The structure will be permanently deformed when

240-404: A coil, are caused by the coiling process. When these conditions are undesirable, it is essential for suppliers to be informed to provide appropriate materials. The presence of YPE is influenced by chemical composition and mill processing methods such as skin passing or temper rolling, which temporarily eliminate YPE and improve surface quality. However, YPE can return over time due to aging, which

300-467: A distinct upper yield point or a delay in work hardening. These tensile testing phenomena, wherein the strain increases but stress does not increase as expected, are two types of yield point elongation. Yield Point Elongation (YPE) significantly impacts the usability of steel. In the context of tensile testing and the engineering stress-strain curve, the Yield Point is the initial stress level, below

360-413: A material with a large plastic deformation range is wet chewing gum , which can be stretched to dozens of times its original length. Under tensile stress, plastic deformation is characterized by a strain hardening region and a necking region and finally, fracture (also called rupture). During strain hardening the material becomes stronger through the movement of atomic dislocations . The necking phase

420-645: A nonlinear fashion. For these materials Hooke's law is inapplicable. This type of deformation is not undone simply by removing the applied force. An object in the plastic deformation range, however, will first have undergone elastic deformation, which is undone simply be removing the applied force, so the object will return part way to its original shape. Soft thermoplastics have a rather large plastic deformation range as do ductile metals such as copper , silver , and gold . Steel does, too, but not cast iron . Hard thermosetting plastics, rubber, crystals, and ceramics have minimal plastic deformation ranges. An example of

480-442: A number of applications including hydrocarbon exploration . In the summer of 1908, at the age of 26, Zoeppritz died after succumbing to an infectious illness of several months. He accomplished all of his work at Göttingen in just two years, but much of it was left unpublished at the time of his death. Much of the unpublished work was revised and published by Wiechert, Gutenberg and Geiger, with his most important work that described

540-578: A plane wave approaching a discontinuity. Zoeppritz was not the first to mathematically describe this phenomenon as the British seismologist Cargill Gilston Knott used a different approach to derive Knott's equations in 1899, but this was still unknown in Germany by the 1920s. The equations that Zoeppritz described are now named after him ( Zoeppritz equations ) and are used extensively in reflection seismology (particularly amplitude versus offset ), for

600-440: A point defining true stress–strain curve is displaced upwards and to the left to define the equivalent engineering stress–strain curve. The difference between the true and engineering stresses and strains will increase with plastic deformation. At low strains (such as elastic deformation), the differences between the two is negligible. As for the tensile strength point, it is the maximal point in engineering stress–strain curve but

660-481: A result, the material is forced out laterally. Internal forces (in this case at right angles to the deformation) resist the applied load. Depending on the type of material, size and geometry of the object, and the forces applied, various types of deformation may result. The image to the right shows the engineering stress vs. strain diagram for a typical ductile material such as steel. Different deformation modes may occur under different conditions, as can be depicted using

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720-526: A smaller elastic range. Linear elastic deformation is governed by Hooke's law , which states: where This relationship only applies in the elastic range and indicates that the slope of the stress vs. strain curve can be used to find Young's modulus ( E ). Engineers often use this calculation in tensile tests. The area under this elastic region is known as resilience. Note that not all elastic materials undergo linear elastic deformation; some, such as concrete , gray cast iron , and many polymers, respond in

780-530: A structural element or specimen will increase the compressive stress until it reaches its compressive strength . According to the properties of the material, failure modes are yielding for materials with ductile behavior (most metals , some soils and plastics ) or rupturing for brittle behavior (geomaterials, cast iron , glass , etc.). In long, slender structural elements — such as columns or truss bars — an increase of compressive force F leads to structural failure due to buckling at lower stress than

840-417: A structure by structural analysis . In the above figure, it can be seen that the compressive loading (indicated by the arrow) has caused deformation in the cylinder so that the original shape (dashed lines) has changed (deformed) into one with bulging sides. The sides bulge because the material, although strong enough to not crack or otherwise fail, is not strong enough to support the load without change. As

900-406: Is a material property and is the stress corresponding to the yield point at which the material begins to deform plastically. The yield strength is often used to determine the maximum allowable load in a mechanical component, since it represents the upper limit to forces that can be applied without producing permanent deformation. For most metals, such as aluminium and cold-worked steel , there

960-409: Is a gradual onset of non-linear behavior, and no precise yield point. In such a case, the offset yield point (or proof stress ) is taken as the stress at which 0.2% plastic deformation occurs. Yielding is a gradual failure mode which is normally not catastrophic , unlike ultimate failure . For ductile materials, the yield strength is typically distinct from the ultimate tensile strength , which

1020-414: Is a measure of a material's work hardening behavior. Materials with a higher n have a greater resistance to necking. Typically, metals at room temperature have n ranging from 0.02 to 0.5. Since we disregard the change of area during deformation above, the true stress and strain curve should be re-derived. For deriving the stress strain curve, we can assume that the volume change is 0 even if we deformed

1080-402: Is experiencing a stress defined to be the ratio of the force to the cross sectional area of the bar, as well as an axial elongation: Subscript 0 denotes the original dimensions of the sample. The SI derived unit for stress is newtons per square metre, or pascals (1 pascal = 1 Pa = 1 N/m ), and strain is unitless . The stress–strain curve for this material is plotted by elongating

1140-555: Is extremely sensitive to the materials processing as well. These mechanisms for crystalline materials include Where deforming the material will introduce dislocations , which increases their density in the material. This increases the yield strength of the material since now more stress must be applied to move these dislocations through a crystal lattice. Dislocations can also interact with each other, becoming entangled. The governing formula for this mechanism is: where σ y {\displaystyle \sigma _{y}}

1200-399: Is indicated by a reduction in cross-sectional area of the specimen. Necking begins after the ultimate strength is reached. During necking, the material can no longer withstand the maximum stress and the strain in the specimen rapidly increases. Plastic deformation ends with the fracture of the material. Usually, compressive stress applied to bars, columns , etc. leads to shortening. Loading

1260-400: Is not a special point in true stress–strain curve. Because engineering stress is proportional to the force applied along the sample, the criterion for necking formation can be set as δ F = 0. {\displaystyle \delta F=0.} This analysis suggests nature of the ultimate tensile strength (UTS) point. The work strengthening effect is exactly balanced by

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1320-432: Is often done to eliminate ambiguity. However, it is possible to obtain stress-strain curves from indentation-based procedures, provided certain conditions are met. These procedures are grouped under the term Indentation plastometry . There are several ways in which crystalline materials can be engineered to increase their yield strength. By altering dislocation density, impurity levels, grain size (in crystalline materials),

1380-462: Is recorded using mechanical or optical extensometers. Indentation hardness correlates roughly linearly with tensile strength for most steels, but measurements on one material cannot be used as a scale to measure strengths on another. Hardness testing can therefore be an economical substitute for tensile testing, as well as providing local variations in yield strength due to, e.g., welding or forming operations. For critical situations, tension testing

1440-445: Is related to the resistance toward the necking. Usually, the value of m {\displaystyle m} is at the range of 0-0.1 at room temperature and as high as 0.8 when the temperature is increased. By combining the 1) and 2), we can create the ultimate relation as below: Where K ″ {\displaystyle K''} is the global constant for relating strain, strain rate and stress. 3) Based on

1500-407: Is significantly lower than the expected theoretical value can be explained by the presence of dislocations and defects in the materials. Indeed, whiskers with perfect single crystal structure and defect-free surfaces have been shown to demonstrate yield stress approaching the theoretical value. For example, nanowhiskers of copper were shown to undergo brittle fracture at 1 GPa, a value much higher than

1560-399: Is strain-hardening coefficient. Usually, the value of n {\displaystyle n} has range around 0.02 to 0.5 at room temperature. If n {\displaystyle n} is 1, we can express this material as perfect elastic material. 2) In reality, stress is also highly dependent on the rate of strain variation. Thus, we can induce the empirical equation based on

1620-402: Is the concentration of solute and ϵ {\displaystyle \epsilon } is the strain induced in the lattice due to adding the impurity. Where the presence of a secondary phase will increase yield strength by blocking the motion of dislocations within the crystal. A line defect that, while moving through the matrix, will be forced against a small particle or precipitate of

1680-533: Is the load-bearing capacity for a given material. The ratio of yield strength to ultimate tensile strength is an important parameter for applications such steel for pipelines , and has been found to be proportional to the strain hardening exponent . In solid mechanics , the yield point can be specified in terms of the three-dimensional principal stresses ( σ 1 , σ 2 , σ 3 {\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}} ) with

1740-420: Is the surface tension between the matrix and the particle, l interparticle {\displaystyle l_{\text{interparticle}}\,} is the distance between the particles. Where a buildup of dislocations at a grain boundary causes a repulsive force between dislocations. As grain size decreases, the surface area to volume ratio of the grain increases, allowing more buildup of dislocations at

1800-465: Is the yield stress, G is the shear elastic modulus, b is the magnitude of the Burgers vector , and ρ {\displaystyle \rho } is the dislocation density. By alloying the material, impurity atoms in low concentrations will occupy a lattice position directly below a dislocation, such as directly below an extra half plane defect. This relieves a tensile strain directly below

1860-440: The 1906 San Francisco earthquake , Zoeppritz's most important early contribution was the construction of travel-time curves – and their associated velocity-depth functions – for P-waves , S-waves and surface waves , recognising for the first time that body waves are reflected and converted at discontinuities. These curves were later used by other members of the research group, Ludwig Carl Geiger and Beno Gutenberg , as well as

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1920-462: The yield point is the point on a stress–strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed, some fraction of the deformation will be permanent and non-reversible and is known as plastic deformation . The yield strength or yield stress

1980-512: The British astronomer and seismologist Herbert Hall Turner at the International Seismological Summary. The related ill-posed inverse problem of inferring a discrete velocity distribution, representing the layers of the crust and mantle, was solved by fellow Göttingen mathematician Gustav Herglotz . Zoeppritz used Wiechert's work and went on to derive a full set of transmission and reflection coefficients for

2040-469: The application of physics in geology, but the field of geophysics was still in its infancy. The only place in Germany where Zoeppritz could specifically study geophysics was at the University of Göttingen , as an assistant in the influential Emil Wiechert 's research group. Using Wiechert's theoretical work and data from earthquakes including the 1905 Kangra earthquake , 1905 Calabria earthquake and

2100-510: The case of engineering strain is applied to materials used in mechanical and structural engineering, such as concrete and steel , which are subjected to very small deformations. Engineering strain is modeled by infinitesimal strain theory , also called small strain theory , small deformation theory , small displacement theory , or small displacement-gradient theory where strains and rotations are both small. For some materials, e.g. elastomers and polymers, subjected to large deformations,

2160-405: The compressive strength. A break occurs after the material has reached the end of the elastic, and then plastic, deformation ranges. At this point forces accumulate until they are sufficient to cause a fracture. All materials will eventually fracture, if sufficient forces are applied. Engineering stress and engineering strain are approximations to the internal state that may be determined from

2220-433: The deformation is negligible, the object is said to be rigid . Occurrence of deformation in engineering applications is based on the following background concepts: The relationship between stress and strain is generally linear and reversible up until the yield point and the deformation is elastic . Elasticity in materials occurs when applied stress does not surpass the energy required to break molecular bonds, allowing

2280-414: The dimensions are instantaneous values. Assuming volume of the sample conserves and deformation happens uniformly, The true stress and strain can be expressed by engineering stress and strain. For true stress, For the strain, Integrate both sides and apply the boundary condition, So in a tension test , true stress is larger than engineering stress and true strain is less than engineering strain. Thus,

2340-428: The dislocation by filling that empty lattice space with the impurity atom. The relationship of this mechanism goes as: where τ {\displaystyle \tau } is the shear stress , related to the yield stress, G {\displaystyle G} and b {\displaystyle b} are the same as in the above example, C s {\displaystyle C_{s}}

2400-486: The engineering definition of strain is not applicable, e.g. typical engineering strains greater than 1%, thus other more complex definitions of strain are required, such as stretch , logarithmic strain , Green strain , and Almansi strain . Elastomers and shape memory metals such as Nitinol exhibit large elastic deformation ranges, as does rubber . However, elasticity is nonlinear in these materials. Normal metals, ceramics and most crystals show linear elasticity and

2460-416: The external forces and deformations of an object, provided that there is no significant change in size. When there is a significant change in size, the true stress and true strain can be derived from the instantaneous size of the object. Consider a bar of original cross sectional area A 0 being subjected to equal and opposite forces F pulling at the ends so the bar is under tension. The material

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2520-462: The grain edge. Since it requires a lot of energy to move dislocations to another grain, these dislocations build up along the boundary, and increase the yield stress of the material. Also known as Hall-Petch strengthening, this type of strengthening is governed by the formula: where The theoretical yield strength of a perfect crystal is much higher than the observed stress at the initiation of plastic flow. That experimentally measured yield strength

2580-400: The lattice energy and move the atoms in the top plane over the lower atoms and into a new lattice site. The applied stress to overcome the resistance of a perfect lattice to shear is the theoretical yield strength, τ max . The stress displacement curve of a plane of atoms varies sinusoidally as stress peaks when an atom is forced over the atom below and then falls as the atom slides into

2640-449: The load is removed, and may have residual stresses. Engineering metals display strain hardening, which implies that the yield stress is increased after unloading from a yield state. Yield strength testing involves taking a small sample with a fixed cross-section area and then pulling it with a controlled, gradually increasing force until the sample changes shape or breaks. This is called a tensile test. Longitudinal and/or transverse strain

2700-410: The material to deform reversibly and return to its original shape once the stress is removed. 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

2760-508: The material. Dislocations can move through this particle either by shearing the particle or by a process known as bowing or ringing, in which a new ring of dislocations is created around the particle. The shearing formula goes as: and the bowing/ringing formula: In these formulas, r particle {\displaystyle r_{\text{particle}}\,} is the particle radius, γ particle-matrix {\displaystyle \gamma _{\text{particle-matrix}}\,}

2820-437: The materials. We can assume that: Then, the true stress can be expressed as below: Additionally, the true strain ε T can be expressed as below: Then, we can express the value as Thus, we can induce the plot in terms of σ T {\displaystyle \sigma _{T}} and ε E {\displaystyle \varepsilon _{E}} as right figure. Additionally, based on

2880-647: The maximum stress, at which an increase in strain occurs without an increase in stress. This characteristic is typical of certain materials, indicating the presence of YPE. The mechanism for YPE has been related to carbon diffusion, and more specifically to Cottrell atmospheres . YPE can lead to issues such as coil breaks, edge breaks, fluting, stretcher strain, and reel kinks or creases, which can affect both aesthetics and flatness. Coil and edge breaks may occur during either initial or subsequent customer processing, while fluting and stretcher strain arise during forming. Reel kinks, transverse ridges on successive inner wraps of

2940-393: The necking appears. Additionally, we can induce various relation based on true stress-strain curve. 1) True strain and stress curve can be expressed by the approximate linear relationship by taking a log on true stress and strain. The relation can be expressed as below: Where K {\displaystyle K} is stress coefficient and n {\displaystyle n}

3000-743: The next lattice point. where b {\displaystyle b} is the interatomic separation distance. Since τ = G γ and dτ/dγ = G at small strains (i.e. Single atomic distance displacements), this equation becomes: For small displacement of γ=x/a, where a is the spacing of atoms on the slip plane, this can be rewritten as: Giving a value of τ max {\displaystyle \tau _{\max }} τ max equal to: The theoretical yield strength can be approximated as τ max = G / 30 {\displaystyle \tau _{\max }=G/30} . During monotonic tensile testing, some metals such as annealed steel exhibit

3060-545: The original cross-section and gauge length is called the engineering stress–strain curve , while the curve based on the instantaneous cross-section area and length is called the true stress–strain curve . Unless stated otherwise, engineering stress–strain is generally used. In the above definitions of engineering stress and strain, two behaviors of materials in tensile tests are ignored: True stress and true strain are defined differently than engineering stress and strain to account for these behaviors. They are given as Here

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3120-614: The reflection and transmission of seismic waves in elastic media – the Zoeppritz equations – not being published until 1919. In 2002, the German Geophysical Society (DGG) began awarding the Zoeppritz Prize to outstanding young geophysicists. Deformation (engineering)#Elastic deformation In engineering , deformation (the change in size or shape of an object) may be elastic or plastic . If

3180-451: The sample and recording the stress variation with strain until the sample fractures . By convention, the strain is set to the horizontal axis and stress is set to vertical axis. Note that for engineering purposes we often assume the cross-section area of the material does not change during the whole deformation process. This is not true since the actual area will decrease while deforming due to elastic and plastic deformation. The curve based on

3240-411: The shrinking of section area at UTS point. After the formation of necking, the sample undergoes heterogeneous deformation, so equations above are not valid. The stress and strain at the necking can be expressed as: An empirical equation is commonly used to describe the relationship between true stress and true strain. Here, n is the strain-hardening exponent and K is the strength coefficient. n

3300-481: The strain rate variation. Where K ′ {\displaystyle K'} is constant related to the material flow stress. ε T ˙ {\displaystyle {\dot {\varepsilon _{T}}}} indicates the derivative of strain by the time, which is also known as strain rate. m {\displaystyle m} is the strain-rate sensitivity. Moreover, value of m {\displaystyle m}

3360-411: The strength of bulk copper and approaching the theoretical value. The theoretical yield strength can be estimated by considering the process of yield at the atomic level. In a perfect crystal, shearing results in the displacement of an entire plane of atoms by one interatomic separation distance, b, relative to the plane below. In order for the atoms to move, considerable force must be applied to overcome

3420-565: The true stress-strain curve and its derivative form, we can estimate the strain necessary to start necking. This can be calculated based on the intersection between true stress-strain curve as shown in right. This figure also shows the dependency of the necking strain at different temperature. In case of FCC metals, both of the stress-strain curve at its derivative are highly dependent on temperature. Therefore, at higher temperature, necking starts to appear even under lower strain value. Yield point In materials science and engineering ,

3480-450: The true stress-strain curve, we can estimate the region where necking starts to happen. Since necking starts to appear after ultimate tensile stress where the maximum force applied, we can express this situation as below: so this form can be expressed as below: It indicates that the necking starts to appear where reduction of area becomes much significant compared to the stress change. Then the stress will be localized to specific area where

3540-642: The universities of Munich and Freiburg , finishing his education with a doctoral dissertation on the geology of part of the Swiss Alps in 1905, at the University of Freiburg. Following the completion of his doctorate, in the summer of 1906 in Karlsruhe , Zoeppritz passed the Oberlehrerexamen , a teaching certificate that allowed him to lecture at a university. Zoeppritz became interested in

3600-405: The yield strength of the material can be fine-tuned. This occurs typically by introducing defects such as impurities dislocations in the material. To move this defect (plastically deforming or yielding the material), a larger stress must be applied. This thus causes a higher yield stress in the material. While many material properties depend only on the composition of the bulk material, yield strength

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