Explain why the stress is considered as a tensor quantity?

What is difference between vector and tensor?

Tensors are simply arrays of numbers, or functions, that transform according to certain rules under a change of coordinates. Scalars and vectors are tensors of order 0 and 1 respectively. So a vector is a type of tensor. An example of a tensor of order 2 is an inertia matrix. And just for fun, the Riemann curvature tensor is a tensor of order 4.

What is post processing in FEA?

After a finite element model has been prepared and checked, boundary conditions have been applied, and the model has been solved, it is time to investigate the results of the analysis. This activity is known as the post-processing phase of the finite element method.Post-processing begins with a thorough check for problems that may have occurred during solution. Most solvers provide a log file, which should be searched for warnings or errors, and which will also provide a quantitative measure of how well-behaved the numerical procedures were during solution. Next, reaction loads at restrained nodes should be summed and examined as a "sanity check". Reaction loads that do not closely balance the applied load resultant for a linear static analysis should cast doubt on the validity of other results. Error norms such as strain energy density and stress deviation among adjacent elements might be looked at next, but for h-code analyses these quantities are best used to target subsequent adaptive remeshing.Once the solution is verified to be free of numerical problems, the quantities of interest may be examined. Many display options are available, the choice of which depends on the mathematical form of the quantity as well as its physical meaning. For example, the displacement of a solid linear brick element's node is a 3-component spatial vector, and the model's overall displacement is often displayed by superposing the deformed shape over the undeformed shape. Dynamic viewing and animation capabilities aid greatly in obtaining an understanding of the deformation pattern. Stresses, being tensor quantities, currently lack a good single visualization technique, and thus derived stress quantities are extracted and displayed. Principal stress vectors may be displayed as color-coded arrows, indicating both direction and magnitude. The magnitude of principal stresses or of a scalar failure stress such as the Von Mises stress may be displayed on the model as colored bands. When this type of display is treated as a 3D object subjected to light sources, the resulting image is known as a shaded image stress plot. Displacement magnitude may also be displayed by colored bands, but this can lead to misinterpretation as a stress plot.An area of post-processing that is rapidly gaining popularity is that of adaptive remeshing. Error norms such as strain energy density are used to remesh the model, placing a denser mesh in regions needing improvement and a coarser mesh in areas of overkill. Adaptivity requires an associative link between the model and the underlying CAD geometry, and works best if boundary conditions may be applied directly to the geometry, as well. Adaptive remeshing is a recent demonstration of the iterative nature of h-code analysis.Optimization is another area enjoying recent advancement. Based on the values of various results, the model is modified automatically in an attempt to satisfy certain performance criteria and is solved again. The process iterates until some convergence criterion is met. In its scalar form, optimization modifies beam cross-sectional properties, thin shell thicknesses and/or material properties in an attempt to meet maximum stress constraints, maximum deflection constraints, and/or vibrational frequency constraints. Shape optimization is more complex, with the actual 3D model boundaries being modified. This is best accomplished by using the driving dimensions as optimization parameters, but mesh quality at each iteration can be a concern.Another direction clearly visible in the finite element field is the integration of FEA packages with so-called "mechanism" packages, which analyze motion and forces of large-displacement multi-body systems. A long-term goal would be real-time computation and display of displacements and stresses in a multi-body system undergoing large displacement motion, with frictional effects and fluid flow taken into account when necessary. It is difficult to estimate the increase in computing power necessary to accomplish this feat, but 2 or 3 orders of magnitude is probably close. Algorithms to integrate these fields of analysis may be expected to follow the computing power increases.

How is math used in engineering?

Many of the calculations done by a working ChE involve Thermodynamics, Physical Chemistry and Kinetics and Catalysis. It's not just adding and subtracting by any means. A firm background in Math is required. I suggest that you find a copy of Perry's Chemical Engineering Handbook in the library, and thumb through it. You will get an feel for some of the work done.

Current carrying capacity of 50sqmm copper wire?

Electrical resistivity (also known as resistivity, specific electrical resistance, or volume resistivity) is a property of a material; it quantifies how strongly the material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm⋅metre (Ω⋅m). It is commonly represented by the Greek letter ρ (rho).Electrical conductivity or specific conductance is the reciprocal quantity, and measures a material's ability to conduct an electric current. It is commonly represented by the Greek letter σ (sigma), but κ (kappa) (especially in electrical engineering) or γ (gamma) are also occasionally used. Its SI unit is siemens per metre (S⋅m−1) and CGSE unit is reciprocal second (s−1).Resistors or conductors with uniform cross-sectionA piece of resistive material with electrical contacts on both ends. Many resistors and conductors have a uniform cross section with a uniform flow of electric current and are made of one material. (See the diagram to the right.) In this case, the electrical resistivity ρ (Greek: rho) is defined as:whereR is the electrical resistance of a uniform specimen of the material (measured in ohms, Ω) is the length of the piece of material (measured in metres, m)A is the cross-sectional area of the specimen (measured in square metres, m²).The reason resistivity is defined this way is that it makes resistivity a material property, unlike resistance. All copper wires, irrespective of their shape and size, have approximately the same resistivity, but a long, thin copper wire has a much larger resistance than a thick, short copper wire. Every material has its own characteristic resistivity-for example rubber's resistivity is far larger than copper's.In a hydraulic analogy, passing current through a high-resistivity material is like pushing water through a pipe full of sand, while passing current through a low-resistivity material is like pushing water through an empty pipe. If the pipes are the same size and shape, the pipe full of sand has higher resistance to flow. But resistance is not solely determined by the presence or absence of sand; it also depends on how wide the pipe is (it is harder to push water through a skinny pipe than a wide one) and how long it is (it is harder to push water through a long pipe than a short one.)The above equation can be transposed to get Pouillet's law:The resistance of a given material will increase with the length, but decrease with increasing cross-sectional area. From the above equations, resistivity has SI units of ohm⋅metre. Other units like ohm⋅cm or ohm⋅inch are also sometimes used.Conductivity is the inverse of resistivity:Conductivity has SI units of Siemens per meter (S/m).General definitionThe above definition was specific to resistors or conductors with a uniform cross-section, where current flows uniformly through them. A more basic and general definition starts from the fact that if there is electric field inside a material, it will cause electric current to flow. The electrical resistivity ρ is defined as the ratio of the electric field to the density of the current it creates: whereρ is the resistivity of the conductor material (measured in ohm⋅metres, Ω⋅m),E is the magnitude of the electric field (in volts per metre, V⋅m−1),J is the magnitude of the current density (in amperes per square metre, A⋅m−2),in which E and J are inside the conductor.Conductivity is the inverse:For example, rubber is a material with large ρ and small σ, because even a very large electric field in rubber will cause almost no current to flow through it. On the other hand, copper is a material with small ρ and large σ, because even a small electric field pulls a lot of current through it.Causes of conductivityBand theory simplifiedElectron energy levels in an insulator Quantum mechanics states that electrons in an atom cannot take on any arbitrary energy value. Rather, there are fixed energy levels which the electrons can occupy, and values in between these levels are impossible[citation needed]. When a large number of such allowed energy levels are spaced close together (in energy-space) i.e. have similar (minutely differing energies) then we can talk about these energy levels together as an "energy band". There can be many such energy bands in a material, depending on the atomic number (number of electrons) and their distribution (besides external factors like environment modifying the energy bands). Two such bands important in the discussion of conductivity of materials are: the valence band and the conduction band (the latter is generally above the former)[citation needed]. Electrons in the conduction band may move freely throughout the material in the presence of an electrical field.In insulators and semiconductors, the atoms in the substance influence each other so that between the valence band and the conduction band there exists a forbidden band of energy levels, which the electrons cannot occupy. In order for a current to flow, a relatively large amount of energy must be furnished to an electron for it to leap across this forbidden gap and into the conduction band. Thus, even large voltages can yield relatively small currents.In metalsA metal consists of a lattice of atoms, each with an outer shell of electrons which freely dissociate from their parent atoms and travel through the lattice. This is also known as a positive ionic lattice[citation needed]. This 'sea' of dissociable electrons allows the metal to conduct electric current. When an electrical potential difference (a voltage) is applied across the metal, the resulting electric field causes electrons to move from one end of the conductor to the other. Near room temperatures, metals have resistance. The primary cause of this resistance is the thermal motion of ions. This acts to scatter electrons (due to destructive interference of free electron waves on non-correlating potentials of ions)[citation needed]. Also contributing to resistance in metals with impurities are the resulting imperfections in the lattice. In pure metals this source is negligible[citation needed].The larger the cross-sectional area of the conductor, the more electrons per unit length are available to carry the current. As a result, the resistance is lower in larger cross-section conductors. The number of scattering events encountered by an electron passing through a material is proportional to the length of the conductor. The longer the conductor, therefore, the higher the resistance. Different materials also affect the resistance.[1]In semiconductors and insulatorsMain articles: Semiconductor and Insulator (electricity) In metals, the Fermi level lies in the conduction band (see Band Theory, above) giving rise to free conduction electrons. However, in semiconductors the position of the Fermi level is within the band gap, approximately half-way between the conduction band minimum and valence band maximum for intrinsic (undoped) semiconductors. This means that at 0 kelvins, there are no free conduction electrons and the resistance is infinite. However, the resistance will continue to decrease as the charge carrier density in the conduction band increases. In extrinsic (doped) semiconductors, dopant atoms increase the majority charge carrier concentration by donating electrons to the conduction band or accepting holes in the valence band. For both types of donor or acceptor atoms, increasing the dopant density leads to a reduction in the resistance, hence highly doped semiconductors behave metallically. At very high temperatures, the contribution of thermally generated carriers will dominate over the contribution from dopant atoms and the resistance will decrease exponentially with temperature.In ionic liquids/electrolytesMain article: Conductivity (electrolytic) In electrolytes, electrical conduction happens not by band electrons or holes, but by full atomic species (ions) traveling, each carrying an electrical charge. The resistivity of ionic liquids varies tremendously by the concentration - while distilled water is almost an insulator, salt water is a very efficient electrical conductor. In biological membranes, currents are carried by ionic salts. Small holes in the membranes, called ion channels, are selective to specific ions and determine the membrane resistance.SuperconductivityMain article: superconductivity The electrical resistivity of a metallic conductor decreases gradually as temperature is lowered. In ordinary conductors, such as copper or silver, this decrease is limited by impurities and other defects. Even near absolute zero, a real sample of a normal conductor shows some resistance. In a superconductor, the resistance drops abruptly to zero when the material is cooled below its critical temperature. An electric current flowing in a loop of superconducting wire can persist indefinitely with no power source.[2]In 1986, it was discovered that some cuprate-perovskite ceramic materials have a critical temperature above 90 K (−183 °C). Such a high transition temperature is theoretically impossible for a conventional superconductor, leading the materials to be termed high-temperature superconductors. Liquid nitrogen boils at 77 K, facilitating many experiments and applications that are less practical at lower temperatures. In conventional superconductors, electrons are held together in pairs by an attraction mediated by lattice phonons. The best available model of high-temperature superconductivity is still somewhat crude. There is a hypothesis that electron pairing in high-temperature superconductors is mediated by short-range spin waves known as paramagnons.[3]Resistivity of various materialsMain article: Electrical resistivities of the elements (data page) A conductor such as a metal has high conductivity and a low resistivity.An insulator like glass has low conductivity and a high resistivity.The conductivity of a semiconductor is generally intermediate, but varies widely under different conditions, such as exposure of the material to electric fields or specific frequencies of light, and, most important, with temperature and composition of the semiconductor material.The degree of doping in semiconductors makes a large difference in conductivity. To a point, more doping leads to higher conductivity. The conductivity of a solution of water is highly dependent on its concentration of dissolved salts, and other chemical species that ionize in the solution. Electrical conductivity of water samples is used as an indicator of how salt-free, ion-free, or impurity-free the sample is; the purer the water, the lower the conductivity (the higher the resistivity). Conductivity measurements in water are often reported as specific conductance, relative to the conductivity of pure water at 25 °C. An EC meter is normally used to measure conductivity in a solution. A rough summary is as follows:MaterialResistivityρ (Ω•m)Metals10−8SemiconductorsvariableElectrolytesvariableInsulators1016Superconductors0This table shows the resistivity, conductivity and temperature coefficient of various materials at 20 °C (68 °F)Materialρ (Ω•m) at 20 °Cσ (S/m) at 20 °CTemperaturecoefficient[note 1](K−1)ReferenceSilver1.59×10−86.30×1070.0038[4][5]Copper1.68×10−85.96×1070.0039[5]Annealed copper[note 2]1.72×10−85.80×107[citation needed]Gold[note 3]2.44×10−84.10×1070.0034[4]Aluminium[note 4]2.82×10−83.5×1070.0039[4]Calcium3.36×10−82.98×1070.0041Tungsten5.60×10−81.79×1070.0045[4]Zinc5.90×10−81.69×1070.0037[6]Nickel6.99×10−81.43×1070.006Lithium9.28×10−81.08×1070.006Iron1.0×10−71.00×1070.005[4]Platinum1.06×10−79.43×1060.00392[4]Tin1.09×10−79.17×1060.0045Carbon steel (1010)1.43×10−76.99×106[7]Lead2.2×10−74.55×1060.0039[4]Titanium4.20×10−72.38×106XGrain oriented electrical steel4.60×10−72.17×106[8]Manganin4.82×10−72.07×1060.000002[9]Constantan4.9×10−72.04×1060.000008[10]Stainless steel[note 5]6.9×10−71.45×106[11]Mercury9.8×10−71.02×1060.0009[9]Nichrome[note 6]1.10×10−69.09×1050.0004[4]GaAs5×10−7 to 10×10−35×10−8 to 103[12]Carbon (amorphous)5×10−4 to 8×10−41.25 to 2×103−0.0005[4][13]Carbon (graphite)[note 7]2.5e×10−6 to 5.0×10−6 //basal plane3.0×10−3 &perp;basal plane2 to 3×105 //basal plane3.3×102 &perp;basal plane[14]Carbon (diamond)[note 8]1×1012~10−13[15]Germanium[note 8]4.6×10−12.17−0.048[4][5]Sea water[note 9]2×10−14.8[16]Drinking water[note 10]2×101 to 2×1035×10−4 to 5×10−2[citation needed]Silicon[note 8]6.40×1021.56×10−3−0.075[4]Deionized water[note 11]1.8×1055.5×10−6[17]Glass10×1010 to 10×101410−11 to 10−15?[4][5]Hard rubber1×101310−14?[4]Sulfur1×101510−16?[4]Air1.3×1016 to 3.3×10163×10−15 to 8×10−15[18]Paraffin1×101710−18?Fused quartz7.5×10171.3×10−18?[4]PET10×102010−21?Teflon10×1022 to 10×102410−25 to 10−23?The effective temperature coefficient varies with temperature and purity level of the material. The 20 °C value is only an approximation when used at other temperatures. For example, the coefficient becomes lower at higher temperatures for copper, and the value 0.00427 is commonly specified at 0 °C.[19]The extremely low resistivity (high conductivity) of silver is characteristic of metals. George Gamow tidily summed up the nature of the metals' dealings with electrons in his science-popularizing book, One, Two, Three...Infinity (1947): "The metallic substances differ from all other materials by the fact that the outer shells of their atoms are bound rather loosely, and often let one of their electrons go free. Thus the interior of a metal is filled up with a large number of unattached electrons that travel aimlessly around like a crowd of displaced persons. When a metal wire is subjected to electric force applied on its opposite ends, these free electrons rush in the direction of the force, thus forming what we call an electric current." More technically, the free electron model gives a basic description of electron flow in metals.Temperature dependenceLinear approximationThe electrical resistivity of most materials changes with temperature. If the temperature T does not vary too much, a linear approximation is typically used: where is called the temperature coefficient of resistivity, is a fixed reference temperature (usually room temperature), and is the resistivity at temperature . The parameter is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, is different for different reference temperatures. For this reason it is usual to specify the temperature that was measured at with a suffix, such as , and the relationship only holds in a range of temperatures around the reference.[20] When the temperature varies over a large temperature range, the linear approximation is inadequate and a more detailed analysis and understanding should be used.MetalsIn general, electrical resistivity of metals increases with temperature. Electron-phonon interactions can play a key role. At high temperatures, the resistance of a metal increases linearly with temperature. As the temperature of a metal is reduced, the temperature dependence of resistivity follows a power law function of temperature. Mathematically the temperature dependence of the resistivity ρ of a metal is given by the Bloch-Grüneisen formula: where is the residual resistivity due to defect scattering, A is a constant that depends on the velocity of electrons at the Fermi surface, the Debye radius and the number density of electrons in the metal. is the Debye temperature as obtained from resistivity measurements and matches very closely with the values of Debye temperature obtained from specific heat measurements. n is an integer that depends upon the nature of interaction:n=5 implies that the resistance is due to scattering of electrons by phonons (as it is for simple metals)n=3 implies that the resistance is due to s-d electron scattering (as is the case for transition metals)n=2 implies that the resistance is due to electron-electron interaction.If more than one source of scattering is simultaneously present, Matthiessen's Rule (first formulated by Augustus Matthiessen in the 1860s) [21][22] says that the total resistance can be approximated by adding up several different terms, each with the appropriate value of n.As the temperature of the metal is sufficiently reduced (so as to 'freeze' all the phonons), the resistivity usually reaches a constant value, known as the residual resistivity. This value depends not only on the type of metal, but on its purity and thermal history. The value of the residual resistivity of a metal is decided by its impurity concentration. Some materials lose all electrical resistivity at sufficiently low temperatures, due to an effect known as superconductivity.An investigation of the low-temperature resistivity of metals was the motivation to Heike Kamerlingh Onnes's experiments that led in 1911 to discovery of superconductivity. For details see History of superconductivity.SemiconductorsMain article: Semiconductor In general, resistivity of intrinsic semiconductors decreases with increasing temperature. The electrons are bumped to the conduction energy band by thermal energy, where they flow freely and in doing so leave behind holes in the valence band which also flow freely. The electric resistance of a typical intrinsic (non doped) semiconductor decreases exponentially with the temperature:An even better approximation of the temperature dependence of the resistivity of a semiconductor is given by the Steinhart-Hart equation:where A, B and C are the so-called Steinhart-Hart coefficients.This equation is used to calibrate thermistors.Extrinsic (doped) semiconductors have a far more complicated temperature profile. As temperature increases starting from absolute zero they first decrease steeply in resistance as the carriers leave the donors or acceptors. After most of the donors or acceptors have lost their carriers the resistance starts to increase again slightly due to the reducing mobility of carriers (much as in a metal). At higher temperatures it will behave like intrinsic semiconductors as the carriers from the donors/acceptors become insignificant compared to the thermally generated carriers.[23]In non-crystalline semiconductors, conduction can occur by charges quantum tunnelling from one localised site to another. This is known as variable range hopping and has the characteristic form of,where n = 2, 3, 4, depending on the dimensionality of the system.Complex resistivity and conductivityWhen analyzing the response of materials to alternating electric fields, in applications such as electrical impedance tomography,[24] it is necessary to replace resistivity with a complex quantity called impeditivity (in analogy to electrical impedance). Impeditivity is the sum of a real component, the resistivity, and an imaginary component, the reactivity(in analogy to reactance). The magnitude of Impeditivity is the square root of sum of squares of magnitudes of resistivity and reactivity. Conversely, in such cases the conductivity must be expressed as a complex number (or even as a matrix of complex numbers, in the case of anisotropic materials) called the admittivity. Admittivity is the sum of a real component called the conductivity and an imaginary component called the susceptivity.An alternative description of the response to alternating currents uses a real (but frequency-dependent) conductivity, along with a real permittivity. The larger the conductivity is, the more quickly the alternating-current signal is absorbed by the material (i.e., the more opaque the material is). For details, see Mathematical descriptions of opacity.Tensor equations for anisotropic materialsSome materials are anisotropic, meaning they have different properties in different directions. For example, a crystal of graphite consists microscopically of a stack of sheets, and current flows very easily through each sheet, but moves much less easily from one sheet to the next.[14] For an anisotropic material, it is not generally valid to use the scalar equationsFor example, the current may not flow in exactly the same direction as the electric field. Instead, the equations are generalized to the 3D tensor form[25][26]where the conductivity σ and resistivity ρ are rank-2 tensors (in other words, 3×3 matrices). The equations are compactly illustrated in component form (using index notation and the summation convention) [27]:The σ and ρ tensors are inverses (in the sense of a matrix inverse). The individual components are not necessarily inverses; for example σxx may not be equal to 1/ρxx.Resistance versus resistivity in complicated geometriesIf the material's resistivity is known, calculating the resistance of something made from it may, in some cases, be much more complicated than the formula above. One example is Spreading Resistance Profiling, where the material is inhomogeneous (different resistivity in different places), and the exact paths of current flow are not obvious. In cases like this, the formulasneed to be replaced withwhere E and J are now vector fields. This equation, along with the continuity equation for J and the Poisson equation for E, form a set of partial differential equations. In special cases, an exact or approximate solution to these equations can be worked out by hand, but for very accurate answers in complex cases, computer methods like finite element analysis may be required.Resistivity density productsIn some applications where the weight of an item is very important resistivity density products are more important than absolute low resistivity- it is often possible to make the conductor thicker to make up for a higher resistivity; and then a low resistivity density product material (or equivalently a high conductance to density ratio) is desirable. For example, for long distance overhead power lines- aluminium is frequently used rather than copper because it is lighter for the same conductance.MaterialResistivity (nΩ•m)Density (g/cm3)Resistivity-density product (nΩ•m•g/cm3)Sodium47.70.9746Lithium92.80.5349Calcium33.61.5552Potassium72.00.8964Beryllium35.61.8566Aluminium26.502.7072Magnesium43.901.7476.3Copper16.788.96150Silver15.8710.49166Gold22.1419.30427Iron96.17.874757Silver, although it is the least resistive metal known, has a high density and does poorly by this measure. Calcium and the alkali metals have the best products, but are rarely used for conductors due to their high reactivity with water and oxygen. Aluminium is far more stable. Two other important attributes, price and toxicity, exclude the (otherwise) best choice: Beryllium. Thus, aluminium is usually the metal of choice when the weight of some required conduction (and/or the cost of conduction) is the driving consideration.

Explain why the stress is considered as a tensor quantity?

Add your answer:

What is difference between vector and tensor?

What is post processing in FEA?

How is math used in engineering?

Current carrying capacity of 50sqmm copper wire?

Is stress is scalar quantity or vector quantity?

Is stress scalar or vector quantity?

Why stress is a tensor?

Why is stress a tensor quantity?

Why stress a tensor quantity?

Is current a tensor quantity?

Is force a tensor?

Why pressure is a scalar quantity while stress is a vector?

Is every vector is a tensor and vice-versa?

An arrow that represets the magnitude and direction of a physical quantity on a diagram is called a tensor?

Is pressure a scaler quantity?

Is centre of gravity a vector quantity or scalar quantity?

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