Misplaced Pages

#526473

93-442: Carbon–carbon is well-suited to structural applications at high temperatures, or where thermal shock resistance and/or a low coefficient of thermal expansion is needed. While it is less brittle than many other ceramics, it lacks impact resistance; Space Shuttle Columbia was destroyed during atmospheric re-entry after one of its RCC panels was broken by the impact of a piece of polyurethane foam insulation that broke off from

186-420: A flow of viscous liquid , the force F may not be perpendicular to S ; hence the stress across a surface must be regarded a vector quantity, not a scalar. Moreover, the direction and magnitude generally depend on the orientation of S . Thus the stress state of the material must be described by a tensor , called the (Cauchy) stress tensor ; which is a linear function that relates the normal vector n of

279-439: 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

372-392: A coordinate system with axes e 1 , e 2 , e 3 {\displaystyle e_{1},e_{2},e_{3}} , the stress tensor is a diagonal matrix, and has only the three normal components λ 1 , λ 2 , λ 3 {\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}

465-457: 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"). Another simple type of stress occurs when

558-454: A differential formula for friction forces (shear stress) in parallel laminar flow . Stress is defined as the force across a small boundary per unit area of that boundary, for all orientations of the boundary. 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

651-473: A non destructive way. The same test-piece can be measured after different thermal shock cycles, and this way the deterioration in physical properties can be mapped out. Thermal shock resistance measures can be used for material selection in applications subject to rapid temperature changes. A common measure of thermal shock resistance is the maximum temperature differential, Δ T {\displaystyle \Delta T} , which can be sustained by

744-743: A result, a commonly chosen material index for thermal shock resistance in the poor heat transfer case is: k K I c E α {\displaystyle {\frac {kK_{Ic}}{E\alpha }}} The temperature difference to initiate fracture has been described by William David Kingery to be: Δ T c = S k σ ∗ ( 1 − ν ) E α 1 h = S h R ′ {\displaystyle \Delta T_{c}=S{\frac {k\sigma ^{*}(1-\nu )}{E\alpha }}{\frac {1}{h}}={\frac {S}{hR^{'}}}} where S {\displaystyle S}

837-448: A surface S to the traction vector T across S . With respect to any chosen coordinate system , the Cauchy stress tensor can be represented as a symmetric matrix of 3×3 real numbers. Even within a homogeneous body, the stress tensor may vary from place to place, and may change over time; therefore, the stress within a material is, in general, a time-varying tensor field . In general,

930-1007: A surface will always be a linear function of the surface's normal vector n {\displaystyle n} , the unit-length vector that is perpendicular to it. That is, T = σ ( n ) {\displaystyle T={\boldsymbol {\sigma }}(n)} , where the function σ {\displaystyle {\boldsymbol {\sigma }}} satisfies σ ( α u + β v ) = α σ ( u ) + β σ ( v ) {\displaystyle {\boldsymbol {\sigma }}(\alpha u+\beta v)=\alpha {\boldsymbol {\sigma }}(u)+\beta {\boldsymbol {\sigma }}(v)} for any vectors u , v {\displaystyle u,v} and any real numbers α , β {\displaystyle \alpha ,\beta } . The function σ {\displaystyle {\boldsymbol {\sigma }}} , now called

1023-434: A surface with normal vector n {\displaystyle n} (which is covariant - "row; horizontal" - vector) with coordinates n 1 , n 2 , n 3 {\displaystyle n_{1},n_{2},n_{3}} is then a matrix product T = n ⋅ σ {\displaystyle T=n\cdot {\boldsymbol {\sigma }}} (where T in upper index

SECTION 10

#1732851350527

1116-413: A system must be balanced by internal reaction forces, which are almost always surface contact forces between adjacent particles — that is, as stress. Since every particle needs to be in equilibrium, this reaction stress will generally propagate from particle to particle, creating a stress distribution throughout the body. The typical problem in stress analysis is to determine these internal stresses, given

1209-434: A system of partial differential equations involving the stress tensor field and the strain tensor field, as unknown functions to be determined. The external body forces appear as the independent ("right-hand side") term in the differential equations, while the concentrated forces appear as boundary conditions. The basic stress analysis problem is therefore a boundary-value problem . Stress analysis for elastic structures

1302-489: 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

1395-1092: Is transposition , and as a result we get covariant (row) vector) (look on Cauchy stress tensor ), that is [ T 1 T 2 T 3 ] = [ n 1 n 2 n 3 ] ⋅ [ σ 11 σ 21 σ 31 σ 12 σ 22 σ 32 σ 13 σ 23 σ 33 ] {\displaystyle {\begin{bmatrix}T_{1}&T_{2}&T_{3}\end{bmatrix}}={\begin{bmatrix}n_{1}&n_{2}&n_{3}\end{bmatrix}}\cdot {\begin{bmatrix}\sigma _{11}&\sigma _{21}&\sigma _{31}\\\sigma _{12}&\sigma _{22}&\sigma _{32}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\end{bmatrix}}} The linear relation between T {\displaystyle T} and n {\displaystyle n} follows from

1488-411: Is a physical quantity that describes forces present during deformation . For example, an object being pulled apart, such as a stretched elastic band, is subject to tensile stress and may undergo elongation . An object being pushed together, such as a crumpled sponge, is subject to compressive stress and may undergo shortening. The greater the force and the smaller the cross-sectional area of

1581-459: Is a shape factor, σ ∗ {\displaystyle \sigma ^{*}} is the fracture stress, k {\displaystyle k} is the thermal conductivity, E {\displaystyle E} is the Young's modulus, α {\displaystyle \alpha } is the coefficient of thermal expansion, h {\displaystyle h}

1674-1181: Is a system constrain constant dependent upon the Poisson's ratio, ν {\displaystyle \nu } , and A {\displaystyle A} is a non-dimensional parameter dependent upon the Biot number , B i {\displaystyle \mathrm {Bi} } . C = { 1 axial stress ( 1 − ν ) biaxial constraint ( 1 − 2 ν ) triaxial constraint {\displaystyle C={\begin{cases}1&{\text{axial stress}}\\(1-\nu )&{\text{biaxial constraint}}\\(1-2\nu )&{\text{triaxial constraint}}\end{cases}}} A {\displaystyle A} may be approximated by: A = H h / k 1 + H h / k = B i 1 + B i {\displaystyle A={\frac {Hh/k}{1+Hh/k}}={\frac {\mathrm {Bi} }{1+\mathrm {Bi} }}} where H {\displaystyle H}

1767-410: Is actually the average of a very large number of atomic forces between their molecules; and physical quantities like mass, velocity, and forces that act through the bulk of three-dimensional bodies, like gravity, are assumed to be smoothly distributed over them. Depending on the context, one may also assume that the particles are large enough to allow the averaging out of other microscopic features, like

1860-583: Is an essential tool in engineering for the study and design of structures such as tunnels, dams, mechanical parts, and structural frames, under prescribed or expected loads. It is also important in many other disciplines; for example, in geology, to study phenomena like plate tectonics , vulcanism and avalanches ; and in biology, to understand the anatomy of living beings. Stress analysis is generally concerned with objects and structures that can be assumed to be in macroscopic static equilibrium . By Newton's laws of motion , any external forces being applied to such

1953-406: Is assumed fixed, the normal component can be expressed by a single number, the dot product T · n . This number will be positive if P is "pulling" on Q (tensile stress), and negative if P is "pushing" against Q (compressive stress). The shear component is then the vector T − ( T · n ) n . The dimension of stress is that of pressure , and therefore its coordinates are measured in

SECTION 20

#1732851350527

2046-478: Is based on the theory of elasticity and infinitesimal strain theory . When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved ( plastic flow , fracture , phase change , etc.). Engineered structures are usually designed so the maximum expected stresses are well within the range of linear elasticity (the generalization of Hooke's law for continuous media); that is,

2139-428: Is extremely resistant to thermal shock, due to graphite 's extremely high thermal conductivity and low expansion coefficient, the high strength of carbon fiber , and a reasonable ability to deflect cracks within the structure. To measure thermal shock, the impulse excitation technique proved to be a useful tool. It can be used to measure Young's modulus, Shear modulus , Poisson's ratio , and damping coefficient in

2232-637: Is given in the article on viscosity . The same for normal viscous stresses can be found in Sharma (2019). 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 . In some situations,

2325-535: 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

2418-507: 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 Euler's equations of motion for continuous bodies (which are consequences of Newton's laws for conservation of linear momentum and angular momentum ) and the Euler-Cauchy stress principle , together with the appropriate constitutive equations. Thus one obtains

2511-408: Is perpendicular to the layer, the net internal force across S , and hence the stress, will be zero. 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. Shear stress is observed also when

2604-636: Is slightly denser than pure carbon-carbon and thought to be more durable. It can be used in the brake disc and brake pads of high-performance road cars. The first car to use it was the Mercedes-Benz C215 Coupe F1 edition. It is standard on the Bugatti Veyron and many Bentleys , Ferraris , Lamborghinis , Porsches , and the Corvette ZR1 and Z06 . They are also offered as an optional upgrade on certain high performance Audi cars, including

2697-412: Is subjected to tension by opposite forces of magnitude F {\displaystyle F} 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,

2790-503: Is the failure stress (which can be yield or fracture stress ), α {\displaystyle \alpha } is the coefficient of thermal expansion, E {\displaystyle E} is the Young's modulus, and B {\displaystyle B} is a constant depending upon the part constraint, material properties, and thickness. B = C A {\displaystyle B={\frac {C}{A}}} where C {\displaystyle C}

2883-521: Is the heat transfer coefficient, and R ′ {\displaystyle R'} is a fracture resistance parameter. The fracture resistance parameter is a common metric used to define the thermal shock tolerance of materials. R ′ = k σ ∗ ( 1 − v ) E α {\displaystyle R'={\frac {k\sigma ^{*}(1-v)}{E\alpha }}} The formulas were derived for ceramic materials, and make

Reinforced carbon–carbon - Misplaced Pages Continue

2976-561: Is the mode I fracture toughness , E {\displaystyle E} is the Young's modulus, α {\displaystyle \alpha } is the thermal expansion coefficient, and H {\displaystyle H} is half the thickness of the plate. A material index for material selection in the fracture mechanics derived perfect heat transfer case is therefore: K I c E α {\displaystyle {\frac {K_{Ic}}{E\alpha }}} For cases with poor heat transfer,

3069-643: Is the thickness, h {\displaystyle h} is the heat transfer coefficient , and k {\displaystyle k} is the thermal conductivity . If perfect heat transfer ( B i = ∞ {\displaystyle \mathrm {Bi} =\infty } ) is assumed, the maximum heat transfer supported by the material is: Δ T = A 1 σ f E α {\displaystyle \Delta T=A_{1}{\frac {\sigma _{f}}{E\alpha }}} A material index for material selection according to thermal shock resistance in

3162-437: 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). Stress analysis is a branch of applied physics that covers the determination of the internal distribution of internal forces in solid objects. It

3255-576: Is too small to be detected. In a solid material, such strain will in turn generate an internal elastic stress, analogous to the reaction force of a stretched spring , tending to restore the material to its original undeformed state. Fluid materials (liquids, gases and plasmas ) by definition can only oppose deformations that would change their volume. If the deformation changes with time, even in fluids there will usually be some viscous stress, opposing that change. Such stresses can be either shear or normal in nature. Molecular origin of shear stresses in fluids

3348-560: Is typically used to perform thermal shock testing. When using single chamber thermal shock equipment, the products remain in one chamber and the chamber air temperature is rapidly cooled and heated. Some equipment uses separate hot and cold chambers with an elevator mechanism that transports the products between two or more chambers. Glass containers can be sensitive to sudden changes in temperature. One method of testing involves rapid movement from cold to hot water baths, and back. Stress (mechanics) In continuum mechanics , stress

3441-505: The (Cauchy) stress tensor , completely describes the stress state of a uniformly stressed body. (Today, any linear connection between two physical vector quantities is called a tensor , reflecting Cauchy's original use to describe the "tensions" (stresses) in a material.) In tensor calculus , σ {\displaystyle {\boldsymbol {\sigma }}} is classified as a second-order tensor of type (0,2) or (1,1) depending on convention. Like any linear map between vectors,

3534-511: The Space Shuttle External Tank . The material is made in three stages: First, material is laid up in its intended final shape, with carbon filament and/or cloth surrounded by an organic binder such as plastic or pitch . Often, coke or some other fine carbon aggregate is added to the binder mixture. Second, the lay-up is heated, so that pyrolysis transforms the binder to relatively pure carbon. The binder loses volume in

3627-610: 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: Galileo Galilei 's rigorous experimental method , René Descartes 's coordinates and analytic geometry , and Newton 's laws of motion and equilibrium and calculus of infinitesimals . With those tools, Augustin-Louis Cauchy

3720-460: The fracture mechanics framework. Lu and Fleck produced criteria for thermal shock cracking based on fracture toughness controlled cracking. The models were based on thermal shock in ceramics (generally brittle materials). Assuming an infinite plate, and mode I cracking, the crack was predicted to start from the edge for cold shock, but the center of the plate for hot shock. Cases were divided into perfect, and poor heat transfer to further simplify

3813-452: The lithium aluminosilicate (LAS) system ) include a controlled proportion of material with a negative expansion coefficient, so that the overall coefficient can be reduced to almost exactly zero over a reasonably wide range of temperatures. Among the best thermomechanical materials, there are alumina , zirconia , tungsten alloys, silicon nitride , silicon carbide , boron carbide , and some stainless steels . Reinforced carbon-carbon

Reinforced carbon–carbon - Misplaced Pages Continue

3906-989: The orthogonal shear stresses . 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 σ {\displaystyle {\boldsymbol {\sigma }}} has three mutually orthogonal unit-length eigenvectors e 1 , e 2 , e 3 {\displaystyle e_{1},e_{2},e_{3}} and three real eigenvalues λ 1 , λ 2 , λ 3 {\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}} , such that σ e i = λ i e i {\displaystyle {\boldsymbol {\sigma }}e_{i}=\lambda _{i}e_{i}} . Therefore, in

3999-457: The principal stresses . If the three eigenvalues are equal, the stress is an isotropic compression or tension, always perpendicular to any surface, there is no shear stress, and the tensor is a diagonal matrix in any coordinate frame. 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

4092-442: 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 . Humans have known about stress inside materials since ancient times. Until

4185-445: The 17th century, this understanding was largely intuitive and empirical, though this did not prevent the development of relatively advanced technologies like the composite bow and glass blowing . 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

4278-519: The Biot number is an important factor in the sustainable temperature jump. Δ T = A 4 K I c E α π H k h H {\displaystyle \Delta T=A_{4}{\frac {K_{Ic}}{E\alpha {\sqrt {\pi H}}}}{\frac {k}{hH}}} Critically, for poor heat transfer cases, materials with higher thermal conductivity, k , have higher thermal shock resistance. As

4371-679: The D3 S8 , B7 RS4 , C6 S6 and RS6 , and the R8 . The material is not used in Formula 1 because of its weight. Carbon brakes became widely available for commercial airplanes in the 1980s, having been first used on the Concorde supersonic transport . A related non-ceramic carbon composite with uses in high-tech racing automotives is the carbotanium carbon–titanium composite used in the Zonda R and Huayra supercars made by

4464-461: The Italian motorcar company Pagani . Thermal shock Thermal shock is a phenomenon characterized by a rapid change in temperature that results in a transient mechanical load on an object. The load is caused by the differential expansion of different parts of the object due to the temperature change. This differential expansion can be understood in terms of strain , rather than stress . When

4557-460: The assumptions of a homogeneous body with material properties independent of temperature, but can be well applied to other brittle materials. Thermal shock testing exposes products to alternating low and high temperatures to accelerate failures caused by temperature cycles or thermal shocks during normal use. The transition between temperature extremes occurs very rapidly, greater than 15 °C per minute. Equipment with single or multiple chambers

4650-429: The bar pushes on the particles immediately below it. When a liquid is in a closed container under pressure , each particle gets pushed against by all the surrounding particles. The container walls and the pressure -inducing surface (such as a piston) push against them in (Newtonian) reaction . These macroscopic forces are actually the net result of a very large number of intermolecular forces and collisions between

4743-451: The body on which it acts, the greater the stress. Stress has dimension of force per area, with SI units of newtons per square meter (N/m ) or pascal (Pa). Stress expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the relative deformation of the material. For example, when a solid vertical bar is supporting an overhead weight , each particle in

SECTION 50

#1732851350527

4836-449: The bulk of the material, varying continuously with position and time. Other agents (like external loads and friction, ambient pressure, and contact forces) may create stresses and forces that are concentrated on certain surfaces, lines or points; and possibly also on very short time intervals (as in the impulses due to collisions). In active matter , self-propulsion of microscopic particles generates macroscopic stress profiles. In general,

4929-498: 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. 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

5022-402: The deformations caused by internal stresses are linearly related to them. In this case the differential equations that define the stress tensor are linear, and the problem becomes much easier. For one thing, the stress at any point will be a linear function of the loads, too. For small enough stresses, even non-linear systems can usually be assumed to be linear. Stress analysis is simplified when

5115-709: 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 . Parts with rotational symmetry , such as wheels, axles, pipes, and pillars, are very common in engineering. Often

5208-434: The elements σ x , σ y , σ z {\displaystyle \sigma _{x},\sigma _{y},\sigma _{z}} are called the orthogonal normal stresses (relative to the chosen coordinate system), and τ x y , τ x z , τ y z {\displaystyle \tau _{xy},\tau _{xz},\tau _{yz}}

5301-424: The external forces that are acting on the system. The latter may be body forces (such as gravity or magnetic attraction), that act throughout the volume of a material; or concentrated loads (such as friction between an axle and a bearing , or the weight of a train wheel on a rail), that are imagined to act over a two-dimensional area, or along a line, or at single point. In stress analysis one normally disregards

5394-439: The fibre scaffold is laid up and the quality/density of the matrix filler. Carbon–carbon materials retain their properties above 2000 °C. This temperature may be exceeded with the help of protective coatings to prevent oxidation. The material has a density between 1.6 and 1.98 g/cm. Carbon fibre-reinforced silicon carbide ( C/SiC ) is a development of pure carbon–carbon that uses silicon carbide with carbon fibre . It

5487-779: The fracture stress derived perfect heat transfer case is therefore: σ f E α {\displaystyle {\frac {\sigma _{f}}{E\alpha }}} For cases with poor heat transfer ( B i < 1 {\displaystyle \mathrm {Bi} <1} ), the maximum heat differential supported by the material is: Δ T = A 2 σ f E α 1 B i = A 2 σ f E α k h H {\displaystyle \Delta T=A_{2}{\frac {\sigma _{f}}{E\alpha }}{\frac {1}{\mathrm {Bi} }}=A_{2}{\frac {\sigma _{f}}{E\alpha }}{\frac {k}{hH}}} In

5580-412: The fundamental laws of conservation of linear momentum and static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations ( Cauchy's equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that

5673-457: The grains of a metal rod or the fibers of a piece of wood . 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 . In a fluid at rest the force is perpendicular to the surface, and is the familiar pressure . In a solid , or in

SECTION 60

#1732851350527

5766-408: 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

5859-429: 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

5952-576: The material for a given thickness. Thermal shock resistance measures can be used for material selection in applications subject to rapid temperature changes. The maximum temperature jump, Δ T {\displaystyle \Delta T} , sustainable by a material can be defined for strength-controlled models by: B Δ T = σ f α E {\displaystyle B\Delta T={\frac {\sigma _{f}}{\alpha E}}} where σ f {\displaystyle \sigma _{f}}

6045-488: The material's high cost. The gray "Reinforced Carbon–Carbon (RCC)" panels on the space shuttle's wing leading edges and nose cone cost NASA $ 100,000/sq ft to produce, although much of this cost was a result of the advanced geometry and research costs associated with the panels. This stage can also include manufacturing of the finished product. C/C is a hard material that can be made highly resistant to thermal expansion, temperature gradients, and thermal cycling, depending on how

6138-519: The medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point. 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

6231-589: The models. The sustainable temperature jump decreases, with increasing convective heat transfer (and therefore larger Biot number). This is represented in the model shown below for perfect heat transfer ( B i = ∞ {\displaystyle \mathrm {Bi} =\infty } ). Δ T = A 3 K I c E α π H {\displaystyle \Delta T=A_{3}{\frac {K_{Ic}}{E\alpha {\sqrt {\pi H}}}}} where K I c {\displaystyle K_{Ic}}

6324-448: The most general case, called triaxial stress , the stress is nonzero across every surface element. 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 T {\displaystyle T} across

6417-420: The nature of the material or of its physical causes. Following the basic premises of continuum mechanics, stress is a macroscopic concept. Namely, the particles considered in its definition and analysis should be just small enough to be treated as homogeneous in composition and state, but still large enough to ignore quantum effects and the detailed motions of molecules. Thus, the force between two particles

6510-433: The particles in those molecules . Stress is frequently represented by a lowercase Greek letter sigma ( σ ). Strain inside a material may arise by various mechanisms, such as stress as applied by external forces to the bulk material (like gravity ) or to its surface (like contact forces , external pressure, or friction ). Any strain (deformation) of a solid material generates an internal elastic stress , analogous to

6603-452: 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 constitutive equations . 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

6696-424: 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

6789-445: The plate). 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

6882-574: The poor heat transfer case, a higher thermal conductivity is beneficial for thermal shock resistance. The material index for the poor heat transfer case is often taken as: k σ f E α {\displaystyle {\frac {k\sigma _{f}}{E\alpha }}} According to both the perfect and poor heat transfer models, larger temperature differentials can be tolerated for hot shock than for cold shock. In addition to thermal shock resistance defined by material fracture strength, models have also been defined within

6975-406: The process, causing voids to form; the addition of aggregate reduces this problem, but does not eliminate it. Third, the voids are gradually filled by forcing a carbon-forming gas such as acetylene through the material at a high temperature, over the course of several days. This long heat treatment process also allows the carbon to form into larger graphite crystals, and is the major reason for

7068-476: The reaction force of a spring , that tends to restore the material to its original non-deformed state. In liquids and gases , only deformations that change the volume generate persistent elastic stress. If the deformation changes gradually with time, even in fluids there will usually be some viscous stress , opposing that change. Elastic and viscous stresses are usually combined under the name mechanical stress . Significant stress may exist even when deformation

7161-657: The same units as pressure: namely, pascals (Pa, that is, newtons per square metre ) in the International System , or pounds per square inch (psi) in the Imperial system . Because mechanical stresses easily exceed a million Pascals, MPa, which stands for megapascal, is a common unit of stress. 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 temperature and phase , and electromagnetic fields) act on

7254-424: The single number τ {\displaystyle \tau } , calculated simply with the magnitude of those forces, F and the cross sectional area, A . τ = F A {\displaystyle \tau ={\frac {F}{A}}} Unlike normal stress, this simple shear stress is directed parallel to the cross-section considered, rather than perpendicular to it. For any plane S that

7347-444: The strain exceeds the tensile strength of the material, it can cause cracks to form, and eventually lead to structural failure. Methods to prevent thermal shock include: Borosilicate glass is made to withstand thermal shock better than most other glass through a combination of reduced expansion coefficient, and greater strength, though fused quartz outperforms it in both these respects. Some glass-ceramic materials (mostly in

7440-407: 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 normal stress ( compression or tension ) perpendicular to the surface, and the shear stress that is parallel to the surface. If the normal unit vector n of the surface (pointing from Q towards P )

7533-501: 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

7626-411: The stress distribution in a body is expressed as a piecewise continuous function of space and time. Conversely, stress is usually correlated with various effects on the material, possibly including changes in physical properties like birefringence , polarization , and permeability . The imposition of stress by an external agent usually creates some strain (deformation) in the material, even if it

7719-475: 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 σ {\displaystyle \sigma } = 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

7812-424: The stress is maximum for surfaces that are perpendicular to a certain direction d {\displaystyle d} , and zero across any surfaces that are parallel to d {\displaystyle d} . 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

7905-399: The stress patterns that occur in such parts have rotational or even cylindrical symmetry . The analysis of such cylinder stresses can take advantage of the symmetry to reduce the dimension of the domain and/or of the stress tensor. 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

7998-684: The stress state of the medium at any point and instant can be specified by only six independent parameters, rather than nine. These may be written [ σ x τ x y τ x z τ x y σ y τ y z τ x z τ y z σ z ] {\displaystyle {\begin{bmatrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{xy}&\sigma _{y}&\tau _{yz}\\\tau _{xz}&\tau _{yz}&\sigma _{z}\end{bmatrix}}} where

8091-411: 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 curvature of the plate. These simplifications may not hold at welds, at sharp bends and creases (where the radius of curvature is comparable to the thickness of

8184-1620: 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 x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} or named x , y , z {\displaystyle x,y,z} , the matrix may be written as [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] {\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}} or [ σ x x σ x y σ x z σ y x σ y y σ y z σ z x σ z y σ z z ] {\displaystyle {\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\\\end{bmatrix}}} The stress vector T = σ ( n ) {\displaystyle T={\boldsymbol {\sigma }}(n)} across

8277-431: The stress tensor is symmetric , that is σ 12 = σ 21 {\displaystyle \sigma _{12}=\sigma _{21}} , σ 13 = σ 31 {\displaystyle \sigma _{13}=\sigma _{31}} , and σ 23 = σ 32 {\displaystyle \sigma _{23}=\sigma _{32}} . Therefore,

8370-423: 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 . A common situation with a simple stress pattern is when a straight rod, with uniform material and cross section,

8463-440: 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 . σ = F A {\displaystyle \sigma ={\frac {F}{A}}} On the other hand, if one imagines the bar being cut along its length, parallel to the axis, there will be no force (hence no stress) between

8556-440: The two halves across the cut. This type of stress may be called (simple) normal stress or uniaxial stress; specifically, (uniaxial, simple, etc.) tensile stress. If the load is compression on the bar, rather than stretching it, the analysis is the same except that the force F and the stress σ {\displaystyle \sigma } change sign, and the stress is called compressive stress. This analysis assumes

8649-400: Was able to give the first rigorous and general mathematical model of a deformed elastic body by introducing the notions of stress and strain. Cauchy observed that the force across an imaginary surface was a linear function of its normal vector; and, moreover, that it must be a symmetric function (with zero total momentum). The understanding of stress in liquids started with Newton, who provided

#526473