In solid mechanics, the tangent modulus is the slope of the stress–strain curve at any specified stress or strain. Below the proportional limit (the limit of the linear elastic regime) the tangent modulus is equivalent to Young's modulus. Above the proportional limit the tangent modulus varies with strain and is most accurately found from test data. The Ramberg–Osgood equation relates Young's modulus to the tangent modulus and is another method for obtaining the tangent modulus.
The tangent modulus is useful in describing the behavior of materials that have been stressed beyond the elastic region. When a material is plastically deformed there is no longer a linear relationship between stress and strain as there is for elastic deformations. The tangent modulus quantifies the "softening" or "hardening" of material that generally occurs when it begins to yield.
Although the material softens it is still generally able to sustain more load before ultimate failure. Therefore, more weight efficient structure can be designed when plastic behavior is considered. For example, a structural analyst may use the tangent modulus to quantify the buckling failure of columns and flat plates.
linear elastic regime) the tangentmodulus is equivalent to Young's modulus. Above the proportional limit the tangentmodulus varies with strain and is...
Dynamic modulus (sometimes complex modulus) is the ratio of stress to strain under vibratory conditions (calculated from data obtained from either free...
of the tangentmodulus of elasticity, Et, which is less than the elastic modulus, in place of the elastic modulus of elasticity. The tangent is equal...
ratio of the longest half sine wave to the actual column length, Et = tangentmodulus at the stress Fcr, and KL = effective length (length of an equivalent...
results in a characteristic three-phase behavior of stiff-soft-stiff tangentmodulus. Phase I corresponds to the small-deformation regime during which the...
us define the angle θ between a vector that is tangent to the polymer at position 0 (zero) and a tangent vector at a distance L away from position 0, along...
complex modulus. The temperature of the sample or the frequency of the stress are often varied, leading to variations in the complex modulus; this approach...
acid)/poly(N,N'-dimethyl acrylamide) or PAMPS/PDMAAm ultimate stress and tangentmodulus increased. However, when using bacterial cellulose and gelatin, it...
The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval...
published until 1693 (in his Traité des Indivisibles). Constructing the tangent of the cycloid dates to August 1638 when Mersenne received unique methods...
endomorphisms in a certain precise sense, roughly that the action on the tangent space at the identity element of A is a direct sum of one-dimensional modules...
manifold M: any holomorphic function on it is constant by the maximum modulus principle. Now if we had a holomorphic embedding of M into Cn, then the...
numbers can be expressed more easily in polar coordinates—the magnitude or modulus of the product is the product of the two absolute values, or moduli, and...
material can store energy in compression ( 1 / elastic modulus {\displaystyle 1/{\text{elastic modulus}}} ) by how well it can stay in the elastic range (...
state is identical to that of the Murnaghan equation of state. The tangent bulk modulus predicted by the MacDonald–Tait model is K = K 0 ( V 0 V ) n . {\displaystyle...
{\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}} means that the slope of the tangent to the graph at each point is equal to its y-coordinate at that point....
{\displaystyle d={\frac {N}{P_{d}\cos \psi }}} in imperial units. Module or modulus, m Since it is impractical to calculate circular pitch with irrational...
distinguished: If a differentiable map f on M has a hyperbolic structure on the tangent bundle, then it is called an Anosov map. Examples include the Bernoulli...
a metric tensor g {\displaystyle g} implies that the modulus of a vector defined on the tangent bundle to a certain point p {\displaystyle p} of the manifold...
skew-symmetric matrix A T = − A {\displaystyle A^{\mathsf {T}}=-A} in the tangent space s o ( n ) {\displaystyle {\mathfrak {so}}(n)} (the special orthogonal...
trigonometric functions of the angle can be defined similarly; for example, the tangent is the ratio between the opposite and adjacent sides. As stated, the values...
Hyperbolic tangent and the superscript of -1, above it, is shorthand notation for the inverse function of the Trigonometric function, tangent. For k > 1...
shear modulus of PDMS varies with preparation conditions, and consequently dramatically varies in the range of 100 kPa to 3 MPa. The loss tangent is very...
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