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In engineering and materials science, a stress–strain curve for a material gives the relationship between stress and strain. It is obtained by gradually applying load to a test coupon and measuring the deformation , from which the stress and strain can be determined (see tensile testing ).
According to this formula, the graph of the applied force F s as a function of the displacement x will be a straight line passing through the origin, whose slope is k. Hooke's law for a spring is also stated under the convention that F s is the restoring force exerted by the spring on whatever is pulling its free end.
Castigliano's method for calculating displacements is an application of his second theorem, which states: If the strain energy of a linearly elastic structure can be expressed as a function of generalised force Q i then the partial derivative of the strain energy with respect to generalised force gives the generalised displacement q i in the direction of Q i.
Instrumented indentation basically indents a sharp tip into the surface of a material to obtain a force-displacement curve. The results provide a lot of information about the mechanical behavior of the material, including hardness, e.g., elastic moduli and plastic deformation.
Young's modulus is defined as the ratio of the stress (force per unit area) applied to the object and the resulting axial strain (displacement or deformation) in the linear elastic region of the material. Although Young's modulus is named after the 19th-century British scientist Thomas Young, the concept was developed in 1727 by Leonhard Euler.
where is the maximum force predicted by the Lennard-Jones potential and is the maximum separation obtained by matching the areas under the Dugdale and Lennard-Jones curves (see adjacent figure). This means that the attractive force is constant for z 0 ≤ z ≤ z 0 + h 0 {\displaystyle z_{0}\leq z\leq z_{0}+h_{0}} .
The so-called force curve is the graph of force (or more precisely, of cantilever deflection) versus the piezoelectric position on the Z axis. An ideal Hookean spring, for example, would display a straight diagonal force curve. Typically, the force curves observed in the force spectroscopy experiments consist of a contact (diagonal) region ...
is the force on the body; is the displacement produced by the force along the same degree of freedom (for instance, the change in length of a stretched spring) Stiffness is usually defined under quasi-static conditions, but sometimes under dynamic loading. [3]