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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 ).
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.
Load-displacement curves can be gathered for a diverse range of materials, and their mechanical properties can be directly inferred from these curves. Additionally, some advanced systems offer the capability to integrate optical imaging with micro-mechanical characterization, enabling a comprehensive understanding of the relationship between ...
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.
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}} .
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 the lattice energy and move the atoms in the top plane over the lower atoms and into a new lattice site.
As for the tensile strength point, it is the maximal point in engineering stress–strain curve but 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.}