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When does Hooke’s law stop accurately describing a material’s deformation?

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Figure 1: The position vector x and the velocity vector v of a material point, the body force fdV acting on an element dV of volume, and the surface force TdS acting on an element dS of surface in a Cartesian coordinate system 1, 2, 3 (see text).
mechanics of solids Figure 1: The position vector x and the velocity vector v of a material point, the body force fdV acting on an element dV of volume, and the surface force TdS acting on an element dS of surface in a Cartesian coordinate system 1, 2, 3 (see text).

Hooke’s law accurately describes a material's deformation only when the applied force and the resulting displacement (or stress and strain) maintain a linear, proportional relationship. This range of behavior is known as linearly elastic behavior.123

Hooke’s law ceases to be an accurate description at the proportional limit. Beyond this point, the stress-strain relationship is no longer linear. It is important to distinguish this from the elastic limit (or yield point), which marks the transition from elastic behavior—where a material returns to its original shape upon the removal of stress—to plastic behavior, where deformation becomes permanent.143

While the proportional limit and elastic limit coincide for some materials, others exhibit a region of nonproportional elasticity between them. In this intermediate region, a material may still behave elastically (returning to its original shape), but it no longer follows the linear proportionality defined by Hooke’s law.142