What happens to a spring's force if you stretch it more?

Whenever a force is applied on a spring, tied at one end, either to stretch it or to compress it, a reaction force comes into play which tries to oppose the change. This force is exerted by the spring on whatever is pulling its free end.

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This is known as Hooke's law and stated mathematically

Reaction Force #F=-kX#,
where #k# is constant which is characteristic of the spring's stiffness, and #X# is the change in the length of the spring. #-ve# sign indicates that restoring force acts opposite to the deformation of the spring.

The relationship holds good so long #X# is small compared to the total possible deformation of the spring.

Alternatively the relationship between applied force and amount of elongation/compression is #F=kX#

In the picture above the red line depicts a Plot of applied force #F# vs. elongation/compression #X# for a helical spring according to Hooke's law.

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Actual plot might look like the dashed line.

Lower part of pictures correspond to various points of the plot. At middle point the spring is in the relaxed state i.e., zero force.

If too much force is applied, one may stretch or compress a spring beyond a certain point that its deformation will occur. On subsequent release of the stress, the spring will return to a permanently deformed shape which will be different from its original shape.

jaumzaum said:

Thanks Chestermiller! I was thinking the linear e quation was just an approximation of the logarithmic one. So by what I understood there are more than one type of spring right? Some of them obey better the logarithm equation while others obey better the linear equation? When one say that one spring have a completely elastic behavior, will this spring be a linear or a logarithm strain spring (what law will it obey better ?). Also, in real life springs, what law they tend to obey better?

There are many strain measures out there, not just these two, and their common characteristic is that they all reduce to the linear strain in the limit of small strains. Fortunately, all materials reduce to linear behavior in the limit of small strains. The other strains are derived from large deformation theory for non-linear elasticity. They satisfy the requirements of isotropy and symmetry in a tensorial 3D framework. But there is no one single strain measure that by itself works for all large strains of real materials (even for uniaxial deformations), and tensorial proper non-linear representations are required. If you want to learn more about this, Google large deformation elasticity of Cauchy-Green deformation tensor.

There are many strain measures out there, not just these two, and their common characteristic is that they all reduce to the linear strain in the limit of small strains. Fortunately, all materials reduce to linear behavior in the limit of small strains. The other strains are derived from large deformation theory for non-linear elasticity. They satisfy the requirements of isotropy and symmetry in a tensorial 3D framework. But there is no one single strain measure that by itself works for all large strains of real materials (even for uniaxial deformations), and tensorial proper non-linear representations are required. If you want to learn more about this, Google large deformation elasticity of Cauchy-Green deformation tensor.

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