Campbell-Ingmanson

Constitutive Equations » Stress-Strain » Campbell-Ingmanson

Description

At concentrations below the gel point, there is no fibre network, and therefore, no fibre network stress. Ingmanson and Whitney (1954) introduced an initial concentration to Campbell relationship (Campbell, 1947), so that it can better describe the fibre network stress-strain behaviour at concentrations close to the gel point.

Application

The Campbell-Ingmanson relationship has a similar applicability region to the Campbell relationship, besides the fact that the former can also describe the fibre network stress-strain behaviour at concentrations close to the gel point.

The Campbell-Ingmanson relationship describes well the stress-strain behaviour of fibre network for individual compression or relaxation, but not for cycles of compression and relaxation. Regarding concentration, it is applicable up to fibre concentrations at where water flows out of the fibre walls (i.e. at approximately 20 percent dryness).

It is therefore suitable in the forming and vacuum sections.

Background

The Campbell-Ingmanson relationship introduces a small modification to the Campbell relationship. According to the Campbell-Ingmanson relationship, there is an initial concentration, $ c_0$ , at which the fibre network start being formed and, therefore, can bear stress:

Campbell-Ingmanson    

where $ c$ is the concentration, $ \sigma$ and $ \sigma_r$ are the applied a reference stress and $ M$ and $ N$ are fitting constants. $ \sigma_r$ is introduced just to avoid complex units in $ M$

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Bibliography

Campbell, W. B., 1947, Pulp and Paper Magazine of Canada 48(3), 103.

Ingmanson, W. L., and R. P. Whitney, 1954, Tappi Journal 37(11), 523.

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Campbell-Ingmason

Campbell-Ingmanson

The Campbell-Ingmanson constitutive equation for stress as a function of concentration (expressed here by the porosity). This relationship assumes that there is a minimum required fibre concentration so that the network still can bear stress.