Appendix: The Considère Condition

Wilko C. Emmens

doi:10.1007/978-3-642-21904-7_19

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Appendix: The Considère Condition

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Wilko C. Emmens

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First Online: 22 July 2011

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Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)

Abstract

In Sect. 4.2 the Considère condition was presented as the basic condition that determines stability in the tensile test. This Appendix discusses that condition in detail.

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The Considère Condition, 1

Consider a part of a tensile test specimen loaded with a tension force F and momentary cross-section area A, the momentary yield stress will be σ = F/A. Suppose that in that part a small section exists that has been strained additionally with a small strain dε ≪1, with resulting cross-section area A′ and stress σ′, see Fig. 19.1. Assuming the volume to remain constant the cross-section area A′ of that part has reduced: A′ = A/(1 + dε) ≈ A.(1 − dε).

The tension force in that section has to be equal to F, so that for the actual stress in that part σ′.A′ = F = σ.A, consequently:

$$ \sigma^{\prime} = \sigma .(1 + d\varepsilon ) $$

(19.1)

The question is now if that section has actually become stronger or weaker. If the section has become stronger, meaning that a force larger than F is required to elongate it more, it will not deform any further but another part of the specimen will start to deform instead; the deformation is stable and no necking will occur. This depends on the change in yield stress, the stress required to deform that part any further. This new yield stress can be expressed as:

$$ \sigma_{\text{yield, \,new}} = \sigma + d\sigma = \sigma + \frac{d\sigma }{d\varepsilon }.d\varepsilon = \sigma .\left( {1 + \frac{1}{\sigma }.\frac{d\sigma }{d\varepsilon }.d\varepsilon } \right) $$

(19.2)

We can now state that no instability will arise, and necking will not occur, if the new yield stress of the section expressed by Eq. 19.2 has become larger than the actual acting stress expressed by Eq. 19.1, this finally yields:

$$ \frac{1}{\sigma }.\frac{d\sigma }{d\varepsilon } > 1\,,\quad \text {or}:\quad \frac{d\sigma }{d\varepsilon } > \sigma $$

(19.3)

This is the well known Considère condition.

The Considère Condition, 2

We will now derive the Considère condition in a more fundamental way, so that it can be expanded easily.

When a strip of thickness t is pulled in tension with a force F, the operation is stable if and only if F increases with elongation. So an instability can start when F reaches a maximum or: dF = 0. Using F = σ.A (A = cross-section area):

$$ {\text{d}}F = 0 = {\text{d}}\left( {\sigma_{1} .A} \right) = {\text{d}}\sigma_{1} .A + \sigma_{1} .{\text{d}}A,\,\,\,\,\frac{{{\text{d}}\sigma_{1} }}{{\sigma_{1} }} = - \frac{{{\text{d}}A}}{A} $$

(19.4)

Using constancy of volume, dV = 0:

$$ {\text{d}}V = 0 = {\text{d}}(l.A) = {\text{d}}l.A + l.{\text{d}}A,\quad \frac{{{\text{d}}A}}{A} = - \frac{{{\text{d}}l}}{l} = - {\text{d}}\varepsilon_{1} $$

(19.5)

Combining both yields the Considère condition.

$$ \frac{{d\sigma_{1} }}{{\sigma_{1} }} = d\varepsilon_{1} ,\quad \frac{{d\sigma_{1} }}{{d\varepsilon_{1} }} = \sigma_{1} $$

(19.6)

Assuming that σ₁ = C.ε ₁ ⁿ :

$$ \frac{{{\text{d}}\sigma_{ 1} }}{{\sigma_{ 1} }} = C.n.\varepsilon_{1}^{n - 1} = C.\varepsilon_{1}^{n} = C.\varepsilon_{ 1} .\varepsilon_{1}^{n - 1} ,\quad \varepsilon_{1} = n $$

(19.7)

Note that if we would get as a condition dσ₁/dε₁ = p.σ₁ instead:

$$ \frac{{{\text{d}}\sigma_{ 1} }}{{\sigma_{ 1} }} = C.n.\varepsilon_{1}^{n - 1} = p.C.\varepsilon_{1}^{n} = p.C.\varepsilon_{ 1} .\varepsilon_{1}^{n - 1} ,\quad \varepsilon_{1} = \frac{n}{p} $$

(19.8)

Hill’s Local Necking Criterion

There are two instabilities in a tensile test, one creating the diffuse neck, and a second one creating the local neck and final fracture. The first one can be analysed as above. The second one is a local neck, meaning that there is no elongation along the neck, only across the neck ( Sect. 5.2). In that case we can ignore any changes in width of the specimen, and apply the same analysis on a piece of unit width, so A = 1.t = t. This yields:

$$ {\text{d}}F = 0 = {\text{d}}(\sigma_{1} .t) = {\text{d}}\sigma_{1} .t + \sigma_{ 1} . {\text{d}}t,\quad \frac{{{\text{d}}\sigma_{ 1} }}{{\sigma_{ 1} }} = - \frac{{{\text{d}}t}}{t} = - {\text{d}}\varepsilon_{3} $$

(19.9)

Or:

$$ \frac{{{\text{d}}\sigma_{1} }}{{{\text{d}}\varepsilon_{3} }} = - \sigma_{1} $$

(19.10)

This is known as Hill’s local necking criterion. In a more general situation we have defined the strain state by a constant β defined as: ε₂ = β.ε₁ ( Sect. 5.1), and consequently: ε₃ = −(1 + β).ε₁ as ε₁ + ε₂ + ε₃ = 0. We can rewrite Eq. 19.10 as:

$$ \frac{{{\text{d}}\sigma_{1} }}{{{\text{d}}\varepsilon_{1} }} = (1 + \beta ).\sigma_{1} $$

(19.11)

Which yields as an instability limit:

$$ \varepsilon_{1} = \frac{n}{1 + \beta },\quad \varepsilon_{3} = - n $$

(19.12)

The limits are graphically presented in Fig. 19.2. For a situation of plane strain (ε₂ = 0) both limits are the same.

Strain Rate Hardening

The effect of strain-rate hardening on the necking limit can also be studied easily. For the following analysis we assume that the tensile test is carried out at a constant strain-rate, so that any strain-rate terms or factors become a constant.

Two cases are distinguished, multiplicative and additive strain-rate hardening. In the first case the total hardening is expressed in general by:

$$ \sigma (\varepsilon ,\dot{\varepsilon }) = \sigma_{1} (\varepsilon ) \cdot \sigma_{2} (\dot{\varepsilon }),\quad \dot{\varepsilon } = \frac{d\varepsilon }{dt} $$

(19.13)

Or more specifically by:

$$ \sigma (\varepsilon ,\dot{\varepsilon } )= C .\dot{\varepsilon }^{m} .\varepsilon^{n} $$

(19.14)

Assuming constant strain-rate relation Eq. 19.14 becomes:

$$ \sigma (\varepsilon )= C' .\varepsilon^{n} $$

(19.15)

which is the same as the relation used in Eq. 19.7. So this does not change the necking limit.

In the second case the total hardening can be expressed by:

$$ \sigma (\varepsilon ,\dot{\varepsilon }) = \sigma_{1} (\varepsilon ) + \sigma_{2} (\dot{\varepsilon }) $$

(19.16)

Assuming a power-law hardening and constant strain rate we get:

$$ \sigma (\varepsilon )= C.\varepsilon^{n} + A $$

(19.17)

Applying the Considère condition as in Eq. 19.7

$$ \begin{gathered} \frac{{{\text{d}}\sigma_{ 1} }}{{\sigma_{ 1} }} = C.n.\varepsilon_{1}^{n - 1} = C.\varepsilon_{1}^{n} + A = C.\varepsilon_{ 1} .\varepsilon_{1}^{n - 1} + A \hfill \\ \varepsilon_{ 1} .\varepsilon_{ 1}^{n - 1} = n.\varepsilon_{1}^{n - 1} - \frac{A}{C} \hfill \\ \end{gathered} $$

(19.18)

For positive values of A this will yield a solution for ε₁ that is lower than n, indicating that a positive strain-rate hardening will lower the necking limit.

Thickness Stress

The case of a non-zero thickness stress σ₃ as discussed in Sect. 8.1 can be analysed similar to the case of additive strain-rate hardening. The Tresca criterion states: σ₁ − σ₃ = σ_f. We can also regard this as a change of the flow stress: σ₁ = σ_f + σ₃. Considering σ₃ to be constant, this is the same situation as analysed above with A = σ₃:

$$ \begin{gathered} \varepsilon_{ 1} .\varepsilon_{ 1}^{n - 1} = n .\varepsilon_{ 1}^{n - 1} - \frac{{\sigma_{ 3} }}{C} \hfill \\ \varepsilon_{ 1} = n - \frac{{\sigma_{ 3} }}{C} \cdot \frac{ 1}{{\varepsilon_{ 1}^{n - 1} }} = n - \frac{{\sigma_{ 3} .\varepsilon_{ 1} }}{{C .\varepsilon_{ 1}^{n} }} \hfill \\ \end{gathered} $$

(19.19)

Still assuming a simple power law hardening with σ_f = C.ε ₁ ⁿ :

$$ \begin{gathered} n = \varepsilon_{1} \cdot \left(1 + \frac{{\sigma_{3} }}{{\sigma_{f} }} \right) = \varepsilon_{1} \cdot \left(1 + \frac{{\sigma_{3} }}{{\sigma_{1} - \sigma_{3} }}\right) = \varepsilon_{1} \cdot \frac{{\sigma_{1} }}{{\sigma_{1} - \sigma_{3} }} \hfill \\ \frac{{\varepsilon_{1} }}{n} = \frac{{\sigma_{1} - \sigma_{3} }}{{\sigma_{1} }} = 1 - \frac{{\sigma_{3} }}{{\sigma_{1} }} \hfill \\ \end{gathered} $$

(19.20)

This is the same relation as shown in Eq. 8.1 as ε₀(σ₃) = ε₁ in Eq. 19.20, and of course ε₀(0) = n. Note however that this is a first order approximation valid for low absolute values of σ_3, as ε₁ is not the effective strain any more.

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Wilko C. Emmens
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- 2
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1.Construerende Technische Wetenschappen, Technische MechanicaUniversity of TwenteEnschedeThe Netherlands
2.Tata Steel RD&TIJmuiden The Netherlands

Appendix: The Considère Condition

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