Tensile Test True Stress–Strain Diagram

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Tensile test true stress–strain diagram

Technical stress–strain diagram

In both conventional and controlled tensile tests (see: tensile test control) on plastics, the apparent, technical or so-called engineering force–elongation diagram is determined. The measured variables in the tensile test are the force and the resulting elongation of a test specimen, whereby the elongation is determined nominally via the traverse path measurement or normatively by means of strain extensometers on the test specimen. The advantage of normative measurement is that it is not the measurement length L₀ but the clamping length L_c (where L₀ < L_c) that is measured, and that influences from the self-deformation of the material testing machine, such as machine compliance, are not included in the measurement signal. As a result of these influences, the nominal elongation or strain is always greater than the normative value, and the modulus of elasticity determined using nominal strain measurement is smaller than the normative value. Since the measured values force F and elongation ΔL depend on the geometry of the test specimens used, these values are related to the initial data A₀ and L₀ (Eq. 1–3), resulting in the technical stress σ and the technical strain as the normative value ε or nominal parameter ε_t.

$\sigma ={\frac {F}{A_{0}}}$

(1)

$\varepsilon ={\frac {\Delta L_{0}}{L_{0}}}={\frac {L-L_{0}}{L_{0}}}$

(2)

$\varepsilon ={\frac {\Delta L_{t}}{L}}={\frac {L_{t}-L}{L}}$

(3)

The parameters of the tensile test that can be derived from this can be used for simple design applications, material selection and material development or quality assurance.

True stress–strain diagram

These characteristic values are not suitable for demanding dimensioning tasks or the design of complex plastic components using the finite element method (FEM), as in this case the true stress–strain diagram and the characteristic values derived from it must be used. The true stress–strain diagram is based on the use of the current, time-varying cross-sectional area and the varying initial measurement length. This can best be illustrated using the example of the rolling or calendering process (Fig. 1).

Fig. 1:

True tension and strain during the calendering process of plastics

If the strand diameter d is measured online during extrusion or the plate thickness d is measured online during calendering, e.g. using shadow image technology, this results in the current cross-section A, which depends on the nozzle geometry and the draw-off speed or the degree of stretching. By measuring the current draw-off force and the strand diameter or sheet thickness, the true stress can then be determined according to Eq. (4).

$\sigma _{w}={\frac {F}{A}}$

(4)

If the current elongation of the product ΔL_t is determined from the constant length between the nozzle and the roller contact point L (Fig. 1), this results in the true normative or nominal strain ε_w or ε_tw according to Eq. (5) or (6).

$\varepsilon _{w}={\frac {L}{L_{0}}}=ln\ (1+\varepsilon )$

(5)

$\varepsilon _{tw}={\frac {L_{t}}{L}}=ln\ (1+\varepsilon _{t})$

(6)

This gives the true withdrawal velocity or strain rate according to Eq. (7).

${\frac {d\varepsilon _{tw}}{dt}}={\frac {1}{L}}\;\;\;{\frac {dL_{t}}{dt}}$

(7)

Nominal and normative strain measurement in tensile testing

If we consider the conventional or controlled tensile test, we can see that the initial cross-sectional area A₀ is only available at the start of the test. The actual cross-sectional area, which constantly decreases in the plane stress state of the test specimen in accordance with the Poisson's ratio of the material, is not measured even in the area of uniform elongation. The occurrence of a necking front in ductile plastics causes even greater problems, as the strain is then recorded with a locally reduced cross-sectional area, which cannot be detected using conventional test methods because the minimum thickness must be tracked by the test technology. In this case, only locally resolving hybrid testing methods such as laser or video extensometry, digital image correlation (DIC) or ESPI (electronic speckle pattern interferometry) or shearography can be used. With regard to the measurement of strain, Figure 2 shows, in comparison to Figure 1, that the initial measurement length L₀ and the reference points 1 and 2 change during the tensile test, which means that it is not possible to directly represent the true strain.

Fig. 2:

Nominal and normative strain measurement in tensile testing

This is only possible by applying Eq. (5) or (6), where the true strain value is calculated from the measured nominal or normative strain. When using conventional strain measurement methods, e.g., strain extensometers or clip-on strain gauges (see: tensile test path measurement technique), the measurement also fails if local strain increases occur as a result of necking fronts, as the initial measurement length then changes constantly and would have to be tracked using local strain measurement methods. Technical possibilities for registering and tracking necking fronts are available with the laser longitudinal–transverse scanner, and the true strain can be measured directly using a laser Doppler scanner (laser anemometer) (Fig. 3).

Fig. 3:

Nominal and normative strain rate of a test specimen

In the latter case, two measurement windows are observed on the surface of the test specimen, in each of which interference patterns from two laser beams are altered as a result of surface roughness on the test specimen during the tensile test. The velocity and path of the speckles can then be calculated from the Doppler frequency, which then gives the true strain in the measurement interval.

True stress–strain diagrams for poly(methyl methacrylate)

For practical purposes, such as the design and dimensioning of plastic components, deformations up to the yield stress should be avoided, which simplifies the determination of true stress-strain diagrams. For this purpose, stress-strain diagrams or stress-time and strain-time diagrams as well as transverse strain-time diagrams are recorded for at least 5 test specimens at a defined test speed until the yield stress or tensile strength is reached. Assuming that the transverse strain behaves identically in the thickness and width directions, the true stress can be calculated from the technical stress using Eq. (8) with knowledge of the variable cross-sectional area A(t), whereby the true strain is obtained from Eq. (5) or (6).

$\sigma _{w}(t)={\frac {A_{0}}{A(t)}}\ \sigma (t)={\frac {1}{(1+\varepsilon _{q}(t))^{2}}}\ \sigma (t)$ mit $\varepsilon _{q}\leq 0$

(8)

Fig. 4:

Technical and true stress-strain diagrams for poly(methyl methacrylate) (abbreviation: PMMA) as a function of test temperature

Using a regression method for the respective 5 test specimens, the true and technical stress–strain diagrams can then be specified (Fig. 4).

See also

• Tensile test event-related interpretation
• Tensile test control
• Laser cross-unit
• Elastic modulus
• Strength

References

[1]	Bierögel, C.: Tensile Tests on Polymers. In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 106–123 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-806-5; see AMK-Library under A 22)
[2]	Lüpke, T.: Fundamental Principles of Mechanical Behaviour. In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 71–86 (ISBN 978-1-56990-807-8; E-Book: ISBN 978-1-56990-806-5; see AMK-Library under A 22)