View source for Tensile Test True Stress–Strain Diagram

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<span style="font-size:1.2em;font-weight:bold;">Tensile test true stress–strain diagram</span>
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==Technical stress–strain diagram==

In both [[Tensile Test|conventional]] and controlled tensile tests (see: [[Tensile Test Control|tensile test control]]) on [[Plastics|plastics]], the apparent, technical or so-called engineering force–elongation diagram is determined. The [[Measured Variable|measured variables]] in the [[Tensile Test|tensile test]] are the force and the resulting elongation of a [[Specimen|test specimen]], whereby the elongation is determined nominally via the traverse path measurement or normatively by means of [[Tensile Test#Tensile test, path measurement technique|strain extensometers]] on the test specimen. The advantage of normative measurement is that it is not the measurement length ''L''<sub>0</sub> but the clamping length ''L''<sub>c</sub> (where ''L''<sub>0</sub> < ''L''<sub>c</sub>) that is measured, and that influences from the self-deformation of the [[Material Testing Machine|material testing machine]], such as [[Machine Compliance|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 [[Elastic Modulus|modulus of elasticity]] determined using nominal strain measurement is smaller than the normative value. Since the [[Measured Value|measured values]] force ''F'' and elongation Δ''L'' depend on the geometry of the test specimens used, these values are related to the initial data ''A''<sub>0</sub> and ''L''<sub>0</sub> ('''Eq. 1–3'''), resulting in the technical stress ''σ'' and the technical strain as the normative value ''ε'' or nominal [[Material Parameter|parameter]] ''ε''<sub>t</sub>.

{|
|-
|width="20px"|
|width="500px" | <math>\sigma =\frac{F}{A_0}</math>
|width="50px" |(1)
|}

{|
|-
|width="20px"|
|width="500px" | <math>\varepsilon =\frac{\Delta L_0}{L_0}= \frac{L-L_0}{L_0}</math>
|width="50px" |(2)
|}

{|
|-
|width="20px"|
|width="500px" | <math>\varepsilon =\frac{\Delta L_t}{L}= \frac{L_t-L}{L}</math>
|width="50px" |(3)
|}

The [[Material Parameter|parameters]] of the [[Tensile Test|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 Component|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''').

[[File:True_TT_Stress-Strain_Diagram_Fig-1.jpg|450px]]
{| 
|- valign="top"
|width="50px"|'''Fig. 1''': 
|width="600px" |True tension and strain during the calendering process of [[Plastics|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)'''.

{|
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|width="20px"|
|width="500px" | <math>\sigma _w =\frac{F}{A}</math>
|width="50px" |(4)
|}

If the current elongation of the product Δ''L''<sub>t</sub> 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''<sub>w</sub> or ''ε''<sub>tw</sub> according to '''Eq. (5)''' or '''(6)'''.

{|
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|width="20px"|
|width="500px" | <math>\varepsilon _w=\frac{L}{L_0}=ln \ (1+\varepsilon )</math>
|width="50px" |(5)
|}


{|
|-
|width="20px"|
|width="500px" | <math>\varepsilon _{tw}=\frac{L_t}{L}=ln \ (1+\varepsilon _t)</math>
|width="50px" |(6)
|}

This gives the true withdrawal velocity or [[Strain Rate Basics|strain rate]] according to '''Eq. (7)'''.

{|
|-
|width="20px"|
|width="500px" | <math>\frac{d\varepsilon _{tw}}{dt}=\frac{1}{L}\;\;\; \frac{dL_{t}}{dt}</math>
|width="50px" |(7)
|}

==Nominal and normative strain measurement in tensile testing==

If we consider the conventional or [[Tensile Test Control|controlled tensile test]], we can see that the initial cross-sectional area ''A''<sub>0</sub> is only available at the start of the test. The actual cross-sectional area, which constantly decreases in the [[Plane Stress and Strain State|plane stress state]] of the [[Specimen|test specimen]] in accordance with the [[Poisson's Ratio|Poisson's ratio]] of the [[Material & Werkstoff|material]], is not measured even in the area of [[Tensile Test Uniform Elongation|uniform elongation]]. The occurrence of a necking front in [[Ductility Plastics|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 Methods|hybrid testing methods]] such as [[Laser Extensometry|laser]] or [[Video Extensometry|video extensometry]], digital image correlation (DIC) or ESPI (electronic speckle pattern interferometry) or [[Shearography|shearography]] can be used. With regard to the measurement of strain, '''Figure 2''' shows, in comparison to '''Figure 1''', that the initial measurement length ''L''<sub>0</sub> 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.

[[File:True_TT_Stress-Strain_Diagram_Fig-2.jpg|350px]]
{| 
|- valign="top"
|width="50px"|'''Fig. 2''': 
|width="600px" |Nominal and normative strain measurement in [[Tensile Test|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#Tensile test, path measurement technique|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|laser longitudinal–transverse scanner]], and the true strain can be measured directly using a [[Laser Doppler Scanner|laser Doppler scanner]] (laser anemometer) ('''Fig. 3''').

[[File:True_TT_Stress-Strain_Diagram_Fig-3.jpg|350px]]
{| 
|- valign="top"
|width="50px"|'''Fig. 3''': 
|width="600px" |Nominal and normative [[Strain Rate Applications|strain rate]] of a test [[Specimen|specimen]]
|}

In the latter case, two measurement windows are observed on the [[Surface|surface]] of the test [[Specimen|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|tensile test]]. The [[Velocity|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 Component|plastic components]], [[Deformation|deformations]] up to the [[Yield Stress|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|yield stress]] or [[Tensile Strength|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)'''.

{|
|-
|width="20px"|
|width="500px" | <math>\sigma _w (t) =\frac{A_0}{A(t)} \  \sigma(t)=\frac{1}{(1+\varepsilon _q (t))^2} \ \sigma (t)</math> mit <math> \varepsilon _q \leq 0 </math>
|width="50px" |(8)
|}

[[File:Z_w_diagramm_4.jpg|450px]]
{| 
|- valign="top"
|width="50px"|'''Fig. 4''': 
|width="600px" |Technical and true stress-strain diagrams for poly(methyl methacrylate) ([[Plastics – Symbols and Abbreviated Terms|abbreviation]]: PMMA) as a function of test temperature
|}

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

==See also==

* [[Tensile Test Event-related Interpretation|Tensile test event-related interpretation]]
* [[Tensile Test Control|Tensile test control]]
* [[Laser Cross-Unit|Laser cross-unit]]
* [[Elastic Modulus|Elastic modulus]]
*[[ Strength]]

==References==

{|
|-valign="top"
|[1]
|[[Bierögel, Christian|Bierögel, C.]]: Tensile Tests on Polymers. In: [[Grellmann,_Wolfgang|Grellmann, W.]], [[Seidler,_Sabine|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)
|-valign="top"
|[2]
|Lüpke, T.: Fundamental Principles of Mechanical Behaviour. In: [https://www.researchgate.net/profile/Wolfgang-Grellmann 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) 
|}

[[Category:Tensile Test]]