It is difficult in practice to identify the proportional limit, i.e. the point at which the linear elastic regime on a stress vs. strain plot transitions to the plastic regime. Conventionally, therefore, yield is defined in terms of the offset yield point (proof stress; strain¼0.002, or 0.2%) construction [11]. This criterion works well when used to compare the yield strengths of samples that have the same elastic moduli – and is therefore useful in its original context of metals, where drawing or other forming processes that lead to work hardening will affect the yield strength but not the stiffness. However, in the case of polymers, drawing affects the stiffness as well as the yield strength, and the offset yield point construction therefore doesnot give a clear-cut view of how yield strength has changed.
In the present context, where reeled samples of silk are susceptible to inadvertent drawing prior to testing, the amount by which the offset yield point exceeds the proportional limit will therefore depend on the stiffness of the sample at the time of the tensile test. To demonstrate and quantify this phenomenon, it is convenient to approximate the nominal stress vs. nominal strain plot by two straight lines: an initial linear segment corresponding to elastic deformation, and a second linear segment of significantly lesser (but still positive) slope that corresponds to plastic drawing. Many worm (lepidopteran) silks exhibit these basic characteristics [12]. We will assume here for simplicity (but not out of necessity) that the transition between the two regimes is abrupt, and that both samples have the same proportional limit.
Figure18.3shows such idealized stress vs. strain plots for samples A and B tested by two hypothetical researchers and stretched differently during reeling and handling – for example, by the researchers identified as “Person 1” and “Person 2” in Table18.1.
18 Untangling a Sticky Problem: The Tensile Properties of Natural Silks 131
Points lying on QC obey the equation
s¼E1;AeþK1
where the constantK1can found by substituting the coordinates of point Q:
0¼0:002E1;AþK1; leading to
s¼E1;Aðe0:002Þ (18.1)
Points lying on SG obey the equation
s¼E2eþK2
where the constantK2can found by substituting the coordinates of point S:
sp¼E2
sp
E1;A
þK2; leading to
s¼E2eþsp 1 E2
E1;A
(18.2) By combining Eqs. 18.1 and 18.2 to eliminatee, we can solve for the value ofsat the intersection of QC and SG, and thus obtain an expression for the offset yield strength of sample A:
sy;A¼spþ0:002E2E1;A
E1;AE2
(18.3) Similarly, by considering the intersection of QD and TH, we can obtain an expression for the offset yield strength of sample B:
sy;B¼spþ0:002E2E1;B
E1;BE2
(18.4) It is immediately apparent that samples A and B will exhibit different values of offset yield strength, even though they both have the same proportional limit,sp.
The amount by which the offset yield strength exceeds the proportional limit depends on the actual stiffness of the sample (and therefore on whether molecular alignment is introduced by plastic deformation when fiber is reeled from the cocoon); it also depends on the slope of the stress vs. strain curve beyond the proportional limit, i.e. it depends on the work hardening rate. B. mori silk reeled by hand and tested by a single researcher (“Researcher A”) under consistent conditions [13]
Fig. 18.3 Idealized nominal stress vs. strain plots, demonstrating – and facilitating quantification of – a limitation of the offset yield criterion
132 C. Viney
exhibitedE1;A22:1 GPa andE215:1 GPa. On that basis, Eq.18.3allows us to estimate that the offset yield strength exceeded the proportional limit by the amount
sy;Aspð0:002Þð15:1 GPaÞð22:1 GPaÞ 22:1 GPa15:1 GPa
ð Þ ¼95 MPa (18.5)
By considering therelativedifferenceRbetween offset yield strength and proportional limit for two samples that have different values of initial stiffness, we can highlight the sensitivity of the offset yield strength to changes in initial stiffness.
From Eqs. 18.3 and 18.4 we obtain:
R¼sy;Bsp
sy;Asp
¼0:002E2E1;B
E1;BE2
0:002E2E1;A
E1;AE2
¼E1;B E1;AE2
E1;A E1;BE2
(18.6)
With reference to the abovementioned “Researcher A”, the quoted data forB. morisilk are related byE20:68E1;A. Consider now a second researcher (“Researcher B”) who reels and prepares samples for whichE1;B¼b E1;Awherebis a constant. Then Eq.18.6gives:
R¼bE1;A E1;A0:68E1;A
E1;A bE1;A0:68E1;A
¼bð10:68Þ
b0:68 ¼ 0:32b
b0:68 (18.7)
The graphical relationship betweenRandbis shown in Fig.18.4a.
Thus if “Researcher B” imparts just a 5% increase in stiffness to the silk (b¼1:05) as it is reeled and prepared for tensile testing, the value ofRis decreased by 10%. Consequently, the offset yield strength measured by “Researcher B” exceeds the proportional limit by 85 MPa instead of 95 MPa (Eq.18.5“Researcher A”), affecting the apparent variability in the yield strength of the material.
It is worth noting that the value ofRcalculated with Eq.18.7does not depend explicitly on the stiffness of the samples.
WhatRdoes, is calculate the factor by whichsysPis changed if we know
1. The factorathat relatesE1;A(the slope of the linear elastic deformation regime on the stress vs. strain plot of a sample chosen as reference) toE2(the slope of the plastic deformation regime for that material; assumed linear and constant), and 2. The factorbthat relatesE1;Ato the initial stiffnessE1;Bof material that differs from the reference material in regard to the
amount of plastic deformation that it has experienced.
0.00 0.20 0.40 0.60 0.80 1.00
1.0 1.5 2.0
R
b a
1.00 1.00 1.20
1.40 1.60
1.80 2.00
0.75 0.50 0.00 0.25
0.20 0.40 0.60 0.80 1.00
a R
b b
0.80–1.00 0.60–0.80 0.40–0.60 0.20–0.40 0.00–0.20
Fig. 18.4 (a) Plot of the relationship between parametersRandbas embodied in Eq.18.7. (b) Plot of the relationship between parametersR,aand bas embodied in Eq.18.8
18 Untangling a Sticky Problem: The Tensile Properties of Natural Silks 133
This generalization of Eq.18.7is expressed in Eq.18.8and illustrated in Fig.18.4b:
R¼bð1aÞ
ba (18.8)