In-Plane Tensile Properties

1. In-Plane Tensile Properties

2. Introduction In-plane mechanical properties in tension are important in papers for printing or other web uses and in packaging papers and boards. Tensile strength is the easiest to understand. Fracture toughness controls the runnability of a paper web. It relates to the tensile strength, elastic modulus and load elongation behavior of paper.

3. The load-elongation curve of paper represents the mechanical equation of the state of the paper, from which all other properties can, in principal be derived using continuum mechanics. The tension of a running web is small and controlled by the elastic modulus. The elastic modulus also controls the bending stiffness and structural rigidity of paper and board sheets.

4. Elastic modulus and load-elongation curve of paper are the characteristics that best help to understand the tensile strength and eventually fracture toughness of paper In discussing tensile properties, we will discuss elastic properties, the load-elongation curve, tensile strength and fracture toughness.

5. Elasticity • The elastic modulus is an important property of paper because paper is seldom loaded near the ultimate failure stress. • It determines how web tension depends on the speed difference in web fed end use. • Through bending stiffness, it also controls performance of paper and board in sheet form. • Elastic modulus can be an indicator of other paper properties, such as strength and dimensional stability.

6. The fundamental importance of the elastic modulus is that is self-averaging and statistically stable as opposed to strength properties that depend on the “weakest link”. • The relationship between elastic properties and structure of paper is better understood than other properties. • The anisotropic elastic modulus of machine made paper is sensitive to drying shrinkage, wet straining of paper and fiber orientation.

7. Elastic constants and their measurement • The usual elastic modulus, Young’s modulus, E measures the force necessary to produce a small elongation. E=ds/de as eÞ 0 where s is the applied stress, or force per unit area, and e is the corresponding strain. • See Figure 1. to slide 12

8. If the stress-strain curve, load elongation curve, is linear, then E=s/e. • At larger elongations, the slope is a tangent modulus. • For the most general anisotropic material, there are three principal directions for which to define a modulus. • For machine made paper it is easy to identify these directions as the machine direction (EMD), the cross machine direction (ECD) and the thickness direction (Ez).

9. For in-plane measurement of the modulus is usually in the tensile mode by stretching the specimen. • In the z-direction, the compression mode is generally used more than tension. • For homogeneous materials the zero strain modulus is the same under tension and compression. • For paper, the apparent elastic modulus may be different in tension and compression. • This is because fibers buckle under compression, making the apparent modulus smaller.

10. For small enough strains the modulus will be the same under tension and compression, but for finite compression, the apparent compression modulus will appear smaller. • Another important elastic property is the Poisson ratio, n, which is the ratio of - the lateral strain to the tensile strain. • For paper, the Poisson ratio of interest is the ratio of CD contraction over MD stretch, called the MD Poisson ratio nMD or nxy.

11. To see the significance of this we consider the biaxial stress/strain state sx=Ex(ex+nyxey)/(1-nxynyx) sy=Ey(ey+nxyex)/(1-nxynyx) • The Maxwell relation (analogous to classical thermodynamics) requires that Exnyx=Eynxy or if x=MD and y=CD EMD/ECD=nMD/nCD • Thus, since EMD>ECD, an elongation in MD therefore causes a larger lateral contraction than an elongation in CD.

12. The other modulus of interest, the shear modulus, gives the stress, , needed for a given skew deformation,  (see Figure 1). G=t/g • Shear stresses arise if the fiber orientation angle is not in MD, or the web doesn’t run straight. • This may cause wrinkles in the web. • For this reason, G is difficult to measure, since wrinkling must be prevented in the measurement.

13. Measurements • Evaluation of elastic moduli from the load-elongation curve requires care. • Good accuracy requires a good linear regression over a long range. • The linear range may be small, as suggested by Figure 2, so that a good estimate of the limiting slope may be difficult. • Values of E range from 2-20 GPa.

14. Typically, E=5 GPa. • Specific modulus of elasticity divided by density, E/=F/Wb, where b is the basis weight, is what is usually measured. • The specific modulus is analogous to the tensile index. • It can be estimated from ultrasonic measurements, E/=c2(1-nxynyx) c2 where it is assumed that nxynyx << 1.

15. This relation follows for an orthotropic continuum material. • Figure 3 shows that the sonic modulus of paper tends to exceed the one measured mechanically. to slide 19

16. Theory • The “self-averaging” nature of elastic properties follows from the relationship of modulus to the average elastic energy. • If U is the average energy per unit volume U=Ee2/2=se/2, or E=2U/e2 for sufficiently small e. • This follows from the definition of work (or energy stored as d(UV)=dW=Fdx

17. Or F=dW/dx=d(UV)/d(L) or =FL/V=dU/d • The energy relation follows from this since s=Ee, with E constant. • If we assume that the elastic energy is due to axial elongation of fibers, then U=rEf/(2rf)<ei2>, where Ef is the elastic modulus of fibers, rf is their density and <ei2> is the average axial strain squared.

18. We assume that the elastic strain is proportional to the fiber strain, i/f=ki. • The “activation constant” ki of a fiber depends on the orientation of fibers. • Then the modulus becomes E=rEf/rf<ki2> • If the actual strain is the axial strain of the fibers, then <ki2> =1/3, for isotropic fiber orientations. • In reality, <ki2> < 1/3, because the fiber strain is zero at the ends.

19. A good semi-empirical relationship is E=E*/3(r-r0)/rw? • E* describers fiber properties and 0 accounts for the inactive fiber ends and their effect on the surrounding network. • As seen in Figure 3, this relationship holds well when density changes through basis weight.

20. Cox's shear-lag theory produces this relationship with r0=wf/lfr/RBA(Ef/2Gf).5 and E*=Ef, where wf and lf are the width and length and Gf is the shear modulus of the fibers. • 0 is constant in data sets where density is linearly related to RBA.

21. The shear lag model is only approximate. • It predicts that r0 is inversely proportional to fiber length. • This holds for low basis weight hand sheets. • For ordinary hand sheets, r0is almost independent of lf, as shown in Figure 4.

22. Papermaking effects through sheet density • We consider elastic modulus versus paper density, implicitly assuming that apparent density and not effective density faithfully reflects changes in RBA. • We will find a compact relationship representation of elastic modulus. • The actual density to RBA relationship will vary from one furnish to another.

23. For a given paper grade at a fixed basis weight, density and modulus depend on beating, wet pressing and different pulp components. • This is shown in Figures 5 and 6.

24. When beating degree varies, as shown in Figure 5, the threshold density 0 appears to vary from 200-400 kg/m3. • The slope E* seems rather independent of wood species. • As shown in Figure 6, the effect of wet pressing is different. • For all but densities below 300 kg/m3, the threshold density, 0, is practically independent of furnish at 0200 kg/m3.

25. These behaviors are qualitatively summarized in Figure 7. • The results can be explained as: • The Threshold density, r0200 kg/m2, is essentially the same for all pulps and • Wood species, pulp type, beating and pressing change E*, density, and RBA. • This plot does not apply to poorly bonded sheets with r≤300 kg/m3.

26. Drying stresses • Figure 8 illustrates that if shrinkage is limited during drying, or wet paper is strained, E increases. • Erestrained drying –Efree shrinkage is large for pulps with a high shrinkage potential. • Chemical pulps are therefore more sensitive to drying shrinkage than mechanical pulps and well-beaten chemical pulps are more sensitive than lightly beaten pulps.

27. Figure 9 shows that the E difference between the two shrinkage levels is larger in MD than CD because the elastic modulus is larger in MD. • The axial elastic modulus of a fiber increases if the fiber dries under tension and decreases if the fiber dries under compression • Compressive drying stresses act on bonded fiber segments and the elastic modulus of fibers decreases locally.

28. If one controls the drying stress so that shrinkage is prevented until the solids level is sufficiently high, then the elastic modulus is linear in drying stress as shown in Figure 10. • To increase drying shrinkage, wet straining increases the elastic modulus of paper, as Figures 8 and 9 show. • Wet straining straightens fibers in the straining direction and removes some of the z-directional “undulations” of fibers.

29. Elastic anisotropy and CD profiles • The Cox theory is useful in expression the effect of fiber orientation on elastic anisotropy. • These relationships are EMD=rFE*(6+4a1+a2)(1-nxynyx)/16 ECD=rFE*(6-4a1+a2)(1-nxynyx)/16 nxy=(2-a2)/(6-4a1+a2) nyx=(2-a2)/(6+4a1+a2) G=rFE*(2-a2)/16 • The Fourier coefficients, an, of the fiber distribution were defined in Chapter 1.

30. The specific elastic modulus, E*, is assumed constant, independent of fiber orientation. • The prefactor,  , is given by =1-0/. • Experiments confirm that the fiber orientation dependence of the elastic moduli, EMD and ECD is consistent with Cox's equations. • Since a2<<1, G, should be only a weak function of fiber orientation.

31. The geometric mean of elastic moduli, Egeom=(EMDECD ).5, is essentially constant if fiber orientation anisotropy varies, but drying restraints remain constant as, shown in Figure 11. • When fiber orientation is constant and only drying shrinkage varies, then Egeom is linearly in the modulus ratio RE=EMD/ECD. • This applies across the paper web as shown in Figure 12.

32. Elastic modulus and tensile strength are smaller and breaking strain is larger at the edges than at the center of the web. • EMD is often 10% higher in the center of the web than at the edges. • For ECD, the corresponding difference is 30%. • As shown in Figure 13, the profile variation of shear modulus is intermediate between MD and CD dependence.

33. Load-elongation behavior • The mechanical response of paper is described by the load-elongation curve. • This is often called the stress-strain curve, but its is only the average, since the local stresses and strains are not constant. • Formation-like variations in local basis weight, fiber orientation, and other factors induce non-uniform stresses and strains in paper. • At the fiber level, the stress field of the fiber network is quite complex.

34. Macroscopic observations • Figure 14 gives a typical load-elongation curve for paper. • In the CD, it extends to large strains and is nonlinear. • The MD behavior is nearly a linear elastic material, while the CD direction is a nearly ideal plastic material. • Ideally plastic means that above a yield threshold, load doesn't increase while elongation grows.

35. The primary difference between MD and CD curves is the effect of drying stress. • Drying stress is the most important single factor that affect the shape of the curves. • The greatest nonlinearity arises from irreversible or plastic elongation. • If a paper specimen is stretched beyond the yield point and then released, it becomes permanently longer than it was originally. • The elastic modulus changes very little during elongation.

36. The reversible, or purely elastic strain, can be obtained by dividing the stress by the modulus el=/E • The irreversible strain component, pl, is given by pl=e-/E • In the load-elongation curve, an initial linear region exists, whose slope is the elastic modulus. • The linear region ends at a yield point and plastic elongation occurs above this point.

37. Figure 15 illustrates how the yield point is only qualitatively defined, since the deviation from linearity grows gradually with elongation. • One criterion for the yield point is a .2% strain deviation from the linear trend, or pl=0.2%. • This choice is arbitrary, but a nonzero yield point occurs for any positive choice of strain offset. • The evolution of the effective elastic modulus outside the linear region can be seen from the load-elongation cycles.

38. The deloading and reloading elastic moduli Ed and Er are defined in Figure 16. • As seen in Figure 17, the MD values of Ed and Er were almost equal and decreased only slightly at 10%-15% over the entire curve • In CD, Ed and Er vary more than in MD. • At macroscopic failure, Ed is typically 10% larger than Er. to slide 43

39. Is some cases in Figure 17, both Ed and Er increased • Increase in CD is due to the straightening of micro-compressions and other structural deformations created in drying. • Ed has increased with elongation by as much as a factor of two. • The area under the load-elongation curve is the work necessary to break the specimen or the tensile energy adsorption (TEA).

40. The shape of the load-elongation curve depends on the rate of elongation. • At high strain rates, the curve is steeper than at low strain rates as shown in Figure 18. • The elongation of paper is not merely a viscoelastic phenomenon, but contains a plastic component. • Paper is therefore a viscoelastic plastic material.

41. Triaxial Deformations • Figure 19 shows the effect of increased beating on in handsheets. • Figure 20 compares machine-made paper. • In isotropic handsheets, in-plane contraction, -yincreases almost linearly with elongation, -x.

42. The ratio y/x is equal to the ordinary Poisson ratio in the limit x -> 0 • In machine-made paper, the ratio y/x generally increases at large stress levels. • In the thickness direction, handsheets contract under in-plane elongation, although the contraction decreases at large elongation. • In machine-made papers no contraction occurs and paper thickness increases even faster when elongation increases. • As bonds open, z-directional fiber undulations in the network relax and thickness increases.

43. Microscopic yielding phenomena • When paper stretches, two things occur; • Fiber segments elongate, in part irreversibly. • Some bonds open gradually. • As shown in Figure 17, the number of fiber segments that bear the load must remain almost constant during the irreversible elongation. • In the elastic region, heat flows into paper when strained as shown in Figure 21.

44. The internal energy of the system increase as a sum of the external work and heat flow. • If the external load is removed in the elastic region, paper releases the same amount of heat that it adsorbed during the elastic elongation, and the internal energy returns to its original value. • The elastic behavior of paper is thermodynamically reversible. • In the plastic region, Figure 21 shows that paper releases heat with decreasing and increasing elongation.

45. Plastic elongation is thermodynamically irreversible. • Since the elastic modulus does not change accordingly, the irreversible heat generation must come from changes in the microscopic or molecular configuration within fibers and bonds. • Influence of drying shrinkage on the load elongation curve comes from an increase in plastic elongation as shown in Figure 22. to slide 53

46. Single fibers show a similar increase in plastic elongation when dried under compression. • However, fibers and paper dried under stress have a more linear load-elongation curve. • The shape of the load elongation curve changes little with drying shrinkage, except for location of the end point. • As seen in Figure 22, changes in the bonding degree have of no systematic effect on the shape of the load-elongation curve.

47. The shape of this curve is independent of changes in the relative bonded area or density induced by beating. • Density and RBA only affect the elastic modulus of paper and the location of macroscopic failure. • Bonding degree has no effect on the shape of the load –elongation curve, neither can the opening of bonds explain the shape of the curve • Effects of drying shrinkage or drying stress is largest on bonded fiber segments.

48. Drying shrinkage induces microcompressions and other deformations into the fiber wall at bond sites. • Drying stress and wet strain pull fibers straight and tight. • The bonded fiber segments show more plastic yielding in freely dried paper than in restrained dried paper as illustrated in Figure 23. • Some bonded segments elongate clearly more than the free segments between bonds.

49. Permanent elongation occurs primarily in bonded segments. • Free segments recover elastically except in freely dried paper, where free segments also undergo irreversible elongation. • When the external stress of paper returns to zero after a loading cycle into the plastic region, bonded fiber segments are under compressive stress and free segments are under tensile stress. • Thus, bonded segments are under compression.

50. The generation of local compressions was verified by Ebeling and are reproduced in Figure 24. • He found that the internal energy of paper increased in a loading cycle into the plastic region. • The internal energy increases because there is elastic energy stored in the compressed and stretched segments, although the average external stress is zero.