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In-Plane Tensile Properties

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## In-Plane Tensile Properties

### 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.

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.

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.

### 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.

• 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.

### Elastic constants and their measurement

• The usual elastic modulus, Youngs modulus, E measures the force necessary to produce a small elongation.

E=ds/de as e 0

where s is the applied stress, or force per unit area, and e is the corresponding strain.

• See Figure 1.

to slide 12

• 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).

• 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.

• 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.

• 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.

• 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 doesnt 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.

### 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.

• 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.

• 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

### 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

• 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.

• 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.

• 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.

• 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.

• 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.

### 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.

• 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.

• 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 0200 kg/m3.

• These behaviors are qualitatively summarized in Figure 7.

• The results can be explained as:

• The Threshold density, r0200 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 r300 kg/m3.

### 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.

• 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.

• 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.

### 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.

• 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.

• 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.

• 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.

• 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.

### 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.

• 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.

• 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.

• 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.

• 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

• 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).

• 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.

### 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.

• 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.

### Microscopic yielding phenomena

• When paper stretches, two things occur;

• Fiber segments elongate, in part irreversibly.

• 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.

• 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.

• 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

• 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.

• 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.

• 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.

• 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.

• 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.

• The opening of bonds is not the cause for the nonlinear curve.

• Most bonds opened gradually and did not break completely even when paper was strained to failure.

• The resulting reduction in bonding area is reflected in the optical properties.

• For a given paper the increase in light scattering coefficient is linearly proportional to the plastic elongation as shown in Figure 25.

### Models

• Examples of load-elongation curves for machine made paper are shown in Figure 26.

• The elastic modulus is sensitive to the relative bonded area or density of paper, beating of chemical pulp, drying shrinkage, and fiber orientation.

• The shape of the load-elongation curve is essentially independent of RBA, beating level and density, as was shown in Figure 22.

• It is also independent of the beating level for chemical pulp.

• The factors that influence the shape of the load-elongation curve are drying shrinkage, wood species, furnish type, and the curl, kinks, and other defects in fibers.

• In machine made papers, the elastic strain versus total strain curves are different in MD and CD as shown in Figure 26.

• This is due to anisotropic drying stresses.

• Mechanical and chemical pulps give different curves.

• Chemical pulps give larger plastic elongation and correspondingly smaller elastic strain than mechanical pulps.

• The apparent difference in the yield strain between machine-made papers and handsheets may relate to the different drying methods.

• The final drying stress decreases if drying time or drying temperature increases.

### Failure in pure tension

• In paper webs, flaws at stress levels that are less than tensile strength usually trigger failure.

• The fracture toughness may be a proper measure of paper performance in such situations.

• In board products, compressive failure is relevant.

• The elastic modulus gives an indirect measure of compressive strength.

### Models of tensile strength

• Paper strength is controlled by gradual failure of inter-fiber bonds.

• In well bonded papers, it is also effected by fiber rupture.

• The simplest estimate of tensile strength is based on the assumption that failure in paper is only a result of fiber strength.

• Thus, the tensile strength, T is

T=Ef

• where, E is the elastic modulus of paper and f is the elastic breaking strain of fibers.

• Paper fails when the axial stress in fibers first exceeds their breaking threshold

• Bond failures result from external load transfer across the network from fiber to fiber.

• As the external load increases a bond fails if the shear force on the bond exceeds its shear strength.

• Figure 27 illustrates the stress transfer mechanism.

• Stress transfers to and from fibers through many bonds.

• Shallhorn and Karris proposed a model similar to those used for fiber reinforced composites.

• They proposed the breaking force per specimen width as

T/Ws=NyRBAtbwflf/2

when no fibers fail (weak bonds) and

T/Ws=NyFf[1-Ff/(2RBAtbwflf)

when some fibers fail (strong bonds),

wheretb is the bending stress

Ff is the fiber strength

lf is the fiber length

wf is the fiber width

Ny is the number of fibers crossing a unit fraction line

• Figure 28 shows how the predicted tensile strength depends on RBA and fiber strength.

• Another model is the Page equation

1/T=9/(8Z)+3wf/(tblfRBA)

where Z is the zero span strength of paper.

• The predictions of the Page equation are shown in Figure 29.

• For the zero span measurement, the nominal gauge length is zero.

• If the fibers are linearly elastic, then their failure strain is f and the zero span strength is

Z=Esf

where Es is the effective modulus for the short test span.

• If we assume that

Es=3/8r/rfEf then

Z=3rFf/(8rfAf)

### Furnish and Papermaking Effects

• The plot of light scattering coefficient against tensile strength in Figure 30 show that for beating, the bonding degree contributes significantly to tensile strength.

• Decreasing light scattering coefficient corresponds to increasing RBA.

• Experimental results simplify significantly when considering breaking strain instead of tensile strength.

• Figures 31-33 show el vs. density for the cases discussed in Figures 3, 5 and 6.

• Within the experimental accuracy, el and E are linearly dependent on density when only one condition such as beating varies.

• When only wet pressing or basis weight varies, the elastic breaking strain is almost constant.

• The mechanical properties of fibers and bonds should remain constant, so that slight variation in el should be due to RBA.

• Beating influences the elastic breaking strain more than wet pressing or basis weight.

• Bonds formed between well-beaten fibers should tolerate more stretching than bonds between lightly beaten fibers.

• The changes in tensile strength with increased beating of chemical pulp or refining of mechanical pulp come partly from increasing fines.

• Figure 34 shows the result of adding two kinds of fines to a a long fiber fraction.

• When adding fines, paper density increases, but elastic modulus is insensitive to the type and content of the fines.

• Fines therefore seem to alter only bond strength and not RBA.

• Thus, when adding mechanical pulp fines, the link between RBA and light scattering no longer holds.

• Figure 35 indicates that changes in breaking strain occur at high levels of shrinkage.

• High drying stress and low drying shrinkage gives brittle bond and low elastic breaking strain

• Different beating levels and furnish type do not effect considerably the trend of el against drying shrinkage

• High drying shrinkage gives low elastic modulus.

• Tensile strength therefore increases with decreasing shrinkage (maybe).

• Figure 36 shows that, in the center of the web, the MD/CD ratio of tensile strength is roughly equal to the elastic modulus ratio.

• The elastic breaking strain appears to be isotropic in that case.

• At the edges of a web, the modulus ratio is higher than the strength ratio.

### Breaking Strain and TEA

• Figures 37 and 38 present data on breaking strain as a function of density when varying beating and wet pressing.

• As seen in the Figures, beating has a strong positive effect on breaking strain, while wet pressing and basis weight have a weak effect.

• Drying shrinkage influences the ordinary breaking strain much the way it influences the elastic breaking strain.

• As shown in Figure 39, for kraft pulp at large shrinkage.

Breaking strain constant - drying shrinkage

• For mechanical pulp, the breaking strain may still be linear, but with a slope > -1.

• For a given drying shrinkage, papers with different fiber orientation still have the same breaking strain.

• The tensile energy absorption, TEA, is the integral of the load-elongation curve:

where ebr is the breaking strain.

• TEA is of units of energy per unit volume.

• Since it is defined as an integral of the stress:

• TEA grows with increasing tensile strength and breaking strain.

• TEA depends on the shape of the load-elongation curve.

### Formation Effects

• Tensile strength decreases if formation becomes more nonuniform as illustrated in Figure 40.

• The average of local strength values measured using strips with a narrow neck was independent of forming concentration.

• Local strength alone does not determine the strength of paper.

• The critical factor for failure is the ratio of local stress or strain over local strength.

• In papers with poor formation, local stresses vary more than in papers with good formation.

• Figure 41 shows the variation of local strains in high and low basis weight areas.

• In the good formation sample, strain difference between low and high basis weight areas is smaller than in a sample of relatively bad formation.

• The more uniform paper tolerates higher local strain in the failure zone than the less uniform paper.

• The non-uniformity of paper leads to size dependence in tensile strength as shown in Figure 42a.

• On the average, the lowest value of local strength is smaller in long strips than in short strips.

• At very short spans, the situation is different, as shown in Figure 42b.

• At zero span, the measured strength should be proportional to fiber strength, but some bond opening does occur at small finite spans.

### Pulp mixtures

• Addition of chemical pulp to mechanical pulp may improve the mechanical properties of paper.

• The change is often nonlinear as a function of the mixing ratio.

• Figure 43 illustrates this for the addition of springwood kraft fibers to a TMP furnish.

• The tensile index increases only if the kraft content exceeds 30%.

• Threshold consistency typically varies between 20% and 50%, but it may be so small that slight amounts of chemical pulp have a positive effect.

• A paper sheet made from a binary pulp mixture differs from a two-ply sheet of the same components in three ways:

• Bonding degrees of the components,

• Load sharing between the components, and

• Breaking strain of the components.

• In a homogeneous sheet, the chemical pulp has lower RBA and the mechanical pulp has higher RBA than in pure layers.

• Figure 44 shows how paper density reflects the nonlinear variation of bonding degree with mixing ratio.

• When mixing fibers of different flexibility, the density of the network may be limited by the stiffer furnish component, if the more flexible component conforms to the structure defined by the stiffer component.

• The degree of fiber collapse may depend on mixing ratio.

• If the complications of load sharing are ignored, the curve for a mixture can be approximated by a properly weighted mean stress of the two components for a common strain.

• It is equally valid to take a mean strain at a common stress, but that gives different results.

• There are some cases where the stress-strain curve of a pulp mixture is not given by the mean stress of the two components.

• Figure 45 demonstrates that the separation between curves is not linear in mixing ratio.

### Fracture Toughness and Tear Resistance

• Failure of a running web is usually due to a defect in the paper or a transient peak in web tension that causes failure at an otherwise harmless defect.

• Fracture toughness measures the ability of paper to resist the growth of small incipient cracks.

• Tear strength has been traditionally used for this purpose, but it is not related to web runnability, experimentally or theoretically.

### Web Breaks and Fracture Toughness

• Web breaks are relatively infrequent.

• If runnability is measured as the mean distance between web breaks, then a typical value is 106 m or more between breaks

• A paper web fails if the local strength is too low or momentary load is too high.

• The frequency of breaks increases with increasing web tension, sometimes even exponentially as shown in Figure 46.

• Significant levels of breaks occur at web tensions that are an order of magnitude lower than the tensile strength of paper.

• Breaks are due to distinct flaws or defects in the paper web.

• In papers made from mechanical furnish, shives can cause breaks.

• Good pulp screening has greatly improved runnability.

• Low moisture content makes paper brittle.

• Local strength minima causing breaks in the web depend on the nature and size of defects.

• Fracture toughness, or fracture resistance, is a material property that describes the ability of paper to resist a flaw from becoming a large crack.

• If an existing crack grows, elastic energy releases because stress vanishes at the crack faces.

• The Griffith criterion says that crack cannot grow unless this released energy is at least equal to the energy necessary to overcome the fracture resistance or toughness of the material.

• In a linearly elastic body, the condition for crack growth is

G=-dP/dA=bs2a/E'Gc=R

whereP is the elastic energy

G=-d/dA is the decrease in elastic energy with crack area increment (or the crack-driving force),

is a geometric factor

is the remote stress far from the crack

a is the crack length

E is an elastic constant, and

R=Gc is the fracture energy of the material.

to slide 83

• A web with a crack releases elastic energy at a rate, G, if the crack grows.

• The energy release rate, G, is proportional to web tension squared.

• If the web tension increases, at some point the energy release rate, G, reaches the fracture energy of the material, Gc=R.

• Fracture toughness Kc is the following:

Kc=(GcE).5

• Thus, the critical web tension is

sc=Kc/(ba).5

• The critical web tension is a function of the fracture energy, crack length, elastic modulus, and geometry

• For flaws of a fixed size, the critical strain level Kc/E can be obtained from Equation 20.

• The crack growth condition (slide 81) is the basis for linear elastic fracture mechanics (LEFM).

• It holds only for materials in the linear elastic regime, generally when the fracture process occurs in a zone of the material that is small compared to the dimensions of the sample.

### Measurement Methods

• Fracture toughness is difficult to measure because the elastic energy release rate, G, usually overshoots the critical value, Gc.

• Often Kc is also known as the critical stress intensity factor, or sometimes "tenacity".

• We will denote Kc as fracture toughness and use the term fracture energy for Gc.

• LEFM has found considerable use with paper, although paper is far from a linear elastic material.

• Figure 47 shows apparent strength values, c, measured in specimens of different notch geometries are in reasonable agreement with the LEFM predictions.

• The J-integral method uses different initial notch lengths,a.

• One determines the energy adsorption, W() as a function of external strain.

• The energy release rate for any given strain is given by

J=-dW(e)/da

• One can determine the fracture energy or "essential work of fracture" (EWF) using specimens with a double edge notch.

• This is illustrated in Figure 48.

• The fracture energy can also be measured using short specimens.

• This is illustrated in Figure 49.

### Tear Tests

• Tearing resistance is the total energy per tear length consumed when a specimen of a given geometry undergoes tearing.

• Two types of tests are possible as shown in Figure 50.

• Figure 50a shows an in-plane measurement, while Figure 50b shows an out of plane test.

• Tearing resistance, therefore has the units of load and is sometimes called tear strength, although it is energy not stress that is measured.

### Furnish Effects

• The main use of fracture toughness is evaluation of the efficiency of reinforcement pulps in improving the runnability of printing papers made form mechanical pulps.

• Figure 51 illustrates the reinforcement effect, when softwood kraft pulp was added to pressure ground wood (PGW).

• The critical value of J-integral and the out of plane tear energy increase when replacing mechanical pulp with kraft.

• Figure 52 illustrates the effect of beating on the in-plane and out of plane tear resistance.

• Beating increases RBA and bond strength.

• The drying shrinkage and moisture content of paper increases the fracture energy.

• The effect of drying shrinkage in machine made papers is shown in Figure 53.

### Microscopic Theories

• When a crack propagates in the fiber network, at least four mechanisms of energy dissipation operate at the fiber level.

• These include:

• Plastic elongation of fibers,

• Breaking of bonds,

• Breaking of fibers, and

• Fiber-to-fiber friction.

• Models of fracture toughness are limited to linear elastic behavior, but consider the other three mechanisms.