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In-Plane Tensile Properties
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.
E=ds/de as e 0
where s is the applied stress, or force per unit area, and e is the corresponding strain.
to slide 12
or if x=MD and y=CD
where b is the basis weight, is what is usually measured.
where it is assumed that nxynyx << 1.
to slide 19
for sufficiently small e.
where Ef is the elastic modulus of fibers, rf is their density and <ei2> is the average axial strain squared.
where wf and lf are the width and length and Gf is the shear modulus of the fibers.
to slide 43
to slide 53
when no fibers fail (weak bonds) and
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
where Z is the zero span strength of paper.
where Es is the effective modulus for the short test span.
Breaking strain constant - drying shrinkage
where ebr is the breaking strain.
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