LectureNote 1-True Stress

Mechanical Behavior of Materials (MATL 362) Lecture Note (1) 10/2/2011

The term mechanical behavior refers to the response of materials to an external force

A material, under load, may either deform (be strained) or break

In

manufacturing, one of the most important groups of processes is plastic deformation, namely, shaping materials by applying forces in various ways (also known as deformation processing). It includes:

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Strain

Small Strain

Large Strain

Time

independent ant

Time dependent

Time independent

Time dependent

Elastic deformation

Anelastic or viscoelastic Plastic

deformation Creep

Fracture or Break

With extensive plastic deformation

Without remarkable deformation

Ductile Material

Brittle Material


In all these processes, the material is subjected to one or more of the basic modes of deformation shown in Fig. 1, namely, tension, compression, and shear.

The degree of deformation to which the material is subjected is defined as strain

Fig. 1 Types of strain (a) tension, (b) compression, (c) simple shear

For tension and compression, the engineering uniaxially strain is defined as:

1f o f

o o

l l le

l l

The simple shear strain is defined as:

tan

In order to change, the geometry of the elements or bodies in Fig. 1, forces must be applied to them as shown by the arrows. The determination of these forces as a function of strain is very important in manufacturing processes. We have to know

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Deformation processing

Bulk deformation

(Forging, rolling, extrusion, and rod and wire drawing)

Sheet-forming processes

(Bending, drawing, spinning, and general press working)


the forces in order to design the proper equipment, select the tool and die materials for proper strength, and determine whether a specific metalworking operation can be accomplished on certain equipment.

Thus, the relation between a force and the deformation it produce is an essential parameter in manufacturing.

Elastic deformation Vs Plastic deformation.

Elastic deformation Plastic deformation

Reversible Not reversible

Depends on initial and final states of stress and strain

Depends on loading path

Stress is proportional to strain No simple relation between stress and strain

No strain hardening effects Strain hardening effect

True stress-strain curve gives a true indication of deformation characteristics because it is based on the instantaneous dimension of the specimen.

The true stress-strain curve is also known as the flow curve. In engineering stress-strain curve, stress drops down after necking since it is based

on the original area. In true stress-strain curve, the stress however increases after necking since the

cross-sectional area of the specimen decreases rapidly after necking.

The true stress is defined as

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σt = F/A (1)

Where; A is the instantaneous cross-sectional area corresponding to the force F. Before necking begins, the true strain is given by

ln f

o

L

L

(2)

The strain defined by equation (2) is the true strain or natural strain. It is a more suitable definition of strain and is particularly useful for large strain analyses. In the case of a sample being compressed to zero thickness, equation (2) would yield

0 0lim ln

ff

fL L

o

L

L

Which is, a more reasonable value than the compressive strain of 100% predicted by definition of equation

o

Le

L

The engineering stress is defined as the force divided by the original area, σt=F/Ao and the engineering strain is defined as the change in length divided by the original length, e =L/Lo. As long as the deformation is uniform along the gauge length, the true stress and true strain can be calculated from the engineering quantities. With constant volume,

LA=LoAo, so that

Ao/A = L/Lo (3)

and thus Ao/A=1+e

Rewriting Equation (1) as σt = (F/Ao) (Ao/A)

and substituting Ao/A = 1+e and σ =F/Ao,

1t e (4)

Substitution of L/Lo =1+e into Equation (2) gives

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ln 1 e (5)

These expressions are valid only if the deformation is uniformly distributed along the gauge section. After necking starts, Equation (1) is still valid for true stress, but the cross-sectional area at the base of the neck must be measured independently. Equation (2) could still be used if L and Lo were known for a gauge section centered on the middle of the neck and so short that the variations of area along its length were negligible.Equation (3) would be valid over such a gauge section, so the true strain can be calculated as

ln oA

A

(6)

Where, A is the area at the base of the neck. Figure below shows the comparison of the engineering and true stress–strain curves for the same material.

Comparison of engineering and true stress–strain curves

Engineering stress-strain curves are useful only for small deformations. In such cases true stress-strain and engineering stress-strain curves coincide within reasonable limits. For large strain, say greater than 1%, the true stress-strain should always be used. It is important to note that equation (1) holds only for uniform deformation, i.e., where stresses in every point at the cross-section are the same. For the nonuniform case, stress is defined as:

0limA

F

A

Before necking, a point on the true stress–strain curve (σt–ε) can be constructed from a point on the engineering stress–strain curve (σ–e) with equations (4) and (5). After necking, the cross-sectional area at the neck must be measured to find the true stress and strain.

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Characteristics of true strain ε :

1. It’s the exact value, not an approximation.

2. Sequential strains can be added: if two strains ε1 and ε2 are executed sequentially, the total strain is

1 2 1 2 21 2

1 1 0

ln ln ln lno o

L L L L Lx

L L L L L

3. This is not the case with engineering strain, where the total strain is

2 0 1 0 2 11 2

1o o

L L L L L Le e

L L L

4. It’s used to characterize materials that deform by large amounts (considerable fractions of their length up to many times their length). A quick look at the literature shows that true strain has been recently used to characterize materials like polyamide yarn, epoxy, rubber, and cartilage.

5. It’s geometrically symmetric: that is, if the strain associated with being stretched to n times the

original length is ε, then the strain associated with being compressed to 1/n the original length is

−ε.

Characteristics of engineering strain e:

1. It’s easier to calculate.

2. It’s overwhelmingly preferred in engineering analyses of materials that experience only small strains (including the common construction materials concrete, wood, and steel, for example, under normal use).

3. It’s symmetric in terms of displacements: that is, if the strain associated with being stretched a

distance L is e, then the strain associated with being compressed a distance L is -e

Instability in Necking

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The formation of a neck in the tensile specimen introduces a complex triaxial state of stress in that

region. This situation produces radial, σr stress and transverse stress σt which raise the value of longitudinal stress required to cause the plastic flow. Therefore, the average true stress at the neck, which is determined by dividing the axial tensile load by the minimum cross-sectional area of the specimen at the neck, is higher than the stress that would be required to cause flow if simple tension prevailed.

Bridgman made a mathematical analysis that provides a correction to the average axial stress to compensate for the introduction of transverse stress. This analysis was based on the following assumptions:

The contour of the neck is approximated by the arc of a circle;

The cross section of the necked region remains circular throughout the test;

The Von Mises criterion for yielding applies;

The strains are constant over the cross section of the neck.

According to this analysis, the ratio of the uniaxial flow stress corresponding to that which would exist in the tensile test if necking had not introduced triaxial stresses to the average true stress at the neck is:

Where R is the radius of curvature e of the neck

a is the minumum radius of the specimen at the neck

The corrected true stress-true strain curve is shown above

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The stress distribution at the neck of a specimen, where there is a triaxial state of stress, is shown in the right figure. The reason for this state is that each element in the region has a different cross-sectional area; the smaller the area, the greater is the tensile stress on the element. Hence element 1 will contract more than element 2, and so on. However, element 1 is restrained from contracting freely by element 2, and element 2 is restrained by element 3, and so on. This restraint causes radial and circumferential tensile stresses in the necked region. This situation results in an axial tensile stress distribution as shown in the attached figure.

Sheet Anisotropy

If the tensile tests performed on specimens cut from sheet material at different orientations to the rolling direction, there may not be much difference between the stress-strain curves. However, the lack of variation of the stress-strain curves with direction does not indicate that the material is isotropic. The parameter that is commonly used to characterize the anisotropy of the sheet metal is the strain ratio: defined as the ratio of the contractile strains measured in a tensile test before necking occurs:

w

t

R

(7)

Direct measurement of the thickness strain in thin sheets is inaccurate. Instead, εt is usually deduced from the width and length strains assuming constancy of volume,

t w l

w

w l

R

(8)

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Some workers suggest that the strains be measured when the total elongation is 15%, if this is less than the necking strain. The change of R during a tensile test is usually quite small and the lateral strains at 15% elongation are great enough to be measured with accuracy.

Tensile strips are cut from a sheet at 0, 45, and 90° to the rolling direction and machined into a test specimen configuration such as ASTM standard E-8 with parallel sides in the test section {see figure}.

The specimens are pulled in a tensile testing machine to an axial strain of 15-20 % or to the strain at maximum load, whichever comes first, and unloaded. The permanent changes in length and width are recorded, then R value is calculated

R > 1 resistant to thinning

R = 1 isotropic

R < 1 easy thinning

∆R =zero no earring

∆R > zero ears will form in 0 and 90° direction

∆R < zero ears will form in 45° direction

The value of R usually depends on the direction of testing. An average R-value is conventionally taken as:

90 45

4oR R R

R

(9)

The angular variation of R is characterized by R, defined as

90 452

2oR R R

R

(10)

Both of these are important in analyzing what happens during sheet metal forming.

Types of Stress-Strain curves

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Schematic illustration of various types of idealized stress strain curves is shown in figure below

Perfectly elastic (a)

A perfectly elastic material behaves like a spring with stiffness E. The behavior of brittle materials such as glass, ceramics, and some cast irons, may be represented by such a curve (a). There is a limit to the stress the material can sustain, after which it breaks. Permanent deformation, if any, is negligible.

Rigid, Perfectly plastic (b)

A rigid, perfectly plastic material has, by definition, an infinite value of E. Once the stress reaches the yield stress σy, it continues to undergo deformation at the same stress level. When the load is released, the material has undergone permanent deformation, with no elastic recovery (b).

Elastic, perfectly plastic (c)

An elastic, perfectly plastic material is a combination of the first two: it has a finite elastic modulus and it undergoes elastic recovery when the load is released (c).

Rigid, linearly strain hardening (d)

A rigid, linearly strain hardening material requires an increasing stress level to undergo further strain. Thus, its flow stress (magnitude of the stress required to maintain plastic deformation at a given strain) increases

Elastic, linearly strain hardening (e)

Elastic, linearly strain hardening curve (e) is an approximation of the behavior of most engineering materials, with the modification that the plastic potion of the curve has a decreasing slop with

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Strain

Stress

tan-1 E

σY

σy

σy

σy


with increasing strain. It has no elastic recovery upon unloading (d).

increasing strain.

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LectureNote 1-True Stress