Editing Compressive strength (section)

== Deviation of engineering stress from true stress ==
[[File:Barelling.svg|75px|thumb|Barrelling]]

When a uniaxial compressive load is applied to an object it will become shorter and spread laterally so its original cross sectional area (<math display="inline">A_o</math>) increases to the loaded area (<math display="inline">A</math>).<ref name=":0" /> Thus the true stress (<math>\acute\sigma = F/A</math>) deviates from engineering stress (<math>\sigma_e=F/A_o</math>). Tests that measure the engineering stress at the point of failure in a material are often sufficient for many routine applications, such as quality control in [[concrete]] production. However, determining the true stress in materials under compressive loads is important for research focused on the properties on new materials and their processing.

The geometry of test specimens and friction can significantly influence the results of compressive stress tests.<ref name=":0" /><ref name=":1" /> Friction at the contact points between the testing machine and the specimen can restrict the lateral expansion at its ends (also known as 'barreling') leading to non-uniform stress distribution. This is discussed in section on [[#Contact with friction|contact with friction]]. {{anchor|Frictionless contact}}

=== Frictionless contact ===
With a compressive load on a test specimen it will become shorter and spread laterally so its cross sectional area increases and the true compressive stress is<math display="block">\acute{\sigma} =F/A</math>and the engineering stress is<math display="block">{\sigma_e} =F/A_o</math>The cross sectional area (<math display="inline">A</math>) and consequently the stress ( <math display="inline">\acute\sigma</math>) are uniform along the length of the specimen because there are no external lateral constraints. This condition represents an ideal test condition. For all practical purposes the volume of a high [[bulk modulus]] material (e.g. solid metals) is not changed by uniaxial compression.<ref name=":0" /> So<math display="block">A l=A_o l_o</math>Using the strain equation from above<ref name=":0" /><math display="block">A=A_o/(1+\epsilon_e)</math>and<math display="block">\acute{\sigma} = \sigma_e(1+\epsilon_e)</math>Note that compressive strain is negative, so the true stress (<math>\acute\sigma</math> ) is less than the engineering stress (<math display="inline">\sigma_e</math>). The true strain (<math>\acute \epsilon</math>) can be used in these formulas instead of engineering strain (<math display="inline">\epsilon_e</math>) when the deformation is large.{{anchor|Contact with friction}}

=== Contact with friction ===
As the load is applied, friction at the interface between the specimen and the test machine restricts the lateral expansion at its ends. This has two effects:

* It can cause non-uniform stress distribution across the specimen, with higher stress at the centre and lower stress at the edges, which affects the accuracy of the result.
* It causes a barreling effect (bulging at the centre) in ductile materials. This changes the specimen's geometry and affects its load-bearing capacity, leading to a higher apparent compressive strength.

Various methods can be used to reduce the friction according to the application:

* Applying a suitable lubricant, such as [[MoS2]], oil or grease; however, care must be taken not to affect the material properties with the lubricant used.
* Use of [[PTFE]] or other low-friction sheets between the test machine and specimen. 
* A spherical or self-aligning test fixture, which can minimize friction by applying the load more evenly across the specimen's surface.
Three methods can be used to compensate for the effects of friction on the test result:

# [[#Correction formula|Correction formulas]]
# [[#Geometric extrapolation|Geometric extrapolation]]
# [[#Finite element analysis|Finite element analysis]]

{{anchor|Correction formula}}

==== Correction formulas ====
[[File:Compression Test Specimen.jpg|thumb]]
Round test specimens made from ductile materials with a high bulk modulus, such as metals, tend to form a barrel shape under axial compressive loading due to frictional contact at the ends. For this case the equivalent true compressive stress for this condition can be calculated using<ref name=":1">{{Cite journal |last1=Ettouney |first1=D. |last2=Hardt |first2=D. E. |date=August 1983 |title=A method for in-process failure prediction in cold upset forging |journal=Journal of Engineering for Industry |volume=105 |issue=3 |pages=161–167|doi=10.1115/1.3185883 }}</ref><math display="block">\acute\sigma= C \sigma_a</math>where
:<math>C= {(1-2 R/d_2) \ln(1-d_2)/(2R))}^{-1}</math>
:<math>R= (l^2+(d_2-d_1)^2)/ (4(d_2-d_1))</math>
:<math>\sigma_a=4F/(\pi d_2^2)</math>
:<math>l</math>   is the loaded length of the test specimen,
:<math>d_1</math>is the loaded diameter of the test specimen at its ends, and
:<math>d_2</math>is the maximum loaded diameter of the test specimen.
Note that if there is frictionless contact between the ends of the specimen and the test machine,  the bulge radius becomes infinite (<math display="inline">R=\infty</math>) and <math display="inline">C=1</math>.<ref name=":1" /> In this case, the formulas yield the same result as  <math display="inline">\acute{\sigma} = \sigma_e(1+\epsilon_e)</math> because <math display="inline">\sigma_a</math> changes according to the ratio <math>(d_o/d_2)^2</math>.

The parameters (<math display="inline">F, d_1, d_2 ,l</math>) obtained from a test result can be used with these formulas to calculate the equivalent true stress <math display="inline">\acute\sigma</math> at failure. 
[[File:Compression Test Specimen Shape Effect.jpg|thumb|Specimen shape effect]]
The graph of [[#Compression Test Specimen Shape Effect.jpg|specimen shape effect]] shows how the ratio of true stress to engineering stress (σ´/σ<sub>e</sub>) varies with the aspect ratio of the test specimen (<math display="inline">d_o/l_o</math>). The curves were calculated using the formulas provided above, based on the specific values presented in the table for [[#Specimen shape effect calculations|specimen shape effect calculations]]. For the curves where end restraint is applied to the specimens, they are assumed to be fully laterally restrained, meaning that the coefficient of friction at the contact points between the specimen and the testing machine is greater than or equal to one (μ ⩾ 1). As shown in the graph, as the relative length of the specimen increases (<math display="inline">d_o/l_o\rightarrow0</math>), the ratio of true to engineering stress (<math>\acute\sigma/ \sigma_e</math>) approaches the value corresponding to [[#Frictionless contact|frictionless contact]] between the specimen and the machine, which is the ideal test condition.   
 
{| class="wikitable"
|+Specimen shape effect calculations 
!
!Frictionless
!Laterally Constrained
|-
|Constant volume
| colspan="2" |<math>\pi l_o d_o^2/4 = \pi l (d_2^2+d_1^2)/12</math>
|-
|Equal diameters
|<math>d_1=d_2</math>
|<math>d_o=d_1</math>
|-
| rowspan="2" |Solve for <math>d_2</math>
|<math>\pi l_o d_o^2/4 = \pi l (d_2^2+d_2^2)/12</math>
|<math>\pi l_o d_o^2/4 = \pi l (d_2^2+d_o^2)/12</math>
|-
|<math>d_2=d_o\sqrt{l_o/l}</math>
|<math>d_2=3d_o\sqrt{(3l_o/l-1)/18 }</math>
|-
|Equivalent stress ratio
|<math>\acute\sigma/\sigma_a= 1</math>
|<math>\acute\sigma/\sigma_a= C</math>
|-
|Engineering stress
| colspan="2" |<math>\sigma_e = 4F/\pi d_o^2</math>
|-
|Average stress
| colspan="2" |<math>\sigma_a = 4F/\pi d_2^2</math>
|-
|Average stress ratio
| colspan="2" |<math>\sigma_a/\sigma_e=(d_o/d_2)^2</math>
|-
|True strain
| colspan="2" |<math>\acute\epsilon=\ln(l/l_o)</math>
|}
{{anchor|Geometric extrapolation}}

==== Geometric extrapolation ====

[[File:Illustration of Extrapolation.jpg|300px|thumb]]As shown in the section on [[#Correction formulas|correction formulas]], as the length of test specimens is increased and their aspect ratio approaches zero (<math>d_o/l_o\longrightarrow 0</math>), the compressive stresses (σ) approach the true value (σ′). However, conducting tests with excessively long specimens is impractical, as they would fail by [[buckling]] before reaching the material's true compressive strength. To overcome this, a series of tests can be conducted using specimens with varying aspect ratios, and the true compressive strength can then be determined through extrapolation.<ref name=":0" />

{{anchor|Finite element analysis}}

==== Finite element analysis ====
{{Expand section|reason=Provide a reason or specify what needs expansion|date=September 2024}}