THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

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Solid Mechanics of Stress and Strain

The Stress - Strain Relationship for Solids

< coordinates to the strains with respect to {x',y',z'} (Figure 23) is performed via the equations ^[7]:

THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

Mohr’s circle for plane strain state

Figure 24

THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

Figure 24 is the Mohr’s circle for plane strain state. The method to construct the circle is similar to that construct Mohr’s circle for plane stress state.

Principal Strains and the directions

THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

Figure 25

Same as principal stresses, principal strains ε₁, ε₂ (shown in Figure 25) are the strains that occur on an inclined planes where shear strains = 0. From the Mohr’s circle for plain strain, we can get ^[7]:

THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

The direction of ε₁, ε₂ are shown in Figure 24, too. The angle can be calculated by:

THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

Maximum shear strain and the directions

Figure 26

The maximum shear strain is ^[7]:

THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

The angle between the plane that maximum shear strain occurs and reference plane is:

THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

Stress-strain relationship

With plane stress assumption, for isotropic material, in elastic region, the stress-strain relationship should be ^[5]:

THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

Note:

Tri-axial stress state elastic analysis

In general, at a point, 3 normal stresses may act on faces of the cube, as well as, 6 components of shear stress (Figure 27). For the 6 shear stresses, only 3 of them are independent, because:

THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

Figure 27

The stain components are shown in Figure 27, too.

The stress on an inclined plane

To simplify the case, choose a set of reference coordinates (Figure 28) coincident to the principle stresses, and then the stresses σ_n and τ_n in any inclined plane with normal direction (l, m, and n) can be calculated by following equtions ^[9]:

THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

Figure 28

Mohr’s circle

Figure 29 is the Mohr’s circle for general tri-axial stress state. The 3 equtions shown above that calculate the stresses on an inclined plane express 3 circles. The 3 circles intersect in point D, which should be located within the 3 reference circles. The value of coordinates of D is the stresses of the inclined plane.

THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

τ_n

THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

σ_n

Figure 29

Stress-strain relationship

In general tri-axial stress state, for isotropic material and in elastic region, if taking into account thermal effect, the stress-strain relationship should be ^[10]:

THE STRESS STRAIN RELATIONSHIP FOR SOLIDS OPTI 521

Reference:

[1] http://zone.ni.com/devzone/cda/ph/p/id/250#toc3

[2] http://www.jwave.vt.edu/crcd/farkas/lectures/mechprop/tsld001.htm

[3] Class notes from Dr. J. H. Burge

[4] Class notes from Dr. Stone’s OPTI 222 class

[5] http://www4.eas.asu.edu/concrete/elasticity2_95/sld001.htm

[6] http://www.egr.msu.edu/classes/me423/aloos/lecture_notes/lecture_4.pdf

[7] http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/calc_principal_strain.cfm

[8] Mechanics of material (Hongwen Liu)

[9] http://www.shodor.org/~jingersoll/weave4/tutorial/tutorial.html

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