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strain relationships
Appendix C
Stress/strain relationships (i) Uniaxial stresses When a material is subjected to a uniaxial stress (Fig. C.1), there will be a strain in the direction of the stress and a strain of the opposite sense in the perpendicular directions. The latter is referred to as the Poisson’s Ratio effect. εx ¼ where E ¼ modulus of material. εy ¼ v
1 sx E
1 , sx E
where v ¼ Poisson’s ratio (typical values are given in , Chapter 2). 1 , sx εz ¼ v E The z-direction is perpendicular to the page. For simplicity the material is assumed to be isotropic, i.e. same properties in all directions. However, in some cases for plastics and almost always for fibre composites, the properties y
x
Vx
Vx
FIG. C.1 Uniaxial stress.
597
598 Appendix C
will be anisotropic. Thus, E and v will have different values in the x, y and z directions. Also, it should be remembered that only at short times can E and v be assumed to be constants. They will both change with time and so for longterm loading, appropriate values should be used.
(ii) Biaxial stresses If the material is subjected to biaxial stresses in both the x and y directions (Fig. C.2) then the strains will be εx ¼
1 1 sx v sy E E
That is, the total strain will be the sum of the tensile strain due to sx and the negative strain due to the Poisson’s ratio effect caused by sy. Similarly εy ¼
1 1 sy v sx E E
εz ¼
v ðsx þ sy Þ E
This situation is sometimes referred to as plane stress because there are only stresses in one plane. It is important to note, however, that there are strains in all three co-ordinate directions.
Vy
y
x
Vx
Vx
Vy FIG. C.2 Biaxial stresses.
Appendix C
599
(iii) Triaxial stresses If there are stresses in the x, y and z directions then the above approach is continued so that εx ¼
1 1 1 1 v sx v sy v sz ¼ sx ðsy þ sz Þ E E E E E εy ¼
1 n sy ðsx þ sz Þ E E
εz ¼
1 n sz ðsy þ sx Þ E E
In the above situation, if the stresses are such that εz¼ 0 then this condition is referred to as plane strain. This is because strains are experienced in only one plane even though there are stresses in all three co-ordinate directions.
Stress/strain relationships (i) Uniaxial stresses When a material is subjected to a uniaxial stress (Fig. C.1), there will be a strain in the direction of the stress and a strain of the opposite sense in the perpendicular directions. The latter is referred to as the Poisson’s Ratio effect. εx ¼ where E ¼ modulus of material. εy ¼ v
1 sx E
1 , sx E
where v ¼ Poisson’s ratio (typical values are given in , Chapter 2). 1 , sx εz ¼ v E The z-direction is perpendicular to the page. For simplicity the material is assumed to be isotropic, i.e. same properties in all directions. However, in some cases for plastics and almost always for fibre composites, the properties y
x
Vx
Vx
FIG. C.1 Uniaxial stress.
597
598 Appendix C
will be anisotropic. Thus, E and v will have different values in the x, y and z directions. Also, it should be remembered that only at short times can E and v be assumed to be constants. They will both change with time and so for longterm loading, appropriate values should be used.
(ii) Biaxial stresses If the material is subjected to biaxial stresses in both the x and y directions (Fig. C.2) then the strains will be εx ¼
1 1 sx v sy E E
That is, the total strain will be the sum of the tensile strain due to sx and the negative strain due to the Poisson’s ratio effect caused by sy. Similarly εy ¼
1 1 sy v sx E E
εz ¼
v ðsx þ sy Þ E
This situation is sometimes referred to as plane stress because there are only stresses in one plane. It is important to note, however, that there are strains in all three co-ordinate directions.
Vy
y
x
Vx
Vx
Vy FIG. C.2 Biaxial stresses.
Appendix C
599
(iii) Triaxial stresses If there are stresses in the x, y and z directions then the above approach is continued so that εx ¼
1 1 1 1 v sx v sy v sz ¼ sx ðsy þ sz Þ E E E E E εy ¼
1 n sy ðsx þ sz Þ E E
εz ¼
1 n sz ðsy þ sx Þ E E
In the above situation, if the stresses are such that εz¼ 0 then this condition is referred to as plane strain. This is because strains are experienced in only one plane even though there are stresses in all three co-ordinate directions.