Hygrothermal Stresses and Strains in a Lamina

1. Hygrothermal Stresses and
Strains in a Lamina

2. Hygrothermal stresses and strains
• Hygrothermal is basically related to both heat and moisture.
• Stresses and strains due to heat and moisture are called hygrothermal
stresses and strains.
• Changing environment conditions (temperature and moisture) have
an important effect on the properties which are matrix dominated.
• Change in temperature and moisture content induces swelling of the
polymer matrix.
• Total strains = mechanical strains + hygrothermal strains

3. Hygrothermal stresses and strains
• Composite materials are generally processed at high temperatures
and then cooled down to room temperatures. For polymeric matrix
composites, this temperature difference is in the range of 200 to
300°C; for ceramic matrix composites, it may be as high as 1000°C.
Due to mismatch of the coefficients of thermal expansion of the fiber
and matrix, residual stresses result in a lamina when it is cooled
down. Also, the cooling down induces expansional strains in the
lamina. In addition, most polymeric matrix composites can absorb or
deabsorb moisture. This moisture change leads to swelling strains and
stresses similar to those due to thermal expansion.

4. Maximum normal tensile stress in the x-direction as a function of angle of lamina using
maximum stress failure theory.

5. Maximum normal tensile stress in the x-direction as a function of angle of lamina using
maximum strain failure theory.

6. Maximum normal tensile stress in the x-direction as a function of angle of lamina using Tsai–Hill
failure theory.

7. Maximum normal tensile stress in the x-direction as a function of angle of lamina using Tsai–Wu
failure theory.

8. Hygrothermal stresses and strains
• Laminates in which laminae are placed at different angles have
residual stresses in each lamina due to differing hygrothermal
expansion of each lamina. The hygrothermal strains are not equal in a
lamina in the longitudinal and transverse directions because the
elastic constants and the thermal and moisture expansion coefficients
of the fiber and matrix are different. In the following sections, stress–
strain relationships are developed for unidirectional and angle
laminae subjected to hygrothermal loads.

9. Hygrothermal Stress–Strain Relationships for a
Unidirectional Lamina
• For a unidirectional lamina, the stress–strain relationship with
temperature and moisture difference gives:
(1)
• Where the subscripts T and C are used to denote temperature and
moisture, respectively.
• No shearing strains are induced in the material axes, so the
temperature and moisture change do not have any shearing strain
terms

10. Hygrothermal Stress–Strain Relationships for a
Unidirectional Lamina
• The thermally induced strains are given by:
• Where α1 and α2 are the longitudinal and transverse coefficients of
thermal expansion, respectively, and ΔT is the temperature change.
• The moisture induced strains are given by:

11. Hygrothermal Stress–Strain Relationships for a
Unidirectional Lamina
• Where β1 and β2 are the longitudinal and transverse coefficients of
moisture, respectively, and ΔC is the weight of moisture absorption
per unit weight of the lamina.
• Equation (1) can be inverted to give:

12. Hygrothermal Stress–Strain Relationships for
an Angle Lamina
• The stress–strain relationship for an angle lamina takes the following
form:
Where
&
• The terms 𝑎 𝑥, 𝑎 𝑦 and 𝑎 𝑥𝑦 are the coefficients of thermal expansion
for an angle lamina

13. Hygrothermal Stress–Strain Relationships for
an Angle Lamina
• The coefficients of thermal expansion for a unidirectional lamina as:
• Similarly, βx, βy, and βxy are the coefficients of moisture expansion
for an angle lamina and are given in terms of the coefficients of
moisture expansion for a unidirectional lamina as: