TMM4175 Polymer Composites

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Plastic behavior¶

Note: Plasticity and plastic behavior are note emphasized topics in the course. Brief knowledge of some basic concepts is however relevant for the interpretation and understanding of strength and failure of polymer matrix composites.

Yield criteria¶

The Von Mises criterion states that yield occurs when the principal stresses $\sigma_{1}$, $\sigma_{2}$ and $\sigma_{3}$ satisfy the relation

\begin{equation} \sqrt{\frac {1}{6}\big((\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\big)} = k \tag{1} \end{equation}

From a uniaxial tensile test where the yield strength is $\sigma_{yield}$:

\begin{equation} k = \frac{\sigma_{yield}}{\sqrt{3}} \tag{2} \end{equation}

Equation (1) can therefore be written

\begin{equation} \sqrt{\frac {1}{2}\big((\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\big)} = \sigma_{yield} \tag{3} \end{equation}

or

\begin{equation} \sigma_v = \sigma_{yield} \tag{4} \end{equation}

where $\sigma_v$ is the von Mises stress.

The von Mises yield criterion is a pressure-independent criterion often used and found valid for metal plasticity. For example, a isostatic state of stress like $\sigma_1 = \sigma_2 = \sigma_3 = P$ gives $\sigma_v = 0$ and $k=0$.

This is usually not the case for plastic behavior of polymers, where the yielding can be significantly influence by the presence of isostatic pressure.

A common pressure-dependent yield criterion used for polymers is the Drucker-Prager yield criterion. The criterion can be expressed by a modification of equation (1) as

\begin{equation} \sqrt{\frac {1}{6}\big((\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\big)} = a + b(\sigma_1 + \sigma_2 + \sigma_3) \tag{5} \end{equation}

where $a$ and $b$ are material parameters that can be found when the tensile yield strength and the compressive yield strength are known.

Computational examples:

# Stress at yielding from tensile testing is:

sy_t = 30

# The left side of equation (5) now:

r_t = ((2/3)*(sy_t**2))**0.5

# Stress at yielding from compressive testing is:

sy_c = -37

# the left side of equation (5) is now

r_c = ((2/3)*(sy_c**2))**0.5

print(r_t, r_c)

24.49489742783178 30.21037349432586

# Shall now solve the set of equations

# a + b * sy_t = r_t
# a + b * sy_c = r_c

import numpy as np

S = np.array([[1, sy_t],
              [1, sy_c]])

R = np.array([r_t, r_c])

a,b = np.linalg.solve(S,R)

print(a,b)

27.054065815814205 -0.08530561293274747

# Verify:

print(a + b*sy_t)
print(a + b*sy_c)

24.49489742783178
30.21037349432586

If the polymer matrix behaves in a plastic manner, what is the consequences regarding the properties and behavior of the fiber composite?

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