Dynamic Mechanical Behaviors PDF

23. Dynamic mechanical behaviors 23.1. Dynamic mechanical testing The situation is most easily analyzed when an oscillating sinusoidal load is applied to a specimen at a particular frequency. If the applied stress varies as a function of time according to...

23. Dynamic mechanical behaviors 23.1. Dynamic mechanical testing The situation is most easily analyzed when an oscillating sinusoidal load is applied to a specimen at a particular frequency. If the applied stress varies as a function of time according to 𝜎 = 𝜎0 𝑠𝑖𝑛𝜔𝑡 where ω is the angular frequency (2πf where f is the frequency), the strain for an elastic material obeying Hooke’s law would vary in a similar manner as 𝑒 = 𝑒0 𝑠𝑖𝑛𝜔𝑡 However, for a viscoelastic material, the strain lags somewhat behind the stress (e.g., during creep). This can be considered as a damping process and the result is that when a stress is applied to the sample the strain varies in a similar sinusoidal manner, but out of phase with the applied stress. Thus, the variation of stress and strain with time can be given by 𝑒 = 𝑒0 sin𝜔𝑡 𝜎 = 𝜎0 sin⁡(𝜔𝑡 + 𝛿) where δ is the ‘phase angle’ or ‘phase lag’, the relative angular displacement of the stress and strain 𝜎 = 𝜎0 sin(𝜔𝑡 + 𝛿) = 𝜎0 cos𝛿sin𝜔𝑡 + 𝜎0 sin𝛿cos𝜔𝑡 The stress, therefore, can be considered as being resolved into two components: one of σ0cosδ which is in phase with the strain and another σ0sinδ which is 90° (π/2 rad) out of phase with the strain. Hence, it is possible to define two dynamic moduli: E1 which is in phase with the strain 1 and E2, which is π/2 rad out of phase with the strain. 𝜎 = 𝜎0 𝐸1 sin 𝜔𝑡 + 𝜎0 𝐸2 cos 𝜔𝑡 𝐸 E1 = (σ0/e0)cosδ, E2 = (σ0/e0)sinδ, and phase angle δ and tan 𝛿 = 𝐸2 1 Complex form: 𝑒 = 𝑒0 exp 𝑖𝑤𝑡 𝜎 = 𝜎0 exp 𝑖(𝑤𝑡 + 𝛿) 𝜎 𝜎0 𝐸∗ = = ⁡(cos 𝛿 + 𝑖 sin 𝛿) = 𝐸1 + 𝑖𝐸2 𝑒 𝑒0 E1 and E2 are sometimes called the real and imaginary parts of the modulus, respectively. 22.2 Frequency dependence of viscoelastic behavior The frequency dependence of the viscoelastic properties of a polymer can be demonstrated very simply using the Maxwell model. 𝑑𝑒 𝑑𝜎 𝐸𝜏0 = 𝜏0 + 𝜎⁡ 𝑑𝑡 𝑑𝑡 τ 0 =η/E if the stress on a viscoelastic material is varied sinusoidally at a frequency ω then the variation of stress and strain with time can be represented by complex equations of the form 𝑒 = 𝑒0 exp 𝑖𝜔𝑡 𝜎 = 𝜎0 exp 𝑖(𝜔𝑡 + 𝛿) Substituting the relationships for σ and e gives 𝑑𝑒 = 𝑖𝜔𝑒0 exp 𝑖𝜔𝑡 = 𝑖𝜔𝑒⁡ 𝑑𝑡 𝑑𝜎 = 𝑖𝜔𝜎0 exp 𝑖(𝜔𝑡 + 𝛿) = 𝑖𝜔 𝜎⁡ 𝑑𝑡 𝐸𝜏0 𝑖𝜔𝑒 = 𝜏0 𝑖𝜔𝜎 + 𝜎 2  ∗ 𝜎 𝐸𝜏0 𝑖𝜔 𝐸𝜏02 𝜔2 𝐸𝜏0 𝜔 𝐸 = = = 2 2 +𝑖 2 2 𝑒 𝜏0 𝑖𝜔 + 1 𝜏0 𝜔 + 1 𝜏0 𝜔 + 1 𝐸𝜏02 𝜔2 𝐸𝜏0 𝜔 𝐸1 = 2 2 ⁡𝑎𝑛𝑑⁡⁡⁡𝐸2 = 2 2 𝜏0 𝜔 + 1 𝜏0 𝜔 + 1 𝐸2 1 tan 𝛿 = =⁡ 𝐸1 𝜏0 𝜔 23.2 Transitions and polymer structure 3 The figure shows the variation of the shear modulus G1 and tanδ with temperature for atactic polystyrene, which is typical for amorphous polymers. The shear modulus decreases as the testing temperature is increased and drops sharply at the glass transition where there is a corresponding large peak (α) in tan δ. Minor peaks at low temperatures correspond to secondary transitions. The position of the β-peak at about 50°C is rather sensitive to testing frequency and it merges with the α-relaxation at high frequencies. It has been assigned to rotation of phenyl groups around the main chain or alternatively the co-operative motion of segments of the main chain containing several atoms. It is thought that the γ-peak may be due to the occurrence of head-to-head rather than head-to-tail repeat sequences. The intensities of the α′- and α-relaxation decrease as the degree of crystallinity is reduced, implying that they are associated with motion within the crystalline regions. γ-relaxation increases with a reduction in crystallinity indicating that it is associated with the amorphous material and it has been tentatively assigned to a glass transition in the amorphous domains. The disappearance of the β-transition with the absence of branching has been taken as a strong indication that it is associated with relaxations at the branch points. 4