4 13 Dynamic loading DMA Laplace plane shear operator G L G R G d s s 1 tau sigma Gd s GL GR s 1 Applied strain in time plane unprotect gamma gamma t gamma 0 cos omega t t 0 cos t Applied strain in laplace plane with inttrans gamma s laplace gamma t t s s 0 s s2 2 Dynamic modulus in laplace plane G bar G L gamma s gamma 0 G bar Gd s G s R 1 s s2 2 Invert for time plane modulus G t invlaplace G bar s t G t Gd e t 2 Gd sin t 2 2 GR cos t 2 2 GR cos t 2 1 1 1 1 Simplifying G t factor collect G t cos omega t G t 2 Gd e t 2 2 2 2 2 Gd cos t 2 2 1 2 2 Gd sin t GR cos t GR cos t Gd cos t 2 2 2 1 2 Simplifying further and rearranging manually t G Gd Gd 2 2 cos t d 2 2 sin t e G R 2 2 2 2 1 1 1 The first term is an initial transient that diminishes with time the second term is varying in phase with the applied strain and is thus the real modulus G and the third term is varyiing 90 deg out of phase G Page 1 and is thus the loss modulus G For the generalized Zener model we sum over the number of Maxwell arms m m G 2 2 G G GR d 2 2 G d 2 2 i 1 1 i 1 1 Page 2
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