The Boltzmann superposition principle states that the strain response of a viscoelastic material is a superposition of the responses to all previous stress histories. Assuming no stress before time $t=0$, the constitutive equation can be written as:<\/p>\n\n
$\\begin{equation} \\gamma(t)=\\int_0^t J(t-t^\\prime)\\dot{\\sigma}(t^\\prime)\\mathrm{d}t^\\prime\n\\end{equation}$<\/p>\n\n
Applying the Laplace transform to this equation yields:<\/p>\n\n
$\\begin{align}\n\\hat{\\gamma}(s)&=\\hat{J}(s)\\hat{\\dot{\\sigma}}(s)\\\\\n&=\\hat{J}(s)\\left(s\\hat{\\sigma}(s)-\\sigma(0^{-})\\right) \\\\\n&=\\hat{J}(s)\\left(s\\hat{G}(s)\\hat{\\dot{\\gamma}}(s)-\\sigma(0^{-})\\right)\\\\\n&=\\hat{J}(s)\\left(s\\hat{G}(s)(s\\hat{\\gamma}(s)-\\gamma(0^-))-\\sigma(0^{-})\\right)\\\\\n&=s^2\\hat{J}(s)\\hat{\\gamma}(s)\\hat{G}(s) \n\\end{align}$<\/p>\n\n
Here, $\\hat{f}(s):=\\mathcal{L}\\lbrace f(t)\\rbrace(s)$ denotes the Laplace transform of $f(t)$. Since the stress starts at time $t=0$, both $\\gamma(0^-)$ and $\\sigma(0^-)$ are zero. The second to third step is derived from the convolution relation $\\sigma(t)=\\int_0^t G(t-t^\\prime)\\dot{\\gamma}(t^\\prime)\\mathrm{d}t^\\prime$. Cancelling out $\\hat{\\gamma}(s)$ and rearranging the last equation gives:<\/p>\n\n
$\\begin{equation}\n\\frac{1}{s^2}=\\hat{J}(s)\\hat{G}(s)\n\\end{equation}$<\/p>\n\n
Taking the inverse Laplace transform of this equation leads to the convolution relation:<\/p>\n\n
$\\begin{equation} t=\\int_0^{t}J(t-t^\\prime)G(t^\\prime)\\mathrm{d}t^\\prime\n\\end{equation}$<\/p>\n\n
Substituting $J(t)=J_e + t\/\\eta$ into the convolution relation and taking the limit as $t\\to\\infty$ gives:<\/p>\n\n
$\\begin{equation}\n\\begin{aligned}\n\\color{blue}{t} &=\\int_0^{t}\\left(J_e+\\frac{t-t^\\prime}{\\eta}\\right)G(t^\\prime)\\mathrm{d}t^\\prime\\\\\n&= J_e\\int_0^\\infty G(t^\\prime)\\mathrm{d}t^\\prime + {\\color{blue}{\\frac{t}{\\eta}\\int_0^\\infty G(t^\\prime)\\mathrm{d}t^\\prime}} – \\int_0^\\infty G(t^\\prime)\\frac{t^\\prime}{\\eta}\\mathrm{d}t^\\prime\n\\end{aligned}\n\\end{equation}$<\/p>\n\n
Since $\\eta:=\\int_0^\\infty G(t^\\prime)\\mathrm{d}t^\\prime$, the blue terms cancel out, resulting in:<\/p>\n\n
\n\\begin{equation}\nJ_e\\int_0^\\infty G(t^\\prime)\\mathrm{d}t^\\prime = J_e\\eta= \\int_0^\\infty G(t^\\prime)\\frac{t^\\prime}{\\eta}\\mathrm{d}t^\\prime = \\frac{1}{\\eta} \\int_0^\\infty G(t^\\prime)t^\\prime\\mathrm{d}t^\\prime\n\\end{equation}\n<\/p>\n\n
Therefore, we obtain:<\/p>\n\n
$J_e = \\frac{1}{\\eta^2}\\int_0^\\infty tG(t)\\mathrm{d}t$<\/p>\n\n
$Q.E.D.$<\/p>\n\n","protected":false},"excerpt":{"rendered":"
The Boltzmann superposition principle states that the strain response of a viscoelastic material is a superposition of the responses to all previous stress histories. Assuming no stress before time $t=0$, the constitutive equation can be written as: $\\begin{equation} \\gamma(t)=\\int_0^t J(t-t^\\prime)\\dot{\\sigma}(t^\\prime)\\mathrm{d}t^\\prime \\end{equation}$ Applying the Laplace transform to this equation yields: $\\begin{align} \\hat{\\gamma}(s)&=\\hat{J}(s)\\hat{\\dot{\\sigma}}(s)\\\\ &=\\hat{J}(s)\\left(s\\hat{\\sigma}(s)-\\sigma(0^{-})\\right) \\\\ &=\\hat{J}(s)\\left(s\\hat{G}(s)\\hat{\\dot{\\gamma}}(s)-\\sigma(0^{-})\\right)\\\\ &=\\hat{J}(s)\\left(s\\hat{G}(s)(s\\hat{\\gamma}(s)-\\gamma(0^-))-\\sigma(0^{-})\\right)\\\\ […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[29],"class_list":["post-228","post","type-post","status-publish","format-standard","hentry","category-notes","tag-polymer-physics"],"_links":{"self":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts\/228","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/comments?post=228"}],"version-history":[{"count":1,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts\/228\/revisions"}],"predecessor-version":[{"id":229,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/posts\/228\/revisions\/229"}],"wp:attachment":[{"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/media?parent=228"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/categories?post=228"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.shirui.me\/blog\/wp-json\/wp\/v2\/tags?post=228"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}