![\"Kelvin-Voigt\"](\"http:\/\/polymodmw.csi.muohio.edu\/wp-content\/uploads\/2013\/04\/Kelvin-Voigt.jpg\")
<\/a><\/p>\nFigure: The Kelvin-Voigt Model<\/p>\n
It is assumed that the strain experienced by the spring is the same as that experienced by the dash pot,<\/p>\n
\\begin{equation} \\label{eq:1}
\n\\[{{\\varepsilon }_{spring}}={{\\varepsilon }_{dashpot}}=\\varepsilon
\n\\end{equation}<\/p>\n
and that the applied external stress equals the sum of the stress within the spring and dashpot elements,<\/p>\n
\\begin{equation} \\label{eq:2}
\n\\[\\sigma ={{\\sigma }_{spring}}+{{\\sigma }_{dashpot}}=E{{\\varepsilon }_{spring}}+\\eta {{\\dot{\\varepsilon }}_{dashpot}}
\n\\end{equation}<\/p>\n
Hence, from Eqs. (\\ref{eq:1}) and (\\ref{eq:2}) the following governing differential equation for the Kelvin Voigt model can be constructed:<\/p>\n
\\begin{equation} \\label{eq:3}
\n\\[\\sigma =E\\varepsilon +\\eta \\dot{\\varepsilon }
\n\\end{equation}<\/p>\n
Equation (\\ref{eq:3}) may now be used for predicting the creep and relaxation behavior associated with the Kelvin-Voigt model by inputting the appropriate initial condition.<\/p>\n
Creep Behavior in Kelvin-Voigt Model<\/strong> \\begin{equation} \\label{eq:4} The homogeneous solution and particular solution of (\\ref{eq:4}) is,<\/p>\n \\begin{equation} \\label{eq:5} where, ${{A}_{1}}$ and ${{A}_{2}}$ are constants. The solution for strain is,<\/p>\n \\begin{equation} \\label{eq:6} Substituting Eq. (\\ref{eq:6}) into Eq. (\\ref{eq:4}) gives,<\/p>\n \\begin{equation} \\label{eq:7} Now by substituting ${{A}_{2}}$ from Eq. (\\ref{eq:7}) and initial strain $\\varepsilon (0)=0$ into Eq. (\\ref{eq:6}), one obtains,<\/p>\n \\begin{equation} \\label{eq:8} Hence from Eqs. (\\ref{eq:6}), (\\ref{eq:7}) and (\\ref{eq:8}), the solution for strain can be obtained as,<\/p>\n \\begin{equation} \\label{eq:9} where, \\[{{J}_{creep}}(t)=\\frac{1}{E}\\left( 1-{{e}^{-\\frac{E}{\\eta }t}} \\right)\\] is the relaxation modulus for the Kelvin-Voigt Model.<\/p>\n If the stress is completely removed, ${{\\sigma }_{0}}=0$ at time $t=\\tau$, as shown in Fig. *, the strain can be computed numerically or by Laplace transform. While assuming the zeros initial conditions, the strain can be computed as,<\/p>\n \\begin{equation} \\label{eq:10} Fig. 8. Creep Behavior of the Kelvin-Voigt Model<\/p>\n Relaxation Behavior in Kelvin-Voigt Model<\/strong> \\begin{equation} \\label{eq:11} << Maxwell Model<\/a> Linear Solid Model >><\/a><\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" [latexpage] The Kelvin (or Kelvin-Voigt) material model consists of a spring having modulus $E$ and a dash pot with viscosity $\\eta $, coupled in parallel, […]<\/p>\n","protected":false},"author":2306,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"footnotes":""},"class_list":["post-316","page","type-page","status-publish","hentry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/sites.miamioh.edu\/polymodmw\/wp-json\/wp\/v2\/pages\/316","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.miamioh.edu\/polymodmw\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.miamioh.edu\/polymodmw\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.miamioh.edu\/polymodmw\/wp-json\/wp\/v2\/users\/2306"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.miamioh.edu\/polymodmw\/wp-json\/wp\/v2\/comments?post=316"}],"version-history":[{"count":0,"href":"https:\/\/sites.miamioh.edu\/polymodmw\/wp-json\/wp\/v2\/pages\/316\/revisions"}],"wp:attachment":[{"href":"https:\/\/sites.miamioh.edu\/polymodmw\/wp-json\/wp\/v2\/media?parent=316"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nIf a sudden constant stress ${{\\sigma }_{0}}$ is applied, then Eq. (\\ref{eq:3}) leads to the following first order differential equation,<\/p>\n
\n\\[{{\\sigma }_{0}}=E\\varepsilon +\\eta \\dot{\\varepsilon }
\n\\end{equation}<\/p>\n
\n\\[0=E{{\\varepsilon }_{h}}+\\eta {{\\dot{\\varepsilon }}_{h}}\\text{ }\\Rightarrow \\text{ }{{\\dot{\\varepsilon }}_{h}}=-\\frac{E}{\\eta }{{\\varepsilon }_{h}}\\text{ }\\Rightarrow \\text{ }{{\\varepsilon }_{h}}(t)={{A}_{1}}{{e}^{-\\frac{E}{\\eta }t}}
\n\\end{equation}<\/p>\n
\n\\[\\varepsilon (t)={{\\varepsilon }_{h}}(t)+{{\\varepsilon }_{p}}(t)={{A}_{1}}{{e}^{-\\frac{E}{\\eta }t}}+{{A}_{2}}
\n\\end{equation}<\/p>\n
\n\\[{{\\sigma }_{0}}=E\\left( {{A}_{1}}{{e}^{-\\frac{E}{\\eta }t}}+{{A}_{2}} \\right)+\\eta \\left( -\\frac{E}{\\eta }{{A}_{1}}{{e}^{-\\frac{E}{\\eta }t}} \\right)\\text{ }\\Rightarrow {{A}_{2}}=\\frac{{{\\sigma }_{0}}}{E}
\n\\end{equation}<\/p>\n
\n\\[\\varepsilon (t)={{A}_{1}}{{e}^{-\\frac{E}{\\eta }t}}+\\frac{{{\\sigma }_{0}}}{E}\\text{ }\\Rightarrow \\varepsilon (0)=0={{A}_{1}}+\\frac{{{\\sigma }_{0}}}{E}\\Rightarrow {{A}_{1}}=-\\frac{{{\\sigma }_{0}}}{E}
\n\\end{equation}<\/p>\n
\n\\[\\varepsilon (t)=\\frac{{{\\sigma }_{0}}}{E}\\left( 1-{{e}^{-\\frac{E}{\\eta }t}} \\right)={{\\sigma }_{0}}{{J}_{creep}}(t)
\n\\end{equation}<\/p>\n
\n\\[\\varepsilon (t)=\\left\\{ \\begin{array}{*{35}{l}}
\n\\frac{{{\\sigma }_{o}}}{E}\\left( 1-{{e}^{-(E\/\\eta )t}} \\right), & 0<t<\\tau \\\\ \\frac{{{\\sigma }_{o}}}{E}{{e}^{-(E\/\\eta )t}}\\left( {{e}^{(E\/\\eta )\\tau }}-1 \\right), & \\quad \\ t>\\tau \\\\
\n\\end{array} \\right
\n\\end{equation}<\/p>\n
\nThis model is not used for predicting relaxation behavior because an instantaneous change in strain $d\\varepsilon ={{\\varepsilon }_{0}}$ with $dt\\to 0$ gives infinite strain $\\sigma (0)\\to \\infty $at time t=0 implying that the dashpot is rigid and an infinite stress is required to obtain a finite value of strain. For $t>0$ the strain is constant implying $\\varepsilon ={{\\varepsilon }_{0}}$ and $\\dot{\\varepsilon }=0$. Hence, from Eq. (\\ref{eq:3}),<\/p>\n
\n\\[\\begin{matrix}
\n\\sigma =E{{\\varepsilon }_{0}}, & t>0 \\\\
\n\\end{matrix}\\
\n\\end{equation}<\/p>\n