Interrelation of creep and relaxation for nonlinearly viscoelastic materials: application to ligament and metal

Abstract

Creep and stress relaxation are known to be interrelated in linearly viscoelastic materials by an exact analytical expression. In this article, analytical interrelations are derived for nonlinearly viscoelastic materials which obey a single integral nonlinear superposition constitutive equation. The kernel is not assumed to be separable as a product of strain and time dependent parts. Superposition is fully taken into account within the single integral formulation used. Specific formulations based on power law time dependence and truncated expansions are developed. These are appropriate for weak stress and strain dependence. The interrelated constitutive formulation is applied to ligaments, in which stiffness increases with strain, stress relaxation proceeds faster than creep, and rate of creep is a function of stress and rate of relaxation is a function of strain. An interrelation was also constructed for a commercial die-cast aluminum alloy currently used in small engine applications.

Appendices

Appendix A

From the proof given below in Appendix B we get

$$ {{\int\limits_{\rm{0}}^{\rm{t}} {{\rm{n}}\tau ^{{ - {\rm{n}}}} {\left( {{\rm{t}} - \tau } \right)}^{{{\rm{n}} - 1}} {\rm{d}}\tau = {\rm{n}}\;\Gamma {\left( { - {\rm{n}} + 1} \right)}} }\Gamma {\left( {\rm{n}} \right)}.} $$

(A1)

However there is a standard identity (Artin 1964) for the product of gamma functions on the right hand side in Eq. (A1) which is given as

$$ {\Gamma {\left( { - {\rm{n}} + 1} \right)}\Gamma {\left( {\rm{n}} \right)} = {\pi \over {\sin \;{\rm{n}}\pi }}} $$

So Eq. (A1)can therefore be written as

$$ {{\int\limits_{\rm{0}}^{\rm{t}} {{\rm{n}}{\rm{.}}\tau ^{{ - {\rm{n}}}} {\left( {{\rm{t}} - \tau } \right)}^{{{\rm{n}} - 1}} {\rm{d}}\tau = } }{{{\rm{n}}\pi } \over {\sin \;{\rm{n}}\pi }}.} $$

(A2)

Appendix B

The method used to solve the integral of type used in the formulation is as follows:

$$ {{\int\limits_{\rm{0}}^{\rm{t}} {\tau ^{{{\rm{m}} - 1}} {\left( {{\rm{t}} - \tau } \right)}^{{{\rm{n}} - 1}} {\rm{d}}\tau } }.} $$

(B1)

Rewriting Eq. (B1) in a different form we get

$$ {{\int\limits_{\rm{0}}^{\rm{t}} {\tau ^{{{\rm{m}} - 1}} {\rm{t}}^{{{\rm{n}} - 1}} {\left( {1 - {\tau \over {\rm{t}}}} \right)}^{{{\rm{n}} - 1}} {\rm{d}}\tau } }.} $$

(B2)

$$ \eqalign{ & {\rm{Let}}\;{\tau \over {\rm{t}}} = {\rm{u}} \cr & {\rm{therefore }}\tau = {\rm{ut}} \cr & {\rm{therefore d}}\tau = {\rm{t}}\;{\rm{du}} \cr} $$

So when τ=0, u=0and when τ=t, u=1

Substituting the above result in Eq. (B2), we get

$$ {\eqalign{ & {\int\limits_{\rm{0}}^{\rm{t}} {{\left( {{\rm{ut}}} \right)}^{{{\rm{m}} - 1}} {\rm{t}}^{{{\rm{n}} - 1}} {\left( {1 - {\rm{u}}} \right)}^{{{\rm{n}} - 1}} {\rm{t}}\;{\rm{du}}} }. \cr & {\rm{t}}^{{{\rm{m}} + {\rm{n}} - 1}} {\int\limits_0^{\rm{t}} {{\rm{u}}^{{{\rm{m}} - 1}} {\left( {1 - {\rm{u}}} \right)}^{{{\rm{n}} - 1}} \;{\rm{du}}.} } \cr} } $$

(B3)

The integral part in Eq. (B3) is nothing but a Beta function (Andrews 1985), the definition of which is given as

$$ {{\rm{B}}{\left( {{\rm{m}}{\rm{,n}}} \right)} = {\int\limits_{\rm{0}}^{\rm{t}} {\tau ^{{{\rm{m}} - 1}} {\left( {1 - \tau } \right)}^{{{\rm{n}} - 1}} {\rm{d}}\tau = {{\Gamma {\left( {\rm{m}} \right)}\Gamma {\left( {\rm{n}} \right)}} \over {\Gamma {\left( {{\rm{m}} + {\rm{n}}} \right)}}}} }.} $$

So the final result can be written as

$$ {{\rm{t}}^{{{\rm{m}} + {\rm{n}} - 1}} {\rm{B}}{\left( {{\rm{m}}{\rm{,n}}} \right)} = {\rm{t}}^{{{\rm{m}} + {\rm{n}} - 1}} {{\Gamma {\left( {\rm{m}} \right)}\Gamma {\left( {\rm{n}} \right)}} \over {\Gamma {\left( {{\rm{m}} + {\rm{n}}} \right)}}}} $$

(B4)