Fully coupled heat conduction and deformation analyses of visco-elastic solids

Abstract

Visco-elastic materials are known for their capability of dissipating energy. This energy is converted into heat and thus changes the temperature of the materials. In addition to the dissipation effect, an external thermal stimulus can also alter the temperature in a visco-elastic body. The rate of stress relaxation (or the rate of creep) and the mechanical and physical properties of visco-elastic materials, such as polymers, vary with temperature. This study aims at understanding the effect of coupling between the thermal and mechanical response that is attributed to the dissipation of energy, heat conduction, and temperature-dependent material parameters on the overall response of visco-elastic solids. The non-linearly visco-elastic constitutive model proposed by Schapery (Further development of a thermodynamic constitutive theory: stress formulation, 1969, Mech. Time-Depend. Mater. 1:209–240, 1997) is used and modified to incorporate temperature- and stress-dependent material properties. This study also formulates a non-linear energy equation along with a dissipation function based on the Gibbs potential of Schapery (Mech. Time-Depend. Mater. 1:209–240, 1997). A numerical algorithm is formulated for analyzing a fully coupled thermo-visco-elastic response and implemented it in a general finite-element (FE) code. The non-linear stress- and temperature-dependent material parameters are found to have significant effects on the coupled thermo-visco-elastic response of polymers considered in this study. In order to obtain a realistic temperature field within the polymer visco-elastic bodies undergoing a non-uniform heat generation, the role of heat conduction cannot be ignored.

Appendices

Appendix A

In this study, using the Gibbs free energy potential that was originally introduced by Schapery (1997) the dissipation function and energy equation for fully coupled non-linearly thermo-visco-elastic analyses are formulated. Explicit expressions also are derived for the mechanical dissipation rate, the rate of change of entropy and specific heat at constant stress, respectively. In deriving the basic constitutive relations the similar notions that were introduced by Schapery (1997) are used. Detail derivation of the constitutive and energy equations can be found in Khan (2011).

Schapery (1997) introduced Gibbs free energy in terms of stresses (σ _i ) and internal state variables (ISVs) ζ _m and temperature T. The Gibbs free energy for a non-linearly thermo-visco-elastic material can be expressed in the following form:

$$ G = G_{0} - A_{m}\zeta_{m} +\frac{1}{2}B_{mn}\zeta_{m}\zeta_{n}$$

(A.1)

where G ₀,A _m and B _mn are function of σ _i and T. The strain, after neglecting terms of second order in ζ _m , can be obtained from the following equation (Schapery 1969, 1997):

$$ \varepsilon_{i} = - \frac{\partial G}{\partial \sigma_{i}} = -\frac{\partial G_{0}}{\partial \sigma_{i}} + \frac{\partial A_{m}}{\partial \sigma_{i}}\zeta_{m}$$

(A.2)

The expression for ζ _m according to Schapery (1969, 1997) is

$$ \zeta_{m} = \frac{1}{C_{m}}\int_{0}^{t}\bigl[ 1 - \exp \bigl[ - \lambda_{m} \bigl( \psi^{t} -\psi^{\tau} \bigr) \bigr] \bigr] \frac{d}{d\tau} \biggl(\frac{A_{m}}{a_{2}} \biggr)\,d\tau $$

(A.3)

where C _m and λ _m are the constant and inverse of positive retardation time, respectively. The a ₂ is a positive scalar quantity that could be a function of σ _i and T. The term ψ ^τ is a function of time τ, time at which the input load is applied. The variable ψ ^t is the reduce time. To facilitate the characterization in terms of master creep functions through reduce time, A _m is given as

$$ A_{m} = C_{mn}\hat{\sigma}_{n} +\alpha_{m}\hat{\phi}$$

(A.4)

where C _mn and α _m are constants. $\hat{\sigma}_{n}$ depend on σ _i and T and $\hat{\phi}$ may depend on temperature but not on stress. $\hat{\phi}$ accounts for the thermal expansion effect; thus $\hat{\phi} = 0$ when T=T _R (reference temperature). Substituting of A _m into Eq. (A.3), neglecting the transient components of thermal expansion coefficients and after some simplification, we can write Eq. (A.2) as

$$ \varepsilon_{i} = - \frac{\partial G}{\partial \sigma_{i}} = -\frac{\partial G_{0}}{\partial \sigma_{i}} + \frac{\partial \hat{\sigma}_{n}}{\partial \sigma_{i}} \biggl( \int_{0}^{t}dD_{nk} \bigl( \psi^{t} - \psi^{\tau} \bigr)\frac{d}{d\tau} \biggl( \frac{\hat{\sigma}_{k}}{a_{2}} \biggr)\,d\tau \biggr)$$

(A.5)

where dD _nk (ψ ^t) represents the transient components of the mechanical creep compliances:

$$ dD_{nk} \bigl( \psi^{t} \bigr) = \sum _{m} \frac{C_{mn}C_{mk}}{C_{m}} \bigl[ 1 - \exp \bigl( -\lambda_{m}\psi{}^{t} \bigr) \bigr]$$

(A.6)

Schapery (1969) derived an expression for strain response due to uniform uniaxial state of stress ($\hat{\sigma} = \sigma$) under an isothermal condition and introduce non-linear parameters to incorporate the contribution of higher orders terms of the Gibbs free energy in terms of the applied stresses. Under isothermal conditions, the uniaxial strain using Eq. (A.5) can be expressed:

$$ \varepsilon(t) = g_{0}\bigl(\sigma^{t}\bigr)D_{0}\sigma^{t} + g_{1}\bigl(\sigma^{t}\bigr)\int_{0}^{t} dD \bigl(\psi^{t} - \psi^{\tau} \bigr) \frac{d}{d\tau} \bigl(g_{2}\bigl(\sigma^{\tau} \bigr)\sigma^{\tau} \bigr)\,d\tau $$

(A.7)

where g _o ,g ₁,g ₂ are the non-linear parameters that depend on the stress (σ ^t). D ₀ and dD are the instantaneous elastic and transient creep compliances, respectively. The Schapery (1969) model has been extended to a three-dimensional model and modified to incorporate the dependence of temperature and stress on the mechanical properties (Khan 2006), which is

(A.8)

The non-linear parameters depend on the current temperature T and effective stress ($\bar{\sigma}^{t}$).

Combining the first and second law of thermodynamics, the following expression for the energy balance in thermovisco-elastic materials is established (Rajagopal and Srinivasa 2011):

$$ T^{t}\dot{\eta} = - T^{t}\frac{d}{dt}\biggl\{ \frac{\partial G}{\partial T} \biggr\} = - q_{i,i} + T^{t}\dot{\gamma}_{\mathrm{int}}$$

(A.9)

where $\dot{\eta}$ is the rate of entropy. The heat generation rate $T\dot{\gamma}_{\mathrm{int}} = f_{m}\dot{\zeta}_{m} \ge 0$ is always positive unless ζ _m =0. The expression for thermodynamic forces from the Gibbs energy in (A.1) is $f_{m} = -\frac{\partial G}{\partial \zeta_{m}} = A_{m} -B_{mn}\zeta_{n}$. The indices of m and n can be associated to the each component of spring and dashpot in a mechanical analog representation (Schapery 1997). Using the above thermodynamic forces and assuming B _m =a ₂ C _m , the entropy production is

$$ T^{t}\dot{\gamma}_{\mathrm{int}} = f_{m}\dot{\zeta}_{m} = ( A_{m} - a_{2}C_{m}\zeta_{n} )\dot{\zeta}_{m}$$

(A.10)

Substituting A _m and ζ _m from Eq. (A.4) and Eq. (A.3), respectively, the heat generation rate can be expressed in terms of engineering stress ($\sigma_{ij}^{t}$) as follows:

(A.11)

For isotropic materials, the thermo-elastic Gibbs potential (G ₀) in Eq. (A.1) is assumed to have the following form:

$$ G_{0} = - C_{\mathrm{th}} - \alpha \bigl(T^{t}\bigr)\theta^{t}\sigma_{kk}^{t}- \frac{1}{2}D_{ijkl}^{(0)}\bigl(\bar{\sigma}^{t},T^{t}\bigr)\sigma_{ij}^{t}\sigma_{kl}^{t}$$

(A.12)

where θ ^t=T ^t−T ₀,T ₀ is the reference temperature. Here $C_{\mathrm{th}}, \alpha(T^{t}),D_{ijkl}^{(0)}$ are the thermal part of the thermo-elastic Gibbs free energy, temperature-dependent coefficient of linear thermal expansion and isotropic elastic compliance, respectively. The left-hand side of Eq. (A.9) can be found using Eq. (A.1) and Eq. (A.12) as

(A.13)

The specific heat capacity is defined as

$$ c_{\sigma} = - T^{t} \biggl(\frac{\partial^{2}G}{\partial T^{2}} \biggr) \bigg \vert _{\sigma} = T^{t}\biggl( \frac{\partial \eta}{\partial T} \biggr) \bigg \vert _{\sigma}$$

(A.14)

The specific heat can also be defined using the internal energy $\mathcal{E}$, i.e.,

$$ c_{\sigma} = \frac{\partial \mathcal{E}}{\partial T}$$

(A.15)

Substituting Eq. (A.1) into Eq. (A.14) and using Eq. (A.12), the expression for specific heat at a constant stress is

(A.16)

The energy equation from Eqs. (A.11) and (A.13) which incorporates visco-elastic dissipation and stress- and temperature-dependent material parameters and thermo-elastic coupling becomes

(A.17)

where C _th in Eq. (A.13) is assumed such that it accounts for the temperature-dependent specific heat at a constant stress. The above equation can be reduced for a linearly visco-elastic material as

(A.18)

Appendix B

This Appendix presents the analytical solutions for strain and temperature evolution in a linearly visco-elastic material under a creep. The standard linear solid (SLS) and Boltzmann integral models are considered (see Flugge 1975, Wineman and Rajagopal 2001, Reddy 2008).

Under a pure shear loading, the governing differential equation for the SLS model is

$$ \dot{\varepsilon}_{12} ( t ) + \frac{\mu_{2}}{\eta}\varepsilon_{12} ( t ) = \frac{1}{\mu_{1}}\dot{\sigma}_{12} (t ) + \frac{ ( \mu_{1} + \mu_{2} )}{\eta \mu_{1}}\sigma_{12} ( t )$$

(B.1)

An instantaneous constant shear stress of magnitude τ ₁ is applied at time t=0:σ ₁₂(t)=τ ₁ H(t), where H(t) is a Heaviside function. The expression for time dependent strain is

$$ \varepsilon_{12} ( t ) = \frac{\tau_{1}}{\mu_{1}} +\frac{\tau_{1}}{\mu_{2}} \biggl( 1 - \exp \biggl[ - \frac{\mu_{2}}{\eta} t\biggr] \biggr)$$

(B.2)

The energy equation under an adiabatic process and no internal heating source, $\rho c_{\sigma} \dot{T} = \dot{w}_{\mathrm{dis}}^{t}$, is

$$ \rho c_{\sigma} \dot{T} = \eta \biggl[ \varepsilon_{12}( t ) - \frac{1}{\mu_{1}}\dot{\sigma}_{12} ( t ) \biggr]^{2}$$

(B.3)

Substituting the expressions of ε ₁₂(t) and σ ₁₂(t) and solving Eq. (B.3) using the initial condition T(0)=0, we can determine the expression for the evolution of temperature:

(B.4)

where δ(t) is the Dirac delta function.

Under a general stress history the shear strain is expressed as

$$ \varepsilon_{12}(t) = J_{0}\sigma_{12}^{t} + \int_{0}^{t}dJ ( t - \tau )\frac{d\sigma_{12}^{\tau}}{d\tau} \,d\tau $$

(B.5)

Here J ₀ and dJ(t) are the shear instantaneous elastic and transient compliances. The transient creep compliance with N arrangements of springs and dashpots is expressed as

$$ dJ^{t} = \sum_{n = 1}^{N} J_{n} \bigl( 1 - \exp \bigl[ -\lambda_{n} ( t - \tau )\bigr] \bigr)$$

(B.6)

where J _n and λ _n are Prony coefficients and retardation times, respectively. With one Prony coefficient, the integral model is equivalent to the SLS model. The energy equation (Appendix A), assuming an adiabatic process and no internal heating source, for an isotropic linearly visco-elastic material can be expressed as

$$ \rho c_{\sigma} \dot{T} = \int_{0}^{t} \exp \bigl[ - \lambda( t - \tau) \bigr] \frac{d\sigma_{12}^{\tau}}{d\tau} \,d\tau\biggl\{\frac{d}{dt}\int_{0}^{t} dJ( t - \tau)\frac{d\sigma_{12}^{\tau}}{d\tau} \,d\tau\biggr\}$$

(B.7)

Using general stress history σ ₁₂(t) and creep compliance dJ(t), solving Eq. (B.7), and satisfying initial condition the evolution of temperature can be obtained.