关于热冲击弹性问题的自由能及变分原理

在讨论耦合热弹性问题的变分原理的一些著作中,以弹性应变e_ij和温度变化值θ为状态参数的自由能φ(e_ij,θ)为自由能的这一表达式只适用于|θ|<0(绝对参考温度)的情况.在热冲击弹性问题中,温度变化值θ很大,甚至可以大过T₀同时,材料常数(λ,μ,γ,c等)随θ而发生变化,不再保持为常数.就这种情况,本文导出自由能的表达式.(0.1)式则为其特殊情况.将自由能的这一表达式引入变分原理,其欧拉方程将成为非线性.为了线性化,将热冲击作用的时间过程划分为若干足够小的时间元△t_k(△t_k=t_k-t_k-1,k=1,2,…,n).在△t_k中,温度变化θ_k很小,材料常数由t_k-1瞬时的温度场T_k-1=T_{x1,x2,x3,tk-1}确定,自由能φ_k可近似地采用(0.1)式的形式,从而得到变分原理的分段近似表达.

On the Free Energy and Variational Theorem of Elastic Problem in Thermal Shock

In some investigations on variational principle for coupled thermoelastic problems, the free energy Φ(e_ij,θ),where the state variables are elastic strain e_ij and temperature increment θ, is expressed by This expression is employed only under the condition of |θ|<0(absolute temperature of reference) But the value of temperature increment is great, even greater than T₀ in thermal shock. And the material properties (λ,μ,ν,c, etc.) will not remain constant, they vary with θ. The expression of free energy for this condition is derived in this paper. Equation (0.1) is its special case.Euler's equations will be nonlinear while this expression of free energy has been introduced into variational theorem. In order to linearise, the time interval of thermal shock is divided into a number of time elements Δt_k, (Δt_k=t_k-t_k-1,k=1,2…,n), which are so small that the temperature increment θ_k within it is very small, too. Thus, the material properties may be defined by temperature field T_k=T(x₁,x₂,x₃,t_k-1) at instant t_k-1, and the free energy Φ_k expressed by eg. (0.1) may be employed in element Δt_k.Hence the variational theorem will be expressed partly and approximately.