Report presents the finite difference equations in time and finite element matrix equations in space for general linear thermovisoelastic problems. The equations are derived for a general three-dimensional body but are applicable to one- and two-dimensional configurations with minor changes

Chen, J. C.

Heer, E.

English

Finite difference and finite element matrix equations for linear thermoviscoelastic materia

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Finite element formulation for linear thermoviscoelastic materials

November 1969	 Brief 69-10660 
.
	
NASA TECH BRIEF 
NASA Tech Briefs are issued to summarize specific innovations derived from the U.S. space program, 
to encourage their commercial application. Copies are available to the public at 15 cents each 
from the Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia 22151. 
Finite Element Formulation for Linear
Thermoviscoelastic Materials 
The finite difference equations in time and finite 
element matrix equations in space have been devel-
oped for general linear thermoviscoelastic problems 
and presented in a report. 
There have been numerous applications of the finite 
element technique mainly to elastic and plastic static 
problems, and to some steady-state dynamic prob-
lems. However, the extension of this technique to 
viscoelastic problems without using the elastic-
viscoelastic correspondence principle has been ac-
complished only in a few relatively simple cases. It 
has been concluded that the elastic-viscoelastic cor-
respondence principle can be evoked for rather special 
cases, i.e., when the material properties are independ-
ent of thermal changes. Since the properties of most 
viscoelastic materials are highly temperature-sensi-
tive, it is concluded that the development of a gen-
eral program should be based on the solution of 
integral equations in real time rather than the cor-
respondence principle. 
A brief statement of the thermoviscoelastic field 
equations is followed by the development of the finite 
difference equations in time and then by the finite 
element formulation in space. The equations are 
derived for a general three-dimensional body but are 
applicable with minor changes to one-and two-dimen-
sional configurations. Some attention is given to the 
experimental determination of material properties 
and their use in analytical work. An expansion of the 
experimentally or analytically determined material 
property functions in terms of exponential series leads 
to recurrence matrix equations, eliminating the prob-
lem of calculating at each time-step the history of 
material response. 
Note: 
The following documentation may be obtained from: 
The Clearinghouse for Federal Scientific 
and Technical Information 
Springfield, Virginia 22151 
Single document price $3.00 
(or microfiche $0.65) 
Reference: TR 32-1381 (N69-31925) 
Finite Element Formulation For 
Linear Thermoviscoelastic Materials 
Patent status: 
No patent action is contemplated by NASA. 
Source: Ewald Heer and Jay-Chung Chen of 
Caltech/JPL 
under contract to 
NASA Pasadena Office 
(N PO- 1 1229) 
Category 03 
This document was prepared under the sponsorship of the National	 Government assumes any liability resulting from the use of the 
Aeronautics and Space Administration. Neither the United States 	 information contained in this document, or warrants that such use 
Government nor any person acting on behalf of the United States 	 will be free from privately owned rights.
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Finite element formulation for linear thermoviscoelastic materials

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