Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 412

F

ð k ¼ ð s 1 þ s t F Þ 1 t F

t x Þ ¼ t x Þ þ t x Þ :

(A.226)

s ð 1 þ s s ð 1 þ s ð 1 þ s

F

Þ
given by (A.226) using inverse transform results in Table A.1 , the creep and
relaxation functions (1.10) and (1.11) are obtained.

Executing the inverse Laplace transforms of x ~ ð s Þ given by (A.225) and ð s

A.17 Direct Integration of First Order Differential Equations

The ordinary differential equation representing the standard linear solid (2.8) may
be solved for F ( t ) given a specified x ( t ) by direct integration of (2.8), recognizing
that it is a first order ordinary differential equation. For the direct integration
method note that all solutions to the linear first order differential equation

d x
d t þ

p ð t Þ x ¼ q ð t Þ (A.227)

are given by

ð q

e R pðtÞ d t e R pðtÞ d t d t

x ð t Þ¼ ð t Þ þ C : (A.228)

where C is a constant of integration (Kaplan 1958). If one sets p ( t ) ¼ (1/ t F ) and q

( t ) t F )(d F /d t ), then (2.8) may be rewritten in the form of the

differential equation (A.227) and it follows from (A.228) that

¼ (1/ k t F ) F þ ( t x / k

2 3

ð

t

þ t x ð

t

e t=t F 1
k

d F
d t e t=t F d t

4 5 þ

e t=t F d t C e t=t F

x ð t Þ¼ F ð t Þ :

t F

0 0

For both the creep and the relaxation functions the constant C is evaluated using
the fact that the dashpot in the standard linear solid cannot extend in the first instant
so the deflection at t ¼ 0, x o , is due only to the deflection of the two springs in the

standard linear solid. The two springs must deflect the same amount, x o , thus the
initial force is F o ¼ ( k þ k R ) x o . This relationship may be rewritten in the form

F o ¼ t x ) x o , using definitions in (2.9). For the creep function the initial force,

F o , is taken to be one unit of force; thus the initial displacement is given by
x (0 + )

( k t F /

t F . For the relaxation function the initial displacement, x o ,is
taken to be one unit of displacement; thus the initial force F o is given by F (0 + )

¼ x o ¼ t x / k

¼

F o ¼ k t F / t x . To obtain the creep function one sets F ( t ) ¼ h ( t ), thus