can be defined by specifying thermal expansion coefficients so that
Abaqus
can compute thermal strains;
can be isotropic, orthotropic, or fully anisotropic;
are defined as total expansion from a reference temperature;
can be specified as a function of temperature and/or field variables;
can be defined with a distribution for solid continuum elements in
Abaqus/Standard;
and
can be specified directly in
Abaqus/Standard
in user subroutine
UEXPAN or in
Abaqus/Explicit
in user subroutine
VUEXPAN if the thermal strains are complicated functions of
temperature, time, field variables, and state variables.
Thermal expansion is a material property included in a material definition
(see
Material Data Definition)
except when it refers to the expansion of a gasket whose material properties
are not defined as part of a material definition. In that case expansion must
be used in conjunction with the gasket behavior definition (see
Defining the Gasket Behavior Directly Using a Gasket Behavior Model).
In an
Abaqus/Standard
analysis a spatially varying thermal expansion can be defined for homogeneous
solid continuum elements by using a distribution (Distribution Definition).
The distribution must include default values for the thermal expansion. If a
distribution is used, no dependencies on temperature and/or field variables for
the thermal expansion can be defined.
Computation of Thermal Strains
Abaqus
requires thermal expansion coefficients, ,
that define the total thermal expansion from a reference temperature,
,
as shown in
Figure 1.
They generate thermal strains according to the formula
where
is the thermal expansion coefficient;
is the current temperature;
is the initial temperature;
are the current values of the predefined field variables;
are the initial values of the field variables; and
is the reference temperature for the thermal expansion coefficient.
The second term in the above equation represents the strain due to the
difference between the initial temperature, ,
and the reference temperature, .
This term is necessary to enforce the assumption that there is no initial
thermal strain for cases in which the reference temperature does not equal the
initial temperature.
Defining the Reference Temperature
If the coefficient of thermal expansion, ,
is not a function of temperature or field variables, the value of the reference
temperature, ,
is not needed. If
is a function of temperature or field variables, you can define
.
Converting Thermal Expansion Coefficients from Differential Form to Total Form
Total thermal expansion coefficients are commonly available in tables of
material properties. However, sometimes you are given thermal expansion data in
differential form:
that is, the tangent to the strain-temperature curve is provided (see
Figure 1).
To convert to the total thermal expansion form required by
Abaqus,
this relationship must be integrated from a suitably chosen reference
temperature, :
For example, suppose is a series of
constant values:
between
and ;
between
and ;
between
and ;
etc. Then,
The corresponding total expansion coefficients required by
Abaqus
are then obtained as
Computing Thermal Strains in Linear Perturbation Steps
During a linear perturbation step, temperature perturbations can produce
perturbations of thermal strains in the form:
where is the temperature perturbation load about the base state,
is the temperature in the base state, and
is the tangent thermal expansion coefficient evaluated in the base state.
Abaqus
computes the tangent thermal expansion coefficients from the total form as
Defining Increments of Thermal Strain in User Subroutines
Increments of thermal strain can be specified in user subroutine
UEXPAN in
Abaqus/Standard
and in user subroutine
VUEXPAN in
Abaqus/Explicit
as functions of temperature and/or predefined field variables. User subroutine
UEXPAN in
Abaqus/Standard
must be used if the thermal strain increments depend on state variables.
Defining the Initial Temperature and Field Variable Values
If the coefficient of thermal expansion, ,
is a function of temperature or field variables, the initial temperature and
initial field variable values,
and ,
are given as described in
Initial Conditions.
Element Removal and Reactivation
If an element has been removed and subsequently reactivated in
Abaqus/Standard
(Element and Contact Pair Removal and Reactivation),
and
in the equation for the thermal strains represent temperature and field
variable values as they were at the moment of reactivation.
Isotropic, orthotropic, and fully anisotropic thermal expansion can be
defined in
Abaqus.
Orthotropic and anisotropic thermal expansion can be used only with
materials where the material directions are defined with local orientations
(see
Orientations).
Isotropic Expansion
If the thermal expansion coefficient is defined directly, only one value of
is needed at each temperature. If user subroutine
UEXPAN is used, only one isotropic thermal strain increment
()
must be defined.
Orthotropic Expansion
If the thermal expansion coefficients are defined directly, the three
expansion coefficients in the principal material directions
(,
,
and )
should be given as functions of temperature. If user subroutines
UEXPAN and
VUEXPAN are used, the three components of thermal strain increment
in the principal material directions (,
,
and )
must be defined.
Anisotropic Expansion
If the thermal expansion coefficients are defined directly, all six
components of
(,
,
,
,
,
)
must be given as functions of temperature. If user subroutine
UEXPAN is used in
Abaqus/Standard,
all six components of the thermal strain increment (,
,
,
,
,
)
must be defined. If user subroutine
VUEXPAN is used in
Abaqus/Explicit,
all six components of the thermal strain increment (,
,
,
,
,)
must be defined.
In an
Abaqus/Standard
analysis if a distribution is used to define the thermal expansion, the number
of expansion coefficients given for each element in the distribution, which is
determined by the associated distribution table (Distribution Definition),
must be consistent with the level of anisotropy specified for the expansion
behavior. For example, if orthotropic behavior is specified, three expansion
coefficients must be defined for each element in the distribution.
Defining Thermal Expansion for a Short-Fiber Reinforced Composite
The thermal expansion coefficient of a short-fiber reinforced composite (for example, an
injection molded composite) can be computed using the orientation averaging described by
Zheng (2011):
where is the orientation-averaged elasticity matrix computed using the
elasticity of the unidirectional (UD) composite and the second-order orientation tensor (see
Defining the Elasticity of a Short-Fiber Reinforced Composite), and is given by:
where and are the elasticity matrix and thermal expansion coefficient of the
unidirectional composite with the 1-direction as the fiber direction, is the second-order orientation tensor, and is the Kronecker delta. The unidirectional composite is assumed to be
transversely isotropic. Similar to elasticity, you must define the material directions with
local orientations (see Orientations), and the axes
of the local system must align with the principal directions of the second-order orientation
tensor.
Thermal Stress
When a structure is not free to expand, a change in temperature will cause
stress. For example, consider a single two-node truss of length
L that is completely restrained at both ends. The
cross-sectional area; the Young's modulus, E; and the
thermal expansion coefficient, ,
are all constant. The stress in this one-dimensional problem can then be
calculated from Hooke's Law as ,
where
is the total strain and
is the thermal strain, where
is the temperature change. Since the element is fully restrained,
.
If the temperature at both nodes is the same, we obtain the stress
.
Constrained thermal expansion can cause significant stress. For typical
structural metals, temperature changes of about 150°C (300°F) can cause yield.
Therefore, it is often important to define boundary conditions with particular
care for problems involving thermal loading to avoid overconstraining the
thermal expansion.
Energy Balance Considerations
Abaqus
does not account for thermal expansion effects in the total energy balance
equation, which can lead to an apparent imbalance of the total energy of the
model. For example, in the example above of a two-node truss restrained at both
ends, constrained thermal expansion introduces strain energy that will result
in an equivalent increase in the total energy of the model.
Material Options
Thermal expansion can be combined with any other (mechanical) material (see
Combining Material Behaviors)
behavior in
Abaqus.
Using Thermal Expansion with Other Material Models
For most materials thermal expansion is defined by a single coefficient or
set of orthotropic or anisotropic coefficients or, in
Abaqus/Standard, by
defining the incremental thermal strains in user subroutine
UEXPAN. For porous media in
Abaqus/Standard,
such as soils or rock, thermal expansion can be defined for the solid grains
and for the permeating fluid (when using the coupled pore fluid
diffusion/stress procedure—see
Coupled Pore Fluid Diffusion and Stress Analysis).
In such a case the thermal expansion definition should be repeated to define
the different thermal expansion effects.
Using Thermal Expansion with Gasket Behaviors
Thermal expansion can be used in conjunction with any gasket behavior
definition. Thermal expansion will affect the expansion of the gasket in the
membrane direction and/or the expansion in the gasket's thickness direction.
Elements
Thermal expansion can be used with any stress/displacement or fluid element
in
Abaqus.
References
Zheng, R., R. I. Tanner, and X. Fan, Injection Molding: Integration of Theory and Modeling Methods, Springer, 2011.