2.5 Data Tables - Implicit Analysis

• Nonlinear Stress-Strain Materials
• Mooney-Rivlin Hyperelastic Materials
• Viscoelastic Materials
• Magnetic Materials
• Anisotropic Elastic Materials
• Piezoelectric Materials
• Creep Equations
• Swelling Equations

See Chapter 8 of the ANSYS Structural Analysis Guide for additional details.

2.5.1 Nonlinear Stress-Strain Materials

Lab	Material Behavior Option
BKIN	Bilinear Kinematic Hardening (Rate-independent plasticity)
MKIN	Multilinear Kinematic Hardening (Rate-independent plasticity)
KINH	Multilinear Kinematic Hardening (Rate-independent plasticity)
MISO	Multilinear Isotropic Hardening (Rate-independent plasticity)
BISO	Bilinear Isotropic Hardening (Rate-independent plasticity)
ANISO	Anisotropic (Rate-independent plasticity)
DP	Drucker-Prager (Rate-independent plasticity)
ANAND	Anand's Model (Rate-dependent plasticity)
MELAS	Multilinear Elastic
USER	User-defined Nonlinear Stress-Strain Material Option

All options except DP, ANAND, and USER require a uniaxial stress-strain curve to be input. All options except ANISO and USER must have elastically isotropic (EX=EY=EZ) materials. Required values that aren't included in the data table are assumed to be zero. If the data table is not defined (or contains all zero values), the material is assumed to be linear. The material behavior options are briefly described below. See Chapter 4 of the ANSYS Theory Reference for more detail.

2.5.1.1 Bilinear Kinematic Hardening

Initialize the stress-strain table with TB,BKIN. For each stress-strain curve, define the temperature [TBTEMP], then define C1 and C2 [TBDATA]. You can define up to six temperature-dependent stress-strain curves (NTEMP=6 maximum on the TB command) in this manner. The constants C1 and C2 are:

Constant	Meaning
C1	Yield stress (Force/Area)
C2	Tangent modulus (Force/Area)

BKIN can be used with the TBOPT option. In this case, TBOPT takes two arguments. For TB,BKIN,,,,0, there is no stress relaxation with an increase in temperature. This option is not recommended for nonisothermal problems. For TB,BKIN,,,,1, Rice's hardening rule is applied (which does take stress relaxation with temperature increase into account). This is the default.

2.5.1.2 Multilinear Kinematic Hardening

The curve defined with the MKIN option is continuous from the origin with a maximum of five total stress-total strain points. The slope of the first segment of the curve must correspond to the elastic modulus of the material and no segment slope should be larger.

The MKIN option has the following restrictions:

• You may define up to five temperature dependent stress-strain curves.
• You may use only five points for each stress-strain curve.
• Each stress-strain curve must have the same set of strain values.

Initialize the stress-strain table with TB,MKIN, followed by a special form of the TBTEMP command (TBTEMP,,STRAIN) to indicate that strains are defined next. The constants (C1-C5), entered on the next TBDATA command, are the five corresponding strain values (the origin strain is not input). The temperature of the first curve is then input with TBTEMP, followed by the TBDATA command with the constants C1-C5 representing the five stresses corresponding to the strains at that temperature. You can define up to five temperature-dependent stress-strain curves (NTEMP=5 max on the TB command) with the TBTEMP command.

MKIN can also be used in conjunction with the TBOPT option (TB,MKIN,,,,TBOPT). TBOPT has the following three valid arguments:

0 - No stress relaxation with temperature increase (this is not recommended for nonisothermal problems); also produces thermal ratcheting.

1 - Recalculate total plastic strain using new weight factors of the subvolume.

2 - Scale layer plastic strains to keep total plastic strain constant; agrees with Rice's model (TB, BKIN with TBOPT=1). Produces stable stress-strain cycles.

TB,KINH,1,2,3
TBTEMP,20.0
TBPT,,0.001,1.0
TBPT,,0.1012,1.2
TBPT,,0.2013,1.3
TBTEMP,40.0
TBPT,,0.008,0.9
TBPT,,0.09088,1.0
TBPT,,0.12926,1.05

Note-The mechanical behavior of the TB,KINH option is the same as TB,MKIN with TBOPT=2.

2.5.1.3 Multilinear Isotropic Hardening

You can specify up to 20 temperature-dependent stress-strain curves. Initialize the curves with TB,MISO. Input the temperature for the first curve [TBTEMP], followed by up to 100 stress-strain points (the origin stress-strain point is not input) [TBPT]. Define up to 20 temperature-dependent stress-strain curves (NTEMP=20 max on the TB command) in this manner. The constants (X,Y) entered on the TBPT command (2 per command) are:

Constant	Meaning
X	Strain value (Dimensionless)
Y	Corresponding stress value (Force/Area)

2.5.1.4 Bilinear Isotropic Hardening

Initialize the stress-strain table with TB,BISO. For each stress-strain curve, define the temperature [TBTEMP], then define C1 and C2 [TBDATA]. Define up to six temperature-dependent stress-strain curves (NTEMP=6 max on the TB command) in this manner. The constants C1 and C2 are:

Constant	Meaning
C1	Yield stress (Force/Area)
C2	Tangent modulus (Force/Area)

2.5.1.5 Anisotropic

The material behavior is described by the uniaxial tensile and compressive stress-strain curves in three orthogonal directions and the shear stress-engineering shear strain curves in the corresponding directions. A bilinear response in each direction is assumed. The initial slope of the curve is taken as the elastic moduli of the material. At the specified yield stress, the curve continues along the second slope defined by the tangent modulus (having the same units as the elastic modulus). The tangent modulus cannot be less than zero or greater than the elastic modulus. Temperature dependent curves cannot be input. All values must be input as no defaults are defined. Input the magnitude of the yield stresses (without signs). No yield stress can have a zero value. The tensile x-direction is used as the reference curve for output quantities SEPL and EPEQ.

Initialize the stress-strain table with TB,ANISO. You can define up to 18 constants with TBDATA commands. The constants (C1-C18) entered on TBDATA commands (6 per command) are:

Constant	Meaning (all units are Force/Area)
C1-C3	Tensile yield stresses in the material x, y, and z directions
C4-C6	Corresponding tangent moduli
C7-C9	Compressive yield stresses in the material x, y, and z directions
C10-C12	Corresponding tangent moduli
C13-C15	Shear yield stresses in the material xy, yz, and xz directions
C16-C18	Corresponding tangent moduli

2.5.1.6 Drucker-Prager

• the cohesion value (must be > 0)
• the angle of internal friction
• the dilatancy angle.

Initialize the constant table with TB,DP. You can define up to three constants with TBDATA commands. The constants (C1-C3) entered on TBDATA are:

Constant	Meaning
C1	Cohesion value (Force/Area)
C2	Angle (in degrees) of internal friction
C3	dilatancy angle (in degrees)

2.5.1.7 Anand's Model

Constant	Meaning	Material Property	Units
C1	so	initial value of deformation resistance	stress
C2	Q/R	Q = activation energy R = universal gas constant	energy /volume energy /(volume temp)
C3	A	pre-exponential factor	1 / time
C4	xi	multiplier of stress	dimensionless
C5	m	strain rate sensitivity of stress	dimensionless
C6	ho	hardening / softening constant	stress
C7	1	coefficient for deformation resistance saturation value	stress
C8	n	strain rate sensitivity of saturation (deformation resistance) value	dimensionless
C9	a	strain rate sensitivity of hardening or softening	dimensionless

1 The coefficient for deformation resistance saturation value is

2.5.1.8 Multilinear Elastic

Specify up to 20 temperature-dependent stress-strain curves. Initialize the curves with TB,MELAS. The temperature for the first curve is input with TBTEMP, followed by TBPT commands for up to 100 stress-strain points (the origin stress-strain point is not input). You can define up to 20 temperature- dependent stress-strain curves (NTEMP=20 max on the TB command) in this manner. The constants (X,Y) entered on TBPT (2 per command) are:

Constant	Meaning
X	Strain value (Dimensionless)
Y	Corresponding stress value (Force/Area)

2.5.1.9 User

2.5.2 Mooney-Rivlin Hyperelastic Material Constants

Constant	Meaning
C1	1st strain energy constant (a10)
C2	2nd strain energy constant (a01)
C3	3rd strain energy constant (a20)
C4	4th strain energy constant (a11)
C5	5th strain energy constant (a02)
C6	6th strain energy constant (a30)
C7	7th strain energy constant (a21)
C8	8th strain energy constant (a12)
C9	9th strain energy constant (a03)

These constants have the units of strain energy per unit volume (Force/Area).

You can use the *MOONEY command to automatically determine two-term, five-term, or nine-term constants from physical test data.

2.5.3 Viscoelastic Material Constants

Constant - Meaning
1 - H/R (activation energy (H) divided by ideal gas constant (R)).
2 - Value of the constant x (0.0x1.0).
3 - No. of Maxwell elements (10 max) in volume decay function M_V.
6-15 - Up to ten values of C_fi (coefficients of the Maxwell element
representing the volume decay function M_V). Used to define the fictive temperature. ( C_fi = 1.0)
16-25 - Up to ten values of _fi (constants associated with a discrete relaxation
spectrum). Used to define the fictive temperature. Each _fi is also known as a relaxation time.
26-30 - Up to five values of C_i (coefficients of thermal expansion for the liquid state). ( = C₁ + C₂T_f + C₃T_f² + C₄T_f³ + C₅T_f⁴, where T_f = fictive temperature)
31-35 - Up to five values of C_gi (coefficients of thermal expansion for the glass state). (_g = C_g1 + C_g2T + C_g3T² + C_g4T³+ C_g5T⁴, where T = actual temperature)
36-45 - Up to ten values of T_fi (fictive temperature). T_f = C_fiT_fi
46 - GXY(0) (shear modulus at time = zero (the full shear modulus)).
47 - GXY() (shear modulus at time = infinity (the residual shear modulus after the full decay)). If no relaxation of the shear modulus,
use GXY() = GXY(0).
48 - K(0) (bulk modulus at time = zero).
49 - K() (bulk modulus at time = infinity). If no relaxation of the bulk
modulus, use K() = K(0).
50 - No. of Maxwell elements (10 max) used to approximate the shear
modulus (GXY(0) - GXY()) relaxation.
51-60 - Up to ten values of C_smi (coefficients for shear modulus relaxation
using Maxwell elements, C_smi = 1.0 if shear modulus relaxes).
61-70 - Up to ten values of _smi (relaxation times for shear modulus relaxation using Maxwell elements).
71 - No. of Maxwell elements (10 max) used to approximate the bulk
modulus (K(0) - K()) relaxation.
76-85 - Up to ten values of C_bmi (coefficients for bulk modulus relaxation
using Maxwell elements, C_bmi = 1.0 if bulk modulus relaxes).
86-95 - Up to ten values of _bmi (relaxation times for bulk modulus relaxation
using Maxwell elements).

2.5.4 Magnetic Materials

Constant	Meaning
X	Magnetic field intensity (H) (Magneto-motive force/length)
Y	Corresponding magnetic flux density (B) (Flux/Area)

Specify the system of units (MKS, CGS, or user defined) with EMUNIT, which also determines the value of the permeability of free space. Free-space permeability is available in elements SOLID5, INFIN9, PLANE13, INFIN47, PLANE53, SOLID62, SOLID96, SOLID97, SOLID98, INFIN110, and INFIN111. This value is used with the relative permeability property values [MP] to establish absolute permeability values. The defaults (also obtained for Lab=MKS) are MKS units and free-space permeability of 4E-7 Henries/meter. If you specify the CGS system of units by Lab=CGS, free-space permeability is set to 1.0 (dimensionless). You can specify Lab=MUZRO to define any system of units, then input free-space permeability.

2.5.5 Anisotropic Elastic Materials

The full 6 x 6 elastic coefficient matrix [D] relates terms ordered x,y,z,xy,yz,xz via 21 constants as shown below.

For 2-D problems, a 4 x 4 matrix relates terms ordered x,y,z,xy via 10 constants (D₁₁, D₂₁, D₂₂, D₃₁, D₃₂, D₃₃, D₄₁, D₄₂, D₄₃, D₄₄). Note, the order of the vector is expected as {x,y,z,xy,yz,xz}, whereas for some published materials the order is given as {x,y,z,yz,xz,xy}. This difference requires the "D" matrix terms to be converted to the expected format. The "D" matrix can be defined in either "stiffness" form (with units of Force/Area operating on the strain vector) or in "compliance" form (with units of the inverse of Force/Area operating on the stress vector), whichever is more convenient. Select a form using the appropriate element KEYOPT. Both forms use the same data table input as described below.

Enter the constants of the elastic coefficient matrix in the data table with the TB commands. Initialize the constant table with TB,ANEL. Define the temperature with TBTEMP, followed by up to 21 constants input with TBDATA commands. For the coupled-field elements, temperature- dependent matrix terms are not allowed. For the SOLID64 element, you can define up to six temperature-dependent sets of constants (NTEMP=6 max on the TB command) in this manner. Matrix terms are linearly interpolated between temperature points. The constants (C1-C21) entered on TBDATA (6 per command) are:

Constant	Meaning
C1-C6	Terms D11, D21, D31, D41, D51, D61
C7-C12	Terms D22, D32, D42, D52, D62, D33
C13-C18	Terms D43, D53, D63, D44, D54, D64
C19-C21	Terms D55, D65, D66

2.5.6 Piezoelectric Materials

Input the elastic coefficient matrix [c] either by specifying the stiffness constants (EX, EY, etc.) with MP commands, or by specifying the terms of the matrix with TB commands as described in Section 2.5.4.

The full 6 x 3 piezoelectric matrix [e] relates terms x,y,z,xy,yz,xz to x,y,z via 18 constants as shown:

For 2-D problems, a 4 x 2 matrix relates terms ordered x,y,z,xy via 8 constants (e₁₁, e₁₂, e₂₁, e₂₂, e₃₁, e₃₂, e₄₁, e₄₂). The order of the vector is expected as {x,y,z,xy,yz,xz}, whereas for some published materials the order is given as {x,y,z,yz,xz,xy}. This difference requires the "e" matrix terms to be converted to the expected format.

Use the TB commands to enter the constants of the piezoelectric [e] matrix in the data table. Initialize the constant table with TB,PIEZ. You can define up to 18 constants (C1-C18) with TBDATA commands (6 per command):

Constant	Meaning
C1-C6	Terms e11, e12, e13, e21, e22, e23
C7-C12	Terms e31, e32, e33, e41, e42, e43
C13-C18	Terms e51, e52, e53, e61, e62, e63

2.5.7 Creep Equations

Three different types of creep equations are available:

• primary creep
• secondary creep
• irradiation induced creep

A linear stepping function is used to calculate the change in the creep strain within a time step (_cr = ()(t)). The creep strain rate is evaluated at the condition corresponding to the beginning of the time interval and is assumed to remain constant over the time interval. Primary equivalent stresses and strains are used to evaluate the creep strain rate. For highly nonlinear creep strain vs. time curves, use a small time step. A creep time step optimization procedure is available for automatically increasing the time step whenever possible. A nonlinear stepping function (based on an exponential decay) is also available (C₁₁=1) but should be used with caution since it can underestimate the total creep strain where primary stresses dominate. This function is available only for creep equations C₆=0,1 and 2. Temperatures used in the creep equations should be based on an absolute scale [TOFFST].

Specify primary creep with constant C₆. Tables 2.5-1 through 2.5-11 show the available equations. An equation is selected with the appropriate value of C₆ (0 to 15). If C₁ 0, or if T + TOFFST 0, no primary creep is computed.

Specify secondary creep with constant C₁₂. Tables 2.5-12 and 2.5-13 show the available equations. An equation is selected with the appropriate value of C₁₂ (0 or 1). If C₇ 0, or if T + TOFFST 0, no secondary creep is computed. Also, primary creep equations C₆=9,10,11,13,14, and 15 bypass any secondary creep equations since secondary effects are included in the primary part.

Specify irradiation induced creep with constant C₆₆. Table 2.5-14 shows the single equation currently available; select it with C₆₆=5. This equation can be used in conjunction with equations C₆=0 to 11. The constants should be entered into the data table as indicated by their subscripts. If C₅₅ 0 and C₆₁ 0, or if T + TOFFST 0, no irradiation induced creep is computed.

Use the BF or BFE commands to enter temperature and fluence values. The input fluence (t) includes the integrated effect of time and time explicitly input is not used in the fluence calculation. Also, for the usual case of a constant flux (), the fluence should be linearly ramp changed.

Initialize the creep table with TB,CREEP. Temperature-dependent creep coefficients are not permitted (TBTEMP is not valid for creep). The constants entered on the TBDATA commands (6 per command) are:

Constant	Meaning
C1-CN	Constants C1, C2, C3, etc. (as defined in Tables 2.5-1 to 2.5-14) These are obtained by curve fitting test results for your material to the equation you choose. Exceptions are defined below.

Table 2.5-1 Primary Creep Equation for C₆=0

where:

= equivalent strain (based on modified total strain)

= equivalent stress

T = temperature (absolute). TOFFST is internally added to all temperatures for convenience.

t = time at end of substep

e = natural logarithm base

Table 2.5-2 Primary Creep Equation for C₆=1

Table 2.5-3 Primary Creep Equation for C₆ =2

where:

Table 2.5-4 Primary Creep Equation for C₆=9

Annealed 304 Stainless Steel:

Double Exponential Creep Equation (C₄ = 0.0)
To use the following Double Exponential creep equation to calculate _c, enter C₄ = 0.0:

_c = _x (1-e^-st) + _t (1-e^-rt) +

where:

_x = 0 for C₂

_x = G + H for C₂ < C3

C₂ = 6000 psi (default), C₃ = 25000 psi (default)

s, r, , G, and H are the functions of temperature and stress as described in the reference.

This double exponential equation is valid for Annealed 304 Stainless Steel over a temperature range from 800 to 1100°F. The equation, known as the Blackburn creep equation when C₁ = 1, is described completely in the Nuclear Systems Material Handbook¹. The first two terms describe the primary creep strain and the last term describes the secondary creep strain.

To use this equation, input a nonzero value for C₁, C₆ = 9.0, and C₇ = 0.0. Temperatures should be in °R (or °F with TOFFST = 460.0). Conversion to °K for the built-in property tables is done internally. If the temperature is below the valid range, no creep is computed. Time should be in hours and stress in psi. The valid stress range is 6,000 - 25,000 psi.

Rational Polynomial Creep Equation with Metric Units (C₄ = 1.0)
To use the following standard Rational Polynomial creep equation (with metric units) to calculate _c, enter C₄ = 1.0:

where:

c = limiting value of primary creep strain

p = primary creep time factor

= secondary (minimum) creep strain rate

This standard rational polynomial creep equation is valid for Annealed 304 SS over a temperature range from 427°C to 704°C. The equation is described completely in the Nuclear Systems Material Handbook¹. The first term describes the primary creep strain. The last term describes the secondary creep strain. The average "lot constant" from Footnote 1 is used to calculate .

To use this equation, input C₁ = 1.0, C₄ = 1.0, C₆ = 9.0, and C₇ = 0.0. Temperature must be in °C and TOFFST must be 273 (because of the built-in property tables). If the temperature is below the valid range, no creep is computed. Also, time must be in hours and stress in Megapascals (MPa).

Various hardening rules governing the rate of change of creep strain during load reversal may be selected with the C₅ value: 0.0 - time hardening, 1.0 - total creep strain hardening, 2.0 - primary creep strain hardening. These options are available only with the standard rational polynomial creep equation.

Rational Polynomial Creep Equation with English Units (C₄ = 2.0)
To use the above standard Rational Polynomial creep equation (with English units), enter C₄ = 2.0.

This standard rational polynomial equation is the same as described above except that temperature must be in °F, TOFFST must be 460, and stress must be in psi. The equivalent valid temperature range is 800 - 1300°F.

Table 2.5-5 Primary Creep Equation for C₆=10

Annealed 316 Stainless Steel:

Double Exponential Creep Equation (C₄ = 0.0)
To use the same form of the Double Exponential creep equation as described for Annealed 304 SS (C₆ = 9.0, C₄ = 0.0) in Table 2.5-4 to calculate _c, enter C₄ = 0.0.

This equation, also described in Ref. 1, differs from the Annealed 304 SS equation in that the built-in property tables are for Annealed 316 SS, the valid stress range is 4000 - 30,000 psi, C₂ defaults to 4000 psi, C₃ defaults to 30,000 psi, and the equation is called with C₆ = 10.0 instead of C₆ = 9.0.

Rational Polynomial Creep Equation with Metric Units (C₄ = 1.0)
To use the same form of the standard Rational Polynomial creep equation with metric units as described for Annealed 304 SS (C₆ = 9.0, C₄ = 1.0) in Table 2.5-4, enter C₄ = 1.0.

This standard rational polynomial equation, also described in Ref. 1, differs from the Annealed 304 SS equation in that the built-in property tables are for Annealed 316 SS, the valid temperature range is 482 - 704°C, and the equation is called with C₆ = 10.0 instead of C₆ = 9.0. The hardening rules for load reversal described for the C₆ = 9.0 standard Rational Polynomial creep equation are also available. The average "lot constant" from Ref. 1 is used in the calculation of .

Rational Polynomial Creep Equation with English Units (C₄ = 2.0)
To use the previous standard Rational Polynomial creep equation with English units, enter C₄ = 2.0.

This standard rational polynomial equation is the same as described above except that the temperatures must be in °F, TOFFST must be 460, and the stress must be in psi (with a valid range from 0.0 to 24220 psi). The equivalent valid temperature range is 900 - 1300°F.

Table 2.5-6 Primary Creep Equation for C₆=11

Annealed 2 1/4 Cr - 1 Mo Low Alloy Steel:

Modified Rational Polynomial Creep Equation (C₄ = 0.0)
To use the following Modified Rational Polynomial creep equation to calculate _c, enter C₄ = 0.0:

A, B, and are functions of temperature and stress as described in the reference.

This modified rational polynomial equation is valid for Annealed 2 1/4 Cr -1 Mo Low Alloy steel over a temperature range of 700 - 1100 °F. The equation is described completely in the Nuclear Systems Material Handbook². The first term describes the primary creep strain and the last term describes the secondary creep strain. No modification is made for plastic strains.

To use this equation, input C₁ = 1.0, C₆ = 11.0, and C₇ = 0.0. Temperatures must be in °R (or °F with TOFFST = 460.0). Conversion to °K for the built-in property tables is done internally. If the temperature is below the valid range, no creep is computed. Time should be in hours and stress in psi. Valid stress range is 1000 - 65,000 psi.

Rational Polynomial Creep Equation with Metric Units (C₄ = 1.0)
To use the following standard Rational Polynomial creep equation (with metric units) to calculate _c, enter C₄ = 1.0:

where:

c = limiting value of primary creep strain

p = primary creep time factor

= secondary (minimum) creep strain rate

This standard rational polynomial creep equation is valid for Annealed 2 1/4 Cr - 1 Mo Low Alloy Steel over a temperature range from 371°C to 593°C. The equation is described completely in the Nuclear Systems Material Handbook². The first term describes the primary creep strain and the last term describes the secondary creep strain. No tertiary creep strain is calculated. Only Type I (and not Type II) creep is supported. No modification is made for plastic strains.

To use this equation, input C₁ = 1.0, C₄ = 1.0, C₆ = 11.0, and C₇ = 0.0. Temperatures must be in °C and TOFFST must be 273 (because of the built-in property tables). If the temperature is below the valid range, no creep is computed. Also, time must be in hours and stress in Megapascals (MPa). The hardening rules for load reversal described for the C₆ = 9.0 standard Rational Polynomial creep equation are also available.

Rational Polynomial Creep Equation with English Units (C₄ = 2.0)
To use the above standard Rational Polynomial creep equation with English units, enter C₄ = 2.0.

This standard rational polynomial equation is the same as described above except that temperatures must be in °F, TOFFST must be 460, and stress must be in psi. The equivalent valid temperature range is 700 - 1100°F.

Table 2.5-7 Primary Creep Equations for C₆=12

where

C1 =	Scaling constant
M,N,K =	Function of temperature (determined by linear interpolation within table) as follows:
	C5	Number of temperature values to describe M, N, or K function (2 minimum, 6 maximum)
	C49	First absolute temperature value
	C50	Second absolute temperature value
	...
	C48+C5	C5th absolute temperature value
	C48+C5+1	First M value
	...
	C48+2C5	C5th M value
	C48+2C5	C5th M value
	...
	C48+2C5	C5th M value
	C48+2C5+1	First N value
	...
	C48+3C5	C5th N value
	C48+3C5+1	First K value
	...

This power function creep law having temperature dependent coefficients is similar to Equation C₆ = 1.0 except with C₁ = f₁(T), C₂ = f₂(T), C₃ = f₃(T), and C₄ = 0. Temperatures must not be input in decreasing order.

Table 2.5-8 Primary Creep Equation for C₆=13

where:

_{acc =} creep strain accumulated to this time (calculated by the program). Internally set to 1 x 10^-5 at the first substep with nonzero time to prevent division by zero.

A = C₁/T

B = C₂/T + C₃

C = C₄/T + C₅

This equation is often referred to as the Sterling Power Function creep equation. Constant C₇ should be 0.0. Constant C₁ should not be 0.0, unless no creep is to be calculated.

Table 2.5-9 Primary Creep Equation for C₆=14

where:

_{c =} cpt/(1+pt) +

ln c = -1.350 - 5620/T - 50.6 x 10^-6 + 1.918 ln (/1000)

ln p = 31.0 - 67310/T + 330.6 x 10^-6 - 1885.0 x 10^-122

ln = 43.69 - 106400/T + 294.0 x 10^-6 + 2.596 ln (/1000)

This creep law is valid for Annealed 316 SS over a temperature range from 800°F to 1300°F. The equation is similar to that given for C₆ = 10.0 and is also described in Ref. 1.

To use equation, input C₁ = 1.0 and C₆ = 14.0. Temperatures should be in °R (or °F with TOFFST = 460). Time should be in hours. Constants are only valid for English units (pounds and inches). Valid temperature range: 800° - 1300°F. Maximum stress allowed for e_c calculation: 45,000 psi; minimum stress: 0.0 psi. If T + TOFFST < 1160, no creep is computed.

Table 2.5-10 Primary Creep Equation for C₆=15

General Material Rational Polynomial:

where:

= C₂ (C₂ must not be negative)

c = C₇

p = C₁₀

This rational polynomial creep equation is a generalized form of the standard rational polynomial equations given as C₆ = 9.0, 10.0, and 11.0 (C₄ = 1.0 and 2.0). This equation reduces to the standard equations for isothermal cases. The hardening rules for load reversal described for the C₆ = 9.0 standard Rational Polynomial creep equation are also available.

Table 2.5-11 Primary Creep Equation for C₆=100

A user-defined creep equation is used. See the Guide to ANSYS User Programmable Features for more information.

Table 2.5-12 Secondary Creep Equation for C₁₂=0

where:

= equivalent stress

T = temperature (absolute). TOFFST is internally added to all temperatures for convenience.

t = time

e = natural logarithm base

Table 2.5-13 Secondary Creep Equation for C₁₂=1

Table 2.5-14 Irradiation Induced Creep Equation for C₆₆=5

where:

B = FG + C₆₃

F =

G = 1 - e^-t/C62a

= equivalent stress

T = temperature (absolute). TOFFST is internally added to all temperatures for convenience.

t = neutron fluence (input on BF or BFE command)

t = time

e = natural logarithm base

This irradiation induced creep equation is valid for 20% Cold Worked 316 SS over a temperature range from 700° to 1300°F. Constants 56, 57, 58 and 62 must be positive if the B term is included.

2.5.8 Swelling Equations

The fluence (which is the flux x time) is input on the BF or BFE command. A linear stepping function is used to calculate the change in the swelling strain within a load step:

where t is the fluence and the swelling strain rate equation is as defined in subroutine USERSW.

Because of the many empirical swelling equations available, the programming of the actual swelling equation is left to the user. In fact, the equation and the "fluence" input may be totally unrelated to nuclear swelling. See the Guide to ANSYS User Programmable Features for user programmable features.

For highly nonlinear swelling strain vs. fluence curves a small fluence step should be used. Note that since fluence (t), and not flux (), is input, a constant flux requires that a linearly changing fluence be input if time is changing. Temperatures used in the swelling equations should be based on an absolute scale [TOFFST]. Temperature and fluence values are entered with the BF or BFE command. Swelling calculations for the current substep are based upon the previous substep results.

Initialize the swelling table with TB,SWELL. The constants entered on the TBDATA commands (6 per command) are:

Constant	Meaning
C1-CN	Constants C1, C2, C3, etc. (as required by the user swelling equations). C72 must equal 10.