김경종
23 KiB
\boldsymbol {a} = C _ {1} \boldsymbol {a} _ {1} + C _ {2} \boldsymbol {a} _ {2} + C _ {3} \boldsymbol {a} _ {3}, \tag {7.69}
where
\boldsymbol {a} _ {1} ^ {T} = \frac {\partial J _ {1}}{\partial \sigma} = \{1, 1, 1, 0, 0, 0 \}
\boldsymbol {a} _ {2} ^ {T} = \frac {\partial (J _ {2} ^ {\prime}) ^ {1 / 2}}{\partial \sigma} = \frac {1}{2 (J _ {2} ^ {\prime}) ^ {1 / 2}} \left\{\sigma_ {x} ^ {\prime}, \sigma_ {y} ^ {\prime}, \sigma_ {z} ^ {\prime}, 2 \tau_ {y z}, 2 \tau_ {z x}, 2 \tau_ {x y} \right\}
\boldsymbol {a} _ {3} ^ {T} = \frac {\partial J _ {3}}{\partial \sigma} = \left\{\left(\sigma_ {y} ^ {\prime} \sigma_ {z} ^ {\prime} - \tau_ {y z} ^ {2} + \frac {J _ {2} ^ {\prime}}{3}\right), \quad \left(\sigma_ {x} ^ {\prime} \sigma_ {z} ^ {\prime} - \tau_ {x z} ^ {2} + \frac {J _ {2} ^ {\prime}}{3}\right), \right.
\left(\sigma_ {x} ^ {\prime} \sigma_ {y} ^ {\prime} - \tau_ {x y} ^ {2} + \frac {J _ {2} ^ {\prime}}{3}\right), \quad 2 \left(\tau_ {x z} \tau_ {x y} - \sigma_ {x} ^ {\prime} \tau_ {y z}\right),
\left. 2 \left(\tau_ {x y} \tau_ {y z} - \sigma_ {y} ^ {\prime} \tau_ {x z}\right), \quad 2 \left(\tau_ {y z} \tau_ {x z} - \sigma_ {z} ^ {\prime} \tau_ {x y}\right) \right\}, \tag {7.70}
and
C _ {1} = \frac {\partial F}{\partial J _ {1}}, \quad C _ {2} = \left(\frac {\partial F}{\partial \left(J _ {2} ^ {\prime}\right) ^ {1 / 2}} - \frac {\tan 3 \theta}{\left(J _ {2} ^ {\prime}\right) ^ {1 / 2}} \frac {\partial F}{\partial \theta}\right),
C _ {3} = \frac {- \sqrt {3}}{2 \cos 3 \theta} \frac {1}{(J _ {2} ^ {\prime}) ^ {3 / 2}} \frac {\partial F}{\partial \theta}. \tag {7.71}
Only the constants C_{1} , C_{2} and C_{3} are then necessary to define the yield surface. Thus we can achieve a simplicity of programming as only these three constants have to be varied between one yield surface and another. The constants C_{i} are given in Table 7.1 for the four yield criteria considered in Section 7.2.1 and other yield functions can be expressed in the same form with equal ease.
Table 7.1 Constants defining the yield surface in a form suitable for numerical analysis.
| Yield Criterion | $C_1$ | $C_2$ | $C_3$ |
| Tresca | 0 | $2\cos\theta(1+\tan\theta\tan3\theta)$ | $\frac{\sqrt{3}}{J_2'}\frac{\sin\theta}{\cos3\theta}$ |
| Von Mises | 0 | $\sqrt{3}$ | 0 |
| Mohr–Coulomb | $\frac{1}{3}\sin\phi$ | $\cos\theta[(1+\tan\theta\tan3\theta)+\sin\phi(\tan3\theta-\tan\theta)/\sqrt{3}]$ | $\frac{(\sqrt{3}\sin\theta+\cos\theta\sin\phi)}{(2J_2'\cos3\theta)}$ |
| Drucker–Prager | $\alpha$ | 1.0 | 0 |
7.5 Basic expressions for two dimensional problems
For two dimensional problems, the general expressions derived so far in this chapter have to be modified. Primarily the main alteration required is the deletion of the stress (and strain) components which vanish under the conditions of plane stress, plane strain or axial symmetry. We have only four non-zero stress or strain components, namely
\sigma^ {T} = \{\sigma_ {x}, \sigma_ {y}, \tau_ {x y}, \sigma_ {z} \}, \quad \sigma_ {z} = 0 \quad \text { for Plane Stress }
\{\sigma_ {x}, \sigma_ {y}, \tau_ {x y}, \sigma_ {z} \}, \quad \epsilon_ {z} = 0 \quad \text { Plane Strain }
\left\{\sigma_ {r}, \sigma_ {z}, \tau_ {r z}, \sigma_ {\theta} \right\} \quad \text { Axial Symmetry. } \tag {7.72}
From Fig. 7.8 it is seen that the z direction is taken as the coordinate independent direction for plane stress and plane strain. It is also found convenient to order the stress components as indicated in (7.72) with the stress in the coordinate independent direction being last.
Fig. 7.8 Two-dimensional applications showing coordinate systems employed.
The explicit form of the elasticity matrix D can be written
\boldsymbol {D} = \frac {E (1 - \nu)}{(1 + \nu) (1 - 2 \nu)} \left[ \begin{array}{c c c c} 1 & \frac {\nu}{1 - \nu} & 0 & \frac {\nu}{1 - \nu} \\ \frac {\nu}{1 - \nu} & 1 & 0 & \frac {\nu}{1 - \nu} \\ 0 & 0 & \frac {1 - 2 \nu}{2 (1 - \nu)} & 0 \\ \frac {\nu}{1 - \nu} & \frac {\nu}{1 - \nu} & 0 & 1 \end{array} \right] \text { for plane strain and axial symmetry, }
\boldsymbol {D} = \frac {E}{1 - \nu^ {2}} \left[ \begin{array}{c c c c c} 1 & \nu & 0 & 0 \\ \nu & 1 & 0 & 0 \\ 0 & 0 & \frac {1 - \nu}{2} & 0 \\ - \frac {2}{0} & 0 & 0 & 1 \end{array} \right] \quad \text { for plane stress. } \tag {7.73}
Note that the components corresponding to the coordinate independent direction have been included for the plane stress and strain cases. These terms will be excluded for element stiffness formulation and only the first 3 \times 3 portion indicated will be employed. By eliminating the appropriate stress terms the expressions developed to date can be readily modified. The flow vector a becomes
\boldsymbol {a} ^ {T} = \left\{\frac {\partial F}{\partial \sigma_ {x}}, \frac {\partial F}{\partial \sigma_ {y}}, \frac {\partial F}{\partial \tau_ {x y}}, \frac {\partial F}{\partial \sigma_ {z}} \right\}, \tag {7.74}
with x, y and z being replaced by r, z and \theta respectively for the case of axial symmetry. The specific form of the vector, a is still given by (7.69) but in this case we have from (7.70)
\boldsymbol {a} _ {1} ^ {T} = \{1, 1, 0, 1 \}
\boldsymbol {a} _ {2} ^ {T} = \frac {1}{2 (J _ {2} ^ {\prime}) ^ {1 / 2}} \left\{\sigma_ {x} ^ {\prime}, \sigma_ {y} ^ {\prime}, 2 \tau_ {x y}, \sigma_ {z} ^ {\prime} \right\}
\boldsymbol {a} _ {3} ^ {T} = \left\{\left(\sigma_ {y} ^ {\prime} \sigma_ {z} ^ {\prime} + \frac {J _ {2} ^ {\prime}}{3}\right), \left(\sigma_ {x} ^ {\prime} \sigma_ {z} ^ {\prime} + \frac {J _ {2} ^ {\prime}}{3}\right), \right.
\left. - 2 \sigma_ {z} ^ {\prime} \tau_ {x y}, \left(\sigma_ {x} ^ {\prime} \sigma_ {y} ^ {\prime} - \tau_ {x y} ^ {2} + \frac {J _ {2} ^ {\prime}}{3}\right) \right\}, \tag {7.75}
and the deviatoric stress invariants become, from (7.5)
J _ {2} ^ {\prime} = \frac {1}{2} \left(\sigma_ {x} ^ {\prime 2} + \sigma_ {y} ^ {\prime 2} + \sigma_ {z} ^ {\prime 2}\right) + \tau_ {x y} ^ {2}
J _ {3} ^ {\prime} = \sigma_ {z} ^ {\prime} (\sigma_ {z} ^ {\prime 2} - J _ {2} ^ {\prime}). \tag {7.76}
To complete the prescription of the elasto-plastic matrix D_{ep} given in (7.47) we require d_{D} . Employing the relevant form of D from (7.73) in (7.47) results in, for plane strain and axial symmetry
\boldsymbol {d} _ {D} = \left\{ \begin{array}{l} d _ {1} \\ d _ {1} \\ d _ {3} \\ d _ {4} \end{array} \right\} = \left\{ \begin{array}{c} \frac {E}{1 + \nu} a _ {1} + M _ {1} \\ \frac {E}{1 + \nu} a _ {2} + M _ {1} \\ G a _ {3} \\ \frac {E}{1 + \nu} a _ {4} + M _ {1} \end{array} \right\}, \quad M _ {1} = \frac {E \nu \left(a _ {1} + a _ {2} + a _ {4}\right)}{(1 + \nu) (1 - 2 \nu)}, \tag {7.77}
where G = E / 2(1 + \nu) is the shear modulus and a_1 \ldots a_4 are the components of \pmb{a} . For plane stress we have
\boldsymbol {d} _ {D} = \left\{ \begin{array}{c} \frac {E}{1 + \nu} a _ {1} + M _ {2} \\ \frac {E}{1 + \nu} a _ {2} + M _ {2} \\ G a _ {3} \\ \frac {E}{1 + \nu} a _ {4} + M _ {2} \end{array} \right\}, \quad M _ {2} = \frac {E \nu \left(a _ {1} + a _ {2}\right)}{1 - \nu^ {2}}. \tag {7.78}
7.6 Singular points on the yield surface
For many yield surfaces the flow vector a is not uniquely defined for certain stress combinations. For example this arises at the corners of the Tresca and Mohr–Coulomb criteria located by \theta = \pm30^{\circ} and the direction of plastic straining there is indeterminate. Koiter ^{(12)} has provided limits within which the incremental plastic strain vector must lie. Numerical difficulties will be encountered as \theta approaches \pm30^{\circ} for the Tresca and Mohr–Coulomb laws since it is seen from Table 7.1 that for these values of \theta both C_{2} and C_{3} become indeterminate. This difficulty can be overcome by returning to the original expressions (7.63) for the Tresca law and (7.65) for the Mohr–Coulomb criterion and rewriting these for the explicit values \theta = \pm30^{\circ} . Thus we have for the Tresca law
\sqrt {(3)} \left(J _ {2} ^ {\prime}\right) ^ {\frac {1}{2}} = Y (\kappa) = \sqrt {(3)} k (\kappa), \tag {7.79}
and thus from (7.71) we have
C _ {1} = 0, \quad C _ {2} = \sqrt {(3)}, \quad C _ {3} = 0 \quad \text {for} \quad \theta = \pm 3 0 ^ {\circ}. \tag {7.80}
Physically, since (7.79) is the Von Mises criterion, this is equivalent to stating that the direction of plastic straining at the corners of the Tresca criterion is that given by the Von Mises circle which also passes through the corner (see Fig. 7.2). Similarly for the Mohr-Coulomb criterion we have
from (7.65),
\frac {1}{3} J _ {1} \sin \phi + \left(J _ {2} ^ {\prime}\right) ^ {1 / 2} \frac {1}{2} \left(\sqrt {3} - \frac {\sin \phi}{\sqrt {3}}\right) - c \cos \phi = 0 \quad \text { for } \quad \theta = + 3 0 ^ {0}
\frac {1}{3} J _ {1} \sin \phi + (J _ {2} ^ {\prime}) ^ {1 / 2} \frac {1}{2} \left(\sqrt {3 + \frac {\sin \phi}{\sqrt {3}}}\right) - c \cos \phi = 0 \quad \theta = - 3 0 ^ {0}, \tag {7.81}
or from (7.71) we have
C _ {1} = \frac {1}{3} \sin \phi , C _ {2} = \frac {1}{2} \left(\sqrt {3} - \frac {\sin \phi}{\sqrt {3}}\right), C _ {3} = 0 \quad \text { for } \quad \theta = + 3 0 ^ {0}
C _ {1} = \frac {1}{3} \sin \phi , C _ {2} = \frac {1}{2} \left(\sqrt {3} + \frac {\sin \phi}{\sqrt {3}}\right), C _ {3} = 0 \quad \theta = - 3 0 ^ {0}. \tag {7.82}
The practical approach adopted in this text is to use the general expressions for C_1, C_2, C_3 given in Table 7.1 for all values of |\theta| \leqslant 29^\circ and to then employ either (7.80) for Tresca or (7.82) for Mohr-Coulomb in the vicinity of the corners. This makes the direction of straining unique, and also satisfies the Koiter requirements. Physically this artifice corresponds to a 'rounding off' of the yield surface corners.
7.7 Finite element expressions and program structure
The basic expressions required for solution can be again obtained by use of the principle of virtual work. Consider the solid, in which the internal stresses \sigma , the distributed loads/unit volume b and external applied forces f form an equilibrating field, to undergo an arbitrary virtual displacement pattern \delta d^{*} which result in compatible strains \delta \epsilon^{*} and internal displacements \delta u^{*} . Then the principle of virtual work requires that
\int_ {\Omega} \left(\delta \epsilon^ {* T} \sigma - \delta u ^ {* T} b\right) d \Omega - \delta d ^ {* T} f = 0. \tag {7.83}
Then the normal finite element discretising procedure leads to the following expressions for the displacements and strains within any element
\delta \boldsymbol {u} ^ {*} = N \delta \boldsymbol {d} ^ {*}, \quad \delta \boldsymbol {\epsilon} ^ {*} = \boldsymbol {B} \delta \boldsymbol {d} ^ {*}, \tag {7.84}
where N and B are respectively the usual matrix of shape functions and the elastic strain matrix. Then the element assembly process gives
\int_ {\Omega} \delta \boldsymbol {d} ^ {* T} (\boldsymbol {B} ^ {T} \boldsymbol {\sigma} - \boldsymbol {N} ^ {T} \boldsymbol {b}) d \Omega - \delta \boldsymbol {d} ^ {* T} \boldsymbol {f} = 0, \tag {7.85}
where the volume integration over the solid is the sum of the individual element contributions. Since this expression must hold true for any arbitrary \delta d^{*} value
\int_ {\Omega} \boldsymbol {B} ^ {T} \sigma d \Omega - \boldsymbol {f} - \int_ {\Omega} \boldsymbol {N} ^ {T} \boldsymbol {b} d \Omega = 0. \tag {7.86}
For the solution of nonlinear problems as described in Chapter 2, (7.86) will not generally be satisfied at any stage of the computation, and
\psi = \int_ {\Omega} \boldsymbol {B} ^ {T} \sigma d \Omega - \left(f + \int_ {\Omega} N ^ {T} \boldsymbol {b} d \Omega\right) \neq 0, \tag {7.87}
where \psi is the residual force vector. For an elasto-plastic situation the material stiffness is continually varying, and instantaneously the incremental stress/strain relationship is given by (7.46). For the purpose of evaluating the material tangential stiffness matrix K_{T} at any stage, the incremental form of (7.87) must be employed. Thus within an increment of load we have
\Delta \psi = \int_ {\Omega} \boldsymbol {B} ^ {T} \Delta \sigma d \Omega - \left(\Delta f + \int_ {\Omega} \boldsymbol {N} ^ {T} \Delta \boldsymbol {b} d \Omega\right). \tag {7.88}
Substituting for \Delta\sigma from (7.46) results in
\Delta \psi = K _ {T} d - \left(\Delta f + \int_ {\Omega} N ^ {T} \Delta b d \Omega\right), \tag {7.89}
where
\boldsymbol {K} _ {T} = \int_ {\Omega} \boldsymbol {B} ^ {T} \boldsymbol {D} _ {e p} \boldsymbol {B} d \Omega . \tag {7.90}
Expression (7.89) is essentially identical to (2.4) and therefore the solution procedures developed in Chapter 2 can be again employed.
The programming philosophy adopted for this application follows that employed in Chapter 3 for one-dimensional elasto-plastic problems. It is suggested that the reader reviews the appropriate sections of Chapter 3 before proceeding to the remainder of this chapter. The solution techniques discussed in Chapters 2 and 3 are utilised and in particular an initial stiffness algorithm, a tangential stiffness algorithm and two options of the combined initial/tangential stiffness approach are included. An outline of the program is provided in Fig. 7.9. Many of the subroutines required are common to the corresponding linear elastic solution program and their function and structure have already been described. In particular, subroutines BMATS, CHECK1, CHECK2, DBE, ECHO, FRONT, GAUSSQ, JACOB2, LOADPS, MODPS, NODEXY and SFR2 have been described in Section 6.4. Also the standard nonlinear subroutines ALGOR, CONVER, INCREM and INPUT have been presented in Section 6.5. We will now formulate the additional subroutines required and assemble them to form a working program.
flowchart
graph TD
A["START"] --> B["DIMEN<br>Presents the variables associated with the dynamic dimensioning process."]
B --> C["INPUT<br>Inputs data defining geometry, boundary conditions and material properties."]
C --> D["LOADPS<br>Evaluates the equivalent nodal forces for pressure loading, gravity loading, etc."]
D --> E["ZERO<br>Sets to zero arrays required for accumulation of data."]
E --> F["INCREM<br>Increments the applied loads according to specified load factors."]
F --> G["ALGOR<br>Sets indicator to identify the type of solution algorithm e.g. initial stiffness, tangential stiffness, etc."]
G --> H["STIFFP<br>Calculates the element stiffnesses for elastic and elasto-plastic material behaviour."]
H --> I["FRONT<br>Solves the simultaneous equation system by the frontal method."]
I --> J["RESIDU<br>Calculates the residual force vector, ψ."]
J --> K["CONVER<br>Checks to see if the solution process has converged."]
K -->|NO| L["END"]
K -->|YES| M["OUTPUT<br>Prints the results for this load increment."]
L --> N["ITERATION LOOP"]
M --> N
Fig. 7.9 Program organisation for two-dimensional elasto-plastic applications.
7.8 Additional program subroutines
A total of eight additional subroutines are required some of which will be common to other nonlinear applications considered in later chapters of this text.
7.8.1 Subroutine DIMEN
The function of this subroutine is to preset the values of variables employed in the program. In particular the variables associated with the dynamic dimensioning process described in Chapter 6 are defined. Thus if it is required to upgrade the magnitude of the maximum problem size which can be solved it is only necessary to modify the dimension statements in the main or master subroutine together with the variables set in subroutine DIMEN. All the variables preset in this subroutine have been previously defined and their specified values are indicated in the following listing.
SUBROUTINE DIMEN(MBUFA,MELEM,MEVAB,MFRON,MMATS,MPOIN,MSTIF,MTOTG,MTOTV,MVFIX,NDOFN,NPROP,NSTRE) DIMN 1
C*************** DIMN 2
C
C**** THIS SUBROUTINE PRESETS VARIABLES ASSOCIATED WITH DYNAMIC DIMN 3
C DIMENSIONING DIMN 4
C
C*************** DIMN 5
MBUFA = 10 DIMN 6
MELEM=40 DIMN 7
MFRON=80 DIMN 8
MMATS = 5 DIMN 9
MPOIN=150 DIMN 10
MSTIF=(MFRON*MFRON-MFRON)/2.0+MFRON DIMN 11
MTOTG = MELEM*9 DIMN 12
NDOFN = 2 DIMN 13
MTOTV = MPOIN*NDOFN DIMN 14
MVFIX=30 DIMN 15
NPROP=7 DIMN 16
MEVAB = NDOFN*9 DIMN 17
RETURN DIMN 18
END DIMN 19
DIMN 20
DIMN 21
DIMN 22
7.8.2 Subroutine ZERO
This subroutine merely sets to zero the contents of several arrays employed in the program. These arrays will be employed to accumulate data as the incremental and iterative process continues and they therefore require to be initialised to zero. This subroutine is self-explanatory and is presented without further comment.
SUBROUTINE ZERO(ELOAD,MELEM,MEVAB,MPOIN,MTOTG,MTOTV,NDOFN,NELEM, ZRO1 1
. NEVAB,NGAUS,NSTR1,NTOTG,EPSTN,EFFST, ZRO1 2
. NTOTV,NVFIX,STRSG,TDISP,TFACT, ZRO1 3
. TLOAD,TREAC,MVFIX) ZRO1 4
C*******************************
C*******************************
C*******************************
C**** THIS SUBROUTINE INITIALISES VARIOUS ARRAYS TO ZERO ZRO1 7
C ZRO1 8
C*******************************
DIMENSION ELOAD(MELEM,MEVAB),STRSG(4,MTOTG),TDISP(MTOTV), ZRO1 10
. TLOAD(MELEM,MEVAB),TREAC(MVFIX,2),EPSTN(MTOTG), ZRO1 11
. EFFST(MTOTG) ZRO1 12
TFACT=0.0 ZRO1 13
DO 30 IELEM=1,NELEM ZRO1 14
DO 30 IEVAB=1,NEVAB ZRO1 15
ELOAD(IELEM,IEVAB)=0.0 ZRO1 16
| 30 | TLOAD(IELEM,IEVAB)=0.0 | ZRO1 | 17 |
| DO 40 ITOTV=1,NTOTV | ZRO1 | 18 | |
| 40 | TDISP(ITOTV)=0.0 | ZRO1 | 19 |
| DO 50 IVFIX=1,NVFIX | ZRO1 | 20 | |
| DO 50 IDOFN=1,NDOFN | ZRO1 | 21 | |
| 50 | TREAC(IVFIX,IDOFN)=0.0 | ZRO1 | 22 |
| DO 60 ITOTG=1,NTOTG | ZRO1 | 23 | |
| EPSTN(ITOTG)=0.0 | ZRO1 | 24 | |
| EFFST(ITOTG)=0.0 | ZRO1 | 25 | |
| DO 60 ISTR1=1,NSTR1 | ZRO1 | 26 | |
| 60 | STRSG(ISTR1,ITOTG)=0.0 | ZRO1 | 27 |
| RETURN | ZRO1 | 28 | |
| END | ZRO1 | 29 |
7.8.3 Subroutine INVAR
The role of this subroutine is to evaluate the various functions of stress used to indicate either initiation of or continuing plastic deformation for the four yield criteria considered in this text. More explicitly we need to calculate the items listed in Table 7.2.
Table 7.2 Effective stress and uniaxial yield stress levels for the yield criteria included in the elasto-plastic computer code.
| Equation No. | Yield criterion | Stress level (effective stress) | Uniaxial (or equivalent yield stress) |
| (7.63) | Tresca | $2(J_2')^{1/2} \cos \theta$ | $\sigma_Y$ |
| (7.64) | Von Mises | $\sqrt{3} (J_2')^{1/2}$ | $\sigma_Y$ |
| (7.65) | Mohr–Coulomb | $\frac{1}{3} J_1 \sin \phi + (J_2')^{1/2} \times (\cos \theta - \sin \theta \sin \phi/\sqrt{3})$ | $c \cos \phi$ |
| (7.66) | Drucker–Prager | $a J_1 + (J_2')^{1/2}$ | $k'$ |
Whether or not plastic deformation takes place at any point is governed by its stress level as monitored by the functions in the third column of Table 7.2. For plastic flow to occur this stress level must achieve the values given in the final column of Table 7.2. For the Tresca and Von Mises criteria this value is precisely the uniaxial yield stress but for the Mohr–Coulomb and Drucker–Prager criteria it is an equivalent value defined by the stress-independent terms in (7.65) and (7.66) respectively. Note that all the values given in the final column of Table 7.2 can be functions of the hardening parameter, \kappa .
Subroutine INVAR merely computes the effective or deviatoric stress components and then evaluates the appropriate function in the third column of Table 7.2 depending on the yield criterion being employed. The choice of yield criterion is governed by the parameter NCRIT, input in subroutine INPUT, and the available options are provided below
NCRIT = 1 Tresca yield criterion
2 Von Mises
3 Mohr-Coulomb
4 Drucker-Prager
Subroutine INVAR is now presented and descriptive notes provided.
SUBROUTINE INVAR(DEVIA,LPROP,MMATS,NCRIT,PROPS,SINT3,STEFF,STEMP,INVR 1
THETA,VARJ2,YIELD) INVR 2
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DIMENSION DEVIA(4),PROPS(MMATS,7),STEMP(4) INVR 9
ROOT3=1.73205080757 INVR 10
SMEAN=(STEMP(1)+STEMP(2)+STEMP(4))/3.0 INVR 11
DEVIA(1)=STEMP(1)-SMEAN INVR 12
DEVIA(2)=STEMP(2)-SMEAN INVR 13
DEVIA(3)=STEMP(3) INVR 14
DEVIA(4)=STEMP(4)-SMEAN INVR 15
VARJ2=DEVIA(3)*DEVIA(3)+0.5*(DEVIA(1)*DEVIA(1)+DEVIA(2)*DEVIA(2) INVR 16
.+DEVIA(4)*DEVIA(4)) INVR 17
VARJ3=DEVIA(4)*(DEVIA(4)*DEVIA(4)-VARJ2) INVR 18
STEFF=SQRT(VARJ2) INVR 19
IF(STEFF.EQ.0.0) GO TO 10 INVR 20
-SINT3=-3.0*ROOT3*VARJ3/(2.0*VARJ2*STEFF) INVR 21
IF(SINT3.GT.1.0) SINT3=1.0 INVR 22
GO TO 20 INVR 23
10 SINT3=0.0 INVR 24
20 CONTINUE INVR 25
IF(SINT3.LT.-1.0) SINT3=-1.0 INVR 26
IF(SINT3.GT.1.0) SINT3=1.0 INVR 27
THETA=ASIN(SINT3)/3.0 INVR 28
GO TO (1,2,3,4) NCRIT INVR 29
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PHIRA=PROPS(LPROP,7)*0.017453292 INVR 37
SNPHI=SIN(PHIRA) INVR 38
YIELD=SMEAN*SNPHI+STEFF*(COS(THETA)-SIN(THETA)*SNPHI/ROOT3) INVR 39
RETURN INVR 40
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YIELD=2.0*COS(THETA)*STEFF INVR 31
RETURN INVR 32
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INVR 11-15 Compute the deviatoric stresses according to (7.7) with the order of the components being as indicated in (7.72).
INVR 16–17 Calculate the second deviatoric stress invariant, J_{2}^{\prime} .
INVR 18 Calculate the third deviatoric stress invariant, J_{3}' .