Abstract

The structure of the isochromatic fringe pattern in a two-dimensional photoelastic model is investigated with the phase diagram method of the theory of dynamic systems. Isochromatics are interpreted as phase paths (or level curves) of a Hamiltonian system. Possible singularities of the fringe pattern are analyzed.

© 2000 Optical Society of America

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References

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  1. H. Aben, J. Josepson, “Strange interference blots in the interferometry of inhomogeneous birefringent objects,” Appl. Opt. 36, 7172–7179 (1997).
    [CrossRef]
  2. H. Aben, L. Ainola, “Interference blots and fringe dislocations in optics of twisted birefringent media,” J. Opt. Soc. Am. A 15, 2404–2411 (1998).
    [CrossRef]
  3. Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns (Bremen, Germany 1997), W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997).
  4. J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. 33, 105–125 (1998).
    [CrossRef]
  5. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer-Verlag, Berlin, 1990).
  6. A. Kuske, G. Robertson, Photoelastic Stress Analysis (Wiley, London, 1974).
  7. D. W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations (Clarendon, New York, 1987).
  8. A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Theory of Dynamical Systems on a Plane (Israel Program of Scientific Translation, Jerusalem, 1973).
  9. S. Lefschetz, Ordinary Differential Equations: Geometric Theory (Interscience, New York, 1957).
  10. D. K. Arrowsmith, C. M. Place, Ordinary Differential Equations: A Qualitative Approach with Applications (Chapman & Hall, London, 1982).
  11. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980).
  12. S. Timoshenko, J. N. Goodier, Theory of Elasticity, 2nd ed. (McGraw-Hill, New York, 1951).
  13. A. J. Durelli, A. Shukla, “Identification of isochromatic fringes,” Exp. Mech. 23, 111–119 (1983).
    [CrossRef]
  14. H. J. Hutchinson, J. F. Nye, P. S. Salmon, “The classification of isotropic points in stress fields,” J. Struct. Mech. 11, 371–381 (1983).
    [CrossRef]
  15. F. de Lamotte, “Analyse détaillée des différents résaux de franges d’interférence en photoélasticité,” Bull. Soc. R. Sci. Liège 38, 523–532 (1969).

1998 (2)

H. Aben, L. Ainola, “Interference blots and fringe dislocations in optics of twisted birefringent media,” J. Opt. Soc. Am. A 15, 2404–2411 (1998).
[CrossRef]

J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. 33, 105–125 (1998).
[CrossRef]

1997 (1)

1983 (2)

A. J. Durelli, A. Shukla, “Identification of isochromatic fringes,” Exp. Mech. 23, 111–119 (1983).
[CrossRef]

H. J. Hutchinson, J. F. Nye, P. S. Salmon, “The classification of isotropic points in stress fields,” J. Struct. Mech. 11, 371–381 (1983).
[CrossRef]

1969 (1)

F. de Lamotte, “Analyse détaillée des différents résaux de franges d’interférence en photoélasticité,” Bull. Soc. R. Sci. Liège 38, 523–532 (1969).

Aben, H.

Ainola, L.

Andronov, A. A.

A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Theory of Dynamical Systems on a Plane (Israel Program of Scientific Translation, Jerusalem, 1973).

Arrowsmith, D. K.

D. K. Arrowsmith, C. M. Place, Ordinary Differential Equations: A Qualitative Approach with Applications (Chapman & Hall, London, 1982).

de Lamotte, F.

F. de Lamotte, “Analyse détaillée des différents résaux de franges d’interférence en photoélasticité,” Bull. Soc. R. Sci. Liège 38, 523–532 (1969).

Durelli, A. J.

A. J. Durelli, A. Shukla, “Identification of isochromatic fringes,” Exp. Mech. 23, 111–119 (1983).
[CrossRef]

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980).

Goodier, J. N.

S. Timoshenko, J. N. Goodier, Theory of Elasticity, 2nd ed. (McGraw-Hill, New York, 1951).

Gordon, I. I.

A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Theory of Dynamical Systems on a Plane (Israel Program of Scientific Translation, Jerusalem, 1973).

Huntley, J. M.

J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. 33, 105–125 (1998).
[CrossRef]

Hutchinson, H. J.

H. J. Hutchinson, J. F. Nye, P. S. Salmon, “The classification of isotropic points in stress fields,” J. Struct. Mech. 11, 371–381 (1983).
[CrossRef]

Jordan, D. W.

D. W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations (Clarendon, New York, 1987).

Josepson, J.

Kuske, A.

A. Kuske, G. Robertson, Photoelastic Stress Analysis (Wiley, London, 1974).

Lefschetz, S.

S. Lefschetz, Ordinary Differential Equations: Geometric Theory (Interscience, New York, 1957).

Leontovich, E. A.

A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Theory of Dynamical Systems on a Plane (Israel Program of Scientific Translation, Jerusalem, 1973).

Maier, A. G.

A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Theory of Dynamical Systems on a Plane (Israel Program of Scientific Translation, Jerusalem, 1973).

Nye, J. F.

H. J. Hutchinson, J. F. Nye, P. S. Salmon, “The classification of isotropic points in stress fields,” J. Struct. Mech. 11, 371–381 (1983).
[CrossRef]

Place, C. M.

D. K. Arrowsmith, C. M. Place, Ordinary Differential Equations: A Qualitative Approach with Applications (Chapman & Hall, London, 1982).

Robertson, G.

A. Kuske, G. Robertson, Photoelastic Stress Analysis (Wiley, London, 1974).

Salmon, P. S.

H. J. Hutchinson, J. F. Nye, P. S. Salmon, “The classification of isotropic points in stress fields,” J. Struct. Mech. 11, 371–381 (1983).
[CrossRef]

Shukla, A.

A. J. Durelli, A. Shukla, “Identification of isochromatic fringes,” Exp. Mech. 23, 111–119 (1983).
[CrossRef]

Smith, P.

D. W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations (Clarendon, New York, 1987).

Timoshenko, S.

S. Timoshenko, J. N. Goodier, Theory of Elasticity, 2nd ed. (McGraw-Hill, New York, 1951).

Verhulst, F.

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer-Verlag, Berlin, 1990).

Appl. Opt. (1)

Bull. Soc. R. Sci. Liège (1)

F. de Lamotte, “Analyse détaillée des différents résaux de franges d’interférence en photoélasticité,” Bull. Soc. R. Sci. Liège 38, 523–532 (1969).

Exp. Mech. (1)

A. J. Durelli, A. Shukla, “Identification of isochromatic fringes,” Exp. Mech. 23, 111–119 (1983).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Strain Anal. (1)

J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. 33, 105–125 (1998).
[CrossRef]

J. Struct. Mech. (1)

H. J. Hutchinson, J. F. Nye, P. S. Salmon, “The classification of isotropic points in stress fields,” J. Struct. Mech. 11, 371–381 (1983).
[CrossRef]

Other (9)

Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns (Bremen, Germany 1997), W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997).

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer-Verlag, Berlin, 1990).

A. Kuske, G. Robertson, Photoelastic Stress Analysis (Wiley, London, 1974).

D. W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations (Clarendon, New York, 1987).

A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Theory of Dynamical Systems on a Plane (Israel Program of Scientific Translation, Jerusalem, 1973).

S. Lefschetz, Ordinary Differential Equations: Geometric Theory (Interscience, New York, 1957).

D. K. Arrowsmith, C. M. Place, Ordinary Differential Equations: A Qualitative Approach with Applications (Chapman & Hall, London, 1982).

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980).

S. Timoshenko, J. N. Goodier, Theory of Elasticity, 2nd ed. (McGraw-Hill, New York, 1951).

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Figures (7)

Fig. 1
Fig. 1

Isochromatic fringe pattern in the xy plane: σx=x-2y+1,σy=-3x-y+2,τxy=x+y+1; center at (0, -1).

Fig. 2
Fig. 2

Isochromatic fringe pattern: σx=x-3y+4,σy=-x-2y+1,τxy=2x-y-4.

Fig. 3
Fig. 3

Isochromatic fringe pattern when the principal stress directions are constant throughout the plane: σx=x+2y+3, σy=-y+1, τxy=x-y+2.

Fig. 4
Fig. 4

Isochromatic fringe pattern: σx=x2-2y2+1, σy=y2+1, τxy=-2xy; degenerate singular point at (0, 0).

Fig. 5
Fig. 5

Isochromatic fringe pattern: σx=x2+2xy-2y2+x+2, σy=-2xy+y2-2y+3, τxy=x2-2xy-y2+2x-y+0.5; centers at (-1.680, 0.016), (-0.305, -0.476), and (0.922, 0.867); saddle points at (-0.959, -0.097) and (0.221, 0.297).

Fig. 6
Fig. 6

Isochromatic fringe pattern: σx=2xy, σy=xy+1, τxy=-0.5x2-y2+1; centers at (-1.306, 0.383), (-0.541, 0.924), (0.0), (0.541, -0.942), and (1.306, -0.383); saddle points at (-0.984, 0.696), (-0.900, -0.637), (0.900, 0.637), and (0.984, -0.696).

Fig. 7
Fig. 7

Isochromatic fringe pattern: σx=xy+1, σy=1, τxy=-y2+1; centers at (0.1), and (0, -1); degenerate singular point at (0, 0).

Equations (60)

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I=I0 sin2(Δ/2),
I=12I0(1-cos Δ),
I=I(x, y).
I(x, y)=const.
dydx=-IxIy.
x˙=f(x, y),
y˙=g(x, y),
y˙x˙=dydx,
dydx=g(x, y)f(x, y).
f(x, y)=Iy,g(x, y)=-Ix,
x˙=Iy,y˙=-Ix.
f(x, y)=0,g(x, y)=0,
Ix=0,Iy=0.
x˙=a(x-x0)+b(y-y0),
y˙=c(x-x0)+d(y-y0),
a=2Ixy(x0, y0),b=2Iy2(x0, y0),
c=-2Ix2(x0, y0),d=-2Ixy(x0, y0).
D=2Ixy2-2Ix22Iy2,
Iy=h(x, y)k(x, y),
-Ix=h(x, y)l(x, y),
x˙=k(x, y),y˙=l(x, y),
h(x, y)=0
Δ=C1(σ1-σ2)l,
σ1-σ2=[(σx-σy)2+4τxy2]1/2,
tan 2ϕ=2τxyσx-σy.
I(x, y)=12I0[1-cos Δ1(x, y)],
Δ1=C12[(σx-σy)2+4τxy2].
sin Δ1Δ1Δ1x=0,sin Δ1Δ1Δ1y=0.
sin Δ1=0,
Δ1(x, y)=(nπ)2,n=0, 1, 2,.
Δ1x=0,Δ1y=0,
D=14I02 sin2 Δ1Δ12Δ1xy2-2Δ1x22Δ1y2.
D1=2Δ1xy2-2Δ1x22Δ1y2,
(σx-σy)σxx-σyx+4τxy τxyx=0,
(σx-σy)σxy-σyy+4τxy τxyy=0.
D1=-16σxx-σyx τxyy-σxy-σyy τxyy2<0.
σx=α1x+α2 y+α3,
σy=β1x+β2 y+β3,
τxy=-β2x-α1 y+γ3,
Δ1(x, y)=ax2+2hxy+by2+2gx+2fy+c,
a=δ12+4β22,h=δ1δ2+4α1β2,
b=δ22+4α12,
g=δ1δ3-4β2γ3,f=δ2δ3-4α1γ3,
c=δ22+4α32,
δ1=α1-β1,δ2=α2-β2,δ3=α3-β3.
ax2+2hxy+by2+2gx+2fy+c=(nπ)2,
n=0, 1, 2,.
ab-h2=4(α1δ1-β2δ2)2.
ax+hy+g=0,
hx+by+f=0.
α1δ2=β2δ1=k.
tan 2ϕ=-2k(δ1x+δ2y)+2γ3δ1x+δ2y+α3.
α1δ2=β2δ1=-γ3α3=k,
tan 2ϕ=-k.
σx=μ1x2+μ2xy-(2μ1+ν1)y2+μ4x+μ5y+μ6,
σy=ν1x2+ν2xy+μ1y2+ν4x+ν5y+ν6,
τxy=-ν22x2-2μ1xy-μ22y2-ν5x-μ4x+ρ6,
Δ1(x, y)=P4(x, y),
P4(x, y)=(nπ)2,(n=0, 1, 2,).
P4(x, y)x=0,P4(x, y)y=0.

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