The objective of this thesis is to present a robust tomographic method for reconstructing 3D stress fields based on 3D photoelasticity. During last few decades, tensor field tomography has received more attention as it has important applications in various fields like medicine, engineering, physics, etc.. Stress field tomography based on 3D photoelasticity is considered one such important application in engineering and physics.However, the inverse problem of 3D photoelasticity has not yet been solved in general case; to identify arbitrary state of stress without any limitation on identifiable state of stress and sensitivity of photoelastic material. A new method capable of reconstructing 3D distribution of general second order tensor fields called load incremental approach is proposed in this thesis for solving the inverse problem of 3D photoelasticity in general case. Based on this, a robust stress field tomographic method has been developed and it has been numerically and experimentally validated.

In 3D photoelasticity, a stressed object made of photoelastic material is observed from different directions using polarized light. While passing through the object the light vectors are changed according to the state of stress along light path and output light carry integrated information on state of stress as changes in phases and amplitudes of light vector. The inverse problem of 3D photoelasticity is to reconstruct the 3D distribution of stress tensor based on these light vector information.

The major difficulties in this inverse problem are its nonlinear and ill-posed natures. Both experimental and numerical techniques have been proposed to solve this nonlinear and ill-posed inverse problem. The experimental approaches degenerates the 3D problem into a series of 2D problems by obtaining photoelastic observations in high spatial resolution along light path, optically or mechanically slicing the object. These methods are either destructive or have practical problems. Instead of solving the original nonlinear and ill-posed inverse problem, most of the analytical approaches are based on a linear approximation valid only for weakly birefringent materials （i.e., less sensitive photoelastic materials）. Less sensitive photoelastic materials have almost linear solution space in a considerable neighborhood of stress free state. Taking this advantage, most of the approaches to solve this inverse problem approximate the relation between output light vectors and stress components to be linear and thereby avoid the difficulties due to nonlinearity and ill-posedness. Since this linear approximation is valid only in limited neighborhood of the stress free state, all the methods based on this linear approximation have an upper limit on identifiable state of stress. Only the analytical methods based on this simplified solution space and experimental based treatments are currently used in practice. All the approaches to solve the general inverse problem of 3D photoelasticity are either limited to theory or numerical simulations.

We developed a load incremental approach for stress component identification as the first attempt to solve this inverse problem in general. As the name suggest, this method is based on a linear relation between increment of light vector and increment of stress components due to small changes in external load. This linear relation is obtained by considering the Taylor expansion of output light vector with respect to externally applied load and neglecting the higher order terms. Unlike the conventional methods, load incremental approach does not use this linear relation to simplify the solution space. Instead, load incremental approach uses the Newton-Raphson algorithm based on the linear relation between increment of light vector and increment of stress component to find the correct solution in the nonlinear solution space. However, the Newton-Raphson algorithm cannot be directly used to find the solution for a given set of photoelastic observations since the solution space geometry is too complicated and the inverse problem is ill-posed. Load incremental approach adopts a suitable experimental procedure to treat the ill-posed nature. Starting from stress free state externally applied load on the object is increased in small steps and photoelastic observations are made at each step. Through this experimental procedure, the nonlinear solution space between output light and stress components is chopped into small segments so that each segment has a unique solution to the photoelastic observations made at that step. Since each these segments are still nonlinear, Newton-Raphson algorithm based on the linear relation between increment of output light and increment of stress components is used to identify the solution after each load increment. As long as the load increments given in the experiment are sufficiently small to chop the solution space in to small segments with a unique solution, the state of stress after each load increment can be successfully identified. State of stress up to arbitrary state can be identified in stepwise by increasing the external load in sufficiently small steps.

Since the equilibrium condition is not used, the solutions of load incremental approach for stress component identification are not necessarily stress fields. Therefore, this method can be considered as a 3D tomography of general second order tensor fields. However, this method is named as load incremental approach for stress component identification, as it is used for identifying distribution of stress components. This method is numerically validated and found that state of stress up to arbitrary state can be reconstructed as long as output light measurements are error free and load increments are sufficiently small. However, this method is highly sensitivity to unavoidable errors in photoelastic measurements like heat noise in CCD chips because each component have freedom to independently change according to the photoelastic observations.

With the aim of reducing the sensitivity to measurement errors in above explained stress component identification method, both the equilibrium condition and linear elasticity are introduced to the inverse problem of 3D photoelasticity. The introduction of equilibrium condition transforms the former 3D tomographic method for general second order tensor fields to 3D stress field tomographic method. This stress field tomographic method should be less sensitive to measurement errors since stress components are bounded together by equilibrium condition. When the equilibrium condition and linear elasticity are introduced, the only unknown is boundary conditions. Therefore, this stress field tomographic method is named boundary condition identification.

In boundary condition identification, the unknown complicated boundary conditions are modeled with independent set of boundary condition modes and state of stress due to unit amplitude of each mode is found using a numerical method like FEM. The new inverse problem in boundary condition identification is to identify the amplitudes of boundary condition modes based on photoelastic measurements. It is shown that this inverse problem is also nonlinear and ill-posed. This nonlinear and ill-posed inverse problem is also solved using the load incremental approach. Through numerical and experimental validations, it has been shown that the load incremental approach for boundary condition identification is less sensitive to unavoidable errors in photoelastic measurements.

The major advantages of boundary condition identification are drastic reduction of number of unknowns and the robustness against measurement errors. Through modeling unknown boundary conditions as an independent set of modes, the number of unknowns in the original inverse problem is drastically reduced; millions of unknown stress components are reduced to few number of boundary condition amplitudes. As an immediate consequence, boundary condition identification needs fairly less computational power. A redundant set of equations can be obtained since the object can be observed using large number of light rays while the number of unknowns is a few. These redundant set of equations have different solution space geometries since different light rays passes through different portions of the object from different directions. Selecting only the equations with favorable solution space geometries for Newton-Raphson method, most of the numerical instabilities due to complicated solution space geometry can be avoided. The number of modes used in boundary condition identification has to be limited to the lowest 20 or lesser since the photoelastic observations do not carry accurate information on higher modes due to unavoidable measurement errors. As a result, boundary condition identification cannot identify the minor details of stress field corresponding to higher modes. However, this cannot be considered as a disadvantage of this method since this loss of minor details is due to experimental limitations. Some disadvantages of this method are necessity of prior knowledge on possible boundary conditions for selection of suitable set of modes and necessity of a numerical method to calculate the state of stress due to each boundary condition modes in models with huge degree of freedom.

本論文の目的は，光弾性の原理を用いて物体内部での応力分布の全視野計測を非破壊で行うこと，すなわち三次元光弾性に基づく応力場トモグラフィーの手法開発を行うことである．光弾性現象とは，高分子・ガラスなど光弾性物質と呼ばれる材料において材料内部に応力が生じると，その応力場と線形関係で結ばれる複屈折場が発生する現象である．光弾性の原理を用いた応力解析では，材料を透過する偏光をプローブとして複屈折場に関する情報を計測し，これを入力とした逆解析により材料内部の応力場を同定する．特に，二次元応力場（平面ひずみ・平面応力状態）が仮定される場合には，材料内部の応力場に対応する光弾性効果を反映する透過光と材料内部の応力状態とが線形関係で結ばれるため，この逆解析は容易なものである．

有限要素法などの数値解析手法が開発される前には，1920年代に手法として確立された二次元光弾性による応力場計測が複雑な形状の物体内での応力分布を知る唯一の方法であり，二次元光弾性による模型実験は応用力学の主要なテーマであった．以来，二次元光弾性の自然な拡張として，三次元光弾性の問題への取り組みが80年以上にわたって続けられてきた．これらの既往研究は，二次元光弾性に落とす方法と支配方程式の線形近似に基づく方法とに大別される．前者の代表例は，供試体を物理的／光学的にスライスして二次元問題に落として解析する応力凍結法／散乱光光弾性法であり，後者の代表例は透過光と応力状態との関係が線形近似できるような条件を満たす応力場のみを扱う Integrated Photoelasticity である．これらの手法には非破壊・全視野といった要件が満たされない，あるいは軸対称などの特殊な対称性をもつ応力場しか取り扱うことができないといった問題があり，任意の応力状態を非破壊で全視野計測するという三次元光弾性の本来のゴールからはかけ離れた結果しか得られていない．結果として，2000年の時点で三次元光弾性による応力場計測に成功した例はなかった．

このような背景の下，論文提出者は三次元光弾性の問題に取り組んだ．論文の第1章では，三次元光弾性に関する既往研究に関する記述とあわせて，問題の難しさの源を非線型性（Nonlinearity）・非適切性（Ill-posedness）・不安定性の3つであることを示した．特に非線型性に関して，透過光と材料内部の応力状態とが，非可換な行列の積の形で表される非線型関係で結ばれること，この非線形性の扱いが三次元光弾性による応力場計測の鍵となることを明示した．第2章では，三次元光弾性の非線型性を扱う手法としてのLoad Incremental Approach （荷重増分型解法）を提案している．通常の数値解析でいう非線型解析と異なり，本論文で提案されている三次元光弾性の増分型解法では荷重増分は実験データにより固定されている．この点をふまえ，実験と解析の両方に配慮した逆解析手法が提案されている．ただし，この時点では逆解析で同定されるものはつりあい条件を満たしていない．つまり，「三次元的な応力場」ではなく，「三次元空間での2階のテンソル場」である．そのため，逆解析の結果は計測誤差に非常に敏感である．第3章で提案される手法は，この「計測誤差に対する敏感さ」の低減を目的としたものであり，上記の「2階のテンソル場」につりあい条件を課して，「応力場」としたものを同定する逆解析手法を提案している．個々の応力成分につりあい条件を課すことにより，計測誤差に対する敏感さは十分に抑制され，実際の計測データを用いた逆解析が可能な安定性が得られることが示された．第4章では三次元光弾性による応力場逆解析の数値解析例と，計測誤差，入射光の波長，光弾性定数，干渉ノイズなど，さまざまな誤差要因が逆解析結果に与える影響の定量的評価の結果を示している．また，この章では実際の計測結果を解析する際に逆解析を不安定にする要因とそれを回避する方法を示しており，第3章で提案された解析手法を計測に結び付け，実用可能な手法に改善している．第5章では実験手法について説明している．本論文で提案されている解析手法では，従来型の光弾性実験で重要視される等傾線・等色線はあまり意味を持たない．むしろ各点での透過光の楕円偏光を計測する必要がある．また，多数の方向から供試体を計測する必要がある．また，制御された荷重増分を段階的に与える必要があり，1回の実験で多数の計測が必要である．この要請を満たすために，偏光板の向き，計測方向，撮影のタイミングを自動制御する計測装置（三次元応力場スキャナー）を開発した．この装置を用いた実験の結果と第4章までに述べた解析結果を比較し，逆解析結果の妥当性の検証も行われている．この結果は，特別な制限なしに三次元光弾性を用いて応力場を同定した世界初の事例である．第6章では本研究で得られた知見と今後の課題・発展の方向がまとめられている．

以上，本研究は三次元光弾性の問題に対して安定な逆解析手法と実験手法をあわせて提案し，およそ1世紀にわたって未解決であった難問に答えを与える独創的な研究成果と評価できる．よって本論文は博士（工学）の学位請求論文として合格と認められる．