In mechanics, a displacement field is the assignment of displacement vectors for all points in a region or body that are displaced from one state to another.[1][2] A displacement vector specifies the position of a point or a particle in reference to an origin or to a previous position. For example, a displacement field may be used to describe the effects of deformation on a solid body.
Before considering displacement, the state before deformation must be defined. It is a state in which the coordinates of all points are known and described by the function: R → → --> 0 : Ω Ω --> → → --> P {\displaystyle {\vec {R}}_{0}:\Omega \to P} where
Most often it is a state of the body in which no forces are applied.
Then given any other state of this body in which coordinates of all its points are described as R → → --> 1 {\displaystyle {\vec {R}}_{1}} the displacement field is the difference between two body states: u → → --> = R → → --> 1 − − --> R → → --> 0 {\displaystyle {\vec {u}}={\vec {R}}_{1}-{\vec {R}}_{0}} where u → → --> {\displaystyle {\vec {u}}} is a displacement field, which for each point of the body specifies a displacement vector.
The displacement of a body has two components: a rigid-body displacement and a deformation.
A change in the configuration of a continuum body can be described by a displacement field. A displacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. The distance between any two particles changes if and only if deformation has occurred. If displacement occurs without deformation, then it is a rigid-body displacement.
Two types of displacement gradient tensor may be defined, following the Lagrangian and Eulerian specifications. The displacement of particles indexed by variable i may be expressed as follows. The vector joining the positions of a particle in the undeformed configuration P i {\displaystyle P_{i}} and deformed configuration p i {\displaystyle p_{i}} is called the displacement vector, p i − − --> P i {\displaystyle p_{i}-P_{i}} , denoted u i {\displaystyle u_{i}} or U i {\displaystyle U_{i}} below.
Using X {\displaystyle \mathbf {X} } in place of P i {\displaystyle P_{i}} and x {\displaystyle \mathbf {x} } in place of p i {\displaystyle p_{i}\,\!} , both of which are vectors from the origin of the coordinate system to each respective point, we have the Lagrangian description of the displacement vector: u ( X , t ) = u i e i {\displaystyle \mathbf {u} (\mathbf {X} ,t)=u_{i}\mathbf {e} _{i}} where e i {\displaystyle \mathbf {e} _{i}} are the orthonormal unit vectors that define the basis of the spatial (lab frame) coordinate system.
Expressed in terms of the material coordinates, i.e. u {\displaystyle \mathbf {u} } as a function of X {\displaystyle \mathbf {X} } , the displacement field is: u ( X , t ) = b ( t ) + x ( X , t ) − − --> X or u i = α α --> i J b J + x i − − --> α α --> i J X J {\displaystyle \mathbf {u} (\mathbf {X} ,t)=\mathbf {b} (t)+\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \qquad {\text{or}}\qquad u_{i}=\alpha _{iJ}b_{J}+x_{i}-\alpha _{iJ}X_{J}} where b ( t ) {\displaystyle \mathbf {b} (t)} is the displacement vector representing rigid-body translation.
The partial derivative of the displacement vector with respect to the material coordinates yields the material displacement gradient tensor ∇ ∇ --> X u {\displaystyle \nabla _{\mathbf {X} }\mathbf {u} \,\!} . Thus we have, ∇ ∇ --> X u = ∇ ∇ --> X x − − --> R = F − − --> R or ∂ ∂ --> u i ∂ ∂ --> X K = ∂ ∂ --> x i ∂ ∂ --> X K − − --> α α --> i K = F i K − − --> α α --> i K {\displaystyle \nabla _{\mathbf {X} }\mathbf {u} =\nabla _{\mathbf {X} }\mathbf {x} -\mathbf {R} =\mathbf {F} -\mathbf {R} \qquad {\text{or}}\qquad {\frac {\partial u_{i}}{\partial X_{K}}}={\frac {\partial x_{i}}{\partial X_{K}}}-\alpha _{iK}=F_{iK}-\alpha _{iK}} where F {\displaystyle \mathbf {F} } is the material deformation gradient tensor and R {\displaystyle \mathbf {R} } is a rotation.
In the Eulerian description, the vector extending from a particle P {\displaystyle P} in the undeformed configuration to its location in the deformed configuration is called the displacement vector: U ( x , t ) = U J E J {\displaystyle \mathbf {U} (\mathbf {x} ,t)=U_{J}\mathbf {E} _{J}} where E i {\displaystyle \mathbf {E} _{i}} are the unit vectors that define the basis of the material (body-frame) coordinate system.
Expressed in terms of spatial coordinates, i.e. U {\displaystyle \mathbf {U} } as a function of x {\displaystyle \mathbf {x} } , the displacement field is: U ( x , t ) = b ( t ) + x − − --> X ( x , t ) or U J = b J + α α --> J i x i − − --> X J {\displaystyle \mathbf {U} (\mathbf {x} ,t)=\mathbf {b} (t)+\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\qquad {\text{or}}\qquad U_{J}=b_{J}+\alpha _{Ji}x_{i}-X_{J}}
The spatial derivative, i.e., the partial derivative of the displacement vector with respect to the spatial coordinates, yields the spatial displacement gradient tensor ∇ ∇ --> x U {\displaystyle \nabla _{\mathbf {x} }\mathbf {U} \,\!} . Thus we have, ∇ ∇ --> x U = R T − − --> ∇ ∇ --> x X = R T − − --> F − − --> 1 or ∂ ∂ --> U J ∂ ∂ --> x k = α α --> J k − − --> ∂ ∂ --> X J ∂ ∂ --> x k = α α --> J k − − --> F J k − − --> 1 , {\displaystyle \nabla _{\mathbf {x} }\mathbf {U} =\mathbf {R} ^{T}-\nabla _{\mathbf {x} }\mathbf {X} =\mathbf {R} ^{T}-\mathbf {F} ^{-1}\qquad {\text{or}}\qquad {\frac {\partial U_{J}}{\partial x_{k}}}=\alpha _{Jk}-{\frac {\partial X_{J}}{\partial x_{k}}}=\alpha _{Jk}-F_{Jk}^{-1}\,,} where F − − --> 1 = H {\displaystyle \mathbf {F} ^{-1}=\mathbf {H} } is the spatial deformation gradient tensor.
α α --> J i {\displaystyle \alpha _{Ji}} are the direction cosines between the material and spatial coordinate systems with unit vectors E J {\displaystyle \mathbf {E} _{J}} and e i {\displaystyle \mathbf {e} _{i}\,\!} , respectively. Thus E J ⋅ ⋅ --> e i = α α --> J i = α α --> i J {\displaystyle \mathbf {E} _{J}\cdot \mathbf {e} _{i}=\alpha _{Ji}=\alpha _{iJ}}
The relationship between u i {\displaystyle u_{i}} and U J {\displaystyle U_{J}} is then given by u i = α α --> i J U J or U J = α α --> J i u i {\displaystyle u_{i}=\alpha _{iJ}U_{J}\qquad {\text{or}}\qquad U_{J}=\alpha _{Ji}u_{i}}
Knowing that e i = α α --> i J E J {\displaystyle \mathbf {e} _{i}=\alpha _{iJ}\mathbf {E} _{J}} then u ( X , t ) = u i e i = u i ( α α --> i J E J ) = U J E J = U ( x , t ) {\displaystyle \mathbf {u} (\mathbf {X} ,t)=u_{i}\mathbf {e} _{i}=u_{i}(\alpha _{iJ}\mathbf {E} _{J})=U_{J}\mathbf {E} _{J}=\mathbf {U} (\mathbf {x} ,t)}
It is common to superimpose the coordinate systems for the deformed and undeformed configurations, which results in b = 0 {\displaystyle \mathbf {b} =0\,\!} , and the direction cosines become Kronecker deltas, i.e., E J ⋅ ⋅ --> e i = δ δ --> J i = δ δ --> i J {\displaystyle \mathbf {E} _{J}\cdot \mathbf {e} _{i}=\delta _{Ji}=\delta _{iJ}}
Thus in material (undeformed) coordinates, the displacement may be expressed as: u ( X , t ) = x ( X , t ) − − --> X or u i = x i − − --> δ δ --> i J X J {\displaystyle \mathbf {u} (\mathbf {X} ,t)=\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \qquad {\text{or}}\qquad u_{i}=x_{i}-\delta _{iJ}X_{J}}
And in spatial (deformed) coordinates, the displacement may be expressed as: U ( x , t ) = x − − --> X ( x , t ) or U J = δ δ --> J i x i − − --> X J {\displaystyle \mathbf {U} (\mathbf {x} ,t)=\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\qquad {\text{or}}\qquad U_{J}=\delta _{Ji}x_{i}-X_{J}}
هذه المقالة يتيمة إذ تصل إليها مقالات أخرى قليلة جدًا. فضلًا، ساعد بإضافة وصلة إليها في مقالات متعلقة بها. (فبراير 2017) إرنست باش (بالألمانية: Ernst Bach) معلومات شخصية الميلاد 20 سبتمبر 1899(1899-09-20)ميندلهايم الوفاة 26 سبتمبر 1944 (45 سنة)ماربورغ الجنسية ألمانيا عضو في كتيبة العا…
مخطط دائرة لمحرك حثي ذي ثغرة هوائية ثابتة وضع التصور الأول لمخطط الدائرة بواسطة هيلاند عام 1894 وبيهرند عام 1895، وهو تمثيل بياني لأداء آلة كهربية.[1][2][3] ويرسم المخطط بمعلومية جهد وتيار الدخل للآلة.[4] يمكن رسم مخطط الدائرة للمنوبات والمحركات التزامنية والمحولات…
Daio Wasabi Stayen Stadium Bijnaam Hel van Stayen Plaats Sint-Truiden, België Capaciteit 14.600 Geopend 1927 Bespelers Sint-Truidense VV Portaal Sport Stayen, vanwege sponsorredenen ook bekend als Daio Wasabi Stayen Stadium, is een voetbalstadion in de Belgische stad Sint-Truiden, in de provincie Limburg, en het thuisstadion van voetbalclub Sint-Truidense VV of afgekort: STVV. Het stadion werd gebouwd in 1927. De naam Stayen is Sint-Truidens dialect voor Staden, een oude wijk van d…
El Reino de Hungría entre 1526 y 1867 existió como un estado fuera del Sacro Imperio Romano,[1] pero como parte de las tierras de la Monarquía de los Habsburgo que se convirtió en el Imperio Austriaco en 1804. Tras la Batalla de Mohács en 1526, el país fue gobernado por dos reyes coronados (Juan I y Fernando I). Al principio, el territorio exacto bajo el dominio de los Habsburgo era objeto de disputa porque ambos gobernantes reclamaban todo el reino. Este período de inestabilidad d…
Italian footballer This article uses bare URLs, which are uninformative and vulnerable to link rot. Please consider converting them to full citations to ensure the article remains verifiable and maintains a consistent citation style. Several templates and tools are available to assist in formatting, such as reFill (documentation) and Citation bot (documentation). (August 2022) (Learn how and when to remove this template message) This biography of a living person needs additional citations for ve…
1976 studio album by Shaun CassidyShaun CassidyStudio album by Shaun CassidyReleased1976 (Europe, Australia)June 1, 1977 (U.S.)GenrePopLength29:16LabelWarnerProducerMichael LloydShaun Cassidy chronology Shaun Cassidy(1976) Born Late(1977) Shaun Cassidy is the debut solo album by American singer Shaun Cassidy. The eponymous album was first released in 1976 in Europe and Australia, where he had top-ten hits with Morning Girl and That's Rock 'n' Roll.[1] It was not until the release…
Jajang C. NoerJajang menerima Piala Citra mewakili Mathias Muchus sebagai Pemeran Pendukung Pria Terbaik Festival Film Indonesia 2015LahirLidia Djunita Pamoentjak28 Juni 1952 (umur 71)Paris, PrancisNama lainJajang C. NoerPekerjaanAktrisTahun aktif1972—sekarangSuami/istriArifin C. Noer (m. 1978; meninggal 1995)Anak2, termasuk Nazyra C. NoerOrang tuaNazir Datuk Pamoentjak (ayah)KeluargaJusuf Kalla (besan) Lidia Djunita Pamoentjak…
Mexican footballer (born 1993) For the Mexican goalkeeper born in 1981, see José de Jesús Corona. In this Spanish name, the first or paternal surname is Corona and the second or maternal family name is Ruíz. Jesús Manuel Corona Corona with Porto in 2018Personal informationFull name Jesús Manuel Corona Ruíz[1]Date of birth (1993-01-06) 6 January 1993 (age 30)[1]Place of birth Hermosillo, Sonora, MexicoHeight 1.73 m (5 ft 8 in)[2]Position(s…
Finnish painter and sculptor This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (December 2014) (Learn how and when to remove this template message) Alpo JaakolaJaakola in 1958.BornAlpo Sakari Jaakola1 April 1929Loimaa, FinlandDied27 February 1997 (aged 67)Loimaa, FinlandNationalityFinnishKnown forPainting, SculptureMovementSurrealismAwardsPro Finlandia, 1974 A…
World Grand Prix 1998 Turnierstatus Turnierart Ranglistenturnier Veranstalter PDC Austragungsort Casino Rooms,Rochester (Kent) Turnierdaten Titelverteidiger - Teilnehmer 16 Eröffnungsspiel 14. Oktober 1998 Endspiel 18. Oktober 1998 Preisgeld (gesamt) £ 38.000 Preisgeld (Sieger) £ 9.000 Sieger England Phil Taylor Statistiken ← - 1999 → Der World Grand Prix 1998 war ein Major-Turnier im Dartsport und wurde vom 14. bis zum 18. Oktober 1998 zum ersten Mal von der Prof…
كوسكينو تقسيم إداري البلد اليونان [1] إحداثيات 36°23′00″N 28°13′00″E / 36.38333333°N 28.21666667°E / 36.38333333; 28.21666667 السكان التعداد السكاني 3175 (إحصاء السكان) (2011) معلومات أخرى التوقيت ت ع م+02:00 (توقيت قياسي)، وت ع م+03:00 (توقيت صيفي) الرمز الجغرافي 6621621 تعديل مصدري…
Дія агрегаційних феромонів Феромони (грец. φέρω — «нести» + ορμόνη — «спонукати, викликати») — біологічно активні речовини, продукти зовнішньої секреції, які виділяють тварини. Є засобами сигналізації між особинами певної популяції (сім'ї). Розрізняють статеві фе…
لمعانٍ أخرى، طالع قناة ديزني (توضيح). هذه المقالة عن قناة ديزني. لتصفح عناوين مشابهة، انظر ديزني (توضيح). قناة ديزني شعار القناة معلومات عامة النوع قناة أطفال القطاع قناة خاصة المالك والت ديزني تاريخ التأسيس 18 أبريل 1983 البلد الولايات المتحدة القنوات الأخوات ديزن…
American politician Lane KirklandBornJoseph Lane Kirkland(1922-03-12)March 12, 1922Camden, South Carolina, USDiedAugust 14, 1999(1999-08-14) (aged 77)Washington, D.C., USOccupationLabor leaderPolitical partyDemocratic Joseph Lane Kirkland (March 12, 1922 – August 14, 1999) was an American labor union leader who served as President of the AFL–CIO from 1979 to 1995. Life and career Kirkland was born in Camden, South Carolina, the son of Louise Beardsley (Richardson) and Randolph Withers K…
第8机械化步兵师제8기계화보병사단第八机械化步兵师徽章存在時期1949年6月20日至今國家或地區 韩国部門韩国陆军種類机械化步兵功能突击部队規模师駐軍/總部京畿道杨州市參與戰役朝鲜战争 第8机械化步兵师(韩语:제8기계화보병사단/第八機動師團)是韓國陸軍的一个战斗单位。 历史 该师最初由第10、16和21团组成。[1]第16步兵团最初于1948年10月28日在马山组建…
В Википедии есть статьи о других людях с такой фамилией, см. Михайловский; Михайловский, Николай. Николай Константинович Михайловский Портрет Н. К. Михайловского работы Н. А. Ярошенко (фрагмент) Псевдонимы Гроньяр, Посторонний, Профан Дата рождения 15 (27) ноября 1842…
2010 film by Irfan Kamal Thanks MaaFilm posterDirected byIrfan KamalWritten byIrfan KamalVishal Vijay KumarProduced byQuantum FilmsStarringMaster Shams PatelMaster SalmanMaster FayaazBaby AlmasRavi MahashabdeMaster JafferBaby SakshiBarry JohnCinematographyAjayan VincentEdited byAmit SaxenaMusic byRanjit BarotDistributed bySPE Films IndiaRelease date 5 March 2010 (2010-03-05) CountryIndiaLanguageHindi Thanks Maa (transl. Thanks mother) is an India Hindi language drama film, d…
Gemeinden des Kantons Freiburg Der Kanton Freiburg umfasst 126 politische Gemeinden (Stand: Januar 2022).[1] Hauptort ist Freiburg (Fribourg). Bezirkshauptorte sind fett hervorgehoben. Siehe auch: Bezirke des Kantons Fribourg und Gemeindefusionen im Kanton Freiburg Inhaltsverzeichnis 1 Liste der Gemeinden 2 Veränderungen im Gemeindebestand 2.1 Fusionen 2.2 Nicht vollzogen oder verschoben 3 Einzelnachweise 4 Weblinks Liste der Gemeinden Inhaltsverzeichnis A B C D E F G H I J K L M N…
Japanese symbol for beginner drivers Wakaba mark Shoshinsha mark displayed on a Suzuki Alto Lapin The shoshinsha mark (初心者マーク) or Wakaba mark (若葉マーク), officially Beginner Drivers' Sign (初心運転者標識, Shoshin Untensha Hyōshiki), is a green and yellow V-shaped symbol that new drivers in Japan must display at the designated places at the front and the rear of their cars for one year after they obtain a standard driver's license. Drivers who consider themselves beginn…
Historical/restoration societyThe logo of The Preservation Society of Chapel Hill, as it appears on its brochures and advertising.[1] The Preservation Society of Chapel Hill (PSCH) is a tax-exempt, nonprofit organization located in Chapel Hill, North Carolina. Founded in 1972 by Ida Friday (wife of William Friday) and Georgia Kyser (wife of Kay Kyser), the society works to save and restore Chapel Hill's natural and man-made, historic artifacts. PSCH is heavily involved in the preservatio…
Lokasi Pengunjung: 13.58.98.167