Lý Thuyế Thuyết Đàn Hồ Hồi MỞ ĐẦU ĐẦU Tài liệ liệu đượ c biên soạ soạn nhằ nhằm mục đích cung cấ cấp cho họ học viên nhữ những khái niệ niệm cơ sở của lý thuyế thuyết phân tích các kế kết cấ cấu công trình nói chung và kế kết cấ cấu tàu thủ thủy nói riêng. Trong tài liệ liệu, các nguyên tắc cơ cơ bbản sau đây cố cố gắng thể thể hiệ hiện: Nguyên tắ tắc đầu đầu tiên là bả bảo đảm đảm tính nhấ nhất quán, tính logic trong toàn định” mơ ở cho mộ bộ tài liệ liệu đượ c trình bày, tránh dùng bấ bất cứ c ứ một “giả “giả định” mơ hhồ nào làm cơ cơ ssở cho một phươ ph ươ ng ng pháp tính nào đó. Mọ Mọi giả giả thiế thiết có tính gầ gần đúng luôn đượ c đi kèm bở bở i nhữ những lờ lờ i giả giải thích về về tính phù hợ hợ p củ của chúng. Nguyên tắ tắc thứ thứ hai là chấ chấp nhậ nhận việ việc giả giải thích thêm khi cầ cần thiế thiết. Nguyên tắ tắc này xuấ xuất phát từ từ đặc đặc điểm củ c ủa quá trình nhậ nhận tcầ tcần có sự sự lặp lạ l ại. Tính chấ chất lặ l ặp lạ l ại là mộ một trong các công cụ cụ quan trọ trọng củ c ủa quá trình nhậ nhận thứ thức. Việ Việc giả giải thích thêm có mụ mục đích tránh hiể hiểu vấn đề một cách sai lệ lệch, tránh ngộ ngộ nhậ nhận. Nguyên tắ tắc thứ thứ 3 là các chủ chủ đề chính củ của các nộ nội dung đượ c trình bày luôn đượ c tóm lượ lượ c, c, nhờ nhờ đđó ngườ ngườ i đọc đọc có đượ c nhậ đầy đủ nhữ đọc qua. nhận thứ thức tổ tổng thể thể vớ i đầy những nộ nội dung chính yế yếu về về nhữ những vấ vấn đề đã đọc Lý Thuyế Thuyết Đàn Hồ Hồi giả giải các bài toán liên quan đến đến việ việc xác định định ứng suấ suất và biế biến dạ dạng, xuấ xuất hiệ hiện trong vậ vật thể thể đàn hồ hồi, dướ dướ i tác dụ dụng củ của lự lực ngoài. Đây cũ cũng chính là vấ vấn đề đã giả giải quyế quyết trong môn họ học sức bề bền vậ vật liệ liệu. Tuy nhiên, trong giáo trình sứ sức bề bền vậ vật liệ liệu, nhiề nhiều giả giả thiế thiết tính toán cụ cụ thể thể khác nhau đã đượ c sử s ử dụng, nhằ nhằm thu đượ c nhữ những lờ i giả giải gần đúng cho các bài toán riêng biệ biệt và do đó, chỉ chỉ áp dụng đượ c cho chính các bài toán này thôi. Lý Thuyế Thuyết Đàn Hồ Hồi đặt đặt ra mụ mục tiêu là tìm nhữ những lờ lờ i giả giải chính xác, dựa trên các giả giả thiế thiết chung về về tính chấ chất củ của vậ vật thể thể khả khảo sát mà không phụ phụ thuộ thuộc gì vào hình dáng vậ vật thể thể cũng như như tính riêng biệ biệt củ của tả tải trọ trọng tác dụ dụng lên vậ vật thể thể. . . Vật thể th ể khả khảo sát trong Lý Thuyế Thuyết Đàn Hồi đượ c giả giả thiế thiết là có tính liên tụ t ục, tứ t ức, vậ v ật thể th ể khả khảo sát luôn điền đầy đầy không gian mà nó chiế chiếm chỗ chỗ, trướ tr ướ c cũ cũng như như sau khi bị bị biế biến dạ d ạng. Ta coi là trong mỗ mỗi thể th ể đến đâu, cũ tích bấ bất kỳ k ỳ, dù nhỏ nhỏ đến cũng chứ chứa vô số số các phân tử tử và tác dụ dụng củ của phầ ph ần vậ v ật thể th ể bị cắt bỏ b ỏ lên phầ phần khả khảo sát có thể thể đánh giá bằ bằng tr ị số trung bình c ủa sự thay đổ i lự c t ươ giữa các phầ phần vậ vật thể thể nằm về về ươ ng ng tác giữ hai phía củ của mặt cắt. Các chuyể chuyển vị là nhữ những hàm liên tụ tục của toạ toạ độ các điểm. Tính chấ chất liên tụ tục cho đại lượ phép ứng dụ dụng giả giải tích các đại l ượ ng ng vô cùng bé vào việ việc nghiên cứ cứu biế bi ến dạ d ạng củ của vậ v ật thể th ể đàn hồ h ồi. Sai số liên quan đến đến việ việc sử sử dụng tính chấ chất nói trên là có thể thể bỏ qua trong các bài toán thự thực tế tế, vì nó chỉ chỉ đáng kể khi xác định định ứng suấ suất trên các diệ diện tích vớ vớ i kích thướ thướ c cỡ cỡ ccủa khoả khoảng cách phân tử tử và khi xác định định các chuyể chuyển vị vị của các điểm mà khoả khoảng cách giữ giữa chúng cũ cũng vào cỡ cỡ kho khoảảng cách giữ giữa các phân tử tử. định luậ ĩ nh động lự Ngoài ra, cũ cũng còn phả phải giả gi ả thiế thiết rằ r ằng, có thể thể áp dụ d ụng các định luật củ c ủa củ c ủa t ĩ nh học và v à động l ực học cho các phân tố tố nhỏ nhỏ tuỳ tuỳ ý, từ từ vật thể thể khả khảo sát. Các vậ vật thể thể đàn hồ hồi, là đối đối tượ tượ ng ng nghiên cứ cứu củ của môn họ học, còn có nhiề nhiều tính chấ chất khác mà ta sẽ sẽ đề đến sau này khi thiế cập đến thiết lậ lập các phươ phươ ng ng trình cơ cơ bbản củ của Lý thuyế thuyết đàn hồ hồi.
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Lý Thuyế Thuyết Đàn Hồ Hồi Chươ Chươ ng ng I TOÁN HỌ HỌC CƠ CƠ S Ở Các ứng dụ dụng củ của Lý Thuyế Thuyết Đàn Hồ Hồi đòi hỏ hỏi sự sự hiể hiểu biế biết về về nhiề nhiều l ĩ ĩ nh nh vự vực toán họ học khác nhau. ở ứng dụ đại lượ đổi (các biế Bản thân Lý Thuyế Thuyết Đàn Hồ Hồi đượ c xây dự dựng trên cơ cơ ssở ứ dụng nhiề nhiều đại lượ ng ng biế biến đổi biến) khác nhau, trong đó có các biế biến vô hướ hướ ng, ng, các vector, các trườ trườ ng ng tensor và phả phải sử sử dụng nhiề nhiều đến đến các phép tính tensor. Vậ Vận dụ dụng các nguyên lý củ của Cơ Cơ H Học Các Môi Trườ Trườ ng ng Liên Tụ Tục, Lý Thuyế Thuyết Đàn Hồ Hồi đượ c xây dựng dướ dướ i dạ dạng mộ một tậ tập hợ hợ p các phươ phươ ng ng trình đạo đạo hàm riêng mà lờ lờ i giả giải củ của chúng đượ c tìm trong các miề miền trùng vớ vớ i không gian mà vậ vật thể thể khả khảo sát chiế chiếm chỗ chỗ. Để giả giải các phươ phươ ng ng trình này, trong nhiề nhiều kỹ kỹ thuậ thuật khác nhau, thườ thườ ng ng phả phải dùng đến đến phươ phươ ng ng pháp Fourier, các kỹ kỹ thuậ thuật biế biến phân, các phép biế biến đổi đổi tích phân, các biế biến số số phứ phức, lý thuyế thuyết thế thế năng, phươ phươ ng ng pháp sai phân hữ hữu hạ hạn, phươ phươ ng ng pháp phầ phần tử tử hữu ở của Lý Thuyế hạn, phầ phần tử tử biên,... Vì thế thế cho nên, để nắm đượ c các cơ cơ ssở c Thuyết Đàn Hồ Hồi, cầ cần có mộ một că căn bả bản toán họ học tươ tươ ng ng xứ xứng. Mụ Mục đích củ của chươ chươ ng ng này là cung cấ cấp cho ngườ ngườ i đọc đọc mộ một số số chuyên đề toán họ học thiế thiết yế yếu, phụ phục vụ vụ cho việ việc l ĩ ĩ nh nh hộ hội các cơ cơ ssở c ở của Lý Thuyế Thuyết Đàn Hồ Hồi. Các chuyên đề toán họ học khác sẽ sẽ đượ c trình bày tóm lượ ở các chươ lượ c ở các chươ ng, ng, nơ nơ i mà chúng đượ c sử sử dụng. §1.1 Các định ngh ĩ a về vô hướ ng, ng, vector, ma tr ận và tensor.
Trong Lý Thuyế Thuyết Đàn Hồ H ồi thườ th ườ ng ng sử sử dụng nhiề nhiều loạ lo ại biế bi ến khác nhau, trong đó có các loạ loại biế bi ến có các biế n vô hướ ng ng. Các biế biến vô hướ hướ ng ng chỉ chỉ biể biểu thị thị về độ lớ n (củ (của mộ một đối đối tượ tượ ng ng nào đó) tạ tại mỗ mỗi mộ một điểm riêng biệ biệt trong không gian (mà vậ vật thể thể khả khảo sát chiế chiếm chỗ chỗ). Các vô hướ hướ ng ng thườ thườ ng ng gặ gặp là: tỉ tỉ trọ trọng vậ vật liệ liệu hồi E , hệ h ệ số Poisson ν , module trượ trượ t G. Một loạ loại biế bi ến khác cũ cũng đượ c sử sử dụng nhiề nhiều, đó ρ , module đàn hồ là các đại lượ ng ng vector . Các biế biến vector có thể thể biể biểu diễ diễn đượ c theo các thành phầ phần củ của nó trong các hệ hệ tọa độ 2 hoặ hoặc 3 chiề chiều. Ví dụ dụ về các biế biến vector là: Chuyể Chuyển vị vị của mộ một điểm vậ vật chấ chất trong vậ vật thể thể đàn hồ hồi, góc xoay củ của mộ một vậ vật thể thể… Các công thứ thức củ của Lý thuyế thuyết Đàn Hồ Hồi còn đượ c diễ diễn đạt đạt vớ vớ i việ việc sử sử dụng các biế n ma tr ận. Các biế biến ma trậ trận thườ th ườ ng ng đượ c dùng khi cầ cần biể bi ểu thị th ị một đại đại lượ l ượ ng ng vớ vớ i nhiề nhiều hơ h ơ n 3 thành phầ phần. Ma trậ trận ứng suấ suất và ma trậ trận biế biến dạ dạng là nhữ những ví dụ dụ về biế biến ma trậ trận. Như Như sẽ thấ thấy trong các chươ chươ ng ng dướ dướ i đây, để biể biểu thị thị ứng suấ suất hoặ hoặc biế biến dạ dạng tạ tại mộ một điểm thuộ thuộc không gian 3 chiề chiều, đòi hỏ hỏi đồng đồng thờ thờ i 9 thành phầ phần (trong đó có 6 thành phầ phần độc độc lậ lập nhau). Trong trườ trườ ng ng hợ hợ p này, chỉ chỉ cần sử sử dụng mộ một biế biến dướ dướ i dạ dạng đủ. ma trậ trận vớ vớ i 3 hàng và 3 cộ cột là đủ. Tóm lạ lại, trong hệ hệ tọa độ Đề Các 3 chiề chiều, các biế biến vô hướ hướ ng, ng, biế biến vector, biế biến ma trậ trận có thể thể đượ c biể biểu diễ diễn (ví dụ dụ) như như sau: Tỉ trọ trọng (vô hướ hướ ng) ng) = ρ ; Vector chuyể chuyển vị vị = u = e1 + ve 2 + e3 ; σ x τ xy τ xz Ma trậ trận ứng suấ suất = σ = [σ] = τ yx σ y τ yz ; τ zx τ zy σ z trong đó, e1 , e 2 , e 3 là 3 vector đơ n vị cơ sở trên 3 trục tọa độ.
Trong việc biểu đạt các vấn đề của Lý Thuyết Đàn Hồi, vớ i một biến nhiều khi còn đòi hỏi số thành phần nhiều hơ n thế nữa. Khi đó phải vận dung đến các phép tính tensor vớ i việc sử dụng các các ký ẫn 1 theo hiệu chỉ số (Index Notation). Điều này cho phép biểu thị tất cả các biến và các phươ ng ng trình d ẫ ơ đồ tiêu chuNn hóa duy nhất. Có thể nói một cách đơ n giản rằng: các biến vô hướ ng, một sơ đồ ng, biến vector, biến ma trận và các biến khác cấp cao hơ n đều có thể biểu diễn bở i các tensor vớ i các cấp khác nhau.
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Lý Thuyết Đàn Hồi ký hiệu aij biểu thị cho cả tập hợ p 9 số: a11, a12, a13, a21, a22, a23, a31, a32, a33. Phạm vi của các chỉ số i và j đều là 1, 2, 3. Có nhiều cách biểu diễn khác nhau đối vớ i tập hợ p các số nói trên, tuy nhiên, cách thông dụng nhất ơ đồ có liên hệ đến các dạng thức vector và dạng thức ma trận như sau: là sử dụng sơ đồ a1 a11 a12 a13 a i = a 2 ; a ij = a 21 a 22 a 23 (1.1) a3 a31 a32 a33 Trong dạng thức ma trận, a1i biểu diễn hàng đầu tiên, ai1 biểu diễn cột đầu tiên. Vớ i các hàng và cột khác cũng có thể biểu diễn theo cách tươ ng ng tự, tức chỉ số đầu tiên thể hiện số thứ tự hàng, chỉ số thứ 2số thứ tự cột. Một cách tổng quát, aij...k vớ i N chỉ số phân biệt biểu thị đượ c đến 3N số phân biệt. Cần thấy rõ rằng ai và a j cùng biểu thị 3 số như nhau, cũng như aij và akl cùng biểu thị một ma trận. ừ , nhân (tích) hay bằ ng • Các phép tính cộng, tr ừ ng nhau của hai ký hiệu chỉ số cũng đượ c định ngh ĩ a theo cách thông thườ ng. ng. Chẳng hạn như, phép cộng và phép trừ đượ c định ngh ĩ a bở ii:: a1 ± b1 a11 ± b11 a12 ± b12 a13 ± b13 ai ± bi = a 2 ± b2 ; aij ± a kl = a 21 ± b21 a 22 ± b22 a 23 ± b23 . (1.2) a3 ± b3 a31 ± b31 a32 ± b32 a33 ± b33 • Nhân v ớ i một vô hướ ng ng đượ c xác định như sau: λ a1 λ a11 λ a12 λ a13 λ a i = λ a 2 ; λ a ij = λ a 21 λ a 22 λ a23 (1.3) λ a3 λ a31 λ a32 λ a33 • Ngoại tích giữa hai ký hiệu vớ i các chỉ số phân biệt trong trườ ng ng hợ p đơ n giản đượ c xác định theo khuôn mẫu sau: a1b1 a1b2 a1b3 a b a3 b2 a3b3 (1.4) i j = a 2 b1 a3 b1 a3b3 a3b3 Các toán tử trên đây tuân thủ luật kết hợ p, p, luật giao hoán và luật phân phối. Các định luật này đượ c minh họa qua các ví dụ sau: a i + bi = bi + a i ; a ij bk = b k a ij ; a i + ( bi + c i ) = ( a i + bi ) + c i ; (1.5) ( jk c l ) = ( a i b jk c ) l ; a i b a ij ( b k + c k ) = a ij b k + a ij c k . Toán tử “bằng” (biểu thị bở i dấu "=") chỉ có thể đặt giữa hai ký hiệu có các chỉ số phân biệt phù hợ p (dồng nhất) vớ i nhau và biểu thị quan hệ bằng nhau của các thành phần tươ ng ng ứng của hai ký hiệu. •
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Lý Thuyết Đàn Hồi Qui ướ c chỉ số câm:
Để cho tiện lợ i trong cách diễn đạt, ta qui ướ c rằng: nếu nh ư một chỉ số dướ i nào đó xu ất hiện hai lần trong một số hạng thì nó có ngh ĩ a là tổng lấy theo chỉ số này khi nó chạy từ 1 đến 3, ví dụ: 3
a ii = ∑ aii = a11 + a 22 + a 33 i =1
3
(1.6)
a ij b j = ∑ a i1b1 + a i 2 b2 + a i 3 b3 j =1
Cần nhắc lại rằng, theo qui ướ c trên, a ii = a jj = a mm = .. . nên các chỉ số lặ p (tức xuất hiện hai lần) có tên là chỉ số câm. Các chỉ số không đượ c ấn định (tức không phải là một con số cụ thể) và không lặp lại đượ c g ọi là chỉ số t ự ự do hoặc chỉ số phân bi ệt . Qui ướ c trên đây đượ c g ọi là qui ướ c chỉ số câm hoặc qui ướ c t ổ ổng. n g. Qui ướ c tổng sẽ không áp dụng nếu như xuất hiện dấu gạch dướ i ở một trong hai chỉ số lặp hoặc, đơ n gi ản hơ n, n, n ếu có chú thích "không tổng" ở bên cạnh. Việc một chỉ số xuất hi ện quá 2 lần trong cùng một s ố hạng (chẳng h ạn: aiii , aijkl bkml clkj) s ẽ có ý ngh ĩ a m ơ hồ và c ần tránh sử dụng. Trong một ký hiệu, tác động của qui ướ c chỉ số câm đối vớ i hai chỉ số giống nhau gọi là phép co. Chẳng hạn như: aii chính là kết quả của aij khi co hai chỉ số i và j . Phép co đượ c sử dụng khi thực hiện ngoại tích giữa hai ký hiệu chỉ số mà một chỉ số trong mỗi ký hiệu (tensor) trùng vớ i chỉ số của ký hiệu kia, làm nảy sinh toán tử nội tích; chẳng h ạn, aij b jk là nội tích thu đượ c t ừ ngoại tích aij bmk nhờ phép co thực hi ện đối vớ i hai chỉ số j và m. Một tensor aij...nm...kl đượ c gọi là đố i xứ ng ng đối vớ i n và m nếu như thỏa mản aij...m..n..kl = aij..n..m..kl (1.7) còn nếu như thỏa mãn aij...m..n..kl = - a aij..n..m..kl (1.8) thì đượ c gọi là đố i xứ ng ng l ệch hoặc phản xứ ng ng. Có thể thấy rằng, vớ i hai tensor mà một trong hai là đối xứng đối v ớ i hai chỉ số nào đó còn tensor kia lại là phản x ứng cũng đối v ớ i hai chỉ số này ngoại tích của hai tensor này bằng 0. Tức: nếu aij..m..n..k là đối xứng đối v ớ i m và n còn bpq...m..n..l là phản x ứng đối vớ i m và n thì aij..m..n..k bpq..m..n..l = 0. (1.9) Ta có thể viết đồng nhất thức thông dụng sau 1 ( ij + a ji ) + 1 ( a ij − a ji ) = a (ij ) + a [ij ] , (1.10) a ij = a 2 2 1 trong đó, số hạng thứ nhất a (ij ) = a ( ij + a ji ) là phần đối x ứng còn số hạng thứ hai a[ij ] = 1 ( a ij − a ji ) là 2 2 phần phản xứng. Vậy là: một tensor aij bất kỳ có thể biểu diễn bằng tổng của phần đối xứng và phần phản xứng. Một tensor đối x ứng aij chỉ có 6 thành phần độc l ập nhau còn nếu aij là phản x ứng, các thành phần trên đườ ng ng chéo aii (không tổng) của nó bằng 0 và như vậy, chỉ có 3 thành phần độc lập. Vì rằng a [ij ] chỉ có 3 thành phần độc lập nên có thể biểu thị nó nhờ ký hiệu vớ i chỉ số đơ n, n, ai , chẳng hạn.
§1.3 Delta Kronecker δij và Ký hiệu hoán vị ε ijk
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Lý Thuyế Thuyết Đàn Hồ Hồi Trong lý thuyế thuyết ma trậ trận, ký hiệ hiệu trên đây chính là ma trậ trận đơ n vị. Delta Kronecker là mộ một tensor đối đối xứ xứng. Các tính chấ chất làm cho Delta Kronecker đượ c ứng dụ dụng rộ rộng rãi bao gồ gồm: δij = δ ji ; δii = 3 ; δi i = 1; (1.12) δij a j = a i ; δij a i = a j ; δij a jk = a ik ; δ jk a ik = a ij ; δij a ij = a ii ; δij δij = 3 Bên cạ cạnh Delta Kronecker (còn gọ gọi là ký hiệ hiệu Kronecker), có mộ một ký hiệ hiệu khác, đượ c gọi là Ký định ngh ĩ a như Hiệu Hoán V ị ( ε ijk ), cũ cũng đượ c sử sử dụng rấ rất rộ rộng rãi. Ký hiệ hiệu hoán vị vị đượ c định như sau: 1 khi i, j, k la hoan vi thuan cua 1 , 2 ,3 ε ijk = - 1 khi i, j, k la hoan vi nghich cua 1 ,2 ,3 (1.13) 0 truong hop con lai vì thế thế cho nên, ε 123 = ε 231 = ε 312 = 1; ε 321 = ε 312 = ε 132 = −1; ε 221 = ε 322 = 0 . Nhận xét: tổng số số 27 ký hiệ hiệu hoán vị vị khác nhau có 3 ký hiệ hiệu nhậ nhận giá trị trị 1, ba ký hiệ hiệu nhậ nhận giá • Trong tổ trị trị -1, còn lạ lại là 0; hiệu hoán vị vị là phả phản xứ xứng đối đối vớ vớ i mộ một cặ cặp 2 chỉ chỉ số bất kỳ kỳ của nó. • Ký hiệ Ký hiệ hiệu đặc đặc biệ biệt này rấ rất tiệ tiện lợ lợ i trong việ việc định định lượ lượ ng ng các định định thứ thức và các tích vector. Định Định thứ thức của mả mảng aij có thể thể đượ c viế viết dướ dướ i hai dạ dạng tươ tươ ng ng đươ ng: ng: a11 a12 a13 [ ij ] = a ij = a21 a22 a23 = ε ijk a 1i a 2j a 3k = ε ijk a i1 a j2 a k 3 (1.14) det a a31 a32 a33 trong đó, biể biểu thứ thức đầu đầu tiên theo ký hiệ hiệu chỉ chỉ số của (1.14) thể thể hiệ hiện khai triể triển theo hàng trong khi biể biểu thứ thức thứ thứ hai là khai triể triển theo cộ cột. Vận dụ dụng tính chấ chất ε ijk ε pqr
δip δiq δir = δ jp δ jq δ jr δ kp δ kq δ kr
(1.15),
có thể thể biể biểu diễ diễn định định thứ thức dướ dướ i dạ dạng khác
[ ij ] = 1 ε ijk ε pqr a ip a jq a kr det a 6
(1.14’)
§1.4 Phép biến đổi tọa độ
ế để
ể
ế
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Lý Thuyế Thuyết Đàn Hồ Hồi độ. Các Hệ 1 (nguồ (nguồn): ( x x1, x2, x3) và hệ hệ 2 (đ (đích): ( x x'1, x'2 , x'3), chỉ chỉ khác nhau về về hướ ng ng củ của các trụ trục tọ tọa độ. vector đơ n vị cơ sở của hệ hệ tọa độ nguồ nguồn là {e i } = {e1 , e 2 , e3 } và củ của hệ hệ tọa độ đích: {ei, } = {e1, , e 2, , e3, }.
Đưa Đưa vào ký hiệ hiệu . (1.16) Sử dụng ký hiệ hiệu này, có thể thể biể biểu diễ diễn các vector cơ cơ ssở trong ở trong hệ hệ tọa độ có dấ dấu phN phNy (đ (đích) theo các ở trong hệ vector cơ cơ ssở trong hệ không có dấ dấu phN phNy (nguồ (nguồn): , e 1 = Q11e1 + Q12 e 2 + Q13 e 3 e 2 , = Q21e1 + Q22 e 2 + Q23 e 3 (1.17) e 3 , = Q31e1 + Q32 e 2 + Q33 e 3 Hay, nế nếu như như dùng ký hiệ hiệu chỉ chỉ số, có thể thể viế viết lạ lại (1.17) dướ dướ i dạ dạng: e ,i = Qij e j (1.18) Q ij = cos x 'i , x j
độ, hay gọ độ). Ma trậ trận Qij gọi là ma tr ận biế n đổ i tọa độ, gọn hơ hơ n, n, là ma tr ận xoay (hệ (hệ tọa độ). Tươ ng ng tự tự, việ việc chuyể chuyển hệ hệ tọa độ ngượ ngượ c lạ lại cũ cũng có thể thể viế viết theo cùng mộ một kiể kiểu như như trên: e i = Q ji e' j (1.19). Bây giờ giờ , vớ vớ i mộ một vector v bất kỳ kỳ, có thể thể biể biểu diễ diễn trong cả cả hai hệ hệ tọa độ như như sau: v = v1 e 1 + v 2 e 2 + v 3 e 3 = v i e i (1.20) = v ' 1 e' 1 + v ' 2 e' 2 + v ' 3 e' 3 = v' j e' j Thay (1.19) vào ph ươ ng ng trình đầu của (1.20) ta đượ c v = v i Q ji e' j , nhưng từ phươ ng ng trình thứ 2 của (1.20), v = v' j e' j nên ta lại có (1.21) v' = Q v
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Lý Thuyết Đàn Hồi một hệ tọa độ, nhờ dùng quan hệ (1.21) hoặc (1.22), ta có th ể tính đượ c các thành phần của vector này trong hệ tọa độ thứ 2, có hướ ng ng xác định so vớ i hệ tọa độ thứ nhất. Vì phép biến đổi nói trên chỉ thực hiện đối vớ i các hệ tọa độ vuông góc nhau nên nó ph ải chịu một số ràng buộc nhất định, đó là những ràng buộc đối vớ i ma trận cosine chỉ phươ ng ng Qij. Ta hãy xác l ập các ràng bu ộc này. Trên cơ sở của các quan h ệ (1.21) và (1.22), có thể viết v i = Q ji v' j = Q ji Q jk v k (1.23). Từ tính chất (1.12) của Delta Kronecker biểu thức trên có thể đượ c viết như sau: δik v k = Q ji Q jk v k hay Q ji Q jk − δik v k = 0. Vì phươ ng ng trình trên đúng vớ i mọi v k nên biểu thức trong dấu ngoặc tròn trong vế bên trái phải bằng 0, tức (1.24) Q ji Q jk = δik . Một cách tươ ng ng tự, dùng (1.21) và (1.22) để loại trừ v i (chứ không phải v' i ), ta đượ c (1.25) Q ij Q kj = δik . Các quan hệ (1.24) và (1.25) chính là các điều ki ện tr t rực giao mà các cosine chỉ phươ ng ng (1.16) phải thỏa mãn và cũng là các ràng buộc cần tìm. Xác định định thức của một trong hai ma trận trên, ta có det Qij = ±1. Các ma trận th ỏa mãn quan hệ trên đây đượ c gọi là ma tr ận tr ự ực giao, còn các phép biến đổi (1.21) và (1.22) thuộc về các phép biế n đổ i tr ự ự c giao. §1.5 Tensor Đề-Các
Các vô hướ hướ ng, ng, vector, ma trậ trận cũ cũng như như các biế biến cấ cấp cao hơ hơ n khác đều đều có thể thể đượ c biể biểu diễ diễn theo một ph p hươ ng ng thứ thức tổ t ổng quát, đó là theo các ký hiệu ch ỉ số . Theo quan niệ niệm này, các biế biến, theo mọ mọi ki k iểu, đều đều đượ c coi như như các tensor vớ vớ i cấp khác nhau. Các tính chấ chất của phép biế biến đổi đổi trình bày trên đây cho vector cũ cũng chính là tính chấ chất củ của phép chuyể chuyển đổi đổi tổ tổng quát đối đối vớ vớ i các tensor này. Ta chỉ chỉ hạn chế chế trong đổi giữ Đề-các. Khi đó, các quan hệ đổi đối đối vớ các phép chuyể chuyển đổi gi ữa các hệ hệ tọa độ Đề-các. hệ của phép biế biến đổi vớ i tensor các cấp khác nhau (giữ (giữa hai hệ hệ tọa độ Đề-Các) Đề-Các) có thể thể đượ c viế viết như như sau: a ' = a − cap zero (vo huong); a' i = Qip a p − cap mot (vector); a' ij = Q ip Q jq a pq − cap hai (ma tran);
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Lý Thuyế Thuyết Đàn Hồ Hồi Cần phân biệ biệt rõ giữ giữa các thành phầ phần củ của mộ một tensor vớ vớ i bả bản thân các tensor này. Trở Trở llại vớ vớ i công thứ thức biể biểu diễ diễn vector v (tensor cấ cấp 1): v = v1 e 1 + v 2 e 2 + v 3 e 3 (1.27) = v1e' 1 + v 2 e' 2 + v 3 e' 3 Cũng tươ tươ ng ng tự tự như như trên, tensor cấ cấp 2 - A có thể thể biể biểu diễ diễn bở bở i r r r r r r A = A 11 e 1 e 1 + A 12 e 1 e 2 + A 13 e 1 e 3 r r r r r r + A 21 e 1 e 1 + A 22 e 1 e 2 + A 23 e 1 e 3 r r r r r r (1.28) + A 31 e 1 e 1 + A 32 e 1 e 2 + A 33 e 1 e 3 = Aij e i e j = A'ij e'i e j' Dạng tươ tươ ng ng tự tự cũng có thể thể dùng để biể biểu diễ diễn tensor cấ cấp cao hơ hơ n. n. Dạ Dạng biể biểu diễ diễn (1.28) củ của tensor có tên là d ạng dyadic (dyadic notation). Nhiề Nhiều khi dạ dạng dyadic còn đượ c biể bi ểu diễ di ễn thông qua tích tensor: e i ⊗ e j . Các quan hệ hệ (1.27) và (1.28) cho thấ thấy, mộ một tensor bấ bất kỳ k ỳ có thể thể biể biểu diễ di ễn theo các thành phầ phần củ c ủa nó trong mộ một hệ hệ tọa độ bất kỳ kỳ. Các thành phầ phần củ của tensor thay đổi đổi theo từ từng hệ hệ tọa độ (còn bả bản thân tensor thì không đổi). đổi). Có thể thể chọ chọn tensor ứng suấ suất, mà chươ chươ ng ng tiế tiếp theo ta sẽ sẽ nghiên cứ cứu chi tiế tiết, làm ví dụ dụ. Các thành phầ phần ứng suấ suất là các thành phầ phần củ của mộ một tensor cấ cấp 2, biể biểu thị thị trạ trạng thái ứng suấ suất tạ tại mộ một điểm. Các thành phầ phần này thay đổi đổi theo (đị (định nh hướ ng ng của) hệ tọa độ đượ c chọ chọn để thể thể hiệ hiện trạ trạng thái ứng suấ suất còn chính bả bản thân tensor ứng suấ suất thì không đổi đổi (vẫ (vẫn là tensor đặc đặc trư trưng cho trạ trạng thái ứng suấ suất khả khảo sát). Ví dụ 1.1: Cho biế biết các thành phầ phần củ của các tensor cấ cấp mộ một và cấ cấp hai như như sau: 1 0 3 1 a i = 4; a ij = 0 2 3 2 3 2 4
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Lý Thuyế Thuyết Đàn Hồ Hồi cos 60 0 Qij = cos 150 0 cos 90 0
cos 30 0 cos 90 0 1 / 2 cos 60 0 cos 90 0 = − 3 2 cos 90 0 cos 0 0 0
3 2 0 1 / 2 0 0 1
Tác động động củ của phép biế biến đổi đổi hệ hệ đối đối vớ vớ i vector xác định định nhờ nhờ ph phươ ươ ng ng trình thứ thứ 2 củ của (1.26) 1 / 2 3 2 0 1 1 / 2 + 2 3 a i' = − 3 2 1 / 2 0. 4 = 2 − 3 2 0 2 0 1 2 Đối Đối vớ vớ i tensor cấ cấp 2, cầ cần dùng phươ phươ ng ng trình thứ thứ 3 củ của (1.26) và thu đượ c kế kết quả quả: 1 / 2 3 2 0 1 0 3 1 / 2 a'ij = Q ip Q jq a pq = − 3 2 1 / 2 0 0 2 3 − 3 2 0 0 1 3 2 4 0 7 / 4 3 / 4 3 / 2 + 3 5 / 4 1 − 3 3 / 2 , = 3 / 4 3 / 2 + 3 1 − 3 3 / 2 4
3 2 0 1 / 2 0 0 1
T
trong đó, chỉ chỉ số trên "T" chỉ chỉ phép chuyể chuyển vị vị đối đối vớ vớ i các thành phầ phần củ của ma trậ trận. §1.6 Trị chính và hướ ng ng chính của tensor đối xứ ng ng cấp 2
Khi xoay hệ hệ tọa độ (Đề-Các), Đề-Các), các thành phầ phần củ của tensor thay đổi đổi theo và tồ tồn tạ tại mộ một hệ hệ tọa độ xác định định nào đó mà, vớ đại). Điều này có vớ i nó, các thành phầ phần củ c ủa tensor có giá trị trị cực trị trị (cự (cực tiể ti ểu hoặ hoặc cực đại). thể thể nhậ nhận thấ th ấy bằ b ằng trự trực giác khi khả khảo sát sự sự thay đổi đổi giá trị trị các thành phầ phần của mộ m ột vector khi xoay hệ hệ tọa độ. độ. Nế Nếu chọ chọn hệ hệ tọa độ đích sao cho trụ trục x' 3 trùng vớ vớ i vector v thì, khi đó, vector này sẽ sẽ có các thành phầ phần trong hệ hệ tọa độ đích là: v = {0 0 v }. Trong trườ trườ ng ng hợ hợ p này, 2 trong 3 thành phầ phần có giá trị trị bằng 0 trong khi thành phầ phần còn lạ lại có giá trị trị lớ n nhấ nhất, bằng độ dài củ của vector. Thuộ Thuộc tính nêu ra trên đây tỏ ra đặc đặc biệ biệt hữu dụng đối đối vớ i tensor đối đối xứ x ứng cấp 2, là loạ loại tensor đượ c sử dụng để biể biểu diễ diễn ứng suấ suất/bi t/biếến dạng tạ tại mộ một điểm trong vậ vật thể thể đàn hồ hồi. Nếu tồ tồn tạ tại số số λ và vector đơ n vị vị n(n1 , n2 , n3 ) sao cho
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Lý Thuyế Thuyết Đàn Hồ Hồi a 1
I = a ii = a11 + a 22 + a 33 a a a a 23 a11 a13 1 I 2a = ( a ii a jj − a ij a ij ) = 11 12 + 22 + a 21 a 22 a 32 a 33 a 31 a33 2
(1.31)
I 3a = det [ a ij ]
Các vô hướ ng ng I 1a , I 2a , I 3a đượ c gọi, tươ ng ng ứng, là các bấ t biế n cơ bản thứ nhấ t t, thứ hai và thứ 3 của tensor a ij , còn quan h ệ (1.30) đượ c gọi là phươ ng ng trình đặ c tr t r ư ư ng ng. Đúng như tên gọi của chúng, các bất biến trên đây không thay đổi giá trị theo phép bi ến đổi h ệ tọa độ. Các nghi ệm c ủa ph ươ ng ng trình đặc tr ưng (1.30) chính là các tr ị riêng λ và vớ i việc thay ngượ c mỗi một trong các nghi ệm này vào ph ươ ng ng trình (1.29), ta tìm đượ c hướ ng ng chính n( n1 , n2 , n3 ) . Nếu các phần tử của a ij là thực thì có thể chứng tỏ rằng các nghiệm λ 1 , λ 2 , λ 3 cũng là thực. Ngoài ra, nếu nh ư các nghiệm là phân biệt nhau thì các h ướ ng ng t ươ ng ng ứng vớ i các trị riêng sẽ vuông góc nhau. Nh ư vậy, có thể kết luận rằng, mỗi một tensor đối xứng cấp 2 có ít nhất 3 hướ ng ng chính vuông góc nhau và có nhi ều nhất 3 trị chính là nghiệm của phươ ng ng trình đặc trưng. Vớ i ký hiệu các hướ ng ng chính là n (1) , n (2 ) , n (3 ) tươ ng ng ứng vớ i các trị riêng λ 1 , λ 2 , λ 3 , có thể xảy ra các kh ả năng sau đây: 1. Cả 3 trị chính phân biệt nhau: khi đó, ba hướ ng ng chính tươ ng ng ứng là duy nh ất; 2. Có 2 trị chính bằng nhau (λ 1 ≠ λ 2 = λ 3 ) : h ướ ng ng chính n (1) là duy nhất; mọi hướ ng ng vuông góc vớ i ng chính tươ ng ng ứng vớ i λ 2 , λ 3 ; n (1) đều có thể coi là hướ ng 3. Cả ba trị chính bằng nhau: mọi hướ ng ng khi đó đều là hướ ng ng chính. Tensor là đẳng hướ ng, ng, như đã đề cập trên đây. Như vậy là vớ i một tensor đối xứng cấp 2 bất kỳ, bao giờ cũng có thể có đượ c một hệ tọa độ Đề-Các thuận sao cho các hướ ng ng chính của tensor nằm dọc theo trục của nó. Hệ trục như trên gọi là hệ trục chính của tensor. Trong trườ ng ng hợ p này, các vector đơ n vị cơ sở chính là các hướ ng ng chính đơ n vị ng chéo n (1) , n (2 ) , n (3 ) , và như vậy, trong h ệ tọa độ chính, tensor nói trên thu g ọn về dạng đườ ng λ 1 0 0 (1.32) a ij = 0 λ 2 0 0 0 λ 3 ở định ngh ĩ a b ở i (1.31) có thể đượ c bi ểu di ễn theo các trị chính Có thể thấy r ằng các bất bi b iến c ơ sở đị như sau
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Lý Thuyế Thuyết Đàn Hồ Hồi định theo công thứ Các bấ bất biế biến xác định thức (1.31): I 1a = aii = 2 + 3 − 3 = 2; 2 0 3 4 2 0 I 2a = + + = 6 − 25 − 6 = −25; 0 3 4 − 3 0 − 3 2 0 0 I 3a = 0 3 4 = 2(− 9 − 16 ) = −50. 0 4 −3 Phươ Phươ ng ng trình đặc đặc trư trưng: det a ij − λδ λδij = −λ 3 + 2λ 2 + 25λ − 50 = 0; ⇒ (λ − 2 )(λ 2 − 25) = 0; ⇒ λ 1 = 5; λ 2 = 2; λ 3 = −5,
trong trườ trườ ng ng hợ hợ p khả khảo sát, cả cả 3 trị trị riêng là phân biệ biệt nhau. Vớ i nghiệ nghiệm λ 1 = 5 , phươ phươ ng ng trình (1.29) trở trở thành thành hệ hệ phươ phươ ng ng trình: − 3n1(1) = 0; (1) (1) − 2n2 + 4n3 = 0; 4n (1) − 8n (1) = 0 2 3 Hệ này cho nghi ệm chuNn hóa sau: n (1) = ±(2e 2 + e 3 ) / 5. Một cách tươ ng ng tự cho hai hướ ng ng chính còn l ại: n ( 2 ) = ±e 1 ; n ( 3 ) = ±(e 2 − 2e 3 ) / 5 Dễ dàng chứng t ỏ đượ c r ằng, các h ướ ng ng này vuông góc nhau. Trên hình H1.3 bi ểu di ễn các hướ ng ng chính trong hệ tọa độ đã cho ban đầu cùng vớ i h ệ tọa độ thuận m ớ i ( x1' , x2' , x3' ). V ớ i h ệ tọa độ mớ i này, ma trận chuyển hệ, định ngh ĩ a bở i (1.16), sẽ là
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Lý Thuyết Đàn Hồi Điểm cần chú ý ở đ ở đây là các hàng c ủa ma trận Qij cũng chính là các vector riêng của tensor aij . Sử dụng ma trận bi ến đổi tọa độ này, ta tính đượ c các thành ph ần của tensor đã cho trong hệ tọa độ mớ ii:: 5 0 0 a' ij = 0 2 0 0 0 − 5
Kết quả này phù hợ p vớ i lý thuyết biểu thị bở i quan hệ (1.32) đã đề ra trên đây rằng, tensor nói trên trong hệ tọa độ chính phải có dạng đườ ng ng chéo, vớ i 3 thành phần trên đườ ng ng chéo chính là các tr ị riêng. §1.7 Các phép tính đối vớ i vector và ma tr ận
Trong Lý Thuyết Đàn Hồi sử dụng rộng rãi các phép tính đối vớ i các vector, ma trận và tensor. Các phép tính này bao gồm: Nội tích (dot product) và ngo ại tích (cross product) giữa các vector cùng nhiều tích ma trận, tensor khác. Tất c ả các toán tử trên đều có thể diễn đạt m ột cách hiệu qu ả nhờ sử dụng ký hiệu chỉ số tensor hoàn chỉnh. 1. Trướ c tiên ta hãy xét các tích thực hiện đối vớ i các vector. Cho các vector a và b , vớ i các thành phần Đề-các là ai , bi . gọi là nộ nội tích, đượ c định định ngh ĩ a bở bở i • Tích vô hướ ng ng (scalar product) hay còn gọ a.b = a1b1 + a 2 b2 + a3b3 = a i bi (1.33) Vì trong biể biểu thứ thức trên, các chỉ chỉ số lặp lạ lại nên kế kết qủ qủa phả phải là mộ một vô hướ hướ ng, ng, hay là mộ một tensor cấ cấp 0. Khi đó, độ dài của vector có thể thể đượ c biể biểu diễ diễn bở bở i 1
1
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Lý Thuyế Thuyết Đàn Hồ Hồi Giữ Giữa hai ma trậ trận A và B tồn tạ tại các tích sau đây: [ ] = Aij B jk ; AB = [ A] B AB T = Aij B kj ; AT B = A ji B jk ;
(1.38)
Tr ( AB ) = Aij B ji ; Tr (AB T ) = Tr (AB T ) = Aij Bij , trong đó, AT là chuyể n vị còn Tr( A định ngh ĩ a bở bở i A) là vế t t của A , đượ c định T Aij = A ji ;
(1.39)
Tr (A) = Aii = A11 + A22 + A33 . đối vớ i tích ma tr ận, khi qui tắ Cũng như như trườ trườ ng ng hợ p tích vevtor , đối tắc tổng theo chỉ chỉ số câm có hiệ hiệu lực, kết quả quả sẽ không phụ phụ thuộ thuộc vào thứ thứ tự liliệệt kê củ của các số số hạng. Điều này không thể thể đượ c hiể hiểu là biết biể biểu thứ thức này nói chung là không đúng. AB = BA . Ta đã biế §1.8 Các phép tính đối vớ i tensor Đề-các
Đa số các biế biến dùng trong Lý Thuyế Thuyết Đàn Hồi đượ c xác định định trong không gian củ của vật thể thể khả khảo sát. Các biế biến này là hàm củ của tọa độ các điểm thuộ thuộc không gian nói trên. Trong các bài toán có sự sự phụ phụ đổi theo thờ thuộ thuộc vào thờ thờ i gian, các biế biến này còn có thể thể biế biến đổi thờ i gian. Vì thế th ế cho nên, các vô hướ hướ ng, ng, ma trậ trận và biế biến tensor tổ tổng quát mà ta nghiên cứ cứu cũng là các hàm củ của các biế biến không gian ( x1 , x2 , x3 ). Vì các phươ phươ ng ng trình củ của Lý Thuyế Thuyết Đàn Hồ Hồi chứ chứa đựng đựng cả cả các toán tử tử vi phân và tích phân nên ngườ ngườ i đọc đọc cầ cần có định về đối vớ Đề-các. nhữ những hiể hiểu biế biết nhấ nhất định về các phép tính liên quan đối vớ i trườ trườ ng ng tensor Đề-các.
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Lý Thuyết Đàn Hồi Vector đơ n vị của hướ ng ng s đượ c biểu thị bở ii:: dx dy dz e1 + e2 + e ds ds ds 3 Vì thế cho nên, đạo hàm định hướ ng ng có thể biểu diễn qua tích vô h ướ ng ng như sau: df (1.40) = n.∇ f ds trong đó, ∇ f gọi là gradien của hàm vô hướ ng ng f , đượ c định ngh ĩ a bở i ∂ f ∂ f ∂ f ∇ f = gradf = e1 + e2 + e3 (1.41) ∂ x ∂ y ∂ z còn toán tử vector ∇ đượ c gọi là toán t ử ử del. n =
∇ = e1
∂ ∂ ∂ + e2 + e3 ∂ x ∂ y ∂ z
(1.42)
(nhắc lại: e i là các vector đơ n vị của hệ tọa độ). Các toán tử trên đây cùng nhiều toán tử hữu dụng khác sẽ đượ c vận dụng trong trườ ng ng hợ p đối vớ i các tensor Đề-Các. Cho φ là hàm vô hướ ng ng còn u là vector, các phép tính toán tử vi phân quen thuộc đều có thể viết dướ i dạng ký hiệu chỉ số như sau : Gradient của một vô hướ ng ng : ∇φ = φ , i e i ; Gradient của một vector : ∇ u = u i , j e i e j ; Laplacian của một vô hướ ng ng : ∇ 2φ = ∇.∇φ = φ , ii Divergence của một vector : ∇.u = u i ,i
(1.43)
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Lý Thuyế Thuyết Đàn Hồ Hồi Giả Giả sử S là một bề mặt liên tụ tục từng phầ phần, bao quanh không gian V . Nếu như như u là một trườ trườ ng ng vector liên tụ tục và có đạo đạo hàm cấ cấp 1 liên tụ tục trong V , thì . dS = ∫∫∫ ∇.udV (1.45) ∫∫ u n S
V
trong đó, n là vector pháp tuyế tuyến ngoài trên mặ mặt S. Kế Kết quả quả này cũ cũng đúng cho các tensor cấ cấp bấ bất kỳ kỳ, tứ tức: ∫∫ a ij...k n k dS = ∫∫∫ a ij ...k , k dV (1.46) S
V
1.8.2 Định lý Stocks
Giả Giả sử S là mộ một mặ m ặt cong hở hở (two-sided (two-sided surface), đượ c bao bở bở i mộ m ột đườ ng ng cong kín, liên tụ tục từ t ừng đoạn C. Nếu như đạo hàm cấ như n liên tụ tục cùng vớ vớ i đạo cấp 1 củ của nó trên S thì: . = ∫∫ (∇ X u) . ndS (1.47) ∫ u dr C
S
trong đó tích phân đườ ng ng là dươ dươ ng ng khi miề miền S nằ n ằm bên trái hướ hướ ng ng di chuyể chuyển dọ dọc theo đườ ng ng cong C theo hướ ng ng củ của n - vector đơ n vị vị pháp tuyế tuyến ngoài củ của mặ mặt S. Và mộ một lầ lần nữ nữa, kế kết quả quả trên cũ cũng đúng cho các tensor cấ cấp bấ bất kỳ kỳ, tứ tức ∫ a ij ...k dx t = ∫∫ ε rst a ij ...k , s n r dS (1.48) C
S
1.8.3 Định lý Green phẳng
định lý Stockes cho miề Áp dụ dụng định miền S phẳ phẳng vớ vớ i trườ trườ ng ng vector chọ chọn u = f e1 + ge 2 , ta thu đượ c kế kết quả quả ∂g ∂ f ∫∫S ∂ x − ∂ y dxdy = C ∫ ( fdx + gdy ).
Tiếp tục, chọn một trong hai hàm f = 0 hoặc g = 0 , có k ết quả
(1.49)
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hệ tọa độ cong, (1.53) ta có
)
)
) e 1 , e 2 , e 3 .
Lý Thuyế Thuyết Đàn Hồ Hồi Sử S ử dụng phươ phươ ng ng pháp tươ tươ ng ng tự như như đã thự thực hiệ hiện trong mụ mục §1.5, kế kết hợ h ợ p vớ i
dx k 1 ∂ xk e k = e ; ds1 h1 ∂ξ 1 k dx 1 ∂ x k ) (1.54) e 2 = k e k = e ; ds 2 h2 ∂ξ 2 k dx 1 ∂ x k ) e 3 = k e k = e ; ds 3 h3 ∂ξ 3 k ) ) Trên cơ cơ ssở c ở của điều kiệ kiện trự trực giao, ei .e j = δij , từ từ (1.54) thu đượ c ∂ x ∂ x (h1 )2 = k k ; )
e1 =
∂ξ 1 ∂ξ 1
(h2 )2 =
∂ xk ∂ xk ; ∂ξ 2 ∂ξ 2
(h3 )2 =
∂ xk ∂ xk . ∂ξ 3 ∂ξ 3
(1.55)
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Lý Thuyế Thuyết Đàn Hồ Hồi Đề-Các xác định định còn a là các thành phầ trong đó, a pq...s là các thành phầ phần trong hệ hệ tọa độ Đề-Các phần trong hệ hệ đến sự ở chỉỉ số là dành cho phầ tọa độ cong. Lư Lưu ý: đến sự có mặ mặt củ của dấ dấu “<.>” ở ch phần tử tử trong hệ hệ tọa độ cong. Bản thân mộ một tensor bấ bất kỳ kỳ có thể thể biể biểu diễ diễn đượ c trong cả cả hai hệ hệ tọa độ: độ: ) ) ) a = a ij ...k e i e j ...e k = a < ij...k > e i e j ...e k . (1.58) Đề-Các có định định hướ định nên ∂e k ∂ x j = ∂e k ∂ξ j = 0. Để tính đạo đạo hàm riêng Vì các cơ cơ ssở e k của hệ hệ Đề-Các hướ ng ng cố cố định ) ở của hệ của các cơ cơ ssở c h ệ tọa độ cong, có thể thể sử dụng quan hệ hệ (1.54) vớ vớ i lư l ưu ý rằ rằng e k là hàm củ của các tọ tọa độ cong ξ r : ) ∂e m 1 ∂hm ) 1 ∂hm ) =− e − e ; m ≠ n ≠ r hn ∂ξ n n hr ∂ξ r r ∂ξ m (1.59) ) ∂e m 1 ∂hn ) = e n ; m ≠ n. ∂ξ n hm ∇ξ m ) Cho vector biể biểu diễ di ễn theo các thành phầ phần trong hệ hệ tọa độ cong: u = u< m > e m . Tính các đạo đạo hàm bậ bậc nhấ nh ất
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Lý Thuyế Thuyết Đàn Hồ Hồi
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Lý Thuyế Thuyết Đàn Hồ Hồi ∂ e r ∂ eθ ∂ e r ∂ eθ = eθ ; = − e r ; = 0. = ∂θ ∂θ ∂r ∂r
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Lý Thuyết Đàn Hồi TÓM LƯỢC CHƯƠNG I
ơ đồ gọn nhất mà nhờ đ ờ đó một tập hợ p các số (là các phần tử hay các thành phần) Ký hiệu chỉ số là sơ đồ có thể đượ c biểu diễn bở i chỉ một ký hiệu có kèm các chỉ số phân biệt, gọi là tensor. Các vô hướ ng, ng, vector, ma trận cùng các bi ến cấp cao hơ n khác đều có thể biểu thị theo một phươ ng ng theo ký hi ệu chỉ số (tức tensor). Delta Kronecker δij và Ký hiệu hoán vị ε ijk là những tensor đặc biệt, thông dụng. Các phép tính đối vớ i tensor đượ c định ngh ĩ a thông qua các định ngh ĩ a đã biết của các phép tính đại số đối vớ i các vô hướ ng. ng. Các thành phần của một tensor nhận các giá trị khác nhau trong các hệ tọa độ khác nhau. Sự thay đổi các thành phần của một tensor khi thay đổi hệ tọa độ đượ c xác định bở i ma trận biến đổi (ma trận xoay) hệ tọa độ, Qij . Hệ tọa độ Đề-Các mà vớ i nó các thành ph ần của một tensor nhận giá trị cực trị đượ c gọi là hệ tọa độ chính. Hướ ng ng của mỗi một trong các trục tọa độ này gọi là hướ ng ng chính.
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Lý Thuyế Thuyết Đàn Hồ Hồi Chươ Chươ ng ng II LÝ THUYẾ THUYẾT Ứ NG NG SUẤ SUẤT §2.1 Phân loại các l ự c
Khi các lự lực từ từ bên ngoài tác dụ dụng lên vậ vật thể thể thì bên trong chúng xuấ xuất hiệ hiện các lự lực tươ tươ ng ng tác giữ giữa các phầ phần củ của vậ vật thể thể vớ i nhau. Theo quan điểm củ của cơ cơ hhọc các môi trườ trườ ng ng liên tụ tục, các lự lực bên trong này phân bố bố liên tụ tục trong lòng vậ vật thể thể. Trong nghiên cứ cứu, để cho tiệ tiện lợ lợ i,i, ngườ ngườ i ta phân các lự lực ra thành hai nhóm chính: Lực khố khối và Lực mặ mặt . 2.1.1. Lự c khối
lực tác dụ dụng lên mỗ mỗi thể tích phân t ố vật thể thể. ố của vậ Lự c khố i là lự
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Lý Thuyế Thuyết Đàn Hồ Hồi đối vớ i lực khố định ngh ĩ a như Cũng như như trướ trướ c đây đối khối, lực mặt đượ c định như là hợ p lực của tất cả các lự lực mặt phân tố tố tác dụ dụng lên bề bề mặt khả kh ảo sát và đượ c tính bằ bằng cách tích phân các lự lực mặ m ặt phân tố tố trên toàn diệ diện tích tác dụ dụng: n (2.2) FS = ∫∫ T ( x )dS S
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Lý Thuyế Thuyết Đàn Hồ Hồi Trên mặ mặt cắ cắt, xét mộ một phân tố tố diệ diện tích vô cùng bé ∆ A , vớ i pháp tuyế tuyến ngoài n. Hợ p lự lực các lự lực mặt tác dụ dụng lên ∆ A đượ c ký hiệ hiệu bở i ∆ P . Trên diệ diện tích vô cùng bé này, không để ý đến đến bất kỳ ngẫ ngẫu lực nào. Vector lự lực mặ mặt đơ n vị vị, hay còn gọ gọi là vector ứ ng định ngh ĩ a bở bở i.i. ng suấ t toàn ph ần, đượ c định T n ( n, x ) =
Ký hiệ hiệu x chỉ chỉ vị trí củ của phân tố tố bề mặt khả khảo sát.
∆ P ∆ A→ 0 ∆ A
lim
(2.3)
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Lý Thuyế Thuyết Đàn Hồ Hồi •
phần củ của lự lực mặ mặt đơ n vị vị trên mặ mặt tọ tọa độ y, T ( n = e2 ) ; σ y ,τ yx ,τ yz là các thành phầ
•
phần củ của lự lực mặ mặt đơ n vị vị trên mặ mặt tọ tọa độ z, T n ( n = e3 ) ; σ z ,τ zy ,τ zx là các thành phầ
n
Các đại đại lượ lượ ng ng σ x , σ y , σ x ,τ xy ,τ yx ,τ yz ,τ zy ,τ xz ,τ zx này đượ c gọ gọi là các thành phầ phần ứng suấ suất. Khi biế biến đổi đổi (xoay) hệ hệ tọa độ, độ, các thành phầ phần ứng suấ suất tạ t ại mộ m ột điểm cũ c ũng thay đổi đổi theo. Sự Sự thay đổi đổi này phả phản ảnh trạ trạng thái ứng suấ suất tạ tại mộ một điểm (chứ (chứ không làm thay đổi đổi trạ trạng thái này).
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Lý Thuyế Thuyết Đàn Hồ Hồi trong đó, n x , n y , n z là các cosine chỉ chỉ phươ phươ ng ng của vector đơ n vị pháp tuyế tuyến ngoài n trong hệ hệ tọa độ cho trướ trướ c còn e1 , e 2 , e3 là các vector đơ n vị vị trên các trụ trục tọ tọa độ của hệ hệ này. Sử dụng điều kiệ kiện hợ p lực của tất cả các lự lực tác dụ dụng lên tứ tứ diệ diện phân tố tố phả phải triệ triệt tiêu (vớ (vớ i lực khố khối đượ c bỏ bỏ qua), ta thu đượ c phươ phươ ng ng trình: n T + n xT (n = − e1 ) + n yT (n = − e2 ) + n zT (n = − e3 ) = 0
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Lý Thuyế Thuyết Đàn Hồ Hồi n 2
Từ đó, có:
2 T xn
2 T yn
2 T zn
(T ) = ( ) + ( ) + ( ) = X n2
+ Y n2
+ Z n2
(2.11)
Các biể biểu thứ thức (2.9) và (2.10) cho phép tính toán trự trực titiếếp ứng suấ suất trên các mặ mặt nghiêng bấ bất kỳ độ. Đây cũ đặt các theo các ứng suấ suất trên các mặ mặt tọa độ. cũng chính là nhữ những công cụ cụ rất hữu ích cho việ việc áp đặt điều kiệ kiện biên tổ tổng quát khi thiế thiết lậ lập và giả giải các bài toán Lý Thuyế Thuyết Đàn hồ hồi.
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Lý Thuyết Đàn Hồi a13 0 0 0 σ x a 23 0 0 0 σ y a33 0 0 0 σ z 0 a 44 0 0 τ xy 0 0 a55 0 τ z 0 0 0 a 66 τ zx
ε x a11 a12 ε a y 21 a 22 ε z a31 a32 = 0 γ xy 0 γ yz 0 0 γ zx 0 0
(4.30)
hay, dạng vô hướ ng: ng: ε x = a11σ x + a12σ y + a13σ z ;
γ xy = a 44τ xy ;
ε y = a 21σ x + a 22σ y + a 23σ z ;
γ yz = a55τ yz ;
ε z = a 31σ x + a 32σ y + a33σ z ;
γ zx = a 66τ zx .
(4.30*)
Các phươ ng ng trình (4.30) nếu đượ c biểu diễn qua các hằng số kỹ thuật sẽ có dạng quen thuộc hơ n: n: 1 E x − ν xy ε x E ε y y − ν xz ε z E = y γ xy 0 γ yz γ zx 0 0
− ν yx
− ν zx
E x 1 E y
E x
0
− ν zy
E y 1 E z
− ν yz
E y
0
0
0
0
0
0
0
1 G xy
0
0
0
1 G yz
0
0
0
0
0
0
0 σ x σ y 0 σ z τ 0 xy τ z 0 τ zx 1 G zx
(4.31)
hay, dướ dướ i dạ dạng vô hướ hướ ng: ng: ε x =
σ x
E x
ε y = − ε z = −
−
ν xy
E x ν xz
E x
ν yx
E y
σ y −
σ x + σ x −
σ y
E y ν yz
E y
ν zx
E z
−
σ z ;
ν zy
E z
σ y +
σ z ; σ z
E z
1 τ xy ; G xy 1 γ yz = τ ; G yz zy 1 γ zx = τ ; G zx zx
γ xy =
;
(4.31*)
Ngoài ra, như như đã nói trên đây, giữ giữa các hệ hệ số Poisson và các module đàn hồ hồi pháp tồ tồn tạ tại quan hệ hệ (4.27).
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Lý Thuyế Thuyết Đàn Hồ Hồi a11 = a22 = a33 ; a44 = a55 = a66 ; a12 = a23 = a13 .
(4.32)
Biể Biểu thứ thức thế thế năng đơ n vị vị khi đó có dạ dạng 1 a W = a11 (σ x2 + σ y2 + σ z2 ) + a12 (σ xσ y + σ yσ z + σ zσ x ) + 44 (τ xy2 + τ yz2 + τ zx2 ). (4.33) 2 2 ượ t bằ ng Từ phươ phươ ng ng trình (4.31) suy ra rằ rằng ứ ng ng suấ t tiế p bằ ng ng 0 khi và chỉ khi biế n d ạng tr ượ ng 0. Như Nhưng vì các phươ phươ ng ng trình này đượ c viế viết (và chỉ chỉ đúng) trong hệ hệ toạ toạ độ mà các mặ mặt phẳ phẳng toạ toạ độ là các đối xứ ếu nh ư ở ư ở vật th t hể đ ể đ àn àn h ồi tr t r ự ự c h ướ ng ng mà mặt đàn hồ h ồi đối x ứng, nên ta có thể thể rút ra kế kết luậ lu ận sau đây: N ế các hướ ng ng chính của ứ ng ng suấ t song song vớ i giao tuyế n gi ữ a các mặ t ph ẳ ng ng đ àn àn hồi đố i xứ ng ng thì các hướ ng ng này cũng là các h ướ ng ng chính của biế n d ạng. §4.5 Vật thể đàn hồi đẳng hướ ng. ng.
V ật thể mà mọi mặ t phẳ ng ng bấ t k ỳ đ i qua đ iể m khảo sát đề u là mặ t phẳ ng ng đố i xứ ng ng đ àn àn h ồi, đượ c gọi là ể đẳ ng tiến hành xác định định số lượ ng ng các hằ hằng số đàn hồi, đặc đặc trư trưng cho tính chấ chất của vật thể đẳ ng hướ ng ng. Ta tiế đẳng hướ vật thể thể đẳng hướ ng. ng. 4.5.1 Các công th ứ c định luật Hooke và th ế năng đơ n vị cho vật thể đàn hồi đẳng hướ ng ng
Giả Giả sử, các trụ trục toạ toạ độ x, y, z trùng vớ i các trụ trục chính củ của ứng suấ suất (Xem §2.5). Khi đó, vớ v ớ i ký hiệ hiệu các hằ hằng số s ố đàn hồi là Aij , có biể biểu thứ th ức thế th ế năng đơ n vị v ị (đượ c biể bi ểu diễ di ễn qua các ứng suấ suất chính) như như sau 1 1 1 W = A11σ 12 + A22σ 22 + A33σ 32 + A12σ 1σ 2 + A23σ 2σ 3 + A13σ 1σ 3 . (4.34) 2 2 2 Đối Đối vớ vớ i vậ vật thể thể đẳng đẳng hướ hướ ng ng thì biể biểu thứ thức trên đây sẽ sẽ phả phải bảo toàn tr ị số khi thay vai trò củ của mộ một
trụ trục ứng suấ suất chính chính bằ bằng mộ một trụ trục ứng suấ suất chính khác. Điều này đòi hỏ hỏi các hằ hằng số số đàn hồ hồi phả phải thỏ thỏa mãn: 1 A = a = const ; 2 ii Aij = b = const (i ≠ j ). Như vậy, đối vớ i vật thể đẳng hướ ng, ng, biểu thức (4.34) có d ạng W = a (σ 12 + σ 22 + σ 32 ) + b(σ 1σ 2 + σ 2σ 3 + σ 3σ 1 ).
(4.35)
Biếu thức (4.35) còn có thể viết dướ i dạng tươ ng ng đươ ng ng sau đây W = A(σ 1 + σ 2 + σ 3 ) 2 + B(σ 1σ 2 + σ 2σ 3 + σ 3σ 1 ),
(4.36)
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Lý Thuyết Đàn Hồi và từ đó, trên cơ sở của công thức (4.18), suy ra ε x = 2 A(σ x + σ y + σ z ) + B(σ y + σ z );
γ xy = −2 Bτ xy ;
ε y = 2 A(σ x + σ y + σ z ) + B(σ z + σ x );
γ yz = −2 Bτ yz ;
ε z = 2 A(σ x + σ y + σ z ) + B(σ x + σ y );
γ zx = −2 Bτ zx .
(4.38)
ng pháp đã sử dụng Để xác định các hằng số A và B theo các h ằng số kỹ thuật, ta phân tích theo ph ươ ng bên trên, vớ i các trạng thái ứng suất đặc biệt. Sau khi thay σ y = σ z = 0 vào công thức (4.38), ta thu đượ c ε x = 2 Aσ x ; ε y = (2 A + B )σ x ;
γ xy = −2 Bτ xy ; γ yz = −2 Bτ yz ;
ε z = (2 A + B )σ x ;
γ zx = −2 Bτ zx .
Từ đó, 2 A =
1 1 ; - 2B = ; E G
Ngoài ra còn có:
(2 A + B ) = −
ν
E
⇒ − B =
1 + ν 1 E . = ; ⇒ G= 2(1 + ν ) E 2G
Kết quả, đối vớ i vật thể đẳng hướ ng, ng, biểu thức thế năng đơ n vị thể tích có dạng 1 2 [σ x + σ y2 + σ z2 − 2ν (σ xσ y + σ yσ z + σ zσ x + ) + 2(1 + ν )(τ xy2 + τ yz2 + τ zx2 )]; 2 E còn định luật Hooke tổng quát sẽ là: W =
1 E ε x − ν ε E y − ν ε z E = γ xy 0 γ yz 0 γ zx 0
hay, dướ i dạng vô hướ ng, ng,
− ν
− ν
E 1 E
− ν
E
0 0
0
0
0
E 0
E 1 E 0
0
0
1 G 0
0
0
0
− ν
0
0 1 G 0
0
0 σ x σ y 0 σ z τ (4.40) 0 xy τ z 0 τ zx 1 G
(4.39)
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Lý Thuyết Đàn Hồi 1 [σ x − ν (σ y + σ z )]; γ xy = 1 τ xy ; E G 1 1 (4.40*) ε y = [σ y − ν (σ x + σ z )]; γ yz = τ yz ; E G 1 1 ε z = [σ z − ν (σ x + σ y )]; γ zx = τ zx . E G Công thức (4.40)/ (4.40*) cho phép tính các thành ph ần biến dạng theo các thành phần ứng suất. Trong nhiều trườ ng ng hợ p, p, cần đến các công thức, mà trong đó, các thành phần ứng suất đượ c biểu diễn qua các thành ph ần biến dạng. Vớ i mục đích này, đảo công thức (4.40), có k ết quả: ε x =
3ν σ x = 2G ε x + ε ; 1 − 2ν 3ν σ y = 2G ε y + ε ; 1 − 2ν 3ν σ z = 2G ε z + ε ; 1 − 2 ν
τ xy = Gγ xy ; τ yz = Gγ yz ;
(4.41)
τ zx = Gγ zx ,
1 (ε + ε + ε ). 3 x y z Công thức (4.41) cho phép ta viết biểu thức thế năng đơ n vị thể tích theo các thành ph ần bi ến dạng: trong đó, như đã biết, ε =
W = G ε x2 + ε y2 + ε z2 +
9ν 2 1 2 ε + (γ xy + γ yz2 + γ zx2 ). 1 − 2ν 2
Phươ ng ng trình (4.41) còn có thể viết lại dướ i dạng σ x = 2Gε x + λϑ ; σ y = 2Gε y + λϑ ; σ z = 2Gε z + λϑ ,
τ xy = 2Gγ xy ; τ yz = 2Gγ yz ; τ zx = 2Gγ zx ;
(4.43)
2Gν - có tên là h ằng số Lame', còn 1 − 2ν ϑ = 3ε - là hệ số giãn nở thể tích (xem §3.4). Từ công thức (4.43), có thể thu đượ c các quan h ệ sau:
trong đó, λ =
σ x − σ y = 2G ε x − ε y ; σ y − σ z = 2G (ε y − ε z ); σ z − σ x = 2G (ε z − ε x ),
và
(4.44)
(4.42)
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Lý Thuyết Đàn Hồi σ x + σ y + σ z =
E (ε + ε + ε ), 1 − 2ν x y z
hay σ =
E ε = k ϑ , (1 − 2ν )
(4.45)
trong đó, 1 (ε + ε + ε ) 3 x y z ; E k = 3(1 − 2ν )
ε =
1 (σ + σ y + σ z ). 3 x Công thức (4.45) cho th ấy rằng, độ co giãn thể tích t ươ ươ ng ng đố i tỉ lệ vớ i ứ ng ng suấ t trung bình (còn gọi là ứ ng ng suấ t thu ỷ t ĩ ĩ nh nh). H ệ số tỉ lệ trong công thức (4.45), tức h ệ số k , đượ c g ọi là l à module đ àn àn h ồi th t hể tích. Thực nghiệm cho thấy bi ểu th ức (4.45) nghiệm đúng cho cả khi ứng suất trung bình rất l ớ n (ví dụ vớ i σ = 15000kG / cm 2 ) và vì thế có thể nói rằng, biến đổi về thể tích luôn là tuyến tính. Lưu ý đến (4.45), có thể viết lại (4.41) dướ i dạng σ =
1 τ xy = 2G γ xy ; 2 1 τ yz = 2G γ yz ; 2 1 τ zx = 2G γ zx . 2
σ x − σ = 2G (ε x − ε ); σ y − σ = 2G (ε y − ε ); σ z − σ = 2G (ε z − ε );
(4.46)
ơ ứng suất lệch, còn v ế bên Chú ý thấy rằng, vế bên trái của (4.46) chính là các thành phần của tensơ ứ phải t ỉ lệ vớ i các thành phần c ủa tensơ biến d ạng lệch, viết dướ i dạng các biến d ạng k ỹ thuật. Và như vậy, có thể viết ~ = 2G ~e (4.47) σ ij
ij
tức, tensơ ứ ơ ứ ng ng suấ t lệch t ỉ ỉ lệ vớ i tensor biế n d ạng lệch (xem §2.6 và §3.5). Dễ nhận th ấy một điều là, trong trạng thái ứng suất m ột tr ục, công thức (4.45) sẽ biến đổi v ề dạng quan hệ Hooke quen thuộc. Thật vậy, vớ i trạng thái ứng suất kéo-nén, ví dụ như, theo trục x, tức vớ i ( σ x ≠ 0; σ y = σ z = τ xy = τ yz = τ zx = 0 ), có: 2 σ i = σ x ; ei = (1 + ν )ε x 3 và biểu thức (4.45) biến đổi thành
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Lý Thuyết Đàn Hồi Các công thức (4.41) và (4.44) cho phép ta xác l ập quan hệ giữa c ườ ng ng độ ứng suất và cườ ng ng độ biến d ạng, là những đại l ượ ng ng đượ c s ử dụng rộng rãi trong lý thuyế t d ẻo. Mu M uốn th t hế, ch ỉ cần thay hiệu số của các thành ph ần ứng suất theo công thức (4.44) và quan h ệ giữa các ứng suất tiếp vớ i các biến dạng trượ t (4.46) vào công thức xác định cườ ng ng độ ứ ng ng suấ t t, σ i : σ i =
1 2
(σ x − σ y )2 + (σ y − σ z )2 + (σ z − σ x )2 + 6(τ xy2 + τ yz2 + τ zx2 ). (4.48),
ta thu đượ c công thức σ i =
E 2 (ε x − ε y )2 + (ε y − ε z )2 + (ε z − ε x )2 + 3 (γ xy2 + γ yz2 + γ zx2 ) . 2(1 + ν ) 2
(4.49)
Cườ ng ng độ biế n d ạng, ei , đượ c định ngh ĩ a bở i
ei =
2 3 2
(ε x − ε y )2 + (ε y − ε z )2 + (ε z − ε x )2 + 3 (γ xy2 + γ yz2 + γ zx2 ). 2
(4.50)
Từ đó, có thể viết công thức biểu thị mối quan hệ giữa cườ ng ng độ ứng suất và cườ ng ng độ biến dạng: σ i =
3 E e. 2 1 + ν i
4.5.2 Giá trị giớ i hạn của các h ằng số kỹ thuật
(4.51)
Giá trị giớ i hạn biến đổi của các hằng số kỹ thuật có thể đượ c xác lập qua việc nghiên cứu biểu thức thế năng đơ n vị trong các trạng thái biến dạng đặc biệt. 1. Tr ạng thái kéo một tr ục: Trong trạng thái này, các thành phần ứng suất đượ c cho bở ii:: σ 0 0 σ ij = 0 0 0 . (4.52) 0 0 0 Tươ ng ng ứng, công thức thế năng đơ n vị của trạng thái kéo một trục sẽ là: 1 + ν ν σ 2
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Lý Thuyết Đàn Hồi 3. Tr ạng thái nén thủ y t ĩ ĩ nh nh: ĩ nh), Trong trạng thái nén đều theo mọi phươ ng ng (gọi là nén thủy t ĩ nh), các thành phần ứng suất cho bở i
0 − p 0 σ ij = 0 − p 0 . 0 0 − p Thế năng đơ n vị trong trạng trhái này đượ c xác định bở i 1 + ν 2 ν 3 p 2 2 (− 3 p ) = (1 − 2ν ) . (4.58) W = 3 p − 2 E 2 E 2 E Sử dụng một lần nữa tính chất thế năng dươ ng ng và E > 0 , từ (4.58) có: 1 (4.59) 1 − 2ν > 0 ⇒ ν < . 2 Kết hợ p (4.59) vớ i (4.57) ta xác định đượ c giớ i hạn của hệ số Poisson: − 1 < ν <
1 . 2
(4.60)
4.5.3 Thế năng biến đổi thể tích và Th ế năng biến đổi hình dáng.
Ta khảo sát thế nă ng ng. Như ng biế n đổ i đơ n vị thể tích (gọi tắt: thế nă ng ng thể tích) của vật thể đẳng hướ ng. đã chỉ ra trên đây, vật th ể đẳng h ướ ng ng có thể chịu đượ c áp suất rất cao, mà vẫn không có bi ến d ạng d ư, do đó, trong lý thuyết d ẻo hi ện đại, trong biểu th ức th ế năng t ổng quát của v ật th t hể, ng ườ i ta tách riêng phần thế năng liên quan đến sự biến đổi hình dáng. Vớ i v ật th ể đàn h ồi lý tưở ng, ng, có thể biểu th ị thế năng d ướ i d ạng t ổng c ủa th ế năng biến đổi th ể tích và thế nă ng ng biế n đổ i hình dáng (gọi tắt: thế nă ng ng hình dáng): (4.61) W = W 0 + W hd , trong đó, W 0 và W hd ng ứng, là công (thế năng) tiêu tốn cho việc thay đổi thể tích và cho thay đổi hd , tươ ng hình dáng. Vì sự biến đổi thể tích liên quan đến (hay, đượ c quyết định bở i) i) ứng suất trung bình σ, cho nên, khi gán σ x = σ y = σ z = σ ; τ xy = τ yz = τ zx = 0 vào công thức (4.39), ta đượ c 3(1 − 2ν ) 2 (4.62) σ W =
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Lý Thuyết Đàn Hồi Vì E > 0 , ta có : 1 − 2ν > 0, 1 + ν > 0. Từ đó suy ra, h ệ số Poisson chỉ có thể biến đổi trong giớ i hạn 1 − 1 < ν < . 2 • • • • • • •
TÓM LƯỢC CHƯƠNG IV Định luật Hooke xác định tính chất vật liệu của mô hình tính toán theo đó, trong điều kiện biến dạng bé, các thành ph ần ứng suất có quan hệ tuyến tính vớ i các thành phần biến dạng. Tính chất c ơ học c ủa v ật th ể đàn h ồi tuyến tính đượ c xác định b ở i các hằng s ố đh c ủa v ật li ệu mô hình. Giữa các hsđh có các quan h ệ xác định Các hsđh có thể biểu th ị qua các hằng s ố kỹ thuật. Các hằng s ố kỹ thuật có c ó đượ c qua đo đạc th ực nghiệm tính chất cơ học của vật liêu. Các hằng số kỹ thuật biến đổi trong những giớ i hạn xác định. Vật liệu đh trực hướ ng ng có 3 MPĐH ĐX vuông góc nhau t ại mỗi điểm. Vớ i vật thể đh đẳng hướ ng, ng, bất kỳ mặt phẳng nào đi qua điểm khảo sát cũng là MPĐHĐX. Thế năng biến dạng đh là công các lực ngoài tích lũy trong quá trình làm vật thể đh bi ến d ạng.Thế năng luôn dươ ng ng khi vật thể có biến dạng Thế năng biến d ạng tổng công đượ c phân tích thành t ổng của thế năng biến đổi hình dang và thế năng biến đổi thể tích
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Lý Thuyết Đàn Hồi Chươ ng ng V THIẾT LẬP BÀI TOÁN LÝ THUYẾT ĐÀN HỒI §5.1 Các ph ươ ng ng trình cơ sở tổng quát của lý thuyết đàn hồi
Trạng thái vật lý tại điểm vật chất trên vật thể đàn hồi đượ c xác định bở i các đại lượ ng: ng: 6 thành phần ng nói trên đượ c g ọi là các biến ứng suất, 6 thành phần bi ến d ạng và 3 thành ph ần chuyển v ị. Các đại l ượ ng cơ bản c ủa bài toán lý thuyết đàn h ồi. Các biến c ơ bản t ại m ột điểm bên trong vật thể phải chịu s ự ràng buộc bở i các quan hệ đã đượ c thiết lập đượ c từ các chươ ng ng II, chươ ng ng III và chươ ng ng IV, bao gồm:
•
•
hệ các phươ ng ng trình cân bằng:
các quan hệ ứng suất-biến dạng :
∂σ x ∂τ yx ∂τ zx + + + F x = 0; ∂ x ∂ y ∂ z ∂τ xy ∂σ y ∂τ zy + + + F y = 0; ∂ x ∂ y ∂ z ∂τ xz ∂τ yz ∂σ z + + + F z = 0. ∂ z ∂ y ∂ z
∂u ε x = ; ∂ x ∂v ε y = ; ∂ y
ε z =
•
hệ phươ ng ng trình các tươ ng ng thích:
∂w ; ∂ z
γ xy =
(5.1)
1 ∂v ∂u + ; 2 ∂ x ∂ y
1 ∂w ∂v + ; γ yz = 2 ∂ y ∂ z 1 ∂u ∂w γ zx = + . 2 ∂ z ∂ x
(5.2)
2 ∂ 2 e xy ∂ 2 e x ∂ e y + =2 ; ∂ x∂ y ∂ y 2 ∂ x 2
∂ 2 e x ∂ ∂e yz ∂e zx ∂e xy ; = − + + ∂ y∂ z ∂ x ∂ x ∂ y ∂ z
∂ 2 e y
∂ 2 e y
∂ z 2
2
∂ e yz ∂ 2 e z + = 2 ; ∂ y∂ z ∂ y 2
∂ 2 e zx ∂ 2 e z ∂ 2 e x + =2 ; ∂ z∂ x ∂ x 2 ∂ z 2
∂ ∂e zx ∂e xy ∂e yz − ; (5.3) + + ∂ z∂ x ∂ y ∂ y ∂ z ∂ x ∂ 2 e z ∂ ∂e xy ∂e yz ∂e zx − . = + + ∂ x∂ y ∂ x ∂ z ∂ x ∂ y =
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Lý Thuyết Đàn Hồi biên lại khác nhau, tùy thu ộc vào từng bài toán. Để có đượ c lờ i giải của các bài toán, cần biết cách xác định đúng các điều kiện biên. Các phươ ng ng trình cơ sở kết hợ p vớ i các điều ki ện biên sẽ hình thành các bài toán giá tr ị biên cơ bản (gọi tắt là bài toán biên) của lý thuyết đàn hồi. Có hai hình th ức bài toán biên thườ ng ng gặp, đó là: bài toán theo chuyển vị và bài toán theo ứng suất. Vì, nói chung, rất khó tìm lờ i giải của bài toán biên nên đã có khá nhiều các chiến thuật giải quyết vấn đề đượ c đề ra phục vụ cho mục đích này. Trướ c khi thảo lu ận về các bài toán biên cơ bản, ta thảo lu ận thêm về các điều kiện biên và cách xác ng nêu ra các chi ến thuật thườ ng ng s ử dụng và giớ i thiệu các định đúng các điều kiện biên. Phần cu ối chươ ng nguyên lý tổng quát đượ c ứng d ụng r ộng rãi khi giải quyết các bài toán lý thuyết đàn h ồi. Các quan hệ đề cập đến trong chươ ng ng này và v ề sau chủ yếu đượ c mô tả dướ i dạng vô hướ ng ng hoặc ma trận. §5.2 Điều kiện biên
Nếu nh ư các giá trị các biến bên trong v ật th ể chỉ chịu s ự ràng buộc c ủa các phươ ng ng trình cơ sở thì các biến cơ bản tại các điểm trên mặt biên còn phải tuân thủ thêm các các điều ki ện biên. Về mặt vật lý, các điều ki ện biên phản ảnh t ươ ng ng tác trên biên giữa vật thể vớ i môi trườ ng ng còn về mặt toán học, các điều kiện biên kết h ợ p v ớ i các phươ ng ng trình cân bằng t ạo để nên các bài toán biên hoàn chỉnh. Thông thườ ng, ng, điều kiện biên cho biết vật thể đượ c giữ cân bằ ng ng trong không gian b ằng cách nào và chịu tác d ụng thế nào từ các đối tượ ng ng bên ngoài. Các điều kiện này đượ c xác định, về mặt toán học, thông qua vi ệc chỉ định (cho biết trướ c) c) các chuyể n vị hoặc lự c mặ t t tại các điểm trên mặt bao vật thể. Nếu ký hiệu bề mặt cho trướ c chuyển vị là Su (gọi tắt là mặ t biên chuyể n vị ) còn bề mặt cho trướ c lực mặt, (mặ t biên lự c), là ST thì bề mặt toàn bộ của vật thể, S , bằng tổng của mặt biên chuyển vị và mặt biên lực: S = Su + ST. Nếu chuyển v ị biết tr ướ c là bằng 0, thì mặt biên đượ c g ọi là biên cố định. Nếu trên mặt biên không có l ực tác dụng và chuyển vị cũng không biết trướ c, c, ta có điều kiện biên tự do.
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Lý Thuyết Đàn Hồi Việc xác lập điều ki ện biên chuyển v ị có thể thực hi ện một cách trực ti ếp và, nói chung, là đơ n gi ản hơ n so vớ i điều kiện biên lực. Việc áp đặt điều kiện biên lực tại một điểm trên biên có ngh ĩ a là áp đặt điều kiện cân bằng của t ứ diện phân tố tại điểm nói trên (xem §2.2). Dựa trên kết quả thu đượ c t ừ chươ ng ng II, điều kiện cân bằng của tứ diện phân tố đượ c biểu diễn (dướ i dạng vô hướ ng) ng) bở i phươ ng ng trình
: T xn = σ x n x + τ yx n y + τ zx n z ; T yn = τ xy n x + σ y n y + τ zy n z ;
(5.5)
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Lý Thuyết Đàn Hồi
§5.3 Các bài toán biên
Giải một bài toán lý thuyết đàn hồi cụ thể là việc xác định phân bố ứng suất, biến dạng và \ hoặc chuyển vị thỏa mãn các phươ ng ng trình cơ sở (5.1) cùng các điều kiện biên cụ thể của bài toán. Như trên đây đã đề cập đến, các phươ ng ng trình cơ sở kết hợ p v ớ i m ột lo ại điều ki ện biên xác định s ẽ hình thành một bài toán giá trị biên tươ ng ng ứng. Ta gọi tắt bài toán giá trị biên là bài toán biên. Có 3 bài toán biên biên c ơ bản
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Lý Thuyết Đàn Hồi cùng một bài toán vớ i lực mặt, lực khối cũng như các điều kiện biên hệt như nhau. Hiệu của hai nghiệm trên đượ c biểu diễn bở ii:: σ ij = σ ij(1) − σ ij( 2) ' eij = eij(1) − eij( 2) ; (5.8) u i = u i(1) − u i( 2) . Vì hai nghiệm (1) và (2) thỏa mãn điều kiện cân bằng vớ i cùng một lực khối như nhau nên hiệu của chúng sẽ phải thỏa mãn điều kiện cân bằng không lực khối, tức: σ ij , j = 0 (5.9) Cũng như vậy, hiệu hai nghiệm sẽ phải thỏa mãn điều kiện biên 0, tức điều kiện không có l ực mặt trên mặt chịu lực ST cũng như không có chuyển vị trên mặt biên cv, hay: T i n = σ ij n j = 0 trên ST (5.10) và cả điều kiện không chuyển vị trên biên chuyển vị Su : u i = 0 trên Su (5.11) Ta có thể biến đổi biểu thức thế năng vật thể ứng trạng thái hiệu của hai nghiệm như sau: 2 ∫ WdV = ∫ σ ij eij dV = ∫ σ ij (u i , j − ω ij )dV V
V
V
∫ V
∫V (σ ij ui )dV −V ∫ σ ij , j ui dV
∫S
V
= σ ij u i, j dV =
(5.12)
∫
= σ ij n j u i dS − σ ij , j u i dV
1 (u − u ) là tensor xoay [xem công thức (3.8)mục §3.1]. Để có kết quả trên, đã vận 2 i , j j , i dụng định lý divergent để chuyển tích phân trên thể tích về tích phân mặt tươ ng ng ứng. Ngoài ra, c ần nhớ
trong đó, ω ij =
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Lý Thuyết Đàn Hồi cách khác, cũng dùng các quan h ệ ứng suất- biến dạng nhưng để loại trừ các thành phần chuyển vị rồi dùng định luật Hooke để biểu diễn các thành phần biến dạng theo các thành phần ứng suất, các phươ ng ng trình cơ sở sẽ trở thành các phươ ng ng trình chỉ chứa các Nn s ố cơ bản là các thành phần ứng suất.Ta có dạng ứng suất của các phươ ng ng trình cơ sở . Trong hai d ạng của phươ ng ng trình cơ sở trên đây, dạng ứng suất ít thông dụng hơ n và chỉ tỏ ra thích hợ p vớ i bài toán biên lự c, c, trong đó, các đ iề u kiện biên chỉ chứ a các thành phần ứ ng ng suấ t.t. Dướ i đây ta tiến hành biến đổi các ứng suất t ổng quát về hai dạng nói trên, chỉ hạn chế trong trườ ng ng hợ p vật thể đàn hồi đàn hồi đẳng hướ ng. ng. 5.4.1 Dạng chuyển vị của các phươ ng ng trình cơ sở chung Để đưa các ứng suất về dạng chuyển vị, trướ c tiên, sử dụng quan hệ ứng suất- biến dạng để biểu diễn các thành phần biến dạng trong quan hệ Hooke (cho vật thể đàn hồi đẳng hướ ng) ng) dướ i dạng (4.43) ∂u ∂v ∂w ∂u + 2G ; + σ x = λ + ∂ x ∂ x ∂ y ∂ z ∂u ∂v ∂w ∂v + 2G ; + σ y = λ + ∂ y ∂ x ∂ y ∂ z ∂u ∂v ∂w ∂w + 2G ; + σ z = λ + ∂ z ∂ x ∂ y ∂ z ∂u ∂v ∂v ∂w ∂w ∂u ; τ zx = G + τ xy = G + ; τ yz = G + . ∂ ∂ ∂ ∂ ∂ ∂ y x z y x z
(5.14)
Tiếp đến, thay các kết quả này vào các ph ươ ng ng trình cân bằng (5.1), biểu diễn tất cả các thành phần ứng suất theo các thành phần chuyển vị, ta thu đượ c hệ 3 phươ ng ng trình cân bằng viết theo các thành ph ần chuyển vị. Hệ phươ ng ng trình kết quả có thể viết gọn dướ i dạng ký hiệu chỉ số như sau: (5.15) Gu i kk + (λ + G ) u k ki + F i = 0 .
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Lý Thuyết Đàn Hồi 5.4.2 Dạng ứ ng ng suất của các phươ ng ng trình cơ sở chung
Để có đượ c các ứng suất d ướ i d ạng chỉ chứa Nn là các thành ph ần ứng suất, c ũng xuất phát từ quan hệ Hooke cho vật th ể đàn h ồi đẳng hướ ng ng để thay thế các thành phần bi ến d ạng trong phươ ng ng trình tươ ng ng thích bở i các thành phần ứng suất rồi kết hợ p v ớ i các phươ ng ng trình cân bằng sẽ thu đượ c các phươ ng ng trình có thể viết gọn theo các ký hi ệu chỉ số như sau: 1 ν (5.18) σ ij , kk + σ kk , ij = δ F − F − F j , i . 1 + ν 1 − ν ij k , k i , j Phươ ng ng trình (5.18) có tên là phươ ng ng trình t ươ ươ ng ng thích Beltrami-Michell. Trong trườ ng ng h ợ p không có l ực khối, các quan h ệ trên có thể biểu diễn dướ i dạng vô hướ ng ng sau đây: ∂2 (1 + ν )∇ σ x + 2 (σ x + σ y + σ z ) = 0; ∂ x ∂2 2 (1 + ν )∇ σ y + 2 (σ x + σ y + σ z ) = 0; ∂ y ∂2 2 (1 + ν )∇ σ z + 2 (σ x + σ y + σ z ) = 0; ∂ z ∂2 2 (σ x + σ y + σ z ) = 0; (1 + ν )∇ τ xy + ∂ x∂ y ∂2 2 (σ x + σ y + σ z ) = 0; (1 + ν )∇ τ yz + ∂ y∂ z ∂2 2 (σ x + σ y + σ z ) = 0. 1 ( + ν )∇ τ zx + ∂ z∂ x 2
(5.19)
Lưu ý rằng, trong 6 phươ ng ng trình trên chỉ có 3 quan hệ độc lập nhau (xem §2.). Vì thế cho nên, để có đượ c ng trình xác định 6 thành phần ứng suất, cần kết hợ p (5.19) vớ i các phươ ng ng trình 3 cân bằng đủ số phươ ng
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Lý Thuyết Đàn Hồi 5.5.1 Nguyên lý c ộng tác dụng
Nguyên lý cộng tác dụng là kỹ thuật khá phổ biến, áp dụng cho các bài toán đượ c dẫn bớ i các phươ ng ng trình tuyến tính. Vớ i giả thiết biến d ạng bé, các ứng suất chuyển vị và ứng suất ứng suất đều là là tuyến tính. Hơ n nữa, các điều ki k iện biên (5.6) và (5.7) thườ ng ng c ũng là tuyến tính. Và như vậy, v ớ i tất cả các phươ ng ng trình dẫn đều là tuyến tính, có thể áp d ụng nguyên lý c ộng tác dụng, đã đượ c ch ứng minh và phát biểu bở i Chou PC và Pagano NJ,1967, như sau [3]: Nguyên lý cộng tác dụng: V ớ ới một bài toán biên cho tr ướ ướ c, c, nế u như tr ạng thái {σ ij(1) , eij(1) , u i(1) } là lờ i giải của các phươ ng ng trình đ àn àn hồi cơ sở , ứ ng ng vớ i lự c khố i F i (1) và lự c mặ t t T i (1) và tr ạng thái {σ ij(2) , eij(2) , ui( 2) } là l ờ i gi ải c ủa các phươ ng ng trình đ àn àn h ồi c ơ sở , ứ ng ng v ớ i l ự c kh ố i F i (2 ) và l ự c m ặ t t T i ( 2 ) thì tr ạng thái {σ ij(1) + σ ij( 2) , eij( 2) + eij( 2) , u i( 2) + u (i 2) } là lờ i gi ải của các phươ ng ng trình đ àn àn hồi cơ sở , ứ ng ng vớ i lự c khố i F i (1) + F i (2 ) và lự c mặ t t T i (1) + T i (2 ) Hình (H5.4) minh họa cho việc áp dụng nguyên lý trên vào bài toán 2D khi không có lực khối.
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Lý Thuyế Thuyết Đàn Hồ Hồi Ví dụ 5.1: Phươ ng ng pháp tích phân tr ự c tiếp. Kéo dầm lăng trụ dướ i tác dụng của tải trọng bản thân
Để minh họ họa cho phươ phươ ng ng pháp tích phân trự trực tiế tiếp, ta xét mộ một ví dụ dụ đơ n giả giản, khả khảo sát trườ trườ ng ng hợ hợ p một thanh lă lăng trụ trụ bị kéo dướ dướ i tác dụ dụng củ của trọ trọng lượ lượ ng ng bả bản thân (H5.5).
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Lý Thuyế Thuyết Đàn Hồ Hồi σ x = Ay; σ y = σ z = τ xy = τ yz = τ zx = 0 ,
(5.25) trong đó, A là hằ hằng số số. Vớ Vớ i trườ trườ ng ng ứng suấ suất tuyế tuyến tính như như trên, các phươ phươ ng ng trình tươ tươ ng ng thích và phươ phươ ng ng trình cân bằ bằng tự tự thỏ thỏa mãn và trườ trườ ng ng ứng suấ suất này chính là nghiệ nghiệm củ của mộ một bài toán đàn hồ hồi nào đó. Vấ Vấn đề đặt đặt ra là bài toán nào nhậ nhận trườ trườ ng ng ứng suấ suất trên làm nghiệ nghiệm? Cách thứ thức chung để trả trả lờ i câu hỏ hỏi trên là khả khảo nghiệ nghiệm các dạ dạng hình khác nhau cùng các điều kiệ kiện biên thự thực tế tế khác nhau, tìm hiể hiểu xem trườ trườ ng ng hợ p nào thì xuấ xuất hiệ hiện trườ trườ ng ng ứng suấ suất cho trướ trướ c trên đây. Vì thế thế, ta khả khảo sát không gian 2D dạ dạng hình chữ chữ nhậ nhật như như trên H5.6
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Lý Thuyế Thuyết Đàn Hồ Hồi Thay kế kết quả quả trên vào phươ phươ ng ng trình cân bằ bằng, thu đượ c phươ phươ ng ng trình thuộ thuộc dạ dạng Navier cho trườ trườ ng ng hợ hợ p khả khảo sát: ∂ 2w ∂ 2w (5.28) + = 0. ∂ x 2 ∂ y 2 Đây chính là phươ phươ ng ng trình giúp ta xác định định phầ phần còn lạ lại củ của nghiệ nghiệm bài toán ban đầu. đầu. Điều kiệ kiện chuyể chuyển vị tại hai mặ mặt biên cũ cũng phả phải đượ c xác định định thuậ thuận theo qui luậ luật (5.26). Cầ Cần nhậ nhận ra rằ rằng, điều kiệ kiện biên đặt miễ đặc biệ ở nhữững vùng trên, dù có bị bị áp đặt miễn cưỡ cưỡ ng ng cũ cũng không làm sai lệ lệch kế kết quả quả đi bao nhiêu, đặc biệt là ở nh xa hai biên, tứ tức ở vùng ở vùng giữ giữa dầ dầm. Như Như vậy là, nhờ nhờ ấấn định định ngay từ từ đầu đầu các chuyể chuyển vị vị dướ i dạ dạng (5.26), bài toán đã đượ c đơ n giả giản hóa rấ rất nhiề nhiều. Việ Việc giả giải phươ phươ ng ng trình Laplace (5.28) thỏ thỏa mãn điều kiệ kiện biên tự tự ở mặt bên củ do ở m của dầ dầm là không mấ mấy khó khă khăn. §5.7 Các phươ ng ng pháp giải phươ ng ng trình bài toán đàn hồi
Có rấ rất nhiề nhiều các phươ phươ ng ng pháphươ pháphươ ng ng pháp toán họ học khác nhau phụ phục vụ v ụ cho việ việc giả giải các hệ hệ phươ phươ ng ng trình củ của bài toán đàn hồ hồi. Mộ Một số số trong đó cho phép thu đượ c các lờ i giải chính xác , số số còn lạ lại cho các lờ i
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Lý Thuyế Thuyết Đàn Hồ Hồi đượ c chọ đạt xấ chọn sao cho điều kiệ kiện biên đượ c thỏ thỏa mãn còn tích phân nă năng lượ lượ ng ng chỉ chỉ đạt xấp xỉ xỉ cực trị trị. Số Số hàm xấp xỉ xỉ chọ chọn càng nhiề nhiều, độ chính xác củ của lờ lờ i giả giải, nói chung, càng đượ c cả cải thiệ thiện. Do khó khă khăn trong việ việc chọ chọn các hàm xấ xấp xỉ xỉ thỏ thỏa mãn các điều kiệ kiện biên trong các bài toán có dạ dạng hình họ học phứ phức tạ tạp nên hiệ hiện nay, phươ phươ ng ng pháp biế biến phân hàm xấ xấp xỉ xỉ ít đượ c sử sử dụng. Tuy vây, Các phươ phươ ng ng pháp biế biến phân nói chung đóng vai trò vô cùng quan trọ trọng trong việ việc áp dụ dụng vào phươ phươ ng ng pháp phầ phần tử tử hữu hạ hạn, là mộ một trong nhữ những kỹ thuậ thuật tính toán mạ mạnh nhấ nhất hiệ hiện nay, đặc đặc biệ biệt là trong việ việc giả giải các bài toán đàn hồ hồ 5.7.3 Các ph ươ ng ng pháp số
Lịch sử sử phát triể triển củ của phươ phươ ng ng pháp số số không lâu đờ i bằ bằng các phươ phươ ng ng pháp khác như nhưng, ngượ ngượ c lạ lại, cùng vớ i sự sự phát triể triển mạ mạnh mẽ mẽ của công nghệ nghệ thông tin, vai trò củ của các phươ phươ ng ng pháp này trong thự thực tế tế hiệ hiện đặc biệ nay không phươ phươ ng ng pháp nào so sánh đượ c. c. Các phươ phươ ng ng pháp số số đặc biệt hữ hữu hiệ hiệu trong các bài toán đàn hồi vớ vớ i dạ dạng hình họ học phứ phức tạ tạp. Chính các phươ phươ ng ng pháp số số là cơ cơ ssở cho ở cho việ việc lậ lập trình củ của các chươ chươ ng ng trình tính toán thông dụ dụng hiệ hiện nay. Ta chỉ chỉ điểm qua mọ mọt cách ngắ ngắn gọ gọn mộ một số số phươ phươ ng ng pháp quan trọ trọng nhấ nhất hiệ hiện nay. 1. Phươ ng ng pháp sai phân h ữ u hạn (FDM):
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Lý Thuyế Thuyết Đàn Hồ Hồi Chươ Chươ ng ng VI BÀI TOÁN PHẲ PHẲNG Như Như đã đượ c nhắ nhắc đến, đến, cho tớ tớ i nay vẫ vẫn chư chưa tìm đượ c lờ l ờ i giả gi ải trự tr ực titiếếp của bài toán đàn hồi cho trườ trườ ng ng hợ p tổ tổng quát, vì thế thế cho nên, lờ lờ i giả giải trong các trườ trườ ng ng hợ hợ p riêng có mộ một giá trị trị hết sứ sức to lớ lớ n. n. Các lờ lờ i giả giải trong các trườ trườ h p này có đượ nhờ nhờ h chế chế bớ t,t, bằ ột cách nào đó, tính tổ tổng quát củ của bài toán
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Lý Thuyế Thuyết Đàn Hồ Hồi đưa về hình lă lăng trụ trụ trên chuyể chuyển vị vị giố giống hệ hệt nhau và như như vậy, bài toán 3D có thể thể đưa về 2D, xác lậ lập trong miề miền R (mặ (mặt phẳ phẳng x-y). Dựa trên các điều kiệ ki ện (6.1) và (6.2), có thể thể thu đượ c các phươ phươ ng ng trình cơ cơ ssở và các công thứ thức chủ chủ yếu của bài toán đàn hồ hồi như như sau: 1. Phươ ng ng trình cân b ằ ng ng: Phươ Phươ ng ng trình cân bằ bằng (5.1) trong trườ trườ ng ng hợ hợ p khả khảo sát có thể thể viế viết: ∂
∂
∂
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Lý Thuyế Thuyết Đàn Hồ Hồi ν 1 − ν 1 σ x = 2G ε x + ε y ; τ xy = 2G γ xy ; 1 − 2ν 1 − 2ν 2 1 − ν ν σ y = 2G ε x + ε ; τ yz = 0; 1 − 2ν y 1 − 2ν ν (ε + ε ) = λ (ε + ε ); τ = 0 σ = 2G
(6.8)
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Lý Thuyết Đàn Hồi hợ p này, vớ i giả thiết chiều dày, 2h, bé, có th ể sử dụng giá trị trung bình (bằng 0) làm cơ sở xác định gần đúng. ng trình cân bằ ng ng: 1. Phươ ng
Sử dụng (6.14) và tính ch ất không phụ thuộc c ủa các thành ph ần ứng suất còn lại vào toạ độ z, t ừ phươ ng ng trình cân bằng (5.1) ta có kết quả: ∂
∂
∂
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Lý Thuyết Đàn Hồi trong bài toán trạng thái ứng suất ph ẳng và để cho bài toán là 2D, cần ph ải xử lý g ần đúng như trình bày trong phần dướ i đây. ươ ng 4. Phươ ng ng trình t ươ ng thích: Sự có mặt không mong muốn của biến d ạng ε z trong bài toán về trạng thái ứng suất ph ẳng làm cho các bề mặt không chịu lực bị vênh chút ít . Tuy nhiên, vớ i các tấm mỏmg, độ vênh nói trên là không đáng kể và có thể bỏ qua. Vì thế cho nên khi lấy gần đúng εz = 0, trên cơ sở phươ ng ng trình tươ ng ng thích (5.3), ta có ấ ẳ ế
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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Trusted by over 1 million members
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