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§ 3.1. Функция напряжений Эри

Ра= 089;смотрим систему координат x, <= i>у, z в твердом теле, находящемсn= 3; в напряженноl= 4; состоянии. В каждой точке можно определить напряжения &= #963;x, σу, σz, τxy, τxz, τyz. В случа = 77; плоского напряженноk= 5;о состояния σz =3D τxz =3D τyz = =3D 0. В случае плоской деформации &= #949;z =3D 0, откуда следует, что σz =3D υ(σx + σу).

Дл= 103; плоских задач уравнения равновесия = 80;меют вид (см. также рис. 6.4)

    (3.1)=

Ес= 083;и перемещениn= 3;ми в направлениn= 3;х x и y являются соответствk= 7;нно величины u и <= i>v, то деформации выражаются через них следующим образом:

   (3.2)

а соотношениn= 3; между напряженияl= 4;и и деформацияl= 4;и имеют такой вид:

         = (3.3)

где модуль сдвига μ связан с модулем Юнг = 72; Е выражением = μ =3D E/[2(1 + υ)], в котором υ — коэффициенm= 0; Пуассона.=

Ур= 072;внения равновесия (3.1) автоматичеl= 9;ки удовлетворn= 3;ются, если

   (3.4)

Фу= 085;кция ψ называется функцией напряжений Эри. Подстав= 083;яя уравнения (3.2) l= 0; (3.4) в (3.3) и выполняя двукратное = 76;ифференцир&#= 1086;вание получим уравнение совместносm= 0;и:

        &= nbsp;    (3.5)

или

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;  (3.6)

В общем случае задача линейной упругости о плоском деформировk= 2;нном состоянии может быть решена, если известна функция ψ, которая удовлетворn= 3;ет уравнению (3.6). Напряжения, = 074;ычисленные из соотношениl= 1; (3.4), должны так = 78;е удовлетворn= 3;ть граничным условиям задачи. Функция напряжений для той или иной конкре = 90;ной задачи може = 90; быть выбран = 72; при наличии соответствm= 1;ющего опыта. Этот материал подробно изложен в любом учебнике по теории упру = 75;ости (например, [1]).        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;           

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;

 

 

§ 3.2. Комплексныk= 7; функции напряжений

Оп= 088;еделим комплекснуn= 2; функцию вид = 72;

        &= nbsp;           &nbs= p;      (3.7)

Дл= 103; того чтобы функция z была аналитичесl= 2;ой, ее производнуn= 2; dZ/dz следует определять однозначно. Это условие приводит к соотношениn= 3;м Коши—Риманk= 2;:

3D"../images/fracturemechanics/IMG00090.gif"=    (3.8)

Дл= 103; решения задач о трещине можно использоваm= 0;ь несколько видов комплексноl= 1; функции Эри [2—9]. Для трещиl= 5; первого тип = 72; удобно испо = 83;ьзовать функцию, предложеннm= 1;ю Вестергардl= 6;м [3]. Си [б], а также Эфтисом и Ли= 073;овицем [7] было показано, чт= 086; функция Вес = 90;ергарда не вполне корректна, что, однако, не влияет на результат, поскольку рассматривk= 2;ются сингулярныk= 7; составляющl= 0;е напряжений.

Фу= 085;кция Вестергардk= 2; имеет вид=

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp; (3.9)

где и заданы соотношениn= 3;ми

        &= nbsp;      (3.10)

Отсюда с учетом соотношениl= 1; Коши—Риманk= 2; (3.8) следует, чт = 86;

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;     (3.11)

это означает, чт= 086; функция (3.9) автоматичеl= 9;ки удовлетворn= 3;ет уравнению (3.6).=

Ис= 087;ользуя уравнения (3.4), выразим напряжения через функцию Z:

   (3.12)

Напряж= ения в форме (3.12) определяютl= 9;я любой анали = 90;ической функцией Z (z= ). Остается найти функцию Z (z), которая удовлетворn= 3;ет также граничным у = 89;ловиям рассматривk= 2;емой задачи. Си [6], а также Эфтисом и Либовицем [7] было отмече = 85;о, что если используетl= 9;я корректная = 92;ункция Вестергардk= 2;, то к уравнениям (3.1= 2) следует добавить постоянные слагаемые. Эти слагаем = 99;е исчезают только для специальныm= 3; условий нагружения = 80; не оказываю = 90; влияния на сингулярныk= 7; составляющl= 0;е напряжений.        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;     

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;           

 

 

§ 3.3. Решение задач о трещине

Ра= 089;смотрим задачу о трещине тип = 72; I (рис. 3.1). На рису= нке изображена бесконечнаn= 3; пластинка, находящаясn= 3; в двухосном напряженноl= 4; состоянии. В этом случае функция напряжений = 80;меет вид

        &= nbsp;           &nbs= p;         (3.13)=

Ри&#= 1089;. 3.1. Трещина типа I в двухосном растягиваюm= 7;ем поле напряжений

Эт= 072; функция аналитична всюду, за исключениеl= 4; области (–a ≤ = x a, y =3D 0). Граничные напряжения получаются из соотноше = 85;ий (3.12). В бесконечноl= 9;ти при |z|→∞ эти формулы даю = 90; σx =3Dσy =3Dσ и &= #964;xy =3D 0 — это означает, чт= 086; граничные условия удо = 74;летворены.

Уд= 086;бнее перейти к системе координат с началом в вершине трещины, заменив при этом z на (z + a). Переходя к задаче общего вида (рис. 3.2), в которой граничные условия еще не удовлетв = 86;рены, получим функцию Z вида

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           (3= .14)

Ри&#= 1089;. 3.2. Задача общего вида = 086; трещине тип = 72; I

где f (z) — достаточн = 86; гладкая функция, принимающаn= 3; в начале координат постоянное вещественнl= 6;е значение. Тогда в соответствl= 0;и с уравнения = 84;и (3.12) величины σу и τxy на пов = 77;рхности трещины равны нулю, т. е. кромки тре&#= 1097;ины свободны от напряжений. Требуемое вещественнl= 6;е и постоянно = 77; значение f (z) в вершине трещины обозначим через КI, откуда

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp; (3.15)

Пе= 088;еходя к полярной системе координат, с= 074;язанной с началом координат исходной си = 89;темы (рис. 3.2), и полагая z =3D rei&= #952; из уравнени = 81; (3.12) и (3.15) вычисляем напряжения, возникающиk= 7; вблизи вершины трещины:

        &= nbsp;   (3.16)

Ве= 083;ичина — σ, как показано Си [6], Эфтисом и Либовицем [7], получается = 74; случае одноосного растяжения, = 077;сли функция Вестергардk= 2; применяетсn= 3; корректно; она не оказы= 074;ает никакого влияния на сингулярныk= 7; члены.

Дл= 103; плоского напряженноk= 5;о состояния σz =3D 0; для плоской деформации &= #963;z =3D n(σx + σy). Параметр КI в этих уравнениях называется коэффициенm= 0;ом интенсивноl= 9;ти напряжений. При r>0 (непосредст = 74;енно в вершине трещины) нап= 088;яжения становятся бесконечныl= 4;и. Поэтому коэффициенm= 0; интенсивноl= 9;ти является мерой сингулярноl= 9;ти напряжений = 74; вершине трещины. Здесь рассматривk= 2;ются упругие напряжения, пропорционk= 2;льные внешней наг = 88;узке. В случае одноосного растяжения, = 082;огда напряжения = 74; бесконечноl= 9;ти становятся равными σ, коэффициенm= 0; КI пропорционk= 2;лен σ. Чтобы обеспечить требуемую размерностn= 0; в выражениях для напряжений (3.1= 6), величина КI должна быть также пропорционk= 2;льна корню квадр = 72;тному из длины. Для бесконечноl= 1; пластинки единственнl= 6;й характерноl= 1; длиной является размер трещ = 80;ны, следователn= 0;но, величину КI следует представитn= 0; в виде

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;   (3.17)

Во= 079;вращаясь к частному случаю двухосного растяжения на рис. 3.1, рассмотрим функцию нап = 88;яжений, заданную соотношениk= 7;м (3.13). Перенеся начало координат в вершину трещины, пол= 091;чим уравнение (3.13) = 74; виде

        &= nbsp;           &nbs= p;            &= nbsp;           (3= .18)

Сравне= ние уравнений (3.15) = 80; (3.18) показываеm= 0;, что

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;  (3.19)

Поскол= ьку можно считать, что поле напряж = 77;ний, параллельнm= 9;х трещине, не претерпеваk= 7;т возмущений вблизи трещ = 80;ны, решение для одноосного случая должно быть таким же, как и в случае двухосного напряженноk= 5;о состояния. Следователn= 0;но, коэффициенm= 0; c в уравнении (3.= 17) для пластин = 99;, находящейсn= 3; под действием одноосного растяжения, равен .

Пе= 088;емещения точек пластины можно опред = 77;лять независимо от напряжений. Из уравнени = 81; (3.2) и (3.12) следует, что для плоского деформировk= 2;нного состояния<= /span>

        &= nbsp;           &nbs= p;  (3.20)

что приводит к соотношениn= 3;м

    (3.21)

Ур= 072;внения (3.16) для поля напряжений являются то = 95;ным решением в окрестностl= 0; вершины тре = 97;ины (r≈0). Их можно использоваm= 0;ь в области, где r мало по сравнению с размером трещины. В общее решение следует включить также слага = 77;мые функции f (z) высших порядков. Общее решение имеет вид=

   (3.22)

или

        &= nbsp;            (3.23)

Коэффи= циент r0 обеспечl= 0;вает приближениk= 7; σх и σу к внешнему напряжению &= #963; на большом расстоянии от трещины. В окрестностl= 0; вершины трещины членами высших поря = 76;ков можно пренебречь, = 072; уравнения (3.16) записать следующим образом:

        &= nbsp;           &nbs= p;         (3.24)=

Об= 097;ий анализ, выполненныl= 1; с помощью рис. 3.2 и уравнk= 7;ния (3.14), показывае = 90;, что поля напряжений вокруг вершины трещины тип = 72; I всегда имею&#= 1090; одинаковую форму. Остается только для заданной конфигурацl= 0;и определить &= #1050;I.

Сх= 086;дную процедуру можно применить для анализа задач о трещинах типов II и III. Соответствm= 1;ющие решения можно найти = 074; [4, 7].

Эт= 080; решения для типа II имеют такой вид:

        &= nbsp;          (3.25)

Дл= 103; бесконечноl= 1; пластины с трещиной, которая на бесконечноl= 4; расстоянии от трещины н= 072;ходится в состоянии однородногl= 6; плоского сдвига,

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;   (3.26)

Ан= 072;логично, для трещины типа III

        &= nbsp;     (3.27)

Коэффи= циенты интенсивноl= 9;ти напряжений были вычислены для многих конфигурацl= 0;й. Методика таких вычислений рассмотренk= 2; в гл. XIII.  &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;    

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;           

 

 

 

 

§ 3.4. Влияние конечных размеров

Тр&= #1077;щины в пластинах конечных размеров представля= 102;т огромный практическ= 080;й интерес, но для таких случаев не существует замкнутых форм решений. Эти задачи сложны из-за граничных условий. Приблизите= 083;ьное решение можно получить для полосы конечной ширины, нагруженно= 081; растягиваю= 097;ими силами с краевой или центрально= 081; трещиной.

Ра&= #1089;смотрим сначала бесконечны= 081; лист, в котор= ом имеется бесконечно= 077; число расположен= 085;ых на одном уровне параллельн= 099;х трещин, как показано на рис. 3.3. Решени= 077; для этого сл&= #1091;чая, полученное Вестергард= 086;м [3], Ирвином [10] и Койтером [11], имеет вид

(3.28)

Ри= с. 3.3. Бесконечна= 103; пластина с параллельн= 099;ми трещинами

Если пластину разрезать вдоль линий <= i>АВ и CD, то получим полосу конечной ширины W, в которой имеется центральна= 103; трещина 2a. Вl= 7;олне вероятно, что решение (3.28) приблизит= 077;льно верно и для полосы. В случае параллельн= 099;х трещин на краях полосы шири&#= 1085;ой W действуют напряжения σ [заметим, что сдвиговые напряжения равны нулю из-за симметрии] (рис. 3.4), тогда как края пла&= #1089;тины конечных размеров АВ= ; и CD свободны от напряжений. Логично предположи= 090;ь, что напряжения, параллельн= 099;е трещине, не оказывают большого влияния на величину коэффициен= 090;а К и, следовател= 100;но, выражение (3.28) можно использова= 090;ь в качестве приближенн= 086;го решения для полосы конечных размеров. Легко видет&#= 1100;, что при стремлении отношения a/W к нулю велич&= #1080;на коэффициен= 090;а KI определяем= 072;я выражением (3.28), будет приближать= 089;я к величине Это означает, что если трещины малы, то поло= са конечных размеров бу&#= 1076;ет вести себя как бесконечна= 103; пластина.

Ри= с. 3.4. Напряжениn= 3; на краях полосы, вырезанной из бесконечно= 081; пластины с параллельн= 099;ми трещинами

Ис&= #1080;да [12] для получения коэффициен= 090;ов интенсивно= 089;ти напряжений развил метод координатн= 099;х функций. Этот метод (см. [4]) можно использова= 090;ь для вычисления коэффициен= 090;ов интенсивно= 089;ти напряжений с любой точностью. В этом случае решение обычно представля= 102;т в виде

(3.29)

где Y есть полином от a/W. = В Y иногда включают коэффициен= 090; Феддерсk= 7;н [13] обнаружил, что решение Исида очень точно аппроксими= 088;уется функцией Поэтому для определени= 103; коэффициен= 090;а интенсивно= 089;ти напряжений в растянутой полосе наиболее удобна формула

        = ;            &n= bsp;            = ;            &n= bsp;  (3.30)

На рис. 3.5 выполнено сравнение корректиро= 074;очный факторов на конечность &#= 1096;ирины пластины Ирвина, Исида и Феддерсена.

Ри= с. 3.5. Коррекции на конечность размеров пл&#= 1072;стины с центрально= 081; трещиной

Ра&= #1079;резая пластину с параллельн= 099;ми трещинами (см. рис. 3.3) вдоль линий <= i>EF и CD, можно также получить полосу с краевой тре&#= 1097;иной. Как и в задаче о центрально= 081; трещине, в качестве аппроксима= 094;ии решения о краевой трещине можно использова= 090;ь выражение (3.28). Как и прежде, для малых значений a/W величина K стремится к <= !--[if gte vml 1]> Однако напряжения, действующи= 077; на краю EF, стремятся несколько закрыть трещину. Отс&= #1091;тствие этих напряжений в полосе кон&= #1077;чных размеров приводит к тому, что в этой полосе раскрытие трещины немного больше. След&= #1086;вательно, в этом случае К не&= #1089;колько больше за счет того, что края пол&= #1086;сы свободны от напряжений. Корректиро= 074;очный коэффициен= 090; отличается от единицы п&= #1088;имерно на 12 % (см. [4]). Таким образом, для маленькой краевой тре&#= 1097;ины величина K задана соотношени= 077;м

        = ;            &n= bsp;            = ;            &n= bsp;            = ;       (3.31)

В табл. 3.1 собраны коэффициен= 090;ы интенсивно= 089;ти напряжений и полиномы учета конечности размеров дл&#= 1103; нескольких практическ= 080; важных конфигурац= 080;й.

 

Та= блица 3.1 Величина K дл= я применяемы= 093; на практике конфигурац= 080;й

Толщи= 085;а B

Толщи= 085;а B

p — сила, приходящая&= #1089;я на единицу толщины

 

 

 

§ 3.5. Специальныk= 7; случаи

По= 089;кольку уравнения, описывающиk= 7; поля напряж = 77;ний вблизи всех трещин типа I, имеют одина = 82;овый вид, коэффициенm= 0; интенсивноl= 9;ти напряжений для ряда нагружающиm= 3; систем можн = 86; получить простым сложением:=

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;  (3.32)

То же самое справедливl= 6; для трещин типов II и III. Для трещин смешанного типа такая суперпозицl= 0;я недопустимk= 2; (см. гл. V).

Пр= 080;нцип суперпозицl= 0;и может быть использоваl= 5; в некоторых случаях для получения коэффициенm= 0;ов интенсивноl= 9;ти напряжений. = 042; качестве пр = 80;мера рассмотрим трещину под внутренним = 76;авлением. На рис. 3.6, а из= 086;бражена пластина бе = 79; трещины, нагружаемаn= 3; одноосным полем растягиваюm= 7;их напряжений. Поскольку трещины в эт= 086;й пластине не = 90;, то коэффициенm= 0; интенсивноl= 9;ти напряжений &= #1050;Ia =3D 0. Сделаем в пластине разрез длиной 2а. Это можно сделать, есл= 080; напряжения, которые первоначалn= 0;но передавалиl= 9;ь на поверхно = 89;ть разреза материалом" пластины, заменить на напряжения, приложенныk= 7; к краям разреза извне (рис. 3.6, б= , где KIб =3D 0). Система (б) представляk= 7;т собой суперпозицl= 0;ю пластины с центральноl= 1; трещиной по = 76; действием о = 76;ноосного растяжения у и пластины с трещиной, нагруженноl= 1; внутреннимl= 0; распределеl= 5;ными силами σ (в и г). Отсюда следует, что

    (3.33)

Случай трещины под внутренним давлением э = 82;вивалентен системе, изображеннl= 6;й на рис. 3.6, г; давление же при этом должно быть направлено = 74; сторону, противополl= 6;жную σ. Поэтому знак K меняется па противополl= 6;жный, т. е.

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;   (3.34)

есть коэффициенm= 0; интенсивноl= 9;ти напряжений для трещины под внутрен = 85;им давлением p.

Тр= 077;щина под действием точечных си = 83;, действующиm= 3; на ее берегах, также имеет большое пра = 82;тическое значение (например, при рассмот = 88;ении вопроса о зарождении трещины в бо= 083;товом или заклепочноl= 4; отверстии под действи = 77;м болта). Эта задача може = 90; быть решена = 089; помощью функций Грина [4]. Общее решение для несимметриm= 5;но расположенl= 5;ой точечной силы, изображеннl= 6;й на рис. 3.7, задано формулами<= /span>

        &= nbsp;       (3.35)

где KIA и KIB — коэффициенm= 0;ы интенсивноl= 9;тей деформаций соответствk= 7;нно в вершинах A и B. Для центральноl= 1; сосредоточk= 7;нной силы (x =3D 0) эти соотношениn= 3; принимают вид

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;       (3.36)

(замети= ;м, что P — это сила, приходящаяl= 9;я на единицу толщины пластины). Из уравнения (3.36) следует, что с увеличени = 77;м размера трещины интенсивноl= 9;ть напряжений = 74; ней уменьшаетсn= 3;. При этом поя= 074;ляется возможностn= 0; торможения = 90;рещины через некоторое время после того, как она начала распростраl= 5;яться при KIA =3D KIc, поскольку с ростом трещины интенсивноl= 9;ть напряжений становится меньшей значения KIc.

Ри&#= 1089;. 3.7. Трещина под действием расклиниваn= 2;щих сил

С помощью принципа суперпозицl= 0;и теперь можн = 86; получить коэффициенm= 0; интенсивноl= 9;ти напряжений для трещины, которая зарождаетсn= 3; в нагруженноl= 4; заклепочноl= 4; отверстии. Отверстие должно быть мало по сравнению с трещиной, в противном случае следует использоваm= 0;ь модифицироk= 4;анные выражения для K (см. гл. XIV).<= /o:p>

В соответствl= 0;и с рис. 3.8 этот случай можн = 86; рассматривk= 2;ть как суперпозицl= 0;ю трех других:

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;         (3.37)=

Ри&#= 1089;. 3.8. Образованиk= 7; трещины в нагруженноl= 4; заклепочноl= 4; отверстии<= span style=3D'font-size:8.5pt;font-family:Arial;color:#330066'>

Поскол= ьку, как легко видеть, KIa = =3D KIг, интенсивноl= 9;ть напряжений задается выражением=

        &= nbsp;       (3.38)

Случай трещины под внутренним давлением м = 86;жно также получить из уравнений (3.35). Это дает возможностn= 0; проверить у = 88;авнение (3.34). Внутренне = 77; давление можно предс = 90;авить в виде множества случайным образом расположенl= 5;ых точечных си = 83;. Поэтому вел = 80;чину K получим, проводя интегрировk= 2;ние вдоль трещи = 85;ы. Точечная сила, действующаn= 3; в интервале = а < х < 0, вносит свой вклад также и в интенсивноl= 9;ть напряжений = 74; точке A, следователn= 0;но,

   (3.39)

Интегр= ирование можно выполнить с помощью подстановкl= 0; x =3D a cos φ. Решени&= #1077; имеет вид=

        &= nbsp;           &nbs= p;         (3.40)=

что действителn= 0;но совпадает с уравнением (3.3= 4).        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;     

§ 3.6. Эллиптичесl= 2;ие трещины

На практике трещины часто зарождаютсn= 3; в углах и на краях тел. Такие трещины распростраl= 5;яются преимущестk= 4;енно внутрь тела = 080; принимают форму четверти ил = 80; половины эл = 83;ипса. Для того чтобы приме = 85;ять положения механики разрушения = 82; «угловым трещинам» (четверть эллипса), а также к «поверхносm= 0;ным выемкам» и «полусквозl= 5;ым трещинам» (половина эллипса), необходимо знать коэффициенm= 0;ы интенсивноl= 9;ти напряжений для трещин с искривленнm= 9;м фронтом. Эта проблема, имеющая большое значение в технике, подробно освещена в литературе [14—26]. Далее рассматривk= 2;ется широко используемl= 6;е приближеннl= 6;е решение.

Сн= 077;ддоном [14] была исследованk= 2; задача о вну= 090;ренней трещине округлой формы, которая нах = 86;дится в бесконечн = 86; протяженно&= #1084; твердом тел = 77; под действием о = 76;нородного растяжения (рис. 3.9). Автор пришел к заключению, что

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp; (3.41)

Ри&#= 1089;. 3.9. Внутренняя трещина округлой формы

Задача о внутренне = 81; раковине эллиптичесl= 2;ой формы точного решения не имеет, однак= 086; на основе поля напряжений вокруг эллиптичесl= 2;ой полости, пол= 091;ченного Грином и Снеддоном [16], k= 2; также Ирвин = 86;м [15], было выведено полезное соотношениk= 7;. Точно так же как в случае сквозных трещин, перемещениn= 3;, полученные из этого решения, был= 080; выражены через коэффициенm= 0; интенсивноl= 9;ти напряжений. = 042; результате, по Ирвину,

        &= nbsp;           &nbs= p;   (3.42)

Здесь Φ — эллиптичесl= 2;ий интеграл второго род = 72;, заданный отношением=

        &= nbsp;           &nbs= p;          (3.43)=

где a и = c можно определить из рис. 3.10. Если a =3D c, то уравнение (3.42), как и следовало ожидать, при= 074;одится к уравнению (3.= 41). Значения Φ находят из математичеl= 9;ких таблиц или и= 079; графика на рис. 3.11. Величи = 85;у Φ можно представитn= 0; в виде ряда:

        (3.44)

3. 10. Эллиптичесl= 2;ая трещина

Ри&#= 1089;. 3.11. Параметр поверхностl= 5;ой раковины

Даже при значениях а/= с, близких к нулю, третий член в квадратных скобках дае = 90; вклад лишь 5 %, поэтому в большинствk= 7; случаях им пренебрегаn= 2;т, что приводи = 90; к формулам<= /p>

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;      (3.45)

 

        &= nbsp;     (3.46)

Со= 086;тношения (3.42) и (3.46) с очень небольшими изменениямl= 0; можно применить к полуэллиптl= 0;ческим поверхностl= 5;ым выемкам и к угловым тре = 97;инам, имеющим форму четверти эллипса (рис. 3= .12). Поэтому эти соотношениn= 3; представляn= 2;т собой большой практическl= 0;й интерес. Величина КI, как выяснилось, меняет свои значения вдоль фронт = 72; трещины. На конце малой оси эллипса (φ =3D π/2) интенсивноl= 9;ть напряжений максимальнk= 2;; на конце большой оси (φ =3D 0) она приним = 72;ет наименьшее значение. Поэтому

        &= nbsp;           &nbs= p;  (3.47)

Обычно к этим выражениям величины K добавляют ряд корректироk= 4;очных коэффициенm= 0;ов. Поверхностl= 5;ая раковина сопоставимk= 2; с краевой трещиной; было показа = 85;о [см. (3.31)], что соответствm= 1;ющую ей величину = К следует изменить с помощью корректироk= 4;очного коэффициенm= 0;а примерно на 12 = %. Эта коррект = 80;ровка называется корректироk= 4;кой на тыльную свободную поверхностn= 0;. Для того чтобы учест = 00; тот факт, что в вершине тр= 077;щины развиваютсn= 3; пластическl= 0;е деформации, часто применяетсn= 3; корректироk= 4;ка на зону пластичносm= 0;и (см. гл. V). Наличие пластическl= 0;х деформаций приводит к тому, что трещина ведет себя так, будто она немного длиннее, чем на самом деле. По этой причине коррекция н = 72; зону пластичносm= 0;и является коррекцией размера трещины:

        &= nbsp;           &nbs= p;  (3.48)

Ри&#= 1089;. 3.12. Угловая трещина, имеющая форму четве = 88;ти эллипса, в кронштейне из высокопрочl= 5;ой стали (а); поверхностl= 5;ые полуэллиптl= 0;ческие выемки в образцах дл = 03; испытания н = 72; разрушение (= б);   эллиптичеl= 9;кая трещина в коленчатом валу авиационноk= 5;о двигателя (в= )

Полага= я (см. гл. V)

        &= nbsp;  (3.49)

получа= ем окончательl= 5;ое выражение для K в виде

   (3.50)

Максимал = 00;ная интенсивноl= 9;ть

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;       (3.51)

Величи= на называет= ся параметром формы раковины. Зн= 072;чения Φ для различных отношений σ/σys представлеl= 5;ы графически на рис. 3.11.

На= 082;онец, часто нужна корректироk= 4;ка для учета близости свободной поверхностl= 0; к фронту трещины. Для этой коррек = 94;ии можно воспользовk= 2;ться формулой (3.28). Однако предпочтитk= 7;льнее использоваm= 0;ь коррекцию н = 72; близость свободной поверхностl= 0; к фронту трещины Кобаяши и др. [= 17], которая приведена в графическоl= 1; форме на рис. 3= .13. Отсюда следует, что максимальнk= 2;я интенсивноl= 9;ть напряжений для поверхностl= 5;ой раковины<= /p>

        &= nbsp;           &nbs= p;            &= nbsp;           (3= .52)

где Mк= — корректир = 86;вочный коэффициенm= 0;, учитывающиl= 1; близость свободной поверхностl= 0; к фронту трещины и определяемm= 9;й из рис. 3.13.

Ри&#= 1089;. 3.13. Коррекция Кобаяши, учитывающаn= 3; близость свободной поверхностl= 0; к фронту трещины

В случае, когд= 072; полуэллиптl= 0;ческая раковина распростраl= 5;яется глубоко внутрь мате = 88;иала, корректироk= 4;очный коэффициенm= 0; учета тыльной свободной поверхностl= 0; следует уменьшить д = 86; единицы. В случае трещ = 80;ны, имеющей форму четверти эллипса, коррекцию н = 72; тыльную свободную поверхностn= 0; следует применить дважды. Однако оказываетсn= 3;, что в этом случае коррекция становится = 89;лишком большой. Поэтому для угловой тре = 97;ины величину корректироk= 4;очного коэффициенm= 0;а на тыльную свободную поверхностn= 0; выбирают ра = 74;ной 1,2.

Предыд= ущие соотношениn= 3; для поверхностl= 5;ых раковин был = 80; получены непосредстk= 4;енно из решения задачи о внутренней эллиптичесl= 2;ой полости (см. [15]). В работах Раиса [18], а также Раиса = 080; Леви [19] проведен прямой анал = 80;з задачи о поверхностl= 5;ой трещине. Пол= 091;ченные ими окончательl= 5;ые уравнения п = 88;иведены к такому виду, чтобы из них можно было получить численные значения величины K, которые особенно полезны, поскольку в этих работа = 93; рассмотрен случай изгиба. Част= 100; этих резуль = 90;атов приведена н = 72; рис. 3.14, а, б. Оказываетсn= 3;, для неглубоких трещин (отно= 096;ение 2c/B велико) величина коэффициенm= 0;а интенсивноl= 9;ти напряжений приближаетl= 9;я к его значен= 080;ю для краевой трещины (2c→∞). Такой же результат получается из уравнени = 03; (3.52), поскольку при а/(2с) =3D 0 значение Q =3D 1. Коэффициенm= 0;ы интенсивноl= 9;ти напряжений = 87;ри изгибе были также вычислены Грандтом и Синклером [20]. Вышеприведk= 7;нные рассуждениn= 3; служат иллюстрациk= 7;й к изменению коэффициенm= 0;а интенсивноl= 9;ти напряжений вдоль фронт = 72; трещины пов = 77;рхностной раковины. Информацию = 86; коэффициенm= 0;ах интенсивноl= 9;ти напряжений для эллипти = 95;еских трещин можн = 86; найти в [17—26].        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;         

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;       

 

 

§ 3.7. Некоторые полезные выражения

Ан= 072;лиз напряжений = 74; задаче о трещине, как оказываетсn= 3;, можно свест = 80; к определениn= 2; коэффициенm= 0;а интенсивноl= 9;ти напряжений &= #1050;. Задачи о трещине тип = 72; III наиболее просты, и ино&#= 1075;да их используют для количествеl= 5;ных оценок поведения трещины тип = 72; I (см. [27]). В практи= ;ческих приложенияm= 3; наиболее важную роль играет задача о трещине тип = 72; I, однако встречаютсn= 3; и другие тип= 099; трещин, а также комбинации = 90;рещин типов I и II (см. гл. V и XIV). В табл. 3.1 приведены в = 99;ражения для коэффициенm= 0;ов К. для различных геометрий. Более полно = 101;ти выражения представлеl= 5;ы в работе Пэр= 080;са и Си [4]. Если пренебречь корректироk= 4;очными коэффициенm= 0;ами учета конечности размера трещины, то во многих случаях справедливk= 2; формула

Ри&#= 1089;. 3.14. Интенсивноl= 9;ти напряжений для поверхностl= 5;ых выемок [18] (по данным ASME):

a= <= b>ра&#= 1089;тяжение; б — изгиб

Зн= 072;я коэффициенm= 0;ы интенсивноl= 9;ти напряжении, сами напряжения можно всегд = 72; определить по следующи = 84; формулам:=

(3.53)

В плоскости θ =3D 0 касательны= 077; напряжения равны нулю. Это означае = 90;, что при θ =3D 0 напряжения &= #963;x и σy являются главными напряженияl= 4;и σ1 и σ2. Третье главное напряжение всегда перпендикуl= 3;ярно плоскости пластины: σz =3D σ3. Главны&#= 1077; напряжения = 74; произвольнl= 6;й точке пласт = 80;ны, как это следует из круга Мора (рис. 3.15), равны:

        &= nbsp;          (3.54)

Ри&#= 1089;. 3.15. Главные напряжения

Подста= новка выражений (3.53) = 74; (3.54) приводит к следующим соотношениn= 3;м:

        &= nbsp;           &nbs= p;             = (3.55)

В конкретных приложенияm= 3; удобно имет = 00; выражения для σr, σθ= , τ. Эти выражения можно получить из соотношениl= 1; (3.53):

        &= nbsp;           &nbs= p;    (3.56)

Аналог= ичные соотношениn= 3; можно получить дл = 03; трещины тип = 72; II:

        &= nbsp;           &nbs= p;  (3.57)

Большу= ю роль играет также перемещениk= 7; краев трещины (относитель = 85;ое перемещениk= 7; краев трещины, или РТ). Для трещины, изображеннl= 6;й на рис. 3.16,

        &= nbsp;           &nbs= p;            &= nbsp;      (3.58)

Ри&#= 1089;. 3.16. Раскрытие трещины

Максим= альное относительl= 5;ое перемещениk= 7; краев трещины в ее центре (x =3D 0)

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;      (3.59)

Решени= е упругих задач о трещине является ос = 85;овой механики разрушения. Однако боль = 96;инство используемm= 9;х в инженерно = 81; практике материалов обладает свойством испытывать пластическl= 0;е деформации, что огранич = 80;вает применимосm= 0;ь упругих решений. В следующей главе рассматривk= 2;ются некоторые случаи ограниченнm= 9;х пластическl= 0;х деформаций.        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;         

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Л¶f{Й[ aиљ‡(Р‡Ѕ,MУЖБ6М ?Н[Ђm6 CРЭТ$QtБР%ЪО9”Т3М{ЏОэ!/щ}з;збuИu8Ж}O!Q¦эr Є°љ RЫЧ\cpe„ЇRф"°¶8-dґББ0А:џЗZTчю}e»¤¤UЭЩќRwІѕg Дќ§jГУ…¤d‘Zёc¦›в–ZF&щ•Ire’eЩ —A.ѓ\†Q.±4№УдN“; §Щ§hЖЗЧqЊG7p,—+—ЪЄMўЭ¶ч0й‚S@№§ЇДэ^ѕмMетдНК€2zѕЬ,Ї–·Л·ЛЃ¬Д–Кkе tyefwг)m+ xЪPn<•Y’ЗSквpz<5єицЬбbЖ­RЕ9fg¦:%¤Uљ¦К¦бDљ2»*3EЮђ/Лї“ѓ2©‘њђv\+·”[Лmд‚зsНњx5З@±rЫ№Ы№`n­ф&T‡кCL(1і|мч ЛИD8Л¬V©Pa€/џBE"9к’FъGBж w%Гѓ(П@Zиі›В*ѓа•E‰F=FўVчµµєbMЕЎ¦ТјSҐйЋ*FvtэGµcэсnyТk=ЫлMЙknтаЧ{+у­CП ' ЕJцж»YOивгWѕ{dщKЮПZїЧ_|±mФjѕб™Ґ’Фр$&HW$o*vz(Зг=фpe$в З€1a#>аЙ|>H|вxnшr†џвЂ_cВіїїWЭ[Ґz¤Mій°5gГi:Wњ}'d;KЋиaг `NMЁ/тыЬ¤ЯЏPп№fІ‰/ЕЖS=@‹ЊYіSй9nтxЋR„!ЋчИН‹TPѓЇ.bз)ХbалњЗМcxЩЉЃѕдфЎBУ`K[36Њ¦qЕШ7BЖµбkoр± @zшi*(/Mm“ЃЋхU¶ Xџj§ќ #с'ё&Xg:ёџ™™ќщЃy°Ц:|ШнЏ„SЙЃ±n–]ДЃЩ‰‰™Vъ±Ѕ\ 'gмЛЇ°MЕЩ„h‚дЂЪRаЧн(џMRр6©µ9‚MЈ–Q р[}ШBdяЎ /л‡шйЊk±vpМ% ч.Eo·ГЅЫ‰тn'К»ИЬЊ–§’Keљ•хдИЭ®_@¬ЉЂфћiЉхУ%aФдфhрn„З€;ЃЮ’%€яЩЩ¶<Оnoocўт”d·wу рвэ’“uS©X±GRья$тљЬЊ6щ%еuнRПлЦеКЫІ\1+Й3кнЊх u][·.‰‘{©–x>тГоЭА®rWј«<РюЩУUХЄFХ*ЫХКјІ)[йК‹Є=bЏж+eVVГЅjѓЅ¤ћґѓГк)vJщPэ—zA{Ює№%яMй‘>Хґ¬#вsЉХ”x,Й•T·%ќ4‚'B+кIнd\2•ББ”uB ¶I›џ6ђT6Sr¦пи{њсs`Y23њГ­ЫЪДI›аҐH,Д”‡XЖ§ДBЧ­”?S%%TЈ=$H: RїЧP&j=сёjZЙ”й‚Рd†d1’’Qg2ГУ™|­љћтB4®ЄЋm%l&Ъ(ы$Њ‰Мl+О‚Q‘UХђK‚ o±ы^Эаї‰Fe рnљ†ќдз№ёПЩm~‡‹|›‹<Їл— f$­ «Ђ0 N>/ёЄ{ХЭvo»Ў%—ќw›®и®•+[м;o§ЯьдИЗ>9»№:{іЯcкжC4?YЅz"Rі8Tќ5сИТp ВјРнЩоп«;єЪ†Њ6ХLЭцЫ 8¶ЇАыЩЬ< хек&[Ґ?б¬pRН›‚ ґI@¶iЌAЮ Ч АS*"F™h%ЉќVQь.вwє·ґ …‰dWШ*Ё"fњЕВh¦о•¤p8N)Ж‹iL5FЭWЕТУІxьо"пJЏІ‹/}іvпЮ+C“ЋщlлрhяXлп¦{ґеОчF•n;Щ;Ў15tсСЖзz8O Љ¶-є3nэй\:Я-;лЌл_`_mЭ^)МqґЁћюbа№Л эЪ0Fљg@€4ЅмЗuФAH\baF1ѓQМ`3З$ ГQ~И;ИQ&1`ЂсЧwp э ‚C\a!"OP„Hф‚CА¦Ѓ¬“пe1гSwџКщ2qТёDЫ8,„0#™bD|4‰К—,о/2|Йв\пы\ЪVЋш*uЈ©oлыz@Gн©О°чUf LїыКф’О<}I_У7ф¦~&†щx*ј8ДЖSRf8‘‰Хв©Д EGIЕeры^Ч•pг•08µ$Тё€{”4ЪCЈ=45ђц°'I(·ю{ЛЯГћА=ач_ј(NµЕцш#А(мcч™щ©#Пcъf/њlx8'Я`ЗлЌWЃЖІґpР9 ПaEѓ5 АyuтµЗЫшЧ4Эя›mЁC xПRїK9^ц тga{Ш=…O6–ГЖБЌЇЩA’‘,•0YтeK5ъUЈ_µ:њг#їmџ.a€о’_ ’с1Ќ–J§лЁси¬wЖїiґ^_9Э&Ћц¤UбЙй‚#tжЅкяШ®єШ¶Ќ;О#i‰ўѕ(Йў(Й©Oљ’b}Тћм4¦гДЋ#+v›/+­wу–­ЙЫE’&Kc#hУµ+6c/{РtЩЂC iќnЭЊ® V`iЊaР—>[ѓЕЭKЂ!C&п'ЙqС жйtw<ҐЯяч1€IР[·WLэЋЩшњЪ W®ЬЖз7’_‚иХ|Х‚FWЙјVы—И,Дk8+Ґнh№‘HСВТ эшFґO зЎcXЈгZxtoФҐ…}ђЉnДТZ8·ВШoД†ґрtЊ±ѓjuи@ша.Nл«e­›ЈМ‰СC‡с“ИШx«ЩДvGGт9ЙЗЧ|ѕЂаЉGr љSк ­¬ Эpцi=йшЧr}h®ЇЮGчб1±zx(>>.W'«фRu№JSUЎJWЎ®ЯнKХ™©Ъ }4kQZAіЇbЭJг0U V=Д©к^уmыѕЭЯЬхw(rь$U"`ДтА—Kmж­vвЉЖmN{"–ЊЫ"]ИбЊ:[®4љџvЅ ЃляД®––Ё$w™}OxdsШј%Џ}Й·Сд¬{Ы·‹‡.xЏяЁ26н|пSЌнћЃ€Џgѓк!эД8M{ыGщсІµ#’™иХчoуз+ЌЃБBЂx\Х‰:УфѓYg25{мҐJе`я…Ж™CЉсМ'Д\“иЌ№CЯcM7*$іЃ*=cy#”йkxЏфгсаАAtф§™HЛЫ(Љщ70Y‘Юd2ќ0YЋшбµЬяаСЛnс°чUф¦еuз§AіltVШ]‘СmпЗЪђСЧ>Ќ›Е?zья@‘EkёќdgШ9v™­і&цЃНЂIГv"Оpxё"Ґч ТХхiмй*хоэ•ъдУG®ЫВcЧevм™#SP¶ЌUЉ…KЮXЕ8<х*А(–кd ч…ыБ-Aj­хўђ;бHТ‰®$џ0%]ОN… ЎЂ‚D ф$3фm:Пџwњwї$ћ–NwqУ5B~ K—а*бтВ—~ЭZЖ;ХўАg§Й‹ЄIЅФЫл‹љLЮN7Ж$(M­]…Яx†/yь”ВhВБE$ж ИхFЫЇЮU0€мДШ1Эi7л0ЮnхбИцЈ'би­™М*aaЉ°0…ќА°аЬЦ бIО6‰vuUё Дљ%€mSл-Є°сш&bЃЗ”p—зъбt·В‘Ґ©&|Ё/Њ 1k ¬к69T щp+> >юҐ…&3ў&cy®5ЙуNЈъ"?ЂСZЖ„g…Ч]ме И T2Пf^pЅђy‘;з:—y…»jѕП=ІШsSЕZйd‰5P–cє5·l•яrФжJЌQjdB S»hwє›a{„^„OB›с™ь’Јђ—щeћћб—шk<ГяSЎ=+иёT”ЙИ\„^Љ *"Dк‘ХИZ¤#2УяaҐf¶ „Цq Y‡ЗZpщКB‹‡ЂэAґ’ХНv.QJЪ’№„n.((k‡¦hйUPЮЪЈPФ&tЃ(禩щiЂ “(z±УБ84ЄmSыћ¤Ћ&a‚Т[F‡FЃдиЏ'Юxnюsпмнн.шК•†впS=^!–Ёdq|wямЋ§џ3¦rЩ8S^шЫ№зOѕт—хџ-zќЫчЏГ‰­щYжлµњдXlјs*Ц?µп[·ю<їOrг¤e@їXоF7[HоN$›dџK%B•dФ \[у‰ЬvrЫ7И3. d™Д'™ ™дІ Њ$ъ а–Ё$АЩ1ЎћRUFн6K6 uзђuH!_сВG·Ы~ЎНВ1ј]о=eYґРШ@2БI њ]$gа3ю‡АYЖ№ 3ој‡зd9Ґ=‘|ШџКЮ№3Ѕ©фAгlgЃ.8 Ъp^bНF K!c‘ёъЛ1UU†’auЕ[S®NE@¬ґdA–І`C¶ГPfрнЗLИ0!SЏњB)К—eYAKКІBSЉ>~UYS:”нЯ#dјйДоН/`o$f ОЌбзnQм®~@WДt7€ыЕЌО‹!ҐКўXф‹X»†к‹иGLґЯD¦mhВ4hCЁжb» JUѓЄ6P^CФyйЅ%йЅЯЭGЗ`Qt№P_ЪT–оЫйMЈ+{XфmцE–fнѕИ6=c~Я¤M[Юб'-ЈEлѓ— ^УUpеєхБ5^[мХ‹ tн6ѕ¤ЕЕvSZTР›ВР©D}Э§%h ЯZr,ц”3+ќ*і|„чшКD}ІЮ©‡ЕzY­c@‘г‹еEыџҐ+е«юЌРЌТЌт-_Pљ~ЫяnяQя:Aџ`:©ЋСЙvrЗыOq‰4ПDcбп_Шр·—cТI%—О땬?ќбg­“Ґ“eA©Ж=·?ZЇ?_yЮ9с»Тkх›МЌ\¬Вm4©ЪDTC4кўк<µt‘С’ы4S_ИљFБ@’aБ›ГќъBwnP”r).„D—T¬‰юJµѕЌ…_Єс#]ЧєМЋV2]Г/–~WAH№T|їшQ‘)v™dKС„8)N‹ЊШE›[єkиAЃCњ?зў wТнёЊеёґыgdQѓИz}ПъЗ1І|и±°+н­OОЇQ{¬Qхџ_EР_ѕэ LШЬ^—–ЧІj¦ЮЃ7]Ћ Йx\8–Є‰ЈТвFI·п,·!iщОrЇMљ=ЅXСш&Є:Fи?зU –$‡#вmёВеа6sTДcsЁGэ/Ѕ„2¬э,т©ф©ь™jЏAр…OкshЋћcж„WгУ©ic:;ќ›ЩрЛТ\ LL¤АзА0ЎVЄ•вП–g}¶=†­ЌмYz#кй Фв4\Y0zу|ГА~Oз<тЙmД$Si&,\ЂРПg¤ТеокНyµQкU1ЁЮRѕ¦цжRzs‰ ,ЎАJГ·ь›Џ[ўГД#Еaќ8ћаг–‡uв0.M&UэрnЖ]Й%М.5pt;height:30.75pt'>         &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;    (5.27)

Ри&#= 1089;. 5. 13. Образец, имеющий форму двухконсолn= 0;ной балки

Уравне= ние (5.26) дает лишь грубое приближениk= 7; величины коэффициенm= 0;а интенсивноl= 9;ти напряжений. Расхождениk= 7; возникает из-за того, что в балке образуются деформации сдвига (это можно учест = 00; при выводе податливосm= 0;и), а также из-за того, что концы балок не жестко закреплены, = 072; опираются н = 72; упругие шарниры.

Из уравнения (5.26) следует, что коэффициенm= 0; интенсивноl= 9;ти напряжений для двухконсолn= 0;ной балки не зав= 080;сит бы от размер= 072; трещины, есл= 080; бы образец имел форму клина: если толщина увеличиваеm= 0;ся пропорционk= 2;льно а, так что отношение а/= В постоянно, то величина = K будет иметь одинаковые значения дл = 03; трещин всех размеров. Эт= 086; дало бы возможностn= 0; исследоватn= 0; процесс роста трещины при постоянных значениях К = или G. Образец переменной толщины не очень практ = 80;чен, поскольку процесс распростраl= 5;ения трещины слишком сильно подвержен влиянию тол = 97;ины образца. Поэтому Мостовой и др. [14] ввели в обращение образец в виде клиновидноl= 1; двухконсолn= 0;ной балки, изображеннl= 6;й на рис. 5.14. Можн= 086; показать, чт= 086; в этом образце при ограниченнl= 6;й области изменения размеров тр = 77;щины величины K и <= i>G (вдоль тех же линий, что и в обычном ДКБ = 211; образце) постоянны.=

Ри&#= 1089;. 5.14. Образец, имеющий форму клиновидноl= 1; консоли: 1 бо&#= 1082;овые выточки

Ка= 082; показано на рис. 5.15, образе = 94; в виде клиновидноl= 1; консоли может быть использоваl= 5; для изучени = 03; роста трещи = 85;ы при постоянном K= . Образец нагружаетсn= 3; вдоль линии = OA силой Р1 пl= 8;и которой достигаетсn= 3; интенсивноl= 9;ть напряжений (= 080; интенсивноl= 9;ть выделения энергии), необходимаn= 3; для роста тр= 077;щины. Трещина немного расширяетсn= 3;, что приводи = 90; к уменьшени = 02; нагрузки. Дл= 103; того чтобы снова началось расширение трещины, нео= 073;ходимо повторное нагружение до той же сил&#= 1099; P1, поскольку при этой нагрузке достигаетсn= 3; то же значение = K (обратите внимание на то, что, например, в образце с краевой трещиной пр = 80; росте трещины то ж= 077; значение К = 074;озникает при меньших значениях P, &= #1087;оскольку с ростом а з&#= 1085;ачение K увеличиваеm= 0;ся по формуле

Пр= 080; использоваl= 5;ии образцов в виде двухко = 85;сольной балки часто оказываетсn= 3;, что путь трещины отклоняетсn= 3; от оси симметрии, к= 072;к показано на рис. 5.14 (трещин = 72; В). Этого можно избежать, если проделать н = 72; боковых поверхностn= 3;х образца выт = 86;чки (см. рис. 5.14), хотя наличие эти = 93; выточек усл = 86;жняет вычисление податливосm= 0;и. Податливосm= 0;ь можно также определить эксперименm= 0;ально следующим образом.

Из= 084;еряя нагрузку и РТ, можно построить диаграмму «нагрузка — перемещениk= 7;» (рис. 5.16, а). Эту работу следует повторить для различных размеров трещин. В соответствl= 0;и с формулой C = =3D ν/P, податливосm= 0;ь определяетl= 9;я наклоном эт = 80;х линий. По измеренным значениям C строится график зависимостl= 0; этой величи = 85;ы от размера трещины (рис. 5= .16, б). Определяя наклон полу = 95;ающейся линии, можно найти производнуn= 2; от податливосm= 0;и ∂C/∂a, на основе которой с помощью уравнения (5.22) можно вычислить величины G и <= i>K (рис. 5.16, в). В случае образца в виде клиновидноl= 1; консольной балки можно найти такую область изм = 77;нения размеров тр = 77;щины (между A и B), в которой G и K фактически не зависят о= 090; длины трещины. Рис. 5= .17 позволяет судить (см. [16, 17]) l= 6; точности измерений податливосm= 0;и путем сравнения и = 93; с результат = 72;ми вычислений.

Ри&#= 1089;. 5.15. Рост трещины в образце, имеющем форму клино = 74;идной консоли

Ри&#= 1089;. 5.16. Измерения податливосm= 0;и

Ри&#= 1089;. 5.17. Измеренные и вычисленныk= 7; величины по = 76;атливости [16, 17]:

1= <= b>ра&#= 1089;чет методом конечных элементов; 2<= b>ка&#= 1083;ибровочный тест

Примеч= ание. В образцах в виде двухконсолn= 0;ной балки подат = 83;ивость определяетl= 9;я раскрытием трещины. Это возможно только для образцов, на= 075;ружаемых по линии трещины. Для других типо = 74; образцов РТ нельзя использоваm= 0;ь для определ = 77;ния податливосm= 0;и, поскольку относительl= 5;ое перемещениk= 7; точек приложения нагрузки не = 088;авно РТ (см. [15]). = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;    

 

 

 

 

§ 5.5. J - интеграл

До сих пор лишь предполагаl= 3;ось, что зона пла= 089;тичности при вершине трещины настолько м = 72;ла, что применима теория упругости. Если это так, то пластическl= 0;е деформации при вершине трещины не оказывают влияния на интенсивноl= 9;ть выделения энергии и величина G определяетl= 9;я упругим полем напряжений. = 052;ожно показать (см. [18]), что если зону пластичносm= 0;и при вершине трещины не считать пренебрежиl= 4;о малой, то она будет оказывать влияние на интенсивноl= 9;ть выделения энергии.

Дл= 103; того чтобы точно вычислить влияние пла = 89;тических деформаций на величину = G, нужно получить точное решение упругопласm= 0;ической задачи о пол= 077; напряжений при вершине трещины. Так= 086;е решение пок = 72; не получено, однако существует косвенный метод, в основе которого ле = 78;ит J – интеграл, определяемm= 9;й выражением (см. [19])

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;     (5.27)

где Γ — замкнутый контур, который нужно обойт = 80; против часовой стрелки, окружающий = 74; напряженноl= 4; твердом тел = 77; некоторую область (рис. 5.18); Т — вектор напряжений, перпендикуl= 3;ярный контуру Г и направленнm= 9;й во внешнюю с= 090;орону, Ti =3D σij nj; <= i>u — перемещениk= 7; в направлениl= 0; оси x; ds — элемент контура Γ. Кроме того,

        &= nbsp;           &nbs= p;          (5.28)=

что является энергией деформаций единицы объема. Можн= 086; показать (см. [19]), что если G — замкнутый контур, то J =3D 0.<= o:p>

Ри&#= 1089;. 5.18. Определениk= 7; J<= b>ин&#= 1090;еграла

Ра= 081;с [19] применил этот интеграл к задачам о трещине. Рассмотрим замкнутый контур ABCDEFA вокруг вершины трещины (рис. 5= .19, а). Интеграл по этому контуру равен нулю. Поскольку н = 72; частях берегов трещины CD и AF значения T =3D 0 и dy=3D0, их вклад &= #1074; интеграл равен нулю. Поэтому инт = 77;грал по контуру ABC должен быть равен (с обратным знаком) инте= 075;ралу по контуру DEF. Это означает, чт= 086; независимо от того, берется ли J – интеграл п= 086; контуру ABC или FED, результат будет один и тот же: интеграл не = 079;ависит от пути интегрировk= 2;ния, т. е. JΓ1 =3D JΓ2 (рис. 5.19, 6). Обратите внимание на то, что этот интеграл, не зависящий о = 90; пути интегрировk= 2;ния, берется не п= 086; замкнутому контуру: пределы инт = 77;грирования лежат на краях трещины.

Ри&#= 1089;. 5.19. Контур обхода вершины трещины

Дл= 103; упругого случая J – интеграл можно вычислить, используя решение упругой задачи о пол= 077; напряжений. Отсюда следует (см. [19]), что

    (5.29)

Райсом было показано [19], что вычисление интеграла (5.29) приводит к соотношениn= 2;

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;  (5.30)

Соверш= енно очевидно, чт= 086; для упругог = 86; случая J – интеграл эквивалентk= 7;н интенсивноl= 9;ти выделения энергии. Рай= 089; также показал, что, вообще говоря,

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;            (5.31)

где V — потенциальl= 5;ая энергия. В упругом случае урав = 85;ения (5.30) и (5.31) эквивалентl= 5;ы.

Др= 091;гими словами, J есть обобщенная функция выделения энергии за счет распростраl= 5;ения трещины; эта функция может быть также справедливk= 2; и в том случа&#= 1077;, когда вблиз = 80; вершины трещины имеются значительнm= 9;е пластическl= 0;е деформации. Поскольку J – интеграл н= 077; зависит от пути интегрировk= 2;ния, его можно определить менее сложным пут = 77;м, выбирая пут = 00; интегрировk= 2;ния, вдоль котор = 86;го интегрировk= 2;ние можно выполнить д = 86;статочно просто (т. е. вд= ;оль краев образца). Таким образом, интеграл дает возможностn= 0; сравнительl= 5;о просто опре = 76;елять интенсивноl= 9;ть выделения энергии для случая, когд= 072; при вершине трещины име = 77;тся большая зон = 72; пластичносm= 0;и. Можно ожида = 90;ь, что существует критическоk= 7; значение JIc, при котором может начаться рост трещин = 99;. Так как это должно имет = 00; силу и в упругом слу = 95;ае, то отсюда следует, что

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;          (5.32)=

Уравне= ние (5.32) утверждаеm= 0;, что процесс роста трещины, связанный с большими пластическl= 0;ми деформацияl= 4;и, можно опред = 77;лить из JIc, зная значение GIc, которое был = 86; определено для случая, когда пласт = 80;ческими деформацияl= 4;и можно прене = 73;речь, и наоборот.

Со= 075;ласно уравнению (5.31), = J – интеграл можно найти из диаграмм = 99; «нагрузка — = 087;еремещение&= raquo;, точно так же как и в упругом слу = 95;ае, определяя податливосm= 0;ь образца. Отл= 080;чие заключаетсn= 3; в том, что в результате = 87;ластичност&#= 1080; на диаграмм = 77; «нагрузка — перемещениk= 7;» может иметь место нелинейный = 91;часток. Этот факт схематичесl= 2;и отображен н = 72; рис. 5.20, а. Площ&#= 1072;дь между двумя кривыми, связывающиl= 4;и нагрузку с перемещениk= 7;м для трещин, имеющих размеры a и a + da, равна (∂V/∂a) da= , т. е. эта площадь равна J. Кривые, связывающиk= 7; нагрузки и перемещениn= 3; в образце, можно получить эксперименm= 0;ально, последоватk= 7;льно увеличивая размер трещины, а пл&#= 1086;щадь между двумя кривыми для трещин с мал= 086; отличающимl= 0;ся размерами — определить = 75;рафически. Полученные таким образом зна = 95;ения J можно построить как функцию = v или а (рис. 5.20, б). Определяя значение v при разрушении для трещин различных размеров, из рис. 5.20, б можн= 086; выяснить, происходит ли разрушение = 74;о всех случая = 93; при одинаковом значении J.

Ри&#= 1089;. 5.20. Эксперименm= 0;альное определениk= 7; J<= b>ин&#= 1090;еграла:

1= <= b>уп&#= 1088;угие деформации; 2 — жесткопла= стические деформации

По= 076;обные эксперименm= 0;ы были выполнены Б = 80;глеем и Лэндисом [20, 21]; некоторые и = 79; результатоk= 4; эксперименm= 0;ов представлеl= 5;ы на рис. 5.21. Они обнаружили, что разрушение действителn= 0;но происходит при постоянном = 79;начении JIc, которое было равно GIc, определеннl= 6;й независимо. Аналогичныk= 7; результаты были получены Кобаяши и др. [= 24].

Пр= 080; использоваl= 5;ии J – интеграла требования, предъявляеl= 4;ые к размерам образца и ег= 086; толщине, не такие строгие, как в случае тестов на оп= 088;еделение GIc или KIc. В последнем случае пластическl= 0;е деформации при вершине трещины должны быть сравнительl= 5;о малы, а поэтому размеры трещины и об= 088;азца должны быть большими (см. гл. VII).

Ри&#= 1089;. 5.21. Значение константы J1c

дл&#= 1103; образца (сталь Ni CrMo V пр&#= 1080; t =3D1196°)

с центральноl= 1; трещиной [80] (пl= 6; да&#= 1085;ным АSТМ)

При использоваl= 5;ии J – интеграла требование ограниченнl= 6;сти пластическl= 0;х деформаций можно опустить. Во= 086;бще говоря, это позволяет определять J= Ic (а также, следователn= 0;но, JIc) из испытаний образца малых размеров. Однако мето = 76; J – интеграла должного развития вс = 77; еще не получил.

Ме= 090;од представляk= 7;тся наиболее многообещаn= 2;щим в тех случаях, когда возникают большие пластическl= 0;е деформации (т. е. при плоск= ;ой деформации). Однако в эти= 093; случаях перед разру = 96;ением идет медленный рост трещин = 99;. В процессе медленного роста трещи = 85;ы за ее вершиной происходит разгрузка материала. Д= 086; сих пор независимоl= 9;ть J – интеграла о = 90; пути интегрировk= 2;ния была доказа = 85;а только при использоваl= 5;ии теории плас = 90;ических деформаций (см. [19, 22, 23]), которая не п= 088;едусматрив&= #1072;ет разгрузку материала. П= 086;этому в настоящее время критерий разрушения, основанный на использоваl= 5;ии J – интеграла, следует применять лишь к проце= 089;су зарождения трещин. Кром= 077; того, до сих п&= #1086;р не существует методики использоваl= 5;ия J – интеграла для описани = 03; процесса стабильногl= 6; роста трещи = 85;ы.

Примеч= ание. Кобаяши и др. [= 24] с помощью инкрементаl= 3;ьной теории пластическl= 6;го течения про = 74;ели исследованl= 0;е задачи о росте трещи = 85;ы.        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;       

 

 

 

 

§ 6.1. Скорость распростраl= 5;ения трещины и кинетическk= 2;я энергия

До сих пор рассматривk= 2;лась задача о медленном росте трещины и о нестабильнl= 6;сти этого процесса перед началом разрушения. = 042; данной глав = 77; рассматривk= 2;ется вопрос о пов= 077;дении трещины после возникновеl= 5;ия нестабильнl= 6;сти. Нестабильнl= 6;сть, предшествуn= 2;щая разрушению, возникает тогда, когда при расширении трещины интенсивноl= 9;ть выделения энергии упругих деформаций G= постоянно превышает сопротивлеl= 5;ие росту трещины R. Избыток выделенной энергии (GR) может перейти в кинетическm= 1;ю энергию. Эта кинетическk= 2;я энергия связана с бы= 089;трым движением точек среды по обе стороны от траектории трещины при ее прохождениl= 0; с большой скоростью. Разница между G и R определяет количество энергии, которое мож = 77;т перейти в кинетическm= 1;ю; следователn= 0;но, эта величин = 72; определяет скорость, с которой эта = 090;рещина будет распростраl= 5;яться в среде. Вели&#= 1095;ины G и R представляn= 2;т собой энергию, связанную с = 088;аспростран&= #1077;нием трещины на Δa= . Следователn= 0;но, общее количество энергии, кот= 086;рое может перейти в кинетическm= 1;ю энергию, пос= 083;е того как размер трещины увеличится = 85;а Δa, определяетl= 9;я интегралом от (GR) на отрезке Δa. Этот интеграл представлеl= 5; на рис. 6.1 заштриховаl= 5;ной областью.=

Из= 086;браженный на рис. 6.1 случай основан на т= 088;ех упрощающих предположеl= 5;иях:

1) процесс распростраl= 5;ения трещины происходит при постоянном напряжении;

2) интенсивноl= 9;ть выделения энергии упругих деформаций не зависит о= 090; скорости распростраl= 5;ения трещины;

3) сопротивлеl= 5;ие росту трещины постоянно.=

Чт= 086; касается третьего предположеl= 5;ия, то в предыду= 097;ей главе было показано, чт= 086; во многих случаях величина R есть возрастающk= 2;я функция, по крайней мер = 77;, во время медленного распростраl= 5;ения трещины. Это= 090; факт не вносит существеннm= 9;х изменений в основные положения данной глав = 99;. Однако на величину R оказывает влияние дру = 75;ое обстоятельl= 9;тво, которым нельзя прен = 77;бречь. Сопротивлеl= 5;ие росту трещины зав = 80;сит от поведени = 03; материала при пластическl= 6;м деформировk= 2;нии вблизи вершины трещины и ег= 086; прочностныm= 3; характерисm= 0;ик. Известно, чт= 086; эти характерисm= 0;ики зависят от скорости деформировk= 2;ния. Поведение многих материалов зависит от скорости деформировk= 2;ния: при более вы= 089;оких скоростях деформировk= 2;ния предел текучести увеличиваеm= 0;ся, а деформаци = 03;, при которой происходит разрушение, уменьшаетсn= 3;. При вершине распростраl= 5;яющейся с большой скоростью трещины скорости деформировk= 2;ния очень велик = 80;, поэтому следует ожидать, что при больших скоростях распростраl= 5;ения трещины материал будет проявлять больше хруп = 82;их свойств. В результате материалы, свойства которых зависят от скорости деформировk= 2;ния, имеют убывающую R – кривую, показанную на рис. 6.1 штриховой линией.

Ри&#= 1089;. 6.1. Графическоk= 7; представлеl= 5;ие кинетическl= 6;й энергии:

1= <= b>R= <= b>кр&#= 1080;вая для материа = 83;ов, свойства которых зависят от скорости деформировk= 2;ния

Вт= 086;рое предположеl= 5;ие означает, чт= 086; решение упругой задачи о статическоl= 4; поле напряж = 77;ний применимо и = 074; динамическl= 6;м случае. В действителn= 0;ности распределеl= 5;ия напряжений = 74; этих двух случаях из-з= 072; введения членов, зависящих о = 90; времени, различны. Эт= 086;й задаче посвящен § 6.2. А в настоящем параграфе предполагаk= 7;тся, что решение статическоl= 1; задачи приблизитеl= 3;ьно верно и для д&#= 1080;намическог= о случая.

Пе= 088;вое предположеl= 5;ие о неизменносm= 0;и напряжения несуществеl= 5;но. Ясно, что нестабильнm= 9;й рост трещин = 99; происходит при постоянной внешней нагрузке. Та= 082; как это является ограничиваn= 2;щим фактором, по= 089;ледующие рассуждениn= 3; приводят к оценке верхней границы скорости распростраl= 5;ения трещины. На практике во время роста трещины величина нагрузки может уменьшатьсn= 3;, что приводи = 90; к уменьшени = 02; G и, следователn= 0;но, к уменьшени = 02; значения (G – = R) при условии, что величин = 72; R постоянна.=

Ос= 085;овываясь на теории размерностk= 7;й, Мотт [1] получил выражение для кинетическl= 6;й энергии трещины. Элемент пла = 89;тины с трещиной, находящийсn= 3; за вершиной = 101;той трещины, перемещаетl= 9;я на расстояния u= и ν (см. гл. III), заданные соотношениn= 3;ми

        &= nbsp;           &nbs= p;   (6.1)

Ес= 083;и вершина тре = 97;ины передвигаеm= 0;ся, то выделенный элемент будет от нее удаляться: расстояние r= от этого элемента до вершины трещины про = 87;орциональн&#= 1086; размеру трещины. Следователn= 0;но, и перемещениn= 3; пропорционk= 2;льны размеру трещины:

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;  (6.2)

Ес= 083;и с течением времени трещина растет, то эт&#= 1080; перемещениn= 3; также увеличиваюm= 0;ся. Скорости этого движения соответствk= 7;нно равны:

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p; (6.3)

в соотношениl= 0; (6.3) точка означает производнуn= 2; по времени. Элемент среды массы = т, движущийся со скорость = 02; V, имеет кинетическm= 1;ю энергию mV2/2. Следователn= 0;но, кинетическk= 2;я энергия мат = 77;риала пластины с т= 088;ещиной, который движется со скоростями и

        &= nbsp;           &nbs= p;            &= nbsp;          (6.4)

гд= 077; r — удельная плотность. Следует заметить, чт= 086; уравнение (6.4) справедливl= 6; для пластин = 99; единичной толщины. Подставляя = 74; уравнение (6.4) соотношениn= 3; (6.3), получим<= /p>

        &= nbsp;           &nbs= p;            (6.5)

В случае бесконечноl= 1; пластины размер трещ = 80;ны а является единственнm= 9;м характерныl= 4; размером, имеющим размерностn= 0; длины. Площадь, вдоль которой выполняетсn= 3; интегрировk= 2;ние, должна быть пропорционk= 2;льна a2. Это означает, чт= 086; результат интегрировk= 2;ния может быть пропорционk= 2;лен 2, где k — константа:=

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;      (6.6)

Ки= 085;етическая энергия пропорционk= 2;льна квадратам размера трещины и напряжения. Совершенно очевидно, чт= 086; она должна быть пропор = 94;иональна удельной массе и квадрату ск = 86;рости распростраl= 5;ении трещины.

Ис= 087;ользуя рис. 6.1, можно получить другое выражение для кинетич = 77;ской энергии:

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;   (6.7)

Ра= 089;сматривая случай, когд= 072; R — константа, а величина G при постоянном напряжении определяетl= 9;я решением статическоl= 1; задачи, полу= 095;аем выражение для кинетическl= 6;й энергии

        &= nbsp;           &nbs= p;            &= nbsp;      (6.8)

Ко= 085;станта R в начале нестабильнl= 6;го процесса равна GIc (величина которого задана соотношениk= 7;м GIc =3D πσ2ас/Е). Подставляя в соотношениk= 7; (6.8) равенство R<= /i> =3D GIc и проводя интегрировk= 2;ние, получаем (дл= 103; двух вершин трещины)

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;       (6.9)

Дв= 072; выражения (6.6) l= 0; (6.9) для кинетическl= 6;й энергии можно приравнять друг к другу, получив

        &= nbsp;       (6.10)

Вы= 088;ажение есть скорость продольных волн в среде, т. е. оно равно скорости звука νs. Значение как оказываетсn= 3;, равно приблизитеl= 3;ьно 0,38. Уравнение (6.10) описывает рост скорости распростраl= 5;ения трещины от нуля при а =3D ас до верхней границы скорости 0,38νs, когда аc/а <= /i>стремится к нулю; это имеет место тогда, когда трещина выр = 72;стает достаточно для того, чтобы выпол = 85;ялось соотношениk= 7; а >> ас. На рис. 6.2 представлеl= 5;о графическоk= 7; изображениk= 7; соотношениn= 3; (6.10). В работе Берри [3] выполнены подобные вычисления для образцо = 74;, имеющих форму двухконсолn= 0;ной балки. То же самое было с= 076;елано Хоаглэндом [4]. Интегрировk= 2;ние соотношениn= 3; (6.7) можно провести и в том случае, если величи = 85;а R является возрастающk= 7;й функцией пр = 80; условии, что эта зависимостn= 0; R от размера трещины известна. Если для апп= 088;оксимации R – кривой принята простая степенная функция (см. гл. VIII), то, как оказываетсn= 3; (см. [5]), результ= 072;т вычислений все равно приводит к о= 075;раничению скорости распростраl= 5;ения трещины величиной .

 =

Ри&#= 1089;. 6.2. Увеличение скорости роста трещины при увеличении ее размера=

 =

Из= 084;еренные скорости распростраl= 5;ения трещин лежа = 90; значительнl= 6; ниже их теоретичесl= 2;их значений, вычисленныm= 3; с помощью уравнения (6.10). = 042; работе Блам = 72; [2] собраны рез= ультаты этих измерений. Некоторые и = 79; этих данных для трещин в хрупких материалах, полученных Робертсом и Уэлсом [6], приведены в табл. 6.1. Значения лежат в пределах 0,20 ̵= 2; 0,37. Причиной расхождениl= 1; может быть то, что одно или более одного пред = 87;оложений, сделанных при выводе уравнения (6.10), не были удовлетворk= 7;ны. С другой стороны, анализ Кэннинема [7], в основу которого была положена модель трещ = 80;ны Дагдейла, показывает, что величин = 72; для вязких трещин в стальных листах имее = 90; порядок 0,1. Скорости распростраl= 5;ения вязких трещин, значения которых приведены в литературе, на самом дел= 077; гораздо ниж = 77;, чем скорост = 80;, приведенныk= 7; в табл. 6.1. Данные, полученные Кэннинемом = 80; др. [8], а также данные, приведенныk= 7; на рис. 6.3, указывают н = 72; то, что скорость распростраl= 5;ения трещины, как и на рис. 6.2, приближаетl= 9;я к постоянномm= 1; значению. Эт= 080; данные хоро = 96;о согласуютсn= 3; с вычисленияl= 4;и, которые был = 80; выполнены Кэннинемом [8] для плоског = 86; напряженноk= 5;о состояния. Даффи и др. [9] сообщают о скоростях распростраl= 5;ения трещин при вязком разрушении стальных труб, имеющи= 093; порядок 200 м/с (), и о скоростя= 093; распростраl= 5;ения трещин при хрупком разрушении, превышающиm= 3; эту величин = 91; в 3 — 4 раза. В заключение следует ска = 79;ать, что общей теории, объясняющеl= 1; динамику развития трещин, пока не существует.

 =

Та&#= 1073;лица 6.1 Результаты измерений скорости ра = 89;пространен&#= 1080;я трещин в различных материалах

Ма&= #1090;ериал

Ст&= #1077;кло

5200=

1500=

0,29=

Ст&= #1072;ль

5000=

1000=

0,20=

Ст&= #1072;ль

5000=

1400=

0,28=

Ац&= #1077;тат целлюлозы

1100=

400<= /span>

0,37=

 =

Ри&#= 1089;. 6.3. Результаты измерений скоростей<= span style=3D'font-size:8.5pt;font-family:Arial;color:#330066'> ра&#= 1089;пространен= ия трещин

в стальной фо&#= 1083;ьге при плоском напряженноl= 4; состоянии [8]<= /b>            =             &nb= sp;            =             &nb= sp;            =             &nb= sp; 

 

 

§ 6.2. Динамическ= ая интенсивноl= 9;ть напряжений = 80; интенсивноl= 9;ть выделения энергии

Из-&#= 1079;а введения членов, зависящих о = 90; времени, распределеl= 5;ие напряжений при вершине = 090;рещины, распростраl= 5;яющейся с большой ск= 086;ростью, отличается от статическоk= 5;о распределеl= 5;ия. Для плоской задачи уравнение движения получается из рис. 6.4, на котором показаны напряжения, действующиk= 7; на элемент среды dxdy. Результируn= 2;щую силу, действующуn= 2; в направлен = 80;и x, можно приравнять = 82; массе элемента, умноженной на ускорени = 77; этого элемента. Если переме = 97;ение элемента в этом направлениl= 0; есть u, а удельная масса материала равна ρ, то уравнение равновесия имеет вид=

        &= nbsp;           &nbs= p;   (6.11)

Ри&#= 1089;. 6.4. Напряжения, действующиk= 7; на элемент в= 077;щества

Уравне= ния (6.11) заменяют уравнения равновесия (3.1= ), в которых правые част = 80; равны нулю. Было предло = 78;ено несколько решений динамическl= 0;х задач о трещине, однако до си= 093; пор не сняты многие вопросы по поводу применимосm= 0;и этих решени = 81;.

Ре= 079;ультаты различных подходов представлеl= 5;ы схематичесl= 2;и на рис. 6.5. Анализ Иофф = 77; [10] показывает (см. [11]), что статическоk= 7; решение в оп= 088;еделенной степени отражает условия динамики. Дл= 103; полосы конечной высоты Нильсон [12] получает только ограниченнm= 9;й эффект при значениях , меньших 0,5, однако при с= 090;ремлении к единице величина динамическl= 6;го коэффициенm= 0;а K стремится к нулю. Аналогичныk= 7; результаты = 73;ыли получены Бробергом [13], k= 2; также Акита = 080; Икеда [14]. Однако в эти= 093; двух работа = 93; получено, чт= 086; величина G стремится к нулю при гораздо меньших зна = 95;ениях . Акита и Икеда вывел = 80; уравнение<= /span>

        &= nbsp;           &nbs= p;     (6.12)

гр= 072;фическое изображениk= 7; которого представлеl= 5;о на рис. 6.5. Сравнение результатоk= 4; вычислений на рис. 6.5 затруднено тем обстоят = 77;льством, что соотнош = 77;ние K2 =3D EG в динамическl= 6;м случае не выполняетсn= 3;.

Ри&#= 1089;. 6.5. Статическаn= 3; и динамическk= 2;я интенсивноl= 9;ти выделения энергии

Со= 075;ласно анализу, проведенноl= 4;у Бэйкером [15], распределеl= 5;ие напряжений = 80; деформаций не зависит существеннl= 6; от скорости распростраl= 5;ения трещины вплоть до значений Фотоупру= гие измерения (см. гл. XIII), выполненныk= 7; Уэлсом и Поу= 089;том [16], вероятно, подтверждаn= 2;т эту точку зр= 077;ния. Динамическl= 0;е фотоупругиk= 7; измерения, выполненныk= 7; Брэдлеем и Кобаяши [17], показали, чт= 086; динамическl= 0;е коэффициенm= 0;ы интенсивноl= 9;ти напряжений при предельной = 89;корости распростраl= 5;ения трещины превышают соответствm= 1;ющие статическиk= 7; значения примерно на 40 = %.

До сих пор отсутствуеm= 0; обобщенный анализ, объе= 076;иняющий различные подходы и объясняющиl= 1; аномалии. На данном этап = 77; развития мо = 78;но лишь утверждать, что динамическk= 2;я интенсивноl= 9;ть выделения энергии опр = 77;деленно меньше ее статическоk= 5;о эквивалентk= 2;, причем разница зависит от скорости ра = 89;пространен&#= 1080;я трещины. Однако не вызывает сомнения то = 90; факт, что влияние скорости ра = 89;пространен&#= 1080;я трещины мал = 86; при скорост = 03;х, меньших 0,3νs.

Далее будем пренебрегаm= 0;ь динамическl= 0;м влиянием на величины K и <= i>G и отдельно исследоватn= 0; вопросы об интенсивноl= 9;ти выделения энергии и кинетическl= 6;й энергии. Это ограничениk= 7;, как полагаю = 90;, не является серьезной помехой для качественнl= 6;го объяснения процесса динамическl= 6;го роста трещины и ег= 086; торможения. = 042; дальнейшем будем предполагаm= 0;ь, что сопротивлеl= 5;ие росту трещины не зависит от скорости ее распростраl= 5;ения, за исключен = 80;ем тех случаев, когда эта зависимостn= 0; от скорости существеннk= 2;. Следует иметь в виду, что реальны = 81; динамическl= 0;й процесс по своей приро = 76;е сложнее, чем его теоретичесl= 2;ая модель, однако суть его, несомненно, та же. В после&= #1076;ующих параграфах, там, где это необходимо, подчеркиваk= 7;тся ограниченнl= 6;сть выдвигаемыm= 3; положений. Ввиду того, что на данно= 084; этапе настоящий предмет находится н = 72; стадии развития, в дальнейшее изложение вводятся некоторые рассуждениn= 3;, справедливm= 9;е лишь для качественнl= 6;й оценки.=             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;  

 

 

§ 6.3. Ветвление трещин

Ра= 089;смотрим еще раз простой случай, когд= 072; значение R постоянно. Этот случай изображен н = 72; рис. 6.6, а. Здес= 100; также предполагаk= 7;тся, что рост тре= 097;ины происходит при постоянном напряжении, т. е. интенсивноl= 9;ть выделения энергии лин = 77;йно зависит от размера тре = 97;ины (если не учитывать поправок на конечность размеров и динамическl= 0;е эффекты). В то&= #1090; момент, когд= 072; размер трещины вдвое превы = 96;ает исходный размер, при котором возникает нестабильнl= 6;сть (А) (т. е. когда = 16;a =3D aс), интенсивноl= 9;ть выделения энергии вдвое превы = 96;ает (В) сопротив&#= 1083;ение росту трещины R. Теоретичесl= 2;и это означае = 90;, что высвобождаk= 7;тся достаточно энергии для роста двух т= 088;ещин. Вследствие этого может произойти в = 77;твление трещины. С дальнейшим ростом трещ = 80;ны до размера а= =3D с a =3D 2a<= sub>с) интенсивноl= 9;ть выделения энергии G становится равной 3R. Это означает, чт= 086; одновременl= 5;о могут расти три трещины, т. е. может возникнуть многократнl= 6;е ветвление.=

Ри&#= 1089;. 6.6. Раздвоение трещин:

а — = бе&#= 1079; учета кинетическl= 6;й энергии; б<= b>с учетом кинетическl= 6;й энергии

Со= 075;ласно рис. 6.6 раздвоение может произойти в том случае, если а/ас = =3D 2; 3 и т. д. С помощью уравнения (6.10.) можно показать, чт= 086; минимальнаn= 3; скорость распростраl= 5;ения трещины, необходимаn= 3; для ветвления, с= 086;ставляет 0,19νs (первая ветвь возникает при ас/а =3D 0= ,5). Ветвление оказывает влияние на скорость распростраl= 5;ения трещин. В момент ветвления увеличение кинетическl= 6;й энергии рез = 82;о замедляетсn= 3; и становитс = 03; равной площ = 72;дям треугольниl= 2;ов ABC и BHF, а не AHL (рис. 6.6, а). Это означает, чт= 086; раздвоенныk= 7; трещины должны двигаться медленнее, чем одиночн = 99;е. Это также означает, чт= 086; уравнения, выведенные = 74; данной глав = 77;, справедливm= 9; только при отсутствии ветвления.=

На= 083;ичие кинетическl= 6;й энергии может привести к ветвлению и при более низкой скорости, чт= 086; отображено на рис. 6.6, б. Общее количество кинетическl= 6;й энергии, кот= 086;рая имеется в то= 090; момент, когд= 072; размер трещ = 80;ны увеличился на величину = 16;a =3D ac/2, представлеl= 5;о на рис. 6.6, б тр= 077;угольником MNP. Эта кинетическk= 2;я энергия может быть использоваl= 5;а для распростраl= 5;ения трещины. Пре= 076;положим, что ветвление возникает, когда Δa =3D ac= /2 и когда энергия, необходимаn= 3; для распрос = 90;ранения трещины, получается при преобра = 79;овании кинетическl= 6;й энергии MNP. Если трещин = 72; раздваиваеm= 0;ся, то энергия, необходимаn= 3; для того, чтобы обе трещины про = 96;ли расстояние [= 6;a, равняется 2R. Пусть обе ветви продвигаютl= 9;я от P к S. Общеk= 7; потреблениk= 7; энергии равно площади чет = 99;рехугольни&#= 1082;а QSTV. Только часть этой энергии мож = 77;т быть получена за счет выделения энергии упругих деформаций, = 072; именно энергия, рав= 085;ая площади NSTV, а оставшаяся часть QSN в момент ветвления уже имеется = 074; виде кинети = 95;еской энергии MNP. Во время дальнейшегl= 6; роста трещины потреблениk= 7; энергии превышает G. Нет притока энергии для преобразовk= 2;ния ее в кинетическm= 1;ю энергию, в то время как им= 077;вшаяся в наличии кинетическk= 2;я энергия уже ушла на распростраl= 5;ение трещин. Посл= 077; того как размер трещины дополнителn= 0;но увеличится на ac/2, кинетическk= 2;я энергия будет полно = 89;тью израсходовk= 2;на на распростраl= 5;ение ответвлениn= 3; трещины (площадь MNP =3D NQS). Нулевой кинетическl= 6;й энергии соответствm= 1;ет нулевая скорость трещины, т. е. скорость распростраl= 5;ения трещины постепенно уменьшаетсn= 3; до нуля в точке S. Поскольку интенсивноl= 9;ть выделения энергии упругих деформаций все еще дост= 072;точна для роста двух трещин, процесс рас = 87;ространени&#= 1103; трещин полностью н = 77; прекращаетl= 9;я: возникает неустойчивl= 6;сть, после чего процесс распростраl= 5;ения раздвоенноl= 1; трещины возобновляk= 7;тся с увеличива = 02;щейся скоростью. Аналогичныl= 4; образом может произойти дальнейшее ветвление.=

Ме= 093;анизм ветвления, изображеннm= 9;й на рис. 6.6, б, может также работать и в том случае, если размер трещины увеличиваеm= 0;ся на величину = 16;a, меньшую ac/2, как показан = 86; на рис. 6.7. Посл = 77; того как трещина прорастет н = 72; некоторое расстояние, кинетическk= 2;я энергия будет равна площади TXV. Эта кинетическk= 2;я энергия может пойти на образова = 85;ие и рост новог= 086; ответвлениn= 3;. За короткий промежуток времени эта кинетическk= 2;я энергия полностью расходуетсn= 3; (площадь TXV ра= ;вна площади XZAB). Это означае = 90;, что скорост = 00; распростраl= 5;ения обеих трещи = 85; становится равной нулю. Скорость выделения энергии упругих деформаций недостаточl= 5;а (точка B) для поддержаниn= 3; процесса распростраl= 5;ения обеих трещи = 85;, но она более чем достаточна для роста одной из них. Это означае = 90;, что одна из в&#= 1077;твей вновь становится неустойчивl= 6;й и продолжае = 90; распростраl= 5;яться, а другая сов= 089;ем прекращает расти. Через некоторое время точно таким же образом может возникнуть другое ответвлениk= 7; (точка F). (Маловероят = 85;о, что первое ответвлениk= 7; снова начне = 90; расти: оно находится позади фронта основной трещины, где напряжения = 74; определеннl= 6;й степени уже уменьшилисn= 0;.) После того как кинетич = 77;ская энергия стала равно = 81; площади BCDF, она может пойти на образованиk= 7; второго отв = 77;твления (HFML).

Ри&#= 1089;. 6.7. Нераспростl= 8;аняющиеся ответвлениn= 3;

Ве= 090;вление трещин иногда наблюдаетсn= 3; на практике, особенно когда трещины образуются = 74; результате разрушения сколом и распростраl= 5;яются с большой скоростью. Хорошо известно ветвление трещин в разбитом оконном стекле (в это&#= 1084; случае задача в определеннl= 6;й степени осл = 86;жняется наличием в стекле внутренних = 85;апряжений). При взрыве емкости большого да = 74;ления в результат = 77; ветвления трещин могу = 90; разорватьсn= 3; на множеств = 86; кусков. Последнее происходит = 89; большей вероятностn= 0;ю, если при раз= 088;ушении величина G продолжает расти, т. е. в газонаполнk= 7;нных емкостях (в которых давление не уменьшаетсn= 3; сразу после образованиn= 3; трещин). На рис. 6.8 показан пример раздвоения трещины.

В вопросе о скорости распростраl= 5;ения трещины, необходимоl= 1; для ее ветвления, все еще мног= 086; неясного. Измеренные скорости ветвления н = 77; согласуютсn= 3; с результатаl= 4;и теоретичесl= 2;их вычислений. Если ветвление п = 88;оисходит без потреблениn= 3; кинетическl= 6;й энергии, то необходимаn= 3; для этого скорость до = 83;жна быть порядк = 72; 0,19νs, как было показано ранее. Если кинетическk= 2;я энергия может быть использоваl= 5;а для роста трещины, то ветвление может наступить п = 88;и меньших скоростях. Д= 083;я того чтобы раздвоеннаn= 3; трещина могла продо = 83;жать распростраl= 5;яться, как показан = 86; на рис. 6.6, б, минимальноk= 7; увеличение размера трещины дол = 78;но быть равно Δa= =3D ac/2, или аc/а =3D 0,66. В этом случае минимальнаn= 3; скорость, пр= 080; которой происходит ветвление, полученная из уравнени = 03; (6.10), равна Однако в случае, когда ответвлениn= 3; не растут, скорость, пр= 080; которой происходит раздвоение, может быть гораздо мен = 00;шей, как показан = 86; на рис. 6.7. Другие расхождениn= 3; между теорией и эксперименm= 0;ом, несомненно, происходят из-за влияни= 103; динамики на = 074;еличины G и R, о котором шла речь в предыдущих параграфах главы. Кроме того, все наши рассуждениn= 3; строились в предположеl= 5;ии, что напряжение является постоянной величиной, т. е. G увеличиваеm= 0;ся пропорционk= 2;льно размеру тре = 97;ины по закону G =3D πσ2a/E. Однако при неподвижныm= 3; захватах в процессе распростраl= 5;ения трещины напряжение = 91;меньшается, поскольку общая жесткость о = 73;разца тем ниже, чем длиннее трещина. В результате величина G увеличиваеm= 0;ся медленнее, чем пропорционk= 2;льно размеру трещины, а для определеннl= 6;й геометрии образца она может даже уменьшитьсn= 3;. Это явление, которое рассматривk= 2;ется в следующем параграфе, приводит к изменениям = 88;ис. 6.6 и 6.7, но не изменяет основные принципы.=

Уг= 086;л между ветвями трещины можно предсказатn= 0; достаточно точно (см. [18]), рассматривk= 2;я рост трещин = 99; смешанного типа. Если трещина отк = 83;оняется от плоскост = 80; перпендикуl= 3;ярно направлениn= 2; растягиваюm= 7;его напряжения, то она также подвергаетl= 9;я действию сд = 74;иговых напряжений, т. е. КII ≠ 0. Поведение трещины при совместном действии КI и KII исследованl= 6; в гл. XIV. В этой главе показано, чт= 086; угол между ветвями должен быть порядка 15°, чт&#= 1086; хорошо согласуетсn= 3; с происходя = 97;им в действителn= 0;ности.

Ри&#= 1089;. 6.8. Теневые фотографии процесса ветвления трещин в стеклянной пластине

[ши&= #1088;ина пластины 100 мм, длина 300 мм, толщина 9 мм;

на&#= 1075;рузка, при которой происходит разрушение, равна 310 кгс;

вр&#= 1077;мя экспозиции одного кадр = 72; 4 мкс

(по данным Калтхоффа и Института механики тв = 77;рдого тела)] &nbs= p; 

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;      

 

 

§ 6.4. Основные принципы торможения роста трещи = 85;ы

До сих пор в данной глав = 77; интенсивноl= 9;ть выделения энергии G была принят = 72; равной πσ2a= /E и все рассуждениn= 3; ограничиваl= 3;ись случаем постоянногl= 6; напряжения. Таким образом, мы полагали, чт= 086; величина G увеличиваеm= 0;ся пропорционk= 2;льно размеру трещины. Однако если = 082;рая пластины закреплены, то в процесс= 077; роста трещины напряжение уменьшаетсn= 3;, а рост велич= 080;ны G при этом замедляетсn= 3;. В некоторых случаях вел = 80;чина G может даже уменьшатьсn= 3;. Последнее чаще всего происходит тогда, когда образец нагружаетсn= 3; расклиниваn= 2;щей силой.

На рис. 6.9 изображен случай, когд= 072; значение G уменьшаетсn= 3;. После начал = 72; распростраl= 5;ения трещины величина G увеличиваеm= 0;ся до точки B, а затем уменьшаетсn= 3;. В точке C скорость выделения энергии становится = 86;пять равной R; если пренебречь кинетическl= 6;й энергией, то в этой точке может произойти остановка т = 88;ещины. В точке С запас кинетическl= 6;й энергии равен площа = 76;и ABC. Эта энергия может пойти на распростраl= 5;ение трещины. Следователn= 0;но, трещина может продо = 83;жать расти, несмо= 090;ря даже на то, что теперь G&l= t;R. Окончательl= 5;ая остановка трещины происходит = 74; точке E, в которой площадь CDE =3D ABC. В точке Е кин= 077;тическая энергия уменьшаетсn= 3; до нуля, следователn= 0;но, скорость распростраl= 5;ения трещины также уменьшаетсn= 3; до нуля. Поск&#= 1086;льку в этот момен= 090; величина G меньше, чем G<= sub>Ic, то трещина после постепенноk= 5;о уменьшения ее скорости до нуля останавливk= 2;ется. Легко видет = 00;, что значени = 77; G при остановке меньше GIc. Это более низкое значение называется вязкостью торможения или динамическl= 6;й вязкостью разрушения. Последний термин неточен (см. § 6= .6).

Ри&#= 1089;. 6.9. Торможение трещины за счет уменьшения G=

Сл= 077;дует отметить, чт= 086; с точки зрения конц = 77;пции, изображеннl= 6;й на рис. 6.9, вязкость разрушения при торможении зависит от размера трещины. Более ясно это показан = 86; на рис. 6.10. Две трещины, имеющие размеры ас= ;1 и ас2, становятся неустойчивm= 9;ми при G =3D G1c. Дл= 103; достижения этого значения G в более длинной тре = 97;ине нужны меньшие напряжения; следователn= 0;но, соответствm= 1;ющая ей G – линия имеет меньший наклон и после точки = 074;озникновен&= #1080;я неустойчивl= 6;сти — форму, отли&#= 1095;ную от формы линии, соответствm= 1;ющей более короткой трещине. Кинетическl= 0;е энергии эти = 93; двух трещин различны, чт= 086; выражается = 74; различных значениях G при торможении этих трещин соответствk= 7;нно в точках A и B. Этот вывод можно принять в качестве объяснения той крайней противоречl= 0;вости данных о значениях G при торможении трещин, которые имеются в ли= 090;ературе. Однако, как будет показано далее, эта задача гора = 79;до сложнее.

Ри&#= 1089;. 6.10. Влияние размера трещины на величину Gl= 6;ст пр&#= 1080; торможении

Пр= 077;жде всего, малов= 077;роятно, что кинетическk= 2;я энергия всегда може = 90; быть использоваl= 5;а для распростраl= 5;ения трещины, особенно в случае очен = 00; длинных трещин, для которых большая часть кинет = 80;ческой энергии заключена в области, леж= 072;щей далеко позади вершины трещины. Вследствие этого Максе = 81; и др. [19, 20] пришли к заключени = 02;, что существ = 91;ет максимальнk= 2;я эффективнаn= 3; длина трещи = 85;ы, после превышения которой трещина вед = 77;т себя так, будто ее размер не изменяется (см. гл. XV). Существует = 80; вторая причина, по к&#= 1086;торой вся кинетическk= 2;я энергия не может пойти на увеличение размера тре = 97;ины. Если величина G становится меньше, чем G<= sub>Ic (и K < KIc), то напряжения при вершине трещины также стано = 74;ятся меньше критическиm= 3; (при условии, что распределеl= 5;ие напряжений то же, что и в статическоl= 4; случае). Пред&#= 1087;олагается, что, как только напряжение (или коэффиц= 080;ент K) становится существеннl= 6; ниже критическоk= 5;о, рост трещин = 99; прекращаетl= 9;я даже в том сл&#= 1091;чае, если для этого имеется достаточно энергии. Отсюда видн = 86;, что энергетичеl= 9;кий критерий является необходимыl= 4; условием роста трещины, хот= 103; и не всегда достаточныl= 4;.

Ри&#= 1089;. 6.11. Возможные случаи торможения трещин в материале<= span style=3D'font-size:8.5pt;font-family:Arial;color:#330066'>

со свойствами, зависящими от скорости = 076;еформирова&= #1085;ия

Пр= 086;цесс распростраl= 5;ения трещины в материалах, свойства которых зависят от скорости деформировk= 2;ния, имеет свои особенностl= 0;. Поскольку при вершине быстро движущейся трещины скорости деформировk= 2;ния велики, сопротивлеl= 5;ие росту трещины уменьшаетсn= 3; (в этом случае напряжение, необходимоk= 7; для роста трещины, также меньш = 77;, чем в начале процесса растрескивk= 2;ния). На рис. 6.11, а пр&#= 1086;иллюстриро= вана задержка трещины в ма= 090;ериале со свойствами, зависящими от скорости деформировk= 2;ния. В изображеннl= 6;м здесь случа = 77; кинетическk= 2;я энергия увеличиваеm= 0;ся до точки М', а затем частично расходуетсn= 3; на распрост = 88;анение трещины; это означает, чт= 086; скорость распростраl= 5;ения трещины уменьшаетсn= 3;. Остановка трещины происходит = 74; точке N' (пред= полагается, что в ней интенсивноl= 9;ть напряжений меньше критическоl= 1;); при этом остается неизрасходl= 6;ванная кинетическk= 2;я энергия. Следователn= 0;но, скорость распростраl= 5;ения трещины должна внезапно уменьшитьсn= 3; от конечног = 86; значения (N') до нуля (N''). В противополl= 6;жном случае (рис. 6.11, 6) вся кинетическk= 2;я энергия расходуетсn= 3; на распростраl= 5;ение трещины, при этом после точки P' скор&= #1086;сть трещины одновременl= 5;о с кинетичес = 82;ой энергией постепенно уменьшаетсn= 3; до нуля в точке Q'. Поск= ольку после точки = P' скорость распростраl= 5;ения трещины уменьшаетсn= 3;, сопротивлеl= 5;ие росту трещины опять начинает возрастать, = 072; в точке остановки трещины эта величина до = 83;жна иметь то же значение, чт= 086; и в начале об&#= 1088;азования нестабильнl= 6;сти, поскольку в = 086;боих случаях скорость трещины равна нулю. Значение G, соответствm= 1;ющее торможению трещины, находится в точке Q, следователn= 0;но, оно значительнl= 6; меньше GIc.

Ви= 076; G – кривой в значительнl= 6;й мере зависи = 90; от типа образца; это означает, чт= 086; величина G при торможении трещины также зависит от р= 072;змера образца, по крайней мер = 77; в тех случая= 093;, когда кинет = 80;ческая энергия может быть движущей трещину силой. Совершенно очевидно, чт= 086; существует множество факторов, влияющих на торможение трещины; есл= 080; на самом дел= 077; кинетическk= 2;я энергия полностью или частичн = 86; может быть израсходовk= 2;на на рост трещины, то универсальl= 5;ого значения G или динамическl= 6;й вязкости разрушения, соответствm= 1;ющих торможению трещины, не существует. Таким образом, до тех пор, пока для анализа эксперименm= 0;альных данных не бу= 076;ет создана общая теори = 03;, эти данные н= 077; могут быть согласованm= 9;.

Из испытаний Х = 72;на и др. [21] следуе = 90;, что кинетическk= 2;я энергия участвует в процессе распростраl= 5;ения трещины после того, как G уменьшилосn= 0; до значения, меньшего R. Хан и другие испытывали образцы, имеющие фор = 84;у двухконсолn= 0;ной балки, нагруженноl= 1; расклиниваn= 2;щей силой. Они провели тщательный анализ этог = 86; образца; нам достаточно провести приближеннm= 9;й анализ, иллюстрируn= 2;щий уменьшение G= в этом образце. Из уравнения (5.25) следует, что &#= 1080;нтенсивнос= ть выделения энергии упругих деформаций = 74; ДКБ–образцk= 7; при постоянной нагрузке ув = 77;личивается пропорционk= 2;льно a2. При нагружении образца расклиниваn= 2;щей силой концы его неподвижны = 80; с ростом трещины нагрузка уменьшаетсn= 3;. С помощью уравнения изгиба бало = 82; нагрузку можно выразить ка = 82; функцию относительl= 5;ого перемещениn= 3; концов образца v:

        &= nbsp; (6.13)

Тогда, подставляя уравнение (6.13) = 74; соотношениk= 7; (5.25), получаем, что интенсивноl= 9;ть выделения энергии

        &= nbsp;           &nbs= p;            &= nbsp;          <= span style=3D'font-family:Arial;color:#330066'>(6.14)

Ле= 075;ко видеть, что движущая трещину сил = 72; убывает как четвертая степень размера тре = 97;ины. Этот случай отображен н = 72; рис. 6.12 линией A= BC. На этом рисунке показано, чт= 086; трещина мгн = 86;венно останавливk= 2;ется, поскольку в самом начал = 77; роста трещины G становится меньше R. Поэтому Хан = 080; др. [21] снабдилl= 0; свои образц = 99; трещинами с тупыми вершинами, чтобы зарождение трещин в точке D происходилl= 6; при начальном значении G, большем GIc (см. § 8.5). Если бы скорость деформировk= 2;ния не влияла на свойства материала и кинетическk= 2;я энергия не расходовалk= 2;сь на распростраl= 5;ение трещины, то торможение трещины произошло б = 99; в точке F. При наличии зависимостl= 0; свойств материала о = 90; скорости деформировk= 2;ния в случае, ког&#= 1076;а кинетическk= 2;я энергия не расходуетсn= 3; на распрост = 88;анение трещины, торможение происходит = 74; точке Н. В случае, когд= 072; кинетическk= 2;я энергия пот = 88;ебляется полностью, трещина останавливk= 2;ется в точке L. Меняя высот = 91; точки D (рис. 6.12), можно было изучать процесс рас = 87;ространени&#= 1103; трещин при различных с = 82;оростях, что связано = 089; различным вкладом кин = 77;тической энергии.

Ри&#= 1089;. 6.12. Торможение трещины в ДКБ–образцk= 7;

Исслед= ование, проведенноk= 7; Ханом и др. [21], показало, чт= 086; около 85 % кинетическl= 6;й энергии, зак= 083;юченной в ДКБ–образцk= 2;х, расходовалl= 6;сь на продвижениk= 7; трещин. Анализ значений ра = 79;личных коэффициенm= 0;ов, встречающиm= 3;ся в литератур = 77;, также показал сильную зависимостn= 0; этих значений от скорости распростраl= 5;ения трещины, что подтверждаk= 7;т идею о том, чт&= #1086; кинетическk= 2;я энергия может быть использоваl= 5;а для роста трещины. Поэтому следует сде = 83;ать заключение, что, несмотр= 103; на частично качественнm= 9;й характер выдвинутых = 74; этой главе положений, имеется некоторое элементарнl= 6;е понимание динамики развития трещин и их остановки, требующее общего теоретичесl= 2;ого анализа. Эти принципы можно применять для торможения трещин в реальных конструкциn= 3;х.        &= nbsp;           &nbs= p;            &= nbsp;       

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;       

 

 

§ 6.5. Торможение трещин на практике<= /h2>

То= 088;можение трещины может произойти, когда G становится меньше R. С другой стороны, оно может произойти, когда R увеличиваеm= 0;ся и становитс = 03; больше G. С ростом трещины в определеннl= 6;й степени увеличиваеm= 0;ся зона пласти = 95;ности. Ясно, что размер этой зоны становится равным по порядку величины толщине пла = 89;тины, так что в пластине образуется плоское напряженноk= 7; состояние. Это приводи = 90; к резкому увеличению R= (см. гл. V); благодаря этому эффек = 90;у величина R может стать больше, чем G, особенно если величина G при неподвижныm= 3; захватах является убывающей функцией. Торможение трещины, соответствm= 1;ющее этому принципу, схематичесl= 2;и изображено на рис. 6.13. В реальных конструкциn= 3;х могут быть получены об = 72; типа торможения трещин (уменьшение = G или увеличение R= ).

Ри&#= 1089;. 6.13. Торможение трещины за счет увеличения R=

То= 088;можение трещины за счет увелич = 77;ния R можно вызвать введением полос из материала большей вязкости. На рис. 6.14 показано поведение трещины при торможении. Когда трещи = 85;а проникает в полосу, сопротивлеl= 5;ие ее росту резко возрастает = 80; возникает ситуация, пр= 080; которой G < R. Если кинетическk= 2;я энергия не расходуетсn= 3; на распростраl= 5;ение трещины, то происходит мгновенная остановка трещины. Есл= 080; же обстоятельl= 9;тва таковы, что кинетическk= 2;я энергия может расхо = 76;оваться в качестве движущей трещину эне = 88;гии, то эта трещина будет прорастать = 74; полосу до те= 093; пор, пока ее скорость не станет равной нулю (на рис. 6.14 этом= 091; моменту соответствm= 1;ет равенство заштриховаl= 5;ных областей). По утверждениn= 2; Бламма [2], Ирвином был = 86; высказано предположеl= 5;ие о том, что остановочнk= 2;я полоса долж = 85;а иметь ширин = 91; порядка 2rp+2B. Такой грубы = 81; подсчет в случае, показанном на рис. 6.14, вряд ли применим, поскольку способностn= 0; полосы тормозить трещину, очевидно, за= 074;исит от размера трещины. Остановочнk= 2;я полоса на рис. 6.14, б имее= 090; те же размеры, что и полоса на рис. 6.14, а, но в момент возникновеl= 5;ия нестабильнl= 6;сти она располо = 78;ена дальше от вершины трещины. Так как площадь = A меньше площади C, то трещина проскакиваk= 7;т через вставку, не з&#= 1072;держиваясь (линия n). В случае мало = 81; трещины (линия т) задержки трещины также не происходит из-за большо= 081; крутизны G – линии. Для убывающих G – линий, конечно, ситуация несколько иная, однако принцип остается те = 84; же.

Ри&#= 1089;. 6.14. Сварная вставка для торможения трещин:

а — = то&#= 1088;можение трещины; б<= b>то&#= 1088;можение трещины не достигнуто

Вр= 103;д ли вставки и= 079; материала большой вяз = 82;ости разрушений можно делат = 00; во многих ко= 085;струкциях, поскольку занимаемое ими простра = 85;ство должно быть небольшим (рис. 6.14, б). Материал вставки обычно имее = 90; свойства, отличные от свойств основного материала (например, бо&#= 1083;ее низкий предел текучести). Если основн = 86;й материал бы = 83; выбран из-за своих статическиm= 3; прочностныm= 3; свойств, то наличие бол = 00;шого числа вставок с гораздо более низко = 81; прочностью недопустимl= 6;. Перспективk= 2; вставки в тр= 091;бопровод колец из низкопрочнl= 6;го материала малопривлеl= 2;ательна. Из-за более низкой стат = 80; ческой прочности стенки вставок должны имет = 00; большую толщину. Помимо возр = 72;стания сопротивлеl= 5;ия течению проходящей по трубопро = 74;оду среды увеличение толщины стенок в мес= 090;ах вставок малоэффектl= 0;вно, поскольку в = 03;зкость зависит от толщины. Эта проблема мо = 78;ет быть разрешена (см. [2]) разрезаниеl= 4; вставок на пластинки. Тонкие пластинки могут быть выполнены и = 79; того же материала, что и основн= 072;я конструкциn= 3;. Наличие пластинок приведет к образованиn= 2; при вершине трещины зон = 99; плоского напряженноk= 5;о состояния, и таким образом сопротивлеl= 5;ие росту трещины рез = 82;о возрастет. Полосы для торможения трещины могут еще быть использоваl= 5;ы так, чтобы даже при постоянной нагрузке значение G уменьшалосn= 0;. Рассмотрим остановочнm= 1;ю полосу, скрепленнуn= 2; с пластиной болтами A и B (рис. 6.15). В отсутствие полосы точк = 80; A и B при приближениl= 0; трещины могут свободно дв = 80;гаться относительl= 5;о друг друга. Наличие полосы ограничиваk= 7;т эти перемещениn= 3;. Это означае = 90;, что от полос= 099; на пластину через болты передаются силы Р. Эти силы уменьшают коэффициенm= 0; интенсивноl= 9;ти напряжений = 80;, таким образом, уменьшают G. Чем ближе вершина трещины приближаетl= 9;я к АВ, тем больше становятся силы P, что делает уменьшение G= более эффективныl= 4;. Наибольший эффект осла = 73;ления достигаетсn= 3; тогда, когда вершина тре = 97;ины немного заходит за линию AB. Схем= атическое изображениk= 7; получающейl= 9;я при ним G – линии представлеl= 5;о на рис. 6.15. (Эта проблема ум = 77;ньшения G c помощью полос и стрингером подробно рассмотренk= 2; в гл. XVI, посвященноl= 1; панелям, усиленным ребрами жесткости.)

На рис. 6.15 изображено торможение трещины, про= 080;сходящее из-за понижения G – кривой. В гл. XVI = 087;риведены многочислеl= 5;ные примеры такого типа задержек трещин. Этот принцип был использоваl= 5; Йошики и др. [22] = 074; испытаниях образцов с навареннымl= 0; полосами. Один из полученных ими результ = 72;тов представлеl= 5; на рис. 6.16. Остановка трещины происходит при G =3D Gc, та = 82; что, по всей вероятностl= 0;, кинетическk= 2;я энергия не участвует в процессе распростраl= 5;ения трещины. Однако следует отметить, чт= 086; величина Gl= 9; имела оценочное значение. Пр= 080; использоваl= 5;ии наваренных полос вмест = 86; полос, скреп= 083;енных болтами, следует обратить внимание на то, чтобы сварные швы проходили п = 86; краям полос. В случае длинных полос следует дел = 72;ть короткие сварные швы через короткие ин = 90;ервалы. Если вдоль всей полосы сделан непр = 77;рывный шов, то трещина может прорасти в с= 074;арочный материал и через него в полосу. В это&#= 1084; случае полоса становится неэффективl= 5;ой.

Со стороны полосы на пластину действуют с = 80;лы P. Если P превысит величину продольной прочности п = 83;астины, то последня = 03; разрушится. = 042; этом случае эффективноl= 9;ть полосы уменьшится = 76;о нуля. Если P достаточно велико, чтоб= 099; в полосе возникли пластическl= 0;е деформации, то эффективноl= 9;ть ее также сильно уменьшаетсn= 3;. Подробное обсуждение этих пробле = 84; приведено в гл. XVI.

Ри&#= 1089;. 6.15. Остановочнk= 2;я полоса на болтах

Рис. 6.16. Испытание н = 72; торможение трещины нав = 72;ренной полосой      

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;      

 

 

§ 6.6. Динамическk= 2;я вязкость разрушения=

Ди= 085;амическая вязкость разрушения представляk= 7;т собой сопротивлеl= 5;ие росту трещи = 85;ы за точкой нестабильнl= 6;сти. Эксперименm= 0;альное определениk= 7; динамическl= 6;й вязкости разрушения связано со значительнm= 9;ми трудностямl= 0;. Из рассужде = 85;ий, приведенныm= 3; в предыдущи = 93; параграфах, можно сделать заключение, что мгновен = 85;ая скорость выделения энергии в динамическl= 6;м процессе, вычисленнаn= 3; по тем же правилам, чт= 086; и ее статическиl= 1; эквивалент, не является мерой вязкости, поскольку в этом случае = 074;озникает излишек энергии. Кроме того, задача дополнителn= 0;но усложняетсn= 3; из-за вклада кинетическl= 6;й энергии. Измерение вязкости пр = 80; остановке трещины (GIa или KIa) не приносит много польз = 99;. Если в процессе ра = 89;пространен&#= 1080;я трещины участвует кинетическk= 2;я энергия, то значение G при ее остановке меньше, чем сопротивлеl= 5;ие ее росту, и это значени = 77; зависит от размера трещины и скорости ее распростраl= 5;ения (см. рис. 6.10 и 6.11). В случае, когд= 072; расходуетсn= 3; вся кинетич = 77;ская энергия, из-за эффект= 072; постепенноk= 5;о понижения скорости до нуля вязкость пр = 80; торможении = 90;рещины, вероятнее всего, равна KIa, а интенсивноl= 9;ть выделения энергии — значительнl= 6; ниже. Понимание того, как надо измеря = 90;ь динамическm= 1;ю вязкость разрушения, = 074; определеннl= 6;й степени улучшит полное реше = 85;ие динамическl= 6;й задачи о распростраl= 5;ении трещины, учи= 090;ывающее энергию упругих деформаций = 80; кинетическm= 1;ю энергию. Динамическk= 2;я вязкость разрушения, или динамическk= 2;я R – кривая, определяетl= 9;я зависимостn= 0;ю свойств материала (и= 079; которых наиболее важным явля = 77;тся предел теку = 95;ести) от скорости деформировk= 2;ния. Динамическm= 1;ю вязкость можно измерять и в квазистатиm= 5;еских испытаниях на разрушение при условии, что скорост = 00; деформировk= 2;ния при вершине трещины (стационарн = 86;й) имеет тот же порядок, что при вершине трещины, движущейся = 89; большой скоростью. Это дало бы возможностn= 0; определять динамическm= 1;ю вязкость KId как значени = 77; KI в точке нестабильнl= 6;сти в обычном испытании н = 72; определениk= 7; KIc, проведенноl= 4; при больших скоростях деформировk= 2;ния. Если величина (или ) для трещины, движущейся = 89; большой скоростью, известна, то соответствm= 1;ющая ей динамическk= 2;я R – кривая может быть установленk= 2; с помощью да= 085;ных о величине KId как функции скорости нагружения ( ). Сложность заключаетсn= 3; в том, что вершина трещины деформируеm= 0;ся и, следовате= 083;ьно, скорость деформировk= 2;ния является функцией предела текучести, в то время как сам предел текучести в значительнl= 6;й мере зависи = 90; от скорости деформировk= 2;ния.

По= 083;уэмпиричес&= #1082;ие решения задачи были предложены Ханом и др. [23], а также Краффтом и др. [24, 25]. Экспери= ментально полученные данные о характерисm= 0;иках разрушения, как оказываетсn= 3;, находятся в определеннl= 6;м согласии с и= 093; вычисленныl= 4;и значениями. Некоторые и = 79; результатоk= 4; представлеl= 5;ы на рис. 6.17. Эти д&#= 1072;нные представляn= 2;т интерес для изучения ди = 85;амики процесса разрушения = 80; его зарожде = 85;ия. Они указывают н = 72; сильную зависимостn= 0; вязкости от скорости деформировk= 2;ния, которую следует принимать в = 86; внимание, им= 077;я дело с конструкциn= 3;ми, деформируюm= 7;имися с большими с= 082;оростями. Расчет зависимостl= 0; вязкости от скорости нагружения (см. [23, 24]) легче всего прове = 89;ти для случая образованиn= 3; трещин в хру= 087;ком материале (см. [26]). Преобразовk= 2;ние этих данных = 074; динамическl= 0;е R – кривые осложняетсn= 3; тем фактом (см. [27]), что расп&= #1088;еделения напряжений при стационарнl= 6;й и движущейс = 03; трещинах могут быть различны.=

Ри&#= 1089;. 6.17. Влияние скорости нагружения на вязкость разрушения стали А302В [24] [&#= 1057; =3D <= /b>(t°F = 32)·1,8]<= /span>

Дл= 103; того чтобы в опыте на разрушение получить высокую скорость нагружения, = 074; некоторых случаях (например, см. [28]) используетl= 9;я усовершенсm= 0;вованный маятник Шарпи. При этом необходимо измерять силы, возникающиk= 7; в образце пр= 080; ударе молотка. Определеннk= 2;я таким образ = 86;м динамическk= 2;я вязкость разрушения = 87;редставлен&#= 1072; на рис. 6.18.

Ри&#= 1089;. 6. 18. Динамическk= 2;я вязкость разрушения,

по&#= 1083;ученная в тестах на усовершенсm= 0;вованном маятнике Шарли [28]

(по данным Пергамона)=             =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =   

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;      

 

 

 

 

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