Gemara

Nedarim 90b (Mishnah): At first, if a woman said to her husband 'let Shamayim judge between us', Beis Din would force him to divorce her and pay her Kesuvah. Chachamim changed this, lest women use this claim in order to marry someone else. Rather, he tries to appease her.

(Rav Hamnuna): If a woman says to her husband (in front of Beis Din) 'you divorced me', we believe her, just like Mishnah Rishonah believes one who says that she was defiled. Mishnah Acharonah suspects her to lie about things her husband does not know about.

Objection (Rava): The Mishnah Rishonah believes her only because she would not lie to disgrace herself (and says that she was defiled)!

Question (Rav Mesharshiya): Mishnah Rishonah believes her when she says 'Shamayim is between us', even though this is no disgrace to herself!

Answer (Rava): This means that Eino Yoreh k'Chetz (his semen does not shoot out at the time of Bi'ah). This is an awesome disgrace to him, she is not so brazen to lie about such a matter.

Question (Mishnah Acharonah): If she says 'Shamayim is between us' she is not believed.

She knows that her husband knows the truth. It takes great audacity to lie about this, yet she is not believed. This refutes Rav Hamnuna!

Answer (Rav Hamnuna): The husband knows that he has Bi'ah, but does not know whether or not he is Yoreh k'Chetz, so she is able to lie about this.

Yevamos 64a (Beraisa): If a couple did not have children in 10 years, he must divorce her and pay a Kesuvah. Perhaps (she is fertile, just) he did not merit to have children through her;

65a (Beraisa): If twice a woman married and did not have children in 10 years, afterwards she may marry only a man who has children. If she married a man without children, she leaves him without a Kesuvah.

(R. Ami): If each of them says that the other is sterile, this is 'matters between the two of them'. She is believed (and receives a Kesuvah). This is because she can tell whether or not he is Yoreh k'Chetz, but he cannot.

Rishonim

Rif (Yevamos 20b): A Gaon says that R. Ami's law is like Mishnah Rishonah. We follow Mishnah Acharonah. When she says 'Shamayim is between us', he tries to appease her. We suspect that she seeks to marry someone else. I disagree. R. Ami's law pertains to the Mishnah and Beraisa (Yevamos 64a) that force one to divorce his wife and pay a Kesuvah after 10 years without children. If they argue about the Kesuvah, i.e. each says that the other is sterile, R. Ami believes her if she says Eino Yoreh k'Chetz. Even if she desires another man, in any case we force her to leave him (for he must fulfill Peru u'Rvu)! The Mishnah Rishonah discusses one who seeks to leave before 10 years.

Rav Acha Gaon (cited in Milchamos Hash-m): After 10 years it suffices for her to claim that she needs children. If he says that she miscarried so 10 years have not passed, or if within 10 years she says that she miscarried three times and he says it was only twice, we honor her claim only if she says that Eino Yoreh k'Chetz. Mishnah Acharonah does not force him to divorce her when she seeks to fulfill Peru u'Rvu. If she wants children for her old age, he must divorce her.

Sefer ha'Zechus: 'Shamayim is between us' can mean that he does not have Bi'ah at all, or that Eino Yoreh k'Chetz. The one who asked against Rava thought that this is not disgraceful. Rava answers that she knows that she will need to explain that Eino Yoreh k'Chetz, and she could not lie about this.

Rambam (Hilchos Ishus 15:15): If a woman comes to Beis Din and says that her husband cannot have normal Bi'ah that can produce children, or that (some texts - because) Eino Yoreh k'Chetz, the judges make a compromise between them. They tell her that it is proper for her to stay with him until 10 years without children, and then she can demand to be divorced. We do not force her to stay, but we do not consider her Moredes. We draw out the matter until they compromise.

(Rosh Yevamos 6:15): R. Ami taught that if each of them says that the other is sterile, this is 'matters between the two of them'; she is believed. This is after 10 years; he has children. We say that he did not merit to have children through her only because she claims Eino Yoreh k'Chetz, i.e. he deteriorated after fathering his children. R. Shmuel says that she waited 10 years with her third husband. This explains his claim that she is sterile. Some say that this is when she wants to force him to divorce her b'Al Korcho. If not for her claim, we would be concerned lest she desires another man.

Poskim

Shulchan Aruch (EH 154:6): If a woman says that she wants a child to support her in her old age (Rema - and she has no children) and her husband is Eino Yoreh k'Chetz, we force him to divorce her. This is if she was with him for 10 years without pregnancy and does not demand a Kesuvah, so there is no motivation for her to (falsely) seek a Get.

Beis Yosef (DH Kosav ha'Rashba): In one Teshuvah (Meyuchas l'Ramban 141), the Rashba says that he is unable (to have proper Bi'ah), even if she is not concerned for children he must divorce her. Everyone knows why a woman goes to Chupah. In another Teshuvah the Rashba was unsure, for the Yerushalmi explains 'Shamayim is between us' to mean that he is unable (and the Mishnah Acharonah does not believe her). However, some Meforshim including the Ri say that it means that Eino Yoreh k'Chetz.

Note: Rashi (Yevamos 112a DH ha'Shamayim) explains 'Shamayim is between us' to mean that he is unable. Tosfos (112a DH Shamayim) says that our Gemara explicitly says that it means that Eino Yoreh k'Chetz. The Rashba (Nedarim 91a DH ul'Fi) cites a Gaon who says that it means that he does not have Bi'ah with her, but she is not afraid to lie because he will assume that she means Eino Yoreh k'Chetz.