Gemara

(Rav Yehudah, citing Rav Asi): We force a husband to divorce his wife only if she is forbidden to him.

(Rav Yehudah citing Shmuel): This applies in cases such as a widow married to a Kohen Gadol... but not a man who has been married for 10 years without children.

(Rav Tachlifa citing Shmuel): We force even a man married for 10 years without children.

(Mishnah): If a husband has any of the following blemishes, we force him to divorce his wife: a leper...

Question: According to Rav Tachlifa, the Mishnah should list a man married for 10 years without children!

Answer #1 (Rav Nachman): The Mishnah discusses cases when we force with lashes. If a man is childless after 10 years, we force only with words.

Objection (R. Aba): "A slave will not fix his ways due to words!"

Answer #2 (R. Aba): In both cases we force with lashes. In our Mishnah, we do not force if she wants to stay with him. We force a childless man in any case.

Objection: The Mishnah lists a leper, even though she may not stay with him!

Answer: She may stay with a leper if there will be witnesses (to ensure that they do not have Bi'ah). We do not allow this for a childless man.

Yevamos 64a (Mishnah): If a man married a woman and she did not give birth in 10 years, he may not neglect Peru u'Rvu. If he divorced her, she may marry another man. The latter man may keep her for 10 years without children.

(Beraisa): If a man married a woman and she did not give birth in 10 years, he must divorce her and pay a Kesuvah. Perhaps (she is not sterile, just) he did not merit to have children through her;

We have no proof from the Torah, but there is a hint - "At the end of 10 years that Avram lived in Eretz Kena'an (he married Hagar)". This teaches that time in Chutz la'Aretz does not count. Therefore, if either of them was sick, or if both were in jail, this does not count towards the 10 years.

Inference: The Mishnah discusses her marrying a second man, but not a third. Our Mishnah is like Rebbi:

(Beraisa #1 - Rebbi): If a woman circumcised two of her sons and they died, she should not circumcise her third son;

R. Shimon ben Gamliel says, she circumcises the third, but not a fourth.

(Rava): Regarding marriage, the Stam Mishnah (and the Halachah) is like Rebbi.

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

Rishonim

Rif and Rosh (7:20): Rav Yehudah says that Shmuel does not force a man married for 10 years without children to divorce his wife. (Rav Tachlifa says that that Shmuel forces him. Some texts of the Rif omit this.) Whenever we force it is with sticks, for words do not suffice.

Ran (DH Ein): The Halachah follows Rav Tachlifa, because the Amora'im discuss his opinion.

Rif (Yevamos 21a): The Halachah follows Rava, who permits a man (married for 10 years without children) to keep his wife and marry another woman, if he can support two wives.

Nimukei Yosef (20a DH Masnisin): She cannot say that he will marry another wife and not divorce her, for perhaps others will not marry a man who already has a wife.

Rambam (Hilchos Ishus 15:7): A man may not marry a woman who is sterile, too old (to conceive), an Ailonis or a minor unless he fulfilled Peru u'Rvu or he had another wife who can conceive. If a man was married 10 years without children, he must divorce her and pay a Kesuvah or marry a fertile woman. If he refuses, we force him with a stick until he does. If he or she said 'we will not have Bi'ah. We will live with witnesses and not be secluded together' we do not accede.

Rambam (11): If he went away on business during the 10 years or one of them was sick or they were in prison, this time does not count towards the 10 years.

Rambam (14): If a woman was divorced after 10 years without children, she may remarry. If she was childless for 10 years with her second husband, she may not marry a third man. If she did, she leaves without a Kesuvah, unless he has another wife or he already fulfilled Peru u'Rvu.

Rosh (Yevamos 6:11 and Sof 15): If a woman did not give birth in 10 years, it seems that we force her husband to divorce her. We find (Kesuvos 77a) that 'Yotzi' connotes forcing to divorce. We force anyone obligated to do something. We use sticks, for words do not suffice. However, Amora'im argue about this there! Therefore, it seems that we force only when he acts improperly with her, but not here. The Yerushalmi connotes that whenever it says Yotzi, we do not force. R. Chananel forces only when the Gemara says so explicitly. Elsewhere, we tell him that Chachamim obligate him to divorce; if he does not, one may call him a transgressor. We force only when he withholds both her Onah (conjugal rights) and food, but not if he fulfills one of them. Rashi and the Rif force him to divorce after 10 years. How can they oppose the Yerushalmi? Perhaps they learn from Kesuvos 77a. Or, the forcing is verbal. R. Chananel's distinction is difficult. Marriage is primarily for Onah; it is sufficient reason to force. We should force also for food alone, or else she will starve! However, since there is an argument we do not force with sticks, lest it be an (improperly) coerced Get, which is Pasul. Presumably, the Rif also forces him with sticks to remarry, for otherwise the divorce did not help. This is like other Mitzvos, which we force people to fulfill.

Rosh (12): Some say that this law of 10 years does not apply in Chutz la'Aretz. We say that childlessness was their punishment for living in Chutz la'Aretz. It is unreasonable to neglect Peru u'Rvu based on this flimsy reasoning. (Even) in Chutz la'Aretz, only one in a thousand is childless for 10 years. Many Amora'im of Chutz la'Aretz ruled about such cases. Will we say that this was for people of Eretz Yisrael?! R. Aba bar Zavda was from Bavel, and he said 'had I merited, I would have had children from my first wife.' Avraham's years in Chutz la'Aretz did not count because he was commanded to go to Eretz Yisrael. He was slothful to come, and even after coming he returned. He was in Eretz Yisrael at the time of Bris Bein ha'Besarim, when he was 70, but he left Bavel at the age of 75. He was punished for returning. We do not count the time when one of them was sick or in prison, even if they were having Bi'ah, for perhaps the Aveirah which caused this punishment also caused barrenness. We would not need to teach that a period in which they were not having Bi'ah does not count towards the 10 years.

Question (Korban Nesan'el): Bereishis Rabah (39) says that Hash-m commanded him to return to Charan!

Poskim

Rema (EH 1:3): Nowadays, the custom is not to force a man to marry to fulfill Peru u'Rvu. Similarly, if he wants to marry an infertile woman because he desires her or her money, really, we should protest. For several generations the custom is not to be particular about marriages. Even if he was married 10 years without children, the custom is not to force him to divorce her.

Shulchan Aruch (154:10): If a man was married 10 years without children, he must divorce her and pay a Kesuvah or marry a fertile woman. If he refuses, we force him until he does. If he or she said 'I will not have Bi'ah with her. I will live with her with witnesses; we will not be secluded together' we do not accede.

Gra (51): The Gemara said 'he may not neglect Peru u'Rvu', not 'he must divorce her'. This shows that he may marry a second wife without divorcing.

Rema: Some say that this is if he has no children at all, but if he has viable seed, even if he has not fulfilled Peru u'Rvu, we do not force him to divorce her.

Rivash (15, cited twice in this Siman in Beis Yosef DH Kosav Rivash): Some say that our Mishnah is only when she had no children. If she had a child, even if he has not fulfilled Peru u'Rvu we do not force him to divorce her if she was not yet established to be sterile. If she is too old to have children he must divorce her. We need not teach this according to the opinion that 'Lo Yoldah' (64a) refers to the Reisha, i.e. if she did not bear to him enough children to fulfill Peru u'Rvu (he must divorce her after 10 years). There are many marriages to which we should protest, but nowadays we do not, e.g. one who is childless after 10 years, marriage to minors, and a Bas Kohen or Bas Chacham to an Am ha'Aretz. We do not prevent or break up such marriages to avoid strife. Every Kesuvah entails disputes.

Bedek ha'Bayis: The Yerushalmi says that if she had children and they died, we count 10 years from when they died.

Mordechai (Siman 50): It seems that the Halachah follows Rav, who forces divorce his wife only if she is forbidden. Even Shmuel forces only in Eretz Yisrael. Nowadays, that we live in Chutz la'Aretz, we rely on this not to force people after 10 years. Really, nowadays (that the kingdom decrees against Mitzvos) we should not have children at all! Chachamim did not decree this, for the Tzibur cannot fulfill it (Bava Basra 60b). All the more so we do not force people to have children!

Rebuttal (Gra 10): All his proofs are wrong. The Halachah follows Shmuel. The Rosh proved that we force even in Chutz la'Aretz. Tosfos (Bava Basra 60b DH Din) explains that we should decree that one who already fulfilled Peru u'Rvu not marry, just the Tzibur could not fulfill it. And this is only when there are royal decrees (against Mitzvos).

Rema: Nowadays we do not force at all. A man is believed to say that he knows that he is sterile, and we do not force him to marry.

Source (Nimukei Yosef 20a DH Gemara, brought in Beis Shmuel 25): Regarding all Mitzvos, we force one only if it is clear that he rebels.

Shulchan Aruch (11): If he went away on business during the 10 years or one of them was sick, even if they were having relations, or they were in prison, this time does not count towards the 10 years.